Chapter 27: 

Crime Trip Generation 

Ned Levine 

Ned Levine & Associates 

Houston, TX 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents 

Background 27.1 

Modeling Trip Generation 27.2 

 Trip Purpose 27.2 

 Aggregated Crime Trips 27.2 

 Correlates of Crime 27.3 

 Theoretical Relevance of the Variables 27.4 

 Spurious correlates 27.4 

 Social Disorganization Variables 27.4 

Statistical Problems with Predictor Variables 27.5 

 Multicollinearity among the independent variables 27.5 

Failure to distinguish origins from destinations 27.5 

 Accuracy and Reliability 27.6 

 Count Model 27.6 

Approaches Toward Trip Generation Modeling 27.6

 Trip Tables 27.6 

 Linear/OLS Regression Modeling 27.8 

 Problems with OLS Regression Modeling 27.9 

 Skewness of crime events 27.9 

 Negative predictions 27.11 

 Non-consistent summation 27.13 

 Non-linear effects 27.13 

 Uneven residual errors 27.13 

 Poisson Regression Modeling 27.14 

Advantages of the Poisson Regression Model 27.16 

Problems with the Poisson Regression Model 27.17 

 Over-dispersion in residual errors 27.17 

 Dispersion correction parameter 27.19 

 Under-dispersion in residual errors 27.20 

Diagnostic Tests 27.20

 Skewness Tests 27.20 

 Likelihood Ratio Test 27.22 

 Adjusted likelihood ratio 27.23 

 R-square Test 27.23 

 R-square for the OLS model 27.24 

 R-square for the Poisson model 27.24 

 Dispersion Parameter 27.25 

Coefficients, Standard Errors, and Significance Tests 27.25 

 Testing for Multicollinearity 27.25 

 Tolerance test 27.26 

 Fixed model v. stepwise variable selection 27.27 

Available Regression Models 27.28 

Adding Special Generators 27.29 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents (continued) 

Adding External Trips 27.30 

Balancing Predicted Origins and Predicted Destinations 27.31 

Summary of the Trip Generation Model 27.32 

The CrimeStat Trip Generation Model 27.32 

Calibrate Model 27.34

 Data File 27.34 

 Type of Model 27.34 

 Dependent Variable 27.34 

 Skewness Diagnostics 27.34 

 Independent Variables 27.35 

 Missing Values 27.35 

 Type of Regression Model 27.35 

 Type of Regression Procedure 27.35 

 Save Estimated Coefficients/Parameters 27.36 

 Save Output 27.36 

 Poisson output 27.36 

 OLS output 27.37 

 Multicollinearity Among Independent Variables 27.37 

Graph 27.38 

Make Trip Generation Prediction 27.38

 Data File 27.38 

 Type of Model 27.38 

 Trip Generation Coefficients/Parameters File 27.38 

 Independent Variables 27.39 

 Matching parameters 27.39 

 Add External Trips 27.39 

 Origin ID 27.39 

 Number of external trips 27.39 

 Type of Regression Model 27.40 

 Save Predicted Values 27.40 

Output 27.40 

Balance Predicted Origins & Destinations 27.40

 Predicted Origin File 27.40 

 Origin variable 27.40 

 Predicted Destination File 27.41 

 Destination variable 27.41 

 Balancing Method 27.41 

Save Predicted Origin/Destination File 27.41 

Output 27.41 

Example of the Trip Generation Model 27.41

 Setting Up the Origin Model 27.42 

 Restructuring the Origin Model 27.44 
 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents (continued) 

 Residual Analysis of Origin Model 27.47 

 Setting Up the Destination Model 27.49 

 Residual Analysis of the Destination Model 27.49 

 Adding in Special Generators 27.52 

 Comparing Different Crime Types 27.53 

Adding External Trips to the Origin Model 27.56 

 Predicting External Trips 27.56 

 Make Prediction 27.58 

Balancing Predicted Origins and Destinations 27.61 

Strengths and Weaknesses of Regression Modeling of Trips 27.64 

Conclusion 27.66 

References 27.67 
 

Chapter 27: 

Crime Trip Generation 

Background 

In this chapter, the theory and mechanics of the trip generation stage will be explained. 

Trip generation is a model of the number of trips that originate and end in each zone for a given 

jurisdiction. Given a set of N destination zones and M origin zones (which include all the 

destination zones and, possibly, zones from adjacent jurisdictions), separate models are produced 

of the number of crimes originating and ending in each of these zones. That is, a separate model 

is produced of the number of crimes originating in each of the M origin zones, and another model 

is produced of the number of crimes ending in each of the N destination zones. The first is a 

crime production model while the second is a crime attraction model. 

Two points should be emphasized. First, the models are predictive. That is, the results of 

the models are a prediction of both the number of crime trips originating in each zone and the 

number of crime trips ending in each zone (i.e., crimes occurring in a zone). Because the models 

are predictions, there is always error between the actual number and that predicted. As long as 

the error is not too large, the models can be useful for both analyzing the correlates of crime as 

well as being useful for forecasting or for simulating policy interventions. 

Second, because the number of crimes attracted to the study jurisdiction will usually be 

greater than the number of crimes predicted for the origin zones, due primarily to crime trips 

coming from outside the origin areas, it is necessary to balance the productions and attractions. 

This is done in two steps. One, an estimate of trips coming from outside the study area (external 

trips) is added to the predicted origins as an >external zone=. Two, a statistical adjustment is done 

in order to ensure that the total number of origins equals the total number of destinations. This is 

called balancing and is essential as an input into the second stage of crime travel demand 

modeling - trip distribution. 

In the following discussion, first, the logic behind trip generation modeling is presented, 

including the calibration of a model, the addition of external trips in making a model, and the 

balancing of predicted origins and predicted destinations. Second, the mechanics of conducting 

the trip generation model within CrimeStat is discussed and illustrated with data from Baltimore 

County. 

27.1 
Modeling Trip Generation 

The process of modeling trip generation is fairly well developed, at least with respect to 

ordinary trips. It proceeds through a series of logical steps that make up the aggregate trip 

generation model. 

Trip Purpose 

Trip generation modeling starts with the reasons behind travel. At an individual level, 

people make trips for a reason - to go to work, to go shopping, to go to a medical appointment, to 

go for recreation, or, in the case of offenders, to commit a crime. These are called trip purposes. 

Since there are a very large number of trip purposes, usually these are categorized into a few 

major groupings. In the case of the usual travel demand forecasting, the distinctions are hometo/from-work (or home-based work trips), home-to/from-non-work (or home-based non-work 

trips, e.g., shopping), and a non-home trip where neither the origin nor the destination are at the 

traveler=s residence location (non-home-based trips). 

Since the model has aggregated trips to a zone, the trip purposes are collections of trips 

from each origin zone to each destination zone. Thus, each zone produces a certain number of 

home-work trips, home-non-work trips, and non-home trips and each zone attracts a certain 

number of home-work trips, home-non-work trips, and non-home trips. This is the usual 

distinction that most transportation modeling organizations make. The trip purposes are 

documented during a large travel survey that asks individuals to fill out travel diaries for one or 

two days of travel. In the travel diaries, detailed information about each trip is documented - 

time of day, destination of trip, purpose of trip, travel modes used in making the trips, 

accompanying passengers, route taken, and time to complete the trip. 

Aggregated Crime Trips 

For crime trips, however, these distinctions are not very meaningful. There is very little 

information on how offenders make trips. One cannot just take a sample of offenders and ask 

them to complete a travel diary about how, when, and where the trip took place. With arrested 

offenders, it might be possible to produce such a diary, but both memory problems as well as 

legal concerns quickly make this an unreliable source of information. Therefore, as indicated in 

Chapter 26, a decision has been made to reference all trips with respect to the residential home 

location. All crime trips are analyzed as home-crime trips. 

However, other distinctions can be made. The most obvious is by type of crime. There 

are robbery trips, burglary trips, vehicle theft trips, and so forth. Similarly, distinctions can be 

made by travel time such as afternoon trips or evening trips. However, the sample size will 

27.2 
decrease with greater distinctions. Logically, one can divide a sample into a very large number of 

important distinctions (e.g., afternoon burglary trips involving two or more offenders). 

However, this reduces the sample size and increases the error in estimation, particularly at the 

trip distribution and subsequent stages. 

An important point that distinguishes the aggregate demand types of travel demand 

models, as is being implemented here, and the newer generation of activity-based trips is that 

there are no linked trips with the aggregate approach (Pribyl & Goulias, 2005). If an offender 

first steals a car, then uses the car to rob a grocery store followed by a burglary, the aggregate 

approach models this as three separate trips, rather than as a series of three linked crime trips 

(which the activity-based models do). This is a deficiency with the aggregate travel demand 

model. In order to make the aggregate models work, each trip is considered independent of any 

other trip. While this is not realistic behaviorally, since we know that many crimes are 

committed in sequence as part of a single journey (or tour), the zonal approach does limit the 

underlying logic of crime trips. Nevertheless, the aggregate approach can be very useful as long 

as it implemented consistently. With the current state of activity-based modeling, there is not yet 

any evidence that they produce more accurate predictions than the cruder, aggregate approach 

(Culp & Lee, 2005). 

Correlates of Crime 

Any trip has contextual correlates associated with it. It is well documented that the 

likelihood of making a trip (crime or otherwise) is not equal across areas of a metropolitan 

region. There are age and gender correlates of travel, socioeconomic correlates of travel, and 

land use correlates of travel; the latter are usually associated with trip purposes (e.g., retail areas 

attract shopping trips). 

The trip generation model being implemented in this version of CrimeStat is an aggregate 

model. Thus, the predictors are aggregate, rather than behavioral, in nature, as discussed in 

Chapter 25. They are correlates of trips, not necessarily the reasons for the trips. For example, 

typically population is the best predictor of trips. Zones with many persons will produce, on 

average, more crime trips than zones with fewer persons. The observation is not a reason, but is 

simply a by-product of the size of the zone. Similarly, low-income zones will tend to produce, on 

average, more crime trips than wealthier zones; again, this is not a reason, but a correlate of the 

characteristics that might contribute to individual likelihoods for committing crimes. 

As mentioned in Chapter 25, there are a number of different variables that could be used 

for prediction, although population (or a proxy for population, such as households), income or 

poverty, and land use variables would be the most common (NCHRP, 1998). 

27.3 
Theoretical Relevance of the Variables 

In general, the variables that are selected should be empirically stable and theoretically 

meaningful. That is, they should be stable variables that do not change dramatically from year to 

year. They should be reliably measured so that an analyst can depend on their values. Finally, 

they should be meaningful in some ways. That is, they should be plausible enough that both 

crime analysts and researchers and informed outsiders should agree that the relationship is 

plausible. The variables either should have been demonstrated to be predictors in earlier 

research or else to be so correlated with known factors as to be considered meaningful proxies. 

Spurious correlates 

On the other hand, if a variable is either a correlate of a known predictor or idiosyncratic, 

then it is liable not be believed. For example, the number of taxis usually correlates with the 

amount of employment since taxis tend to ply commercial areas for their trade. Adding the 

number of taxis in a predictive model is liable to produce significant statistical effects in 

predicting crime destinations. However, few persons are going to believe that this is a real factor 

since it is understood to be a correlate of a more structural variable. 

Idiosyncratic variables are those that appear in unique situations. For example, in some 

cities, adjacency to a freeway is a correlate of crime origins (e.g., in Baltimore County where low 

income populations live) whereas in other cities, it is a correlate of crime destinations (e.g., in 

Houston where there are frontage roads with major commercial strips that attract crimes). The 

variables may be real predictors. However, the analyst or researcher will have difficulty 

persuading others to believe in the model, at least until the results can be replicated. 

In other words, what is required for the model is a set of reasonable correlates of crime 

trips that would be plausible and stable over time. It is an ecological model, not a behavioral 

one. 

Social Disorganization Variables 

There is a very large literature on the predictors of crime, typically following from the 

social disorganization literature (for example, Park &Burgess, 1924; Thrasher, 1927; Shaw & 

McKay, 1942; Newman, 1972; Ehrlich, 1975; Cohen & Felson, 1979; Wilson & Kelling, 1982; 

Stack, 1984; Messner, 1986; Chiricos, 1987; Kohfeld & Sprague, 1988; Bursik & Grasmick, 

1993; Hagan & Peterson, 1994; Fowles & Merva, 1996; Bowers & Hirschfield, 1999 among 

many other studies). Much of this literature identifies correlates that are associated with crime 

incidents. Among the factors that have been associated with crime and delinquency at an 

aggregate geographical level are poverty, low income households, overcrowding, substandard 

27.4 
housing, low education levels, single-parent households, high unemployment, minority and 

immigrant populations.1 

Statistical Problems with Predictor Variables 

Multicollinearity among the independent variables 

There are two statistical problems associated with using these variables as predictors. 

The first is the high degree of overlap between the variables. Zones that have high poverty levels 

typically also have low household income levels, higher population densities, substandard 

housing, a high percentage of renters, and higher proportion of minority and immigrant 

populations. In a regression model, this overlap causes a condition known as multicollinearity. 

Essentially, the independent variables correlate so highly among themselves that they produce 

ambiguous, and sometimes strange, results in a regression model. For example, if two 

independent variables are highly correlated, frequently one will have a positive coefficient with 

the dependent variable while the other will have a negative coefficient; conversely, they 

sometimes can cancel each other out. Chapter 17 discussed multicollinearity and provided an 

example that showed correlated independent variables can cancel each other out. Thus, in spite 

of the correlates with crime levels, in a model it is usually best to eliminate co-linear variables. 

The result is that simple variables usually end up being the most straightforward to use 

(population, median household income) with many of the subtle, but theoretically relevant, 

variables typically dropping out of the equation. 

Failure to distinguish origins from destinations 

Second, in much of this literature, however, there is not a clear distinction between origin 

predictors and destination predictors. That is, in most cases, the correlates of crimes were 

identified but it is often unclear whether these correlates are associated with the neighborhoods 

of the offenders (origins) or the locations where the crimes occur (destinations). This can result 

in a set of vague correlates without clear direction about whether the variables are associated 

with producing or attracting conditions. In fact, in much of the early literature on social 

disorganization, it was implicitly assumed that crimes are produced in the neighborhoods where 

the offenders lived, a linkage that is increasingly becoming disconnected. For modeling crime 

trips, however, it is essential that the predictors of origins be kept separate from the predictors of 

destinations. 

Note that a correlation at an aggregate level does not necessarily imply a correlation at the individual level. 

As has been noted frequently, the vast majority of people do not commit serious crimes and that most 

crimes are committed by a small proportion of the population (Ratcliffe, 2008). 

27.5 

1 
Accuracy and Reliability 

A trip generation model should be accurate and reliable. Accuracy means that the model 

should replicate as closely as possible the actual number of trips originating or ending in zones 

and that there should be no bias (which is a systematic under- or over-estimating of trips). 

Reliability means that the amount of error is minimized. 

These criteria have two implications which are somewhat at odds. First, we have to 

choose models that replicate as closely as possible the number of trips originating or ending in a 

zone. In general, this would be a model that had the highest overall predictability. But, second, 

we have to choose models that minimize total prediction errors. This allows a model to replicate 

the number of trips for as many zones as possible. The two criteria are somewhat contradictory 

because crime trips are highly skewed. That is, a handful of zones will have a lot of crimes 

originating or ending in them while most zones will have few or no crimes. The zones with the 

most crimes will have a disproportionate impact on the final model. Thus, a model that obtains 

as high a prediction as possible (i.e., highest log-likelihood or R2

) may actually only predict 

accurately for a few zones and may be very wrong for the majority. 

The strategy, therefore, is to obtain a model that balances high predictability but by 

keeping the total prediction error low. 

Count Model 

Another element of the model is that the trip generation model is for counts (or volumes), 

not for rates. The model predicts the number of crimes originating in each origin zone and the 

number of crimes occurring in each destination zone. The model could be constructed to predict 

rates, but normally it is not done. For most travel demand modeling, as mentioned in Chapter 25, 

the model predicts the number of trips originating or ending in a zone. Thus, there is a crime 

production model that predicts the number of crimes originating in each zone and a crime 

attraction model that predicts the number crimes 

Approaches Toward Trip Generation Modeling 

Trip Tables 

There are two classic approaches to trip generation modeling. The first uses a trip table 

(sometimes called a cross-classification table or a category analysis). A trip table is a crossclassification matrix. Several predictive variables are divided into categories (e.g., three level of 

household income; four levels of vehicle ownership; three levels of population density) and a 

mean number of trips is estimated for each cell, usually from a survey. For example, a survey of 

27.6 
 

household income might show the relationship between household income and the number of 

trips taken by individuals of the households. Based on a sample, estimates of the average 

number of trips per person can be obtained for each income level (e.g., 3.4 trips per day for 

persons from low income households; 4.5 trips per day for persons from median income 

households; 6.7 trips per day for persons from high income households). These variables are 

further subdivided into two-way or three-way cross-tabulation tables (e.g., low income and 

medium vehicle ownership; low income and high vehicle ownership). Table 27.1 illustrates a 

possible trip table model involving two variables. In practice, three or four variables are used. 

The main reason that trip tables are used in a trip generation model is because of the nonlinear nature of trips. Predictive variables are usually not linear in their effects on the number of 

trips. Thus, unless a sophisticated non-linear model is used, sizeable error can be introduced in a 

prediction. It is usually safer to use a trip table approach (Ortuzar & Willumsen, 2001). There 

are some major handbooks on the topic (Henscher & Button, 2002; ITE, 2003). In fact, the 

Institute of Transportation Engineers publishes a large handbook that gives extensive trip 

production and trip attraction tables by detailed land uses (ITE, 2003). These tables are often 

used in formal environmental review processes for site analysis and are frequently accepted by 

courts in litigation. They are not without their problems, however, and there have been 

numerous critiques of the tables (Shoup, 2002; NCHRP, 1998). They also cannot be used in a 

travel demand model and will produce erroneous results. 

The problem for crime analysis, however, is that it is impossible to obtain these data. 

One cannot ask a sample of offenders how many crimes they undertake each day in order to 

estimate the mean expectations for a table. Thus, one has to adopt a more indirect approach in 

modeling crime productions and attractions. 

A second problem with the trip table approach is its use of zonal data. While it could be 

applied to zonal data (e.g., using median household income and average vehicle ownership in 

Table 27.1 instead of individual household income and vehicle ownership), this type of approach 

is prone to ecological inference errors and could be very wrong (Freedman, 1999; Langbein & 

Lichtman, 1978). There is no guarantee that the splitting of two aggregate variables (essentially, 

the cross-product of their marginal probabilities) will produce an accurate trip estimate; often, 

such an approach leads to very wrong results. 

Further, such an approach requires interpretation and some degree of arbitrariness. For 

example, how does one subdivide median household income? One person might interpret it 

slightly differently than another; unlike simple numerical counts (e.g., 0 vehicle ownership; 

27.7 
   

 

 

 

 

 

 

Table 27.1: 

Illustration of Possible Trip Table Approach to Trip Generation 

Average Trips per Adult, Age 16+ 

Household income

3.2 4.6 6.7

5.4 7.8 8.1

Low Medium High

Vehicle 0‐1

Ownership 2+

1 vehicle ownership; 2 vehicle ownership), there is too much variability in categorizing variables 

at the zonal level.2 

Linear/OLS Regression Modeling 

The second approach is to use a regression framework. In this approach, the number of 

crimes either originating or ending in each zone is estimated from zone characteristics using a 

regression model. This can be written in a generalized linear model (‘link’ function) form (see 

Chapter 16): 

 (27.1) 

This equation says that some function of the mean number of crimes, f(Yi), either 

originating or ending in zone I, is a linear function of a number of independent variables, X1, X2, 

X3,....XK for these zones; there are K independent variables plus a possible constant. There is also 

an error term which represents the discrepancy between the actual observation and what the 

model predicts. This is sometimes called residual error since it is the difference between the 

observed and predicted values (Oi - Yi). The function is unspecified and can be non-linear.3 

The traditional approach to regression modeling assumed that the independent variables 

are linear in their effect on the dependent variable. Thus, 

2 There is also subjectivity in subdividing variables at an individual level. For example, household income 

levels can be subdivided in different ways. However, with aggregate data, all variables have to be 

subdivided arbitrarily whereas with individual level data, typically only income is done this way. 3 Some statisticians often refer to the number of parameters that have to be estimated in an equation, not just 

the number of independent variables. In most regression models, for example, there are K+1 parameters 

that are estimated - coefficients for the K independent variables and a constant term. In this text, K refers 

to the number of independent variables, not estimated parameters. 

27.8 
 (27.2) 

In this model, there are K independent variables and one constant term (β0, sometimes 

called α) that needs to be estimated. For each zone, i, each of the independent variables has a 

weight associated with it (the coefficient, β). The product of the value of the independent variable 

times its weight represents its effect. The individual effects of each of the K independent 

variables are summed to produce an overall estimate of the dependent variable, Y. 

The method for estimating this equation usually minimizes the sum of the squares of the 

residual errors. Hence, the procedure is called Ordinary Least Squares (or OLS). If the equation 

is correctly specified (i.e., the dependent variable is normally distributed and all relevant variables 

have been included), the error term, ε, will be normally distributed with a mean of 0 and a 

constant variance, σ

2

. 

Problems with OLS Regression Modeling 

However, there are a number of major problems associated with OLS regression 

modeling. These were discussed in Chapter 15 (15.16-15.19). To repeat, there are five major 

problems with the OLS model. 

Skewness of crime events 

First, crime events are extremely statistically skewed. Some locations have a much higher 

likelihood of a crime event (either an origin or a destination) than others. Figure 27.1 below 

shows the number of crimes from 1993 to 1997 in Baltimore County that occurred at each 

location. That is, the graph shows the number of incidents that occurred at every location, 

plotted in decreasing order of frequency. Thus, there were 7,965 locations where only one crime 

occurred between 1993 and 1997. There were 2,878 locations where two crimes occurred in that 

period. There were 1,138 locations where three crimes occurred in that period. At the other end 

of the spectrum, there were 332 locations that had 10 or more crimes during the period and there 

were 97 locations that had 30 or more crimes occur. If we add to this the very large number of 

locations where no crimes occurred, the unequal likelihoods of crime by location is even more 

dramatic. In other words, the data are highly skewed with respect to the frequency of crimes. 

Most locations either had no crimes occur or very few, while a few locations had many crimes 

occur. 

Aggregating crimes into zones tends to reduce some of the skewness. For example, 

grouping the crimes by origin traffic analysis zone (TAZ) reduced it a little bit. Nineteen of the 

525 origin zones in Baltimore County and Baltimore City did not have any crimes occur in them 

while 15 zones had only one crime occur. Six zones had two crimes originate from them while 8 

27.9 
Figure 27.1: 
 

 

zones had three crimes originate from them. At the other end, 1 zone had 738 crimes originate 

from it and another zone had 533 originate from it. Of the 525 origin zones, 155 had 100 or more 

crime events. Similar results are found for the destination zones. Figure 27.2 graphs the 

distribution of origins and destinations by TAZ’s in bins of 50 incidents each. 

Skewness in the dependent variable usually makes the final model biased and unreliable. 

Particularly if the skewness is positive (i.e., a handful of cases have very large values), the 

resulting regression coefficients will reflect the cases with the highest values rather than represent 

all the cases with approximately equal weights. These so-called ‘outliers’ can overwhelm a 

regression equation. In an extreme case, a very large outlier may totally determine the model.4 

Skewness makes prediction difficult. The OLS model assumes that each independent 

variable contributes to the dependent variable at an arithmetic rate; there is a constant slope such 

that a one unit change in the independent variable is associated with a constant change in the 

dependent variable. With skewness, on the other hand, such a relationship will not be found. 

Large changes in the independent variable will be necessary to produce small changes in the 

dependent variable, but the effect is not constant. In other words, the OLS model typically cannot 

explain the non-linear changes in the dependent variable.5 

Negative predictions 

A second problem with OLS is that it can have negative predictions. With a count 

variable, such as the number of crimes originating or ending in a zone, the minimum number is 

zero. That is, the count variable is always positive, being bounded by 0 on the lower limit and 

some large number on the upper limit. The OLS model, on the other hand, can produce negative 

predicted values since it is additive in the independent variables. This clearly is illogical and is a 

major problem with data that are skewed. If the most common value is close to zero, it is very 

possible for an OLS model to predict a negative count. 

4 For example, an experiment with 100 cases was created with a progressing dependent variable and a 

random independent variable (i.e., the independent variable had its value selected randomly). The 

dependent variable progressed from 1 to 100. For the first 99 cases, the independent variable took values 

from 0.12 to 9.9, randomly assigned. The correlation between these two variables for the first 49 cases was 

0.04. However, for the 100th case, the independent variable was given a value of 100. The correlation 

between the two variables now shot up to 0.17. Even though the F-test for this was not significant, it 

represented a sizeable jump. Replacing one other independent value with a 50 caused the correlation to jump 

to 0.23, which was statistically significant. In other words, two outliers caused a random series to appear 

significant! 5 It is possible to transform the independent variable into a non-linear predictor, for example by taking the log 

of the independent variable or raising it to some power (e.g., X2

). However, this will not solve the other 

problems associated with OLS, namely negative and non-summative predictions. 

27.11 
Figure 27.2: 
 

 

 

 

 

 

Non-consistent summation 

A third problem with OLS models is that the sum of the input data values do not 

necessarily equal the sum of the predicted values. Since the estimate of the constant and 

coefficients is obtained by minimizing the sum of the squared residual errors, there is no 

balancing mechanism to require that they add up to the same as the input values. For a trip 

generation model in which the number of predicted origins must equal the number of predicted 

destinations (after adding in the number of predicted external trips), this can be a big problem. In 

calibrating the model, adjustments can be made to the constant term to force the sum of the 

predicted values to be equal to the sum of the input values. But in applying that constant and 

coefficients to another data set, there is no guarantee that the consistency of summation will hold. 

In other words, the OLS method cannot guarantee a consistent set of predicted values. 

Non-linear effects 

A fourth problem with the OLS model is that it assumes the independent variables are 

linear in their effect. If the dependent variable was normal or relatively balanced, then a linear 

model might be appropriate. But, when the dependent variable is highly skewed, as is seen with 

these data, typically the additive effects of each component cannot usually account for the nonlinearity. Independent variables have to be transformed to account for the non-linearity and the 

result is often a complex equation with non-intuitive relationships.6 It is far better to use a nonlinear model for a highly skewed dependent variable. 

Uneven residual errors 

The final problem with an OLS model and a skewed dependent variable is that the model 

tends to over- or under-predict the correct values, but rarely comes up with the correct estimate. 

With skewed data, typically an OLS equation produces non-constant residual errors. That is, one 

of the major assumptions of the OLS model is that all relevant variables have been included. If 

that is the case, then the errors in prediction (the residual errors - the difference between the 

observed and predicted values) should be uncorrelated with the predicted value of the dependent 

variable. Violation of this condition is called heteroscedasticity because it indicates that the 

residual variance is not constant. The most common type is an increase in the residual errors with 

For example, to account for a skewed dependent variable, one or more of the independent variables have to 

be transformed with a non-linear operator (e.g., log or exponential term). When more than one independent 

variable is non-linear in an equation, the model is no longer easily understood. It may end up making 

reasonable predictions for the dependent variable, but it is not intuitive and not easily explained to nonspecialists. It is possible to transform the independent variable into a non-linear predictor, for example by 

taking the log of the independent variable or raising it to some power (e.g., X2

). However, this will not 

solve the other problems associated with OLS, namely negative and non-summative predictions. 

27.13 

6 
 

higher values of the predicted dependent variable. That is, the residual errors are greater at the 

higher values of the predicted dependent variable than at lower values (Draper & Smith, 1981, 

147). 

A highly skewed distribution tends to encourage this. Because the least squares procedure 

minimizes the sum of the squared residuals, the regression line balances the lower residuals with 

the higher residuals. The result is a regression line that neither fits the low values or the high 

values. For example, motor vehicle crashes tend to concentrate at a few locations (crash hot 

spots). In estimating the relationship between traffic volume and crashes, the hot spots tend to 

unduly influence the regression line. The result is a line that neither fits the number of expected 

crashes at most locations (which is low) nor the number of expected crashes at the hot spot 

locations (which are high). The line ends up over-estimating the number of crashes for most 

locations and under-estimating the number of crashes at the hot spot locations. 

Poisson Regression Modeling 

Poisson regression is a non-linear modeling method that overcomes some of the problems 

of OLS regression. It is particularly suited to count data (Cameron & Trivedi, 1998). In the 

model, the number of events is modeled as a Poisson random variable: 

(27.3) 

where Yi is the count for one group or class, i, λ is the mean count over all groups, and e is the 

base of the natural logarithm. The distribution has a single parameter, λ, which is both the mean 

and the variance of the function. 

The ‘law of rare events’ assumes that the total number of events will approximate a 

Poisson distribution if an event occurs in any of a large number of trials but the probability of 

occurrence in any given trial is small (Cameron & Trivedi, 1998). Thus, the Poisson distribution 

is very appropriate for the analysis of rare events such as crime incidents (or motor vehicle 

crashes or rare diseases or any other rare event). The Poisson model is not particularly good if the 

probability of an event is more balanced; for that, the normal distribution is a better model as the 

sampling distribution will approximate normality with increasing sample size. Figure 27.3 

illustrates the Poisson distribution for different expected means. 

27.14 
Figure 27.3: 
 

 

The mean can, in turn, be modeled as a function of some other variables (the independent 

variables). Given a set of observations on dependent variables, Xki (X1, X2, X3,...,XK), the 

conditional mean of Yi can be specified as an exponential function of the X’s: 

(27.4)

where Xki is a set of independent variables, β is a set of coefficients, and e is the base of the 

natural logarithm.. Now, the conditional mean (the mean controlling for the effects of the 

independent variables) is non-linear. Equation 27.4 is sometimes written as: 

and is known as the loglinear model. In more familiar notation, this is written as: 

That is, the natural log of the mean is a function of K random variables and a constant. 

(27.5) 

 (27.6) 

Note, that in this formulation, there is not a random error term. The data are assumed to 

reflect the Poisson model. There can be residual errors, but these are assumed to reflect an 

incomplete specification (i.e., not including all the relevant variables. Also, since the variance 

equals the mean, it is expected that the residual errors should increase with the conditional mean. 

That is, there is inherent heteroscedasticity (Cameron & Trivedi, 1998). This is very different 

than an OLS where the residual errors are expected to be constant. 

The model is estimated using a maximum likelihood procedure, typically the NewtonRaphson method. In Appendix B, Luc Anselin presents a more formal treatment of both the OLS 

and Poisson regression models 

Advantages of the Poisson Regression Model 

The Poisson model overcomes some of the problems of the OLS model. First, the Poisson 

model has a minimum value of 0. It will not predict negative values. This makes it ideal for a 

distribution in which the mean or the most typical value is close to 0. Second, the Poisson is a 

fundamentally skewed model; that is, it is non-linear with a long ‘right tail’. Again, this model is 

appropriate for counts of rare events, such as crime incidents. 

Third, because the Poisson model is estimated by either maximum likelihood or Markov 

Chain Monte Carlo (MCMC; see chapters 16 and 17), the estimates are adapted to the actual data. 

27.16 
In practice, this means that the sum of the predicted values is virtually identical to the sum of the 

input values, with the exception of very slight rounding off error. In the subsequent balancing of 

the predicted origins and the predicted destinations, this leads to a more stable estimate since the 

only difference between the predicted origins and predicted destinations is the number of trips that 

come from outside the study area (external trips). Since the external trips are added to the 

predicted origins, the balancing operation is less prone to adjustment error. 

Fourth, compared to the OLS model, the Poisson model generally gives a better estimate of 

the number of crimes for each zone. The problem of over- or under-estimating the number of 

incidents for most zones with the OLS model is usually lessened with the Poisson. When the 

residual errors are calculated, generally the Poisson has a lower total error than the OLS. 

In short, the Poisson model has some desirable statistical properties that make it very useful 

for predicting crime incidents (origins or destinations). 

Problems with the Poisson Regression Model 

On the other hand, the Poisson model is not perfect. The primary problem is that count 

data are usually over-dispersed but occasionally can be under-dispersed. 

Over-dispersion in residual errors 

In the Poisson distribution, the mean equals the variance. In a Poisson regression model, 

the mathematical function, therefore, equates the conditional mean (the mean controlling for all the 

predictor variables) with the conditional variance. However, most real data are over-dispersed; the 

variance is generally greater than the mean. Figure 27.4 shows the distribution of Baltimore 

County and Baltimore City crime origins and Baltimore County crime destinations by TAZ (repeat 

of Figure 27.2) and also indicates the variance-to-mean ratio of each variable. For the origin 

distribution, the ratio of the variance to the mean is 14.7; that is, the variance is 14.7 times that of 

the mean! For the destination distribution, the ratio is 401.5! 

In other words, the variance is many times greater than the mean. Most real-world count 

data are similar to this; the variance will usually be a lot greater than the mean. What this means in 

practice is that the residual errors - the difference between the observed and predicted values for 

each zone, will be greater than what is expected. The Poisson model calculates a standard error as 

if the variance equals the mean. Thus, the standard error will be underestimated using a Poisson 

model and, therefore, the significance tests (the coefficient divided by the standard error) will be 

greater than it really should be. This would have the effect of identifying variables as being more 

statistically significant in a model than what they actually should be. In other words, in a Poisson 

27.17 
Figure 27.4: 
 

regression model, we would end up selecting variables that really should not be selected because 

we think they are statistically significant when, in fact, they are not. 

Another problem with the Poisson, which is true for most of the common regression 

methods, is the lack of a spatial predictor component. For these, the MCMC Poisson models, 

discussed in Chapter 17 can be used. Also, in the crime travel demand model, spatial interaction is 

explicitly incorporated during the second stage of the model - trip distribution. Thus, any errors 

introduced in the first stage - trip generation, are usually compensated for during the second. 

Nevertheless, the inclusion of a spatial component in a regression model will generally improve 

the prediction. 

Dispersion correction parameter 

There are a number of methods for correcting the over-dispersion in a count model. Most 

of them involve modifying the assumption of the conditional variance equal to the conditional 

mean. For example, the negative binomial model assumes a Poisson mean but a gammadistributed variance term (Cameron & Trivedi, 1998, 62-63; Venables & Ripley, 1997, 242-245). 

That is, there is an unobserved variable that affects the distribution of the count. There are several 

interpretations of the negative binomial (see Boswell & Patil, 1970) but the most common is to 

assume that there are mixtures of distinct Poisson distributions that make up the real distribution. 

 The negative binomial model has a Poisson mean but with a ‘longer tail’ variance function 

and is usually preferred for over-dispersed data sets, such as typical with crime data. In Appendix 

C, Dominique Lord and Byung-Jung Park present a formal treatment of the negative binomial 

model. Other adjustments that can be made include the Poisson-Lognormal model (which can be 

estimated in CrimeStat; see Chapter 17) and the zero-inflated Poisson model assumes a Poisson 

function combined with a degenerate function with a probability of 1 for zero counts (Hall, 2000). 

These generally produce better estimates that the simple Poisson especially if a spatial component 

is added. 

There is a simple linear correction for over-dispersion that frequently works, called the 

NB1 model (Cameron & Trivedi, 1998, 63-65). The model proceeds in two steps. In the first, the 

Poisson model is fitted to the data and the degree of over- (or under-) dispersion is estimated. The 

dispersion parameter is defined as: 

(27.7)

where N is the sample size, K is the number of independent variables, Yi is the observed number of 

events that occur in zone ݅, and Pi is the predicted number of events for zone ݅. The test is similar 

27.19 
 

 

to an average Chi-square in that it takes the square of the residuals (Yi - Pi) and divides it by the 

predicted values, and then averages it by the degrees of freedom. The dispersion parameter is a 

standardized number. A value greater than 1.0 indicates over-dispersion while a value of less than 

1 indicates under-dispersion (which is rare, though possible). A value of 0 indicates 

equidispersion (or the variance equals the mean). 

In the second step, the Poisson standard error is multiplied by the square root of the 

dispersion parameter to produce an adjusted standard error: 

(27.8) 

The new standard error is then used in the t-test to produce an adjusted t-value. Cameron and 

Trivedi (1998) have shown that this adjustment produces results that are almost identical to that of 

the negative binomial, but involving fewer assumptions. Chapter 16 discussed the NB1 model in 

more depth. 

The point is that the Poisson model needs to be adjusted for over-dispersion. CrimeStat 

provides a number of regression tools for accounting over-dispersion and which can also include a 

spatial autocorrelation adjustment. Chapters 16 and 17 provide information on these models. 

Under-dispersion in residual errors 

Occasionally, a data set will be under-dispersed, meaning that the conditional variance is 

substantially lower than the mean. As a rough approximation, Cameron and Trivedi (1998) suggest 

that if the raw variance-to-mean ratio is less than 2.0, then most likely the model will show underdispersion with the conditional mean. If the under-dispersion is slight, then the NB1 model can be 

used to adjust the standard errors. If it is substantial, however, then other models have to be 

considered. See Chapter 17 for more details. 

Diagnostic Tests 

There are a number of diagnostics tests that are used in a regression framework. 

Skewness Tests 

First, there are tests of skewness in the dependent variable. As mentioned above, the OLS 

model cannot be applied to data that are highly skewed. If they are skewed, a non-linear model, 

such as the Poisson, must be used. Therefore, it is essential to evaluate the degree of skewness. 

27.20 
An approximate Z-test can be obtained from: 

 

 

 

 

 

 

 

 

 

 

A commonly used measure of skewness is the g statistic (Microsoft, 2012): 

(27.9) 

where N is the sample size, Xi is observation ݅, ܺത is the mean of X, and s is the sample standard 

deviation (corrected for degrees of freedom): 

(27.10)

The standard error of skewness (SES) can be approximated by (Tabachnick & Fidell, 

1996): 

(27.11) 

(27.12)

Thus, if Z is greater than +1.96 or smaller than -1.96, then the skewness is significant at the p≤.05 

level. 

As an example, for the data on the origins of crimes by TAZ in Baltimore County, we have: 

ܺത = 75.108 (27.13) 

s = 96.017 (27.14) 

N = 325 (27.15) 

(27.16)

Therefore, 

(27.17) 

27.21 
In the Poisson model, the log-likelihood is: 

 

 

ఉ 

 

 

(27.18)

Z(g)=20.51 (27.19) 

The Z of the g value shows the data are highly skewed as was, of course, already known. 

Likelihood Ratio Test 

Second, there are tests of the overall model. In a maximum likelihood framework, the first 

test is of the log-likelihood function. A likelihood function is the joint density of all the 

observations, given a value for the parameters, β, and the variance, σ

2

. The log-likelihood is the 

natural log of this product, or the sum of the logs of the individual densities. For the OLS model, 

the log-likelihood is: 

(27.20)

where N is the sample size, σ

2

 is the variance, Yi is the observed number of events for zone ݅, and 

Xkiβk is a series of K independent predictors multiplied by their coefficients. 

(27.21) 

where λi is the conditional mean for zone ݅, Yi is the observed number of events for zone ݅, and Yi 

Xkiβk is a cross-product of the observed events times the K independent predictors multiplied by 

their coefficients. As mentioned above, Luc Anselin provides a more detailed discussion of these 

functions in Appendix B. 

Since the maximum likelihood method achieves the model with the highest log-likelihood, 

the log-likelihood is a negative number. Even though the model with the highest log-likelihood is 

considered ‘best’, it is not an intuitive number. Consequently, the Likelihood Ratio compares the 

log-likelihood of the regression model with the log-likelihood that would be obtained if only the 

mean number of counts was taken. This latter log-likelihood is: 

(27.22) 

27.22 
 

 

 

 

 

The Likelihood Ratio test is: 

ܮܴ ൌ 2ሺܮ െ ܮோሻ (27.23) 

where L is the model log-likelihood and LR is the log-likelihood of the mean count. The 

Likelihood Ratio is twice the difference between log-likelihood values of the regression and mean 

models respectively. It follows a χ2

 distribution with K degrees of freedom (where K is the 

number of independent variables).7 

Adjusted likelihood ratio 

The Likelihood Ratio is a more intuitive index since it is a Chi-square test. However, it is 

prone to the problem of all regression methods of over-fitting - the more independent variables are 

added to the model, the higher is the Likelihood Ratio. Consequently, there are several methods 

that adjust for the number of parameters fit. One is the Akaike Information Criterion (AIC) which 

is defined as: 

ܥܫܣ ൌ െ2ܮ ൅ 2ሺܭ ൅ 1ሻ (27.24) 

where L is the log-likelihood and K is the number of independent variables. A second one is the 

Bayesian Information Criterion/Schwartz Criterion (BIC/SC), which is defined as: 

BIC/SC		ൌ		‐2L൅ሾሺK൅1ሻlnሺNሻሿ (27.25) 

These two measures penalize the number of parameters added in the model, and reverse the 

sign of the log-likelihood (L) so that the statistics are more intuitive. The model with the lowest 

BIC/SC value is ‘best’. 

R-square Test

The most familiar test of an overall model is the R-square (or R2

) test. This is the percent 

of the total variance of the dependent variable accounted for by the model. More formally, it is 

defined as: 

(27.26)

Note, in Appendix B Luc Anselin uses K for the number of parameters (coefficients + intercept) whereas we 

use it for the number of independent variables. Readers should be aware of this difference. 

27.23 

7 
 

 

where Yi is the observed number of events for a zone ݅, Pi is the predicted number of events given 

a set of K independent variables, and ܻത is the mean number of events across zones. The R2 is a 

number from 0 to 1; 0 indicates no predictability while 1 indicates perfect predictability. 

R-square for OLS model

For an OLS model, R2

 is a very consistent estimate. It increases in a linear manner with 

predictability and is, therefore, a good indicator of how effective one model is compared to 

another. As with all diagnostic tests, the value of the R2

 increases with more independent 

variables. Consequently, R2

 is usually adjusted for degrees of freedom: 

(27.27)

where N is the sample size and K is the number of independent variables. 

R-square for Poisson model

With the Poisson model, however, the R2 value (whether adjusted or not) is not a good 

measure of overall fit. While the Poisson R2 varies from 0 to 1, similar to the OLS, it is not 

monotonic. That is, the addition of a new variable to an equation often has unpredictable effects; 

sometimes it will increase substantially and sometimes it will increase only a little independent of 

how strong is a variable’s association with the dependent variable (Miaou, 1996). This 

inconsistency comes from the decomposition of the total sum of squares: 

(27.28)

The first term in the equation is the residual sum of squares (or error term) while the second term is 

the explained sum of squares. In an OLS model, the third term is zero if an intercept is included 

(Cameron & Trivedi, 1998, 153). Hence, the total sum of squares is broken into two parts - that 

which is explained and that which is unexplained. However, for the Poisson and other non-linear 

regression methods, the last term is not zero. Consequently, a test that compares the explained 

sum of squares to the total sum of squares will not produce consistent results. 

Other measures have been proposed, such as the deviance R-square which measures the 

reduction in the Likelihood Ratio due to the inclusion of predictor variables (Cameron & 

Windmeijer, 1996). It produces a slightly different R-square, one that is typically higher than the 

traditional R-square. Nevertheless, it has problems, too. Miaou (1996) argues that there is not a 

27.24 
 

 

single R-square index that is perfectly consistent. The AIC, BIC/SC and Deviance statistics 

(discussed in Chapter 16) are better indicators of goodness of fit. 

Dispersion Parameter 

Finally, in the Poisson model only, the dispersion parameter indicates the extent to which 

the variance is different from the mean. This was defined in equation 27.7 above. 

Coefficients, Standard Errors, and Significance Tests 

The second type of diagnostic test is those for the individual predictors in the model. In 

both the OLS and Poisson models, there are three tests: 

1. The coefficient. This indicates the change in the dependent variable associated with

the change in the independent variable. In the case of the OLS, it is a linear term

(i.e., the value of the dependent variable is multiplied by the coefficient) while in

the Poisson model, the change in the dependent variable is estimated by

exponentiating the coefficient (i.e., eβX).

2. The standard error. Each estimated coefficient in a model accounts for some of the

variance in the dependent variable. This variance is the contribution of the

particular independent variable to the variance of the dependent variable. The

square root of that variance is the standard error.

3. The significance level. The ratio of the coefficient to the standard error produces a

significance test of the coefficient. In the OLS model, it is a t-test with N-K-1

degrees of freedom whereas in the Poisson model it is an asymptotic t-test, which is

effectively a Z-test. The appropriate tables (t-test or standard normal) produce

approximate probability levels of a Type I error (the likelihood of falsely rejecting a

true null hypothesis of no relationship).

Testing for Multicollinearity 

One of the major problems with any regression model, whether OLS or Poisson, is 

multicollinearity among the independent variables. In theory, each independent variable should be 

statistically independent of the other independent variables. Thus, the amount of variance for the 

dependent variable that is accounted for by each independent variable should be a unique 

contribution. In practice, however, it is rare to obtain completely independent predictive variables. 

More likely, two or more of the independent variables will be correlated. The effect is that the 

estimated standard error of a predictor variable is no longer unique since it shares some of the 

27.25 
 

variance with other independent variables. The greater communality of shared variance, the more 

ambiguous will be the predicted effects. If two variables are highly correlated, it is not clear what 

contribution each makes towards predicting the dependent variable. In effect, multicollinearity 

means that variables are measuring the same effect. 

Multicollinearity among the independent variables can produce very strange effects in a 

regression model. Among these effects are: 1) If two independent variables are highly correlated, 

but one is more correlated with the dependent variable than the other, the stronger one will usually 

have a correct sign while the weaker one will sometimes get flipped around (e.g., from positive to 

negative, or the reverse); 2) Two variables can cancel each other out; each coefficient is significant 

when it alone is included in a model but neither are significant when they are together; 3) One 

independent variable can inhibit the effect of another correlated independent variable so that the 

second variable is not significant when combined with the first one; and 4) If two independent 

variables are virtually perfectly correlated, many regression routines break down because the 

matrix cannot be inverted. 

All these effects indicate that there is non-independence among the independent variables. 

Aside from producing confusing coefficients, multicollinearity can overstate the amount of 

predictability in a model. Since every independent variable accounts for some of the variance of 

the dependent variable, with multicollinearity the overall model will appear to improve when it 

probably has not. 

Tolerance test 

A user has to be aware of the problem of multicollinearity and seek to minimize it. The 

simplest solution is to drop variables that are co-linear with other independent variables already in 

the equation. A relatively simple test for assessing this is called tolerance. Tolerance is defined as 

lack of predictability of each independent variable by the other independent variables, or: 

(27.29) 

where (Rjk..)

2

 is the R2 of an OLS equation where independent variable, ݅, is predicted by the other 

independent variables, j, k, l, and so forth. That is, each independent variable in turn is regressed 

against the other independent variables in the equation. The R2 associated with that model is 

subtracted from 1. The higher the tolerance level, the less a particular independent variable shares 

its variance with the other independent variables. 

Note that the tolerance test uses an OLS model; it assumes the dependent variable in the 

test (i.e., one of the independent variables) is normally distributed, which may or may not be true. 

Thus, in a Poisson or other non-linear model, one has to be careful about interpretation based on 

27.26 
 

the tolerance test. Nevertheless, the test can be a good indicator of whether two variables are 

collinear. As a rough guideline, a tolerance value of 0.7 or less usually indicates substantial 

multicollinearity. This level means that there is overlap of 50% or more in the variance of the 

tested variable with the other independent variables. A more strict and conservative approach uses 

a tolerance level of 0.8 or less as indicating multicollinearity. This level means that there is 

overlap of 36% or more in the variance of the tested variable with the other independent variables. 

An even stricter criterion is to use a tolerance level of 0.9 or less, essentially allowing 18% overlap 

in the variance of the tested variable with the other independent variables. In general, it is better to 

have a stricter model that has little multicollinearity. The interpretation of the coefficients will be 

cleaner and the model will generally be more reliable with other data sets. 

Fixed model v. stepwise variable selection 

There are several strategies designed to reduce multicollinearity in a model. One is to start 

with a defined model and eliminate those variables that have a low tolerance. The total model is 

estimated and the coefficients for each of the variables are estimated at the same time. This is 

sometimes called a fixed model. Then, variables that are co-linear are removed from the equation, 

and the model is re-run. 

Another strategy is to estimate the coefficients a step at a time, a procedure known as 

stepwise regression (Der & Everitt, 2002, 88-89). There are several standard stepwise procedures. 

In the first procedure, variables are added one at a time (a forward selection model). The 

independent variable having the strongest linear correlation with the dependent variable is added 

first. Next, the independent variable from the remaining list of independent variables with the 

highest correlation with the dependent variable, controlling for the one variable already in the 

equation, is added next and the model is re-estimated. In each step, the independent variable with 

the highest correlation with the dependent variable controlling for the variables already in the 

equation is added to the model, and the model is re-estimated. This proceeds until either all the 

independent variables are added to the equation or else a stopping criterion is met. The usual 

criterion is only variables with a specified significance level are allowed to enter (called a p-toenter). 

A backward elimination procedure works in reverse. All independent variables are initially 

added to the equation. The variable with the weakest coefficient (as defined by the significance 

level) is removed, and the model is re-estimated. Next, the variable with the weakest coefficient in 

the second model is removed, and the model is re-estimated. This procedure is repeated until 

either there are no more independent variables left in the model or else a stopping criterion is met. 

The usual criterion is that all remaining variables pass a specified significance level (called a p-toremove). 

27.27 
There are combinations of these procedures, for example adding variables in a forward 

selection but then removing any that are no longer significant or using a backward elimination 

procedure but allowing new variables to enter the model if they suddenly become significant. 

There are advantages to each approach. A fixed model allows defined variables to be all 

included. If either theory or previous research has indicated that a particular combination of 

variables is important, then the fixed model allows that to be tested. A stepwise procedure might 

drop one of those variables. On the other hand, a stepwise procedure usually can obtain the same 

or higher predictability than a fixed procedure (whether predictability is measured by a loglikelihood or an R-square). 

Within the stepwise procedures, there are also advantages and disadvantages to each 

method, though the differences are generally very small. A forward selection procedure adds 

variables one at a time. Thus, the contribution of each new variable can be seen. On the other 

hand, a variable that is significant at an early stage could become not significant at a later stage 

because of the unique combinations of variables. Similarly, a backward elimination procedure will 

ensure that all variables in the equation meet a specified significance level. But, the contribution 

of each variable is not easily seen other than through the coefficients. In practice, one usually 

obtains the same model with either procedure, so the differences are not that critical. 

A stepwise procedure will not guarantee that multicollinearity will be removed entirely. 

However, it is a good procedure for narrowing down the variables to those that are significant. 

Then, any co-linear variables can be dropped manually and the model re-estimated. In the 

CrimeStat trip generation, both a fixed model and a backward elimination procedure are allowed. 

Available Regression Models 

CrimeStat has 10 different regression models that can be used for trip generation and which 

can be estimated with either maximum likelihood (MLE) or Markov Chain Monte Carlo (MCMC): 

MLE Normal (OLS) 

MCMC Normal 

MCMC Normal-spatial autocorrelation component (CAR or SAR) 

MLE Poisson 

MLE Poisson with Linear Correction (NB1) 

MLE Negative Binomial (Poisson-Gamma) 

MCMC Poisson-Gamma 

MCMC Poisson-Gamma-spatial autocorrelation component (CAR or SAR) 

MCMC Poisson-Lognormal 

MCMC Poisson-Lognormal-spatial autocorrelation component (CAR or SAR) 

27.28 
Users should consult Chapters 16 and 17 for details of these models. There are other 

methods for estimating the likely value of a count given a set of independent predictors. Among 

these are the zero-inflated Poisson (or ZIP; Hall, 2000), the Weibul function, the Cauchy function, 

and the lognormal function (see NIST 2004 for a list of common non-linear functions). There are 

also other spatial regression type models that correct for spatial autocorrelation in the dependent 

variable, such as geographically-weighted regression using a Poisson function (Fotheringham, 

Brunsdon, & Charlton, 2002). These are not included in this version of CrimeStat. 

Adding Special Generators 

In a travel demand model, there are special generators. These are unique land uses or 

environments that produce an extra large number of trips. For regular travel demand modeling, 

stadiums, airports, train stations, large parks, and mega-malls generate more than their share of 

trips, or at least than what would be predicted by the amount of permanent employment at those 

locations. They are usually attractors, not producers. In a normal transportation travel demand 

model, these zones are excluded from the cross-classification and independent estimates are made 

of them. 

For crime trips, there are also special generators. Typically, these are zones that have more 

crimes being attracted to them than are expected on the basis of the population and employment at 

those locations. Since we are using a regression model to estimate the productions and attractions, 

a simple way to model a special generator is to create a simple dummy variable. This is a variable 

where zones with the special generator get a value ‘1' and zones without the special generator get a 

‘0'. Essentially, the variable is a cross-classification of the special generator versus every other 

zone. 

One has to be cautious is doing this, however. Typically, special generators are identified 

by having a greater number of crimes being attracted to a zone than is predicted by the model. In 

other words, they have a greater positive residual error (observed - predicted) and are ‘outliers’ in 

the residual error distribution. By adding a variable to explain those cases, the residual error 

decreases. But, in doing so, we are not really explaining why the zone has more crimes than 

expected, but simply accounting for it by putting in an empirical variable. In re-running the model, 

there will be, usually, new outliers that have a greater positive residual error. If this logic is to be 

repeated, then we would create new special generators for those zones and re-estimate the model. 

If continued without limits, eventually there would not be a model anymore but just a collection of 

dummy variables, one for each zone. 

Therefore, a user should be cautious in introducing special generators. It is generally alright 

to introduce a few for the truly exceptional zones. These are zones where it is logical to treat them 

as special generators and where one would expect continuity over time. In other words, they 

27.29 
 

should be used if the special generator status is expected to last over time. For example, a stadium 

or an airport or a train station is liable to remain at its location for many years. A particular 

shopping mall, on the other hand, may attract crimes at one particular point in time but not 

necessarily in the future. Unless a mall is so much larger than other malls in the region (a megamall), it should not be assigned a special generator status. 

Adding External Trips 

External trips are, by definition, trips that come from outside the region. They are part of 

the origin/production model in that these are trips that are not accounted for by the model. There 

are also trips that originate within the study area, but end outside the area; however, those are 

usually not modeled since the focus will be on the study area itself. In the usual travel demand 

framework, external trips are those coming from major corridors into the region. Estimates of the 

travel on these corridors are obtained by cordon counts, counts of vehicles coming into the region 

and leaving the region (net inflow). Estimates of future growth of those external trips has to based 

on expectations of future population growth the metropolitan region and in nearby regions. 

For crime trips, external trips are defined as trips that originate outside the study area. But 

they must be estimated by the difference between the total number of crimes occurring in the 

destination study area and the total originating in the origin zones. That is, of all the crimes 

occurring in the study area, the origin zones are modeled. Those trips that originate from outside 

the origin zones are external trips. They must be added to the predicted number of origin trips to 

produce an adjusted estimate of total origins, or: 

(27.30) 

where Oj is the total number of crime origins for crimes committed in study area, ݆, Opi is the total 

number of crimes originating in the origin zones, ݅, and Oe is the total number of crimes 

originating outside the region, e. 

In other words, for the production (origin) model only, we add an external zone to account 

for crime trips that originated outside the modeled region. If that is not done, in the balancing step 

the number of crimes originating in each zone will be overestimated because the predicted origins 

will be multiplied by a factor to ensure that the total number of origins equals the total number of 

destinations. 

Not including the external trips can lead to bias in the model. If the number of external 

trips is a sizeable percentage of all crime origins occurring in the study area, then the coefficients 

of the origin model could be misleading. In practice, most travel demand modelers assume that if 

27.30 
 

 

 

 

 

the percentage of external trips is not greater than 5%, there usually is little bias introduced 

(Ortuzar & Willumsen, 2001). If it is greater than 5%, then origin zones from adjacent 

jurisdictions need to be included in the origin model. 

Balancing Predicted Origins and Predicted Destinations 

The trip generation ‘model’ is actually two separate models: 1) a model of trips produced 

by every zone and 2) a model of trips attracted to every zone. Since a trip has an origin and a 

destination (by definition), then the total number of productions must equal the total number of 

attractions, 

(27.31)

where O is a trip origin, D is a trip destination, and ݅ and ݆ are zone numbers. Note that in equation 

27.31, there are M origin zones and N destination zones. This implies that M and N do not have to 

be equal. In fact, including an external zone guarantees that M and N will not be equal (M will be 

at least 1 greater than N). If an entire metropolitan area is being modeled, the M and N will be 

almost identical (differing only in the external zone). However, if the study area being modeled is 

a sub-set of the metropolitan region, then M will be much greater than N. For example, in 

modeling crime trips in Baltimore County, there are 532 origin zones (including those from 

Baltimore County and from the City of Baltimore) and only 325 destination zones (only those in 

Baltimore County). 

To ensure that this equality is true, a balancing operation is conducted. Essentially, this 

means multiplying either the number of predicted origins in each origin zone or the number of 

predicted destinations in each destination zone by a constant which is the ratio of either the total 

destinations to the total origins (to multiply the number of predicted origins) or the ratio of the total 

origins to the total destinations (to multiply the number of predicted destinations). 

With crime analysis, the number of destinations would generally be considered a more 

reliable data set than the number of origins. Because crimes are enumerated where they occur, the 

number of crimes occurring at any one location is more accurate than the location of the offenders. 

Thus, we adjust the predicted origins so that they equal the predicted destinations.8 

In the usual travel demand modeling, on the other hand, modelers usually adjust the predicted destinations 

since the origin data is more reliable. These numbers are obtained from the census or from the sample of 

households who are interviewed to produce a sample from which data on destinations are obtained. 

27.31 

8 
Summary of the Trip Generation Model 

In summary, the trip generation model is estimated in four steps: 

1. A model of the predictors of the number of crimes origins (a crime production 

model); 

2. A model of predictors of the number of crime destinations (a crime attraction 

model); 

3. External trips are estimated and added to the number of predicted origins as an 

external zone; and 

4. The total number of predicted crime origins is balanced to be equal to the total 

number of predicted crime destinations. 

The CrimeStat Trip Generation Model 

In this section, we describe the trip generation model implemented in CrimeStat. As 

mentioned above, this step involves calibrating a regression model against the zonal data. Two 

separate models are developed, one for trip origins and one for trip destinations. The dependent 

variable is the number of crimes originating in a zone (for the trip origin model) or the number of 

crimes ending in a zone (for the trip destination model). The independent variables are zonal 

variables that may predict the number of origins or destinations. 

There are three steps to the model, each corresponding to a separate tab in CrimeStat: 

1. Calibrate the model 

2. Make a prediction 

3. Balance the predicted origins and the predicted destinations 

Figure 27.5 shows an image of the trip generation model page within CrimeStat. The trip 

generation model is made up of three separate pages (or tabs): 

1. A Calibrate model page in which a regression model can be run to estimate either 

an origin (production) model or a destination (attraction) model; 

2. A Make prediction page in which the estimated coefficients can be applied to the 

same or a different data set and in which the external trips can be added to the 

predicted origins; and 

27.32 
Figure 27.5: 

Trip Generation Module 
3. A Balance predicted origins & destinations page in which the total predicted 

origins can be adjusted to equal the total predicted destinations. 

Calibrate Model 

In the first step, models are calibrated using the input data. There is a model for the origin 

zones and another model for the destination zones. The user should indicate what type of model is 

being run in order to make the output more clear. 

Data File 

The data file is input as either the primary or secondary file. Specify whether the data file 

is the primary or secondary file. 

Type of Model 

Specify whether the model is for origins or destinations. This will be printed out on the 

output header. 

Dependent Variable 

Select the dependent variable from the list of variables. There can be only one dependent 

variable per model. 

Skewness Diagnostics 

If checked, the routine will test for the skewness of the dependent variable. The output 

includes: 

1. The ‘g’statistic 

2. The standard error of the ‘g’ statistic 

3. The Z value for the ‘g’ statistic 

4. The probability level of a Type I error for the ‘g’ statistic 

5. The ratio of the sample variance to the sample mean 

Error messages indicate whether there is probable skewness in the dependent variable. If 

there is skewness, use a Poisson regression model. 

27.34 
Independent variables 

Select independent variables from the list of variables in the data file. Up to 15 variables 

can be selected. 

Missing values 

Specify any missing value codes for the variables. Blank records will automatically be 

considered as missing. If any of the selected dependent or independent variables have missing 

values, those records will be excluded from the analysis. 

Type of Regression Model 

Specify the type of regression model to be used. The default is a Poisson regression with 

over-dispersion correction (NB1). Other alternatives are: 

1. Ordinary Least Squares regression; 

2. Poisson regression; 

3. MLE Poisson-Gamma (negative binomial; NB2); 

4. MCMC Poisson-Gamma; 

5. MCMC Poisson-Lognormal; and 

6. MCMC Conway-Maxwell Poisson. 

Each of the MCMC models can be run with a spatial autocorrelation component added, 

either a CAR or a SAR. See Chapters 16 and 17 for more details. 

Type of Regression Procedure 

If the model being run is an MLE routine (Poisson, Poisson with linear correction (NB1), 

or Poisson-Gamma (NB2), specify whether a fixed model (all selected independent variables are 

used in the regression) or a backward elimination stepwise model is used. The default is a fixed 

model. If a backward elimination stepwise model is selected, choose the P-to-remove value 

(default is .01). The backward elimination starts with all selected variables in the model (the fixed 

procedure). However, it proceeds to drop variables that fail the P-to-remove test, one at a time. 

Any variable that has a significance level in excess of the P-to-remove value is dropped from the 

equation. 

With MCMC routines, however, only fixed models can be run. 

27.35 
Save Estimated Coefficients/Parameters 

The estimated coefficients of the final model can be saved as a ‘dbf’ file. Specify a file 

name. This would be useful in order to repeat the regression while adding in external trips to the 

predicted origins (see Make trip generation prediction below) or to apply the coefficients to 

another dataset (e.g., future values of the independent variable). 

Save Output 

The output is saved as a ‘dbf’ file under a different file name. The output includes all the 

variables in the input data set plus two new ones: 1) the predicted values of the dependent variable 

for each observation (with the name PREDICTED); and 2) the residual error values, representing 

the difference between the actual /observed values for each observation and the predicted values 

(with the name RESIDUAL). 

Poisson output 

The output of the Poisson regression routines includes 13 fields for the entire model: 

1. The dependent variable 

2. The type of model 

3. The sample size (N) 

4. The degrees of freedom (N - # dependent variables B 1) 

5. The type of regression model (Poisson, Poisson with over-dispersion 

correction) 

6. The log-likelihood value 

7. The Likelihood Ratio 

8. The probability value of the Likelihood Ratio 

9. The Akaike Information Criterion (AIC) 

10. The Bayesian Information Criterion/Schwartz Criterion (BIC/SC) 

11. The Dispersion Multiplier 

12. The approximate R-square value 

13. The deviance R-square value 

and 5 fields for each estimated coefficient: 

14. The estimated coefficient 

15. The standard error of the coefficient 

16. The pseudo-tolerance value of the coefficient (see below) 

27.36 
17. The Z-value of the coefficient 

18. The p-value of the coefficient. 

OLS output 

The output of the Ordinary Least Square (OLS) routine includes 9 fields for the entire 

model: 

1. The dependent variable 

2. The type of model 

3. The sample size (N) 

4. The degrees of freedom (N - # dependent variables B 1) 

5. The type of regression model (Norma/Ordinary Least Squares) 

6. Squared multiple R 

7. Adjusted squared multiple R 

8. F test of the model 

9. p-value of the model 

and 5 fields for each estimated coefficient: 

10. The estimated coefficient 

11. The standard error of the coefficient 

12. The tolerance value of the coefficient (see below) 

13. The t-value of the coefficient 

14. The p-value of the coefficient. 

Multicollinearity Among Independent Variables 

To test multicollinearity, a tolerance test is run (see equation 27.29 above). There is not a 

simple test of whether a particular tolerance is meaningful or not. In CrimeStat, several qualitative 

categories are used and error messages are output: 

1. If the tolerance value is 0.80 or greater, then there is little multicollinearity (No 

apparent multicollinearity); 

2. If the tolerance is between 0.60-0.79, there is some multicollinearity (possible 

multicollinearity); 

3. If the tolerance is between 0.25-0.59, there is probable multicollinearity (probable 

multicollinearity. Eliminate variable with lowest tolerance and re-run); and 

27.37 
 

4. If tolerance is less than 0.25, there is definite multicollinearity (Definite 

multicollinearity. Results are not reliable. Eliminate variable with lowest tolerance 

and re-run). 

Graph 

While the output page is open, clicking on the graph button will display a graph of the 

residual errors (on the Y axis) against the predicted values (on the X axis). 

Make Trip Generation Prediction 

This routine applies an already-calibrated regression model to a data set. This would be 

useful for several reasons: 1) if external trips are to be added to the model (which is normally 

preferred); 2) if the model is applied to another data set; and 3) if variations on the coefficients are 

being tested with the same data set. The model will need to be calibrated first (see Calibrate Trip 

Generation Model) and the coefficients saved as a parameters file. The coefficient parameter file 

is then re-loaded and applied to the data. 

Data File 

The data file is input as either the primary or secondary file. Specify whether the data file 

is the primary or secondary file. 

Type of Model 

Specify whether the model is for origins or destinations. This will be printed out on the 

output header. 

Trip Generation Coefficients/Parameters File 

This is the saved coefficient parameter file. It is an ASCII file and can be edited if 

alternative coefficients are being tested (be careful about editing this without making a backup). 

Load the file by clicking on the Browse button and finding the file. Once loaded, the variable 

names of the saved coefficients are displayed in the ‘Matching parameters’ box. 

27.38 
Missing Values

Independent Variables 

Select independent variables from the list of variables in the data file. Up to 15 variables 

can be selected. 

Matching parameters 

The selected independent variables need to be matched to the saved variables in the trip 

generation parameters file in the same order. Add the appropriate variables one by one in the order 

in which they are listed in the matching parameters box. It is essential that the order by the same 

otherwise the coefficients will be applied to the wrong variables. 

Hint: With your cursor placed in the list of independent variables, typing the first 

letter of the matching variable name will take you to the first variable that starts with 

that letter. Repeating the letter will move down the list to the second, third, and so 

forth until the desired variable is reached. 

Specify any missing value codes for the variables. Blank records will automatically be 

considered as missing. If any of the selected dependent or independent variables have blank 

values, those records will be excluded from the analysis. 

Add External Trips 

External trips are those that start outside the modeled study area. Because they are crimes 

that originate outside the study area, they were not included in the zones used for the origin model. 

Therefore, they have to be independently estimated and added to the origin zone total to make the 

number of origins equal to the number of destinations. Click on the ‘Add external trips’ button to 

enable this feature. 

Origin ID 

Specify the origin ID variable in the data file. The external trips will be added as an extra 

origin zone, called the ‘External’ zone. Note: the ID’s used for the destination file zones should be 

the same as in the origin file. This will be necessary in subsequent modeling stages. 

Number of external trips 

Add the number of external trips to the box. This number will be added as an extra origin 

zone (the External zone). 

27.39 
 

Type of Regression Model 

Specify the type of regression model to be used. The default is a Poisson regression and the 

other alternative is an Ordinary Least Squares regression. 

Save Predicted Values 

The output is saved as a >dbf= file under a different file name. The output includes all the 

variables in the input data set plus the predicted values of the dependent variable for each 

observation (with the name PREDICTED). In addition, if external trips are added, then there 

should be a new record with the name EXTERNAL listed in the Origin ID column. This record 

lists the added trips in the PREDICTED column and zeros (0) for all other numeric fields. 

Output 

The tabular output includes summary information about file and lists the predicted values 

for each input zone. 

Balance Predicted Origins & Destinations 

Since, by definition, a ‘trip’ has an origin and a destination, the number of predicted origins 

must equal the number of predicted destinations. Because of slight differences in the data sets of 

the origin model and the destination model, it is possible that the total number of predicted origins 

(including any external trips) may not equal the total number of predicted destinations. This step, 

therefore, is essential to guarantee that this condition will be true. The routine adjusts either the 

number of predicted origins or the number of predicted destinations so that the condition holds. 

The trip distribution routines will not work unless the number of predicted origins equals the 

number of predicted destinations (within a very small rounding-off error). 

Predicted Origin File 

Specify the name of the predicted origin file by clicking on the Browse button and locating 

the file. 

Origin variable 

Specify the name of the variable for the predicted origins (e.g., PREDICTED). 

27.40 
Predicted Destination File 

Specify the name of the predicted destination file by clicking on the Browse button and 

locating the file. 

Destination variable 

Specify the name of the variable for the predicted origins (e.g., PREDICTED). 

Balancing Method 

Specify whether origins or destinations are to be held constant. The default is >Hold 

destinations constant=. 

Save Predicted Origin/Destination File 

The output is saved as a ‘dbf’ file under a different file name. The output includes all the 

variables in the input data set plus the adjusted values of the predicted values of the dependent 

variable for each observation. If destinations are held constant, the adjusted variable name for the 

predicted trips is ADJORIGIN. If origins are held constant, the adjusted variable name for the 

predicted trips is ADJDEST. 

Output 

The tabular output includes file summary information plus information about the number of 

origins and destinations before and after balancing. In addition, the predicted values of the 

dependent variable are displayed. 

Example of the Trip Generation Model 

To illustrate this model, an example from Baltimore County. In the case of Baltimore 

County, MD, will be used. The zonal geography is traffic analysis zones (TAZ). Two data sets 

were produced, one for the crime origins and one for the crime destinations. For Baltimore 

County, the origin data set had 532 zones covering both Baltimore County and the City of 

Baltimore with the total number of crime origins for each zone (sub-divided into different crime 

types - robberies, burglaries, vehicle theft) and a number of possible predictor variables 

(population, retail and non-retail employment, median household income, poverty levels, and 

vehicle ownership). Similarly, the destination data set had 325 zones with the number of crime 

destinations for each zone (again, sub-divided into different crime types) and number of possible 

predictor variables (population, retail and non-retail employment, median household income, and 

27.41 
several land use categories - acreage allocated for retail, residential, office space, and conservation 

uses). Sample data sets are provided on the CrimeStat download page. 

Setting Up the Origin Model 

In the first step, an origin model is created. Figure 27.6 shows the selection of the 

dependent variable and some possible independent variables. The type of model is an ordinary 

Poisson regression. The dependent variable is the number of crimes occurring between 1993 and 

1997 in each origin zone (BCORIG). Eight possible independent variables have been selected: 

1. 1996 population of each zone (POPULATION) 

2. 1990 median household income of the zone relative to the zone with the highest median 

household income (INCOME EQUALITY) 

3. Number of 1996 non-retail employees in each zone (NON-RETAIL EMPLOYMENT) 

4. Number of 1996 retail employees in each zone (RETAIL EMPLOYMENT) 

5. Total linear miles of arterial roads in each zone (ARTERIAL ROADS) 

6. A dummy variable for whether the Baltimore Beltway (I-695) passed through the zone 

or not (BELTWAY) 

7. Linear distance of the zone from Baltimore harbor in the CBD (DISTANCE FROM 

CENTER); and 

8. 1990 Number of households without automobiles (HOUSEHOLDS WITH NO 

AUTOMOBILES) 

The model is set up to run an ordinary Poisson regression (without an adjustment to the 

dispersion). It is a fixed model in which all independent variables are included. The coefficients 

are saved under ‘Save estimated coefficients’ dialogue box and the output (the predicted values) 

are saved under the ‘Save output’ dialogue box. Both boxes ask for a file name. 

Table 27.2 shows the results. The format is simplified from that shown in Chapter 16. Key 

statistics are highlighted. The overall model is highly significant. The log likelihood is shown as 

are the AIC and BIC/SC adjusted log likelihood. The deviance and Pearson are highly significant, 

indicating that the model predicts significantly better than chance. The coefficients for each of the 

variables are all significant. 

However, there are two major problems. First, the dispersion multiplier (parameter) is very 

large (36.09) and significant, indicating that the conditional variance is more than 36 times greater 

than the conditional mean. Second, while all of the coefficients are significant, several show 

sizeable multicollinearity as evidenced by the pseudo-tolerance value (POPULATION, 

DISTANCE, HOUSEHOLDS WITH NO AUTOMOBILES as well as INCOME EQUALITY). 

This indicates that these variables are essentially measuring the same thing. 

27.42 
 

 

 

 

 

 

 

 

 

 

 

 Z-value 

 4.1890 

0.0003 

Tolerance 

 

 

 

 -18.91 

 

0.00002 

 

 0.1510 

0.0343 

 

Predictor Coefficient Stand Error p-value 

CONSTANT 0.0202 . 207.03 0.001 

POPULATION 0.000003 0.46 121.13 0.001 

INCOME 

EQUALITY -0.0330 0.0007 0.61 -48.85 0.001 

NON-RETAIL 

EMPLOYMENT -0.0002 0.000005 0.84 -36.87 0.001 

RETAIL 

EMPLOYMENT -0.0004 0.96 0.010 

ARTERIAL ROAD -0.1083 0.0059 0.77 0.001 

BELTWAY 0.0193 0.96 7.84 

 -18.49 

0.001 

DISTANCE 

FROM CENTER 0.0016 0.49 21.09 0.001 

HOUSEHOLDS 

WITH NO 

AUTOMOBILES -0.0005 0.00002 0.36 -18.95 0.010 

 

---------------------------------------------------------------------------------------------------------------------- 

---------------------------------------------------------------------------------------------------------------------- 

Table 27.2: 

Full Origin Model: Poisson 

Model result: 

Data file: BaltOrigins.dbf 

Type of model: Origin 

DepVar: BCORIG 

N: 532 

Df: 522 

Type of regression model: MLE Poisson 

Likelihood statistics 

Log Likelihood: -10,678.05 

AIC: 21,376.10 

BIC/SC: 21,418.87 

Deviance: 18,547.38 p≤.0001 

Pearson Chi-square: 19,396.48 p≤.0001 

Model error estimates 

Mean absolute deviation: 38.70 

Mean squared predicted error: 3,920.66 

Dispersion tests 

Dispersion multiplier: 36.09 p≤.01 

27.43 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Predictor Coefficient Stand Error Tolerance Z-value p-value 

CONSTANT 3.4832 0.131 . 26.61 0.001 

POPULATION 0.0004 0.00003 0.95 17.10 0.001 

INCOME 

EQUALITY -0.0178 0.003 0.91 -5.46 0.001 

NON-RETAIL 

EMPLOYMENT -0.0001 0.00002 0.87 -6.71 0.001 

RETAIL 

EMPLOYMENT -0.0002 0.0001 0.96 -2.17 0.05 

--------------------------------------------------------------------------------------------------------------------- 

---------------------------------------------------------------------------------------------------------------------

Restructuring the Origin Model 

Consequently, the model was restructured in three ways (Figure 27.7). First, to correct for 

over-dispersion, an MLE Poisson-Gamma (negative binomial) model was run. This is the most 

common approach to handling over-dispersion (see Chapter 16). Second, two co-linear variables - 

DISTANCE and ZEROAUTO, were dropped from the model. Third, a stepwise backward 

elimination procedure is used with the probability for keeping a variable in the equation (p-toremove) being 0.01; that is, unless the probability that a coefficient could be obtained by chance is 

less than 1 in 100, the variable was dropped. 

Table 27.3: 

Reduced Origin Model: Poisson-Gamma 

Model result: 

Data file: BaltOrigins.dbf 

Type of model: Origin 

DepVar: BCORIG 

N: 532 

Df: 526 

Type of regression model: MLE Poisson-Gamma 

Likelihood statistics 

Log Likelihood: -2,627.65 

AIC: 5,267.30 

BIC/SC: 5,292.96 

Deviance: 623.49 p≤.0001 

Pearson Chi-square: 500.59 p≤.0001 

Model error estimates 

Mean absolute deviation: 57.28 

Mean squared predicted error: 18,143.78 

Dispersion tests 

Dispersion multiplier: 0.74 n.a. 

27.44 
Figure 27.6: 

Origin Poisson Model Setup 
Figure 27.7: 

Origin Poisson-Gamma Model Setup 
 

 

The result is a model with four significant variables. Note that the full Poisson model 

(Table 27.3) has a greater negative log likelihood and a much greater AIC and BIC/SC value than 

the reduced model. This is because the reduced model was tested with a Poisson-Gamma mixed 

function and has a different probability structure. To properly compare it, the full model was run 

as a Poisson-Gamma (not shown). In the reduced model, the log likelihood was -2,627, which is 

even stronger than -2,618 for the full model, while the AIC was 5,267 and the BIC/SC was 5,293, 

compared to 5,526 and 5,299 for the full model respectively. In other words, the reduced model 

produced likelihood values very similar to the full model. More importantly, the overall fit of the 

model was almost as good as the full model. The mean absolute deviation was 57, compared to 55 

for the full model, while the mean squared predicted error was 18,144, compared to 17,881 for the 

full model. Most importantly, the dispersion parameter is now less than 1.0.9 

In other words, we have a simpler model that predicts almost as well as the full model but 

with coefficients that are less ambiguous. Such a model is liable to be more stable because the 

Poisson-Gamma has adjusted for over-dispersion in the Poisson while collinear and less significant 

variables have been removed. 

Looking at the model, we see four variables that significantly predict the number of crime 

origins. Population is the strongest, as indicated by its Z-test. Non-retail employment is the next 

strongest with a negative coefficient (i.e., zones with less non-retail employment generate more 

crime trips). This is followed by relative income equality is the next strongest, also with a negative 

coefficient (i.e., zones with low relative income equality produce more crime origins). The fourth 

variable is retail employment and, like non-retail employment, the coefficient is also negative. In 

other words, zones with less overall employment produce more crime trips. 

Residual Analysis of Origin Model 

The CrimeStat output includes a graph of the residual errors (actual values minus the 

predicted values) on the Y-axis by the predicted values on the X-axis. It is important to examine 

the residual errors as these can indicate outliers, problems in the data, and violation of 

assumptions. Figure 27.8 shows an image of the residual graph screen. As seen, the errors 

increase with the value of the predicted dependent variable. With the Poisson model, this is 

expected and does not indicate the violation of the independent errors assumption, as it does with 

the OLS. The errors are reasonably symmetrical and do not indicate differences in over- and 

under-estimation across the band of the predicted values. 

There are some outliers. There are two zones in which the predicted number of crimes 

originating from the zones substantially exceeded the number that actually originated from those 

A test of the dispersion parameter is not appropriate since it only tests for over-dispersion, not underdispersion. 

27.47 

9 
Figure 27.8: 

Plot of Residual Errors and Predicted Values 
zones and there is one zone that had more crimes originate from it than was predicted by the 

model. But, in general, the model appears to be reasonably balanced. 

Setting Up the Destination Model 

The same logic was applied for the destination model. In this case, the destination file has 

data on 325 zones within Baltimore County only. Similar possible predictor variables are included 

in the file. Aside from population, retail and non-retail employment, and the roadway variables, 

more detailed analysis on land uses were included (acreage of commercial, residential, office 

space, recreational, and conservation lands). The model that was run was a Poisson-Gamma 

(negative binomial) because the simple Poisson showed very high over-dispersion. Again, a 

backward elimination procedure was adopted. Once a final model was selected, it was re-run as a 

fixed model to ensure that the coefficients were consistently estimated. Table 27.4 presents the 

results. 

Four variables ended up in the final model. Again, population was significantly related to 

the number of crimes attracted to a zone, but was not the strongest predictor as indicated by the Ztest. The strongest relationship was for the number of retail employees. This suggests that 

retail/commercial areas attract many crimes. Two other variables are in the equation. Relative 

income equality was, again, negatively related to crime destinations/attractions; zones with low 

income tend to attract more crimes. Also, there was a negative association with distance from the 

CBD. The farther away from the CBD was the zone, the lower the number of crimes. Overall, the 

model suggests that zones with commercial activities, which are closer to the city center, and 

which have households with relatively lower incomes are those that attract the most crimes. 

The overall model was highly significant, as indicated by the Deviance and the Pearson 

Chi-square. The amount of multicollinearity is very low, which is ideal. Even though a model 

with more negative log likelihood (and more positive AIC and BIC/SC) could be produced by 

adding more variables, the amount of multicollinearity would be substantial. The philosophy 

expressed here is that a simpler model, but with little multicollinearity, is to be preferred over a 

more complex model but where the coefficients are less stable and more ambiguous. Generally, 

simpler models hold up better with new data sets (Radford, 2006; Nannen, 2003). 

Residual Analysis of Destination Model 

As with the origin model, an analysis was conducted of the residual errors. This time, the 

output >dbf= file was brought into Excel and a nicer graph created (Figure 27.9). Unlike the best 

origin model, the dispersion of the residuals is not symmetrical. There are several major outliers, 

both on the negative end of the residuals (over-estimation of crime attractions) and on the positive 

end (under-estimation of crime attractions). In particular, there are two zones that seem to stand 

27.49 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Predictor Coefficient Stand Error Tolerance Z-value p-value 

CONSTANT 4.5208 0.153 . 29.53 0.001 

POPULATION 0.0003 0.00003 0.94 12.06 0.001 

INCOME 

EQUALITY -0.0213 0.003 0.90 -7.87 0.001 

RETAIL 

EMPLOYMENT 0.0020 0.0001 0.94 17.49 0.001 

DISTANCE 

FROM CENTER -0.0714 0.010 0.88 -7.46 0.001 

---------------------------------------------------------------------------------------------------------------

---------------------------------------------------------------------------------------------------------------

out. Both of them have shopping malls (Golden Ring Mall and Eastpoint Mall) but the amount of 

crime in those zones was much greater than the model predicted. This is seen as high positive 

residuals (i.e., there were more actual crimes than predicted). They both are older malls, but are 

located in relatively high crime areas. Golden Ring Mall was demolished some years ago, but 

after the data used in this example were collected. 

Table 27.4: 

Reduced Destination Model: Poisson-Gamma 

Model result: 

Data file: BaltOrigins.dbf 

Type of model: Origin 

DepVar: BCDEST 

N: 325 

Df: 319 

Type of regression model: MLE Poisson-Gamma 

Likelihood statistics 

Log Likelihood: -1,697.01 

AIC: 3,406.03 

BIC/SC: 3,428.73 

Deviance: 350.74 p≤.0001 

Pearson Chi-square: 379.87 p≤.0001 

Model error estimates 

Mean absolute deviation: 167.43 

Mean squared predicted error: 2,893,931.60 

Dispersion tests 

Dispersion multiplier: 0.43 n.a. 

27.50 
Figure 27.9: 
 

Adding in Special Generators 

Since the number of crime incidents (attractions) in those two zones was much higher than 

expected, they were treated as ‘special generators’. Keeping in mind the caution that one does not 

want to over-use this category, a demonstration of how it works will be illustrated. Two new 

variables were created for the data set. One was for the Golden Ring Mall and one was for the 

Eastpoint Mall. For the Golden Ring Mall, the zone that included it received a ‘1' for this variable 

while all other zones received a ‘0'. Similarly, for the Eastpoint Mall variable, the zone in which it 

occurred received a ‘1' while all other zones received a ‘0'. These dummy variables were then 

included in the model (Table 27.5). 

Adding the two special generators produces a model that, on the face of it, has not 

improved the predictability.. The log likelihood value is less negative than without the special 

generators and the AIC and BIC/SC statistics are also lower. The deviance values are about the 

same. 

However, the Pearson Chi-square is quite a bit higher with the special generators. Also, the 

Mean Absolute Deviation (MAD) and the Mean Squared Predictive Error (MSPE) are 

substantially better with the special generators. This indicates that the new model which includes 

the special generators fit the data much better. 

The coefficients for the two zones, treated as special generators, are both significant though 

not as strongly as the other variables. All other variables have the same relationships as in the first 

run. There is little multicollinearity. In other words, adding dummy variables for the two zones 

with higher than expected numbers of crime committed has produced a closer fitting model than 

not including the dummy variables. 

This brings up an issue over the status of a special generator. In this example, the two 

zones were treated as special generators in the model. While the model fit increased substantially, 

one has to wonder whether this was a meaningful operation or not? That is, if this model were 

applied to data for a later time period (e.g., 2010-2012 crime data), would the relationships still 

hold? In the case of the Golden Ring Mall, it would not since that mall has since been demolished 

The value of a special generator is that it identifies a land use that would be expected to be 

relatively permanent (e.g., a stadium or a train station or an airport). If it is a high visibility 

‘regional’ mall, then treating it as a special generator is probably a good idea. If it is a smaller, 

older mall, on the other hand, the analysis is guessing that the mall will maintain its status as a high 

crime attraction location. Clearly, judgment and knowledge of the particular mall is essential. 

27.52 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Predictor Coefficient Stand Error Tolerance Z-value p-value 

CONSTANT 4.4625 0.149 . 29.91 0.001 

POPULATION 0.0004 0.00003 0.93 12.72 0.001 

INCOME 

EQUALITY -0.0205 0.003 0.90 -7.84 0.001 

RETAIL 

EMPLOYMENT 0.0018 0.0001 0.90 16.44 0.001 

DISTANCE 

FROM CENTER -0.0686 0.009 0.87 -7.30 0.001 

GOLDEN RING 

MALL 1.5163 0.645 0.98 2.35 0.05 

EASTPOINT 

MALL 1.6111 0.648 0.97 2.49 0.05 

-------------------------------------------------------------------------------------------------------------------------------

-------------------------------------------------------------------------------------------------------------------------------

Table 27.5: 

Destination Model with Special Generators: Poisson-Gamma 

Model result: 

Data file: BaltOrigins.dbf 

Type of model: Origin 

DepVar: BCDEST 

N: 325 

Df: 319 

Type of regression model: MLE Poisson-Gamma 

Likelihood statistics 

Log Likelihood: -1,688.30 

AIC: 3,392.60 

BIC/SC: 3,422.87 

Deviance: 350.89 p≤.0001 

Pearson Chi-square: 398.73 p≤.0001 

Model error estimates 

Mean absolute deviation: 116.54 

Mean squared predicted error: 1,017,064.23 

Dispersion tests 

Dispersion multiplier: 0.40 n.a. 

Comparing Different Crimes Types 

With or without special generators, a trip generation model is an ecological model that 

predicts crime origins and crime destinations. A point was made in Chapter 25 that these models 

are not behavioral, but are correlates of crimes. That is, the variables that end up predicting the 

number of crimes are not reasons (or explanations) for the crimes. Population almost always 

27.53 
 

CONSTANT 

 

enters the equation because, all other things being equal, zones with larger numbers of persons will 

have more crimes, both originating and ending in them. Similarly, low income status is frequently 

associated with high crime areas. It does not follow that low income persons will be more prone to 

commit crimes; it may be true but these models do not test that proposition (Ratcliffe, 2008). 

These are only correlates with crime in those environments. As was mentioned earlier, these 

variables are often correlated with many specific conditions that may be predictors of individual 

crime - poverty, drug use, substandard housing, and lack of job opportunities. 

To see this, three separate models of specific crime types were run for robbery, burglary, 

and vehicle theft. For each crime type, the general model was tested for both the origin and the 

destination models. If a variable was not significant, it was dropped and the model was re-run. 

Table 27.6: 

Models for Specific Crime Types: Poisson-Gamma Origin Model

 Veh

Theft 

icle 

All Crimes Robbery Burglary 

 3.483 1.1165 1.1165 -1.4994 

POPULATION 0.0004 0.0004 0.0004 0.0005 

INCOME 

EQUALITY -0.0178 - - -0.0214 

NON-RETAIL 

EMPLOYMENT -0.0001 -0.0002 -0.0002 -0.0001

RETAIL 

EMPLOYMENT -0.0002 - - -

27.54 
 

 

 

 

 

 

Table 27.7: 

Models for Specific Crime Types: Poisson-Gamma Destination Model 

Vehicle 

All Crimes Robbery Burglary Theft 

CONSTANT 4.5208 2.7489 0.7977 2.2903 

POPULATION 0.0003 0.0003 0.0004 0.0004 

INCOME 

EQUALITY -0.0213 -0.0295 -0.0220 -0.0131 

RETAIL 

EMPLOYMENT 0.0020 0.0019 - 0.0009 

DISTANCE 

FROM CBD -0.0714 -0.0965 -0.0365 -0.1013 

The population variable appears in every single model. As mentioned, all other things 

being equal, the larger the number of persons in a zone, the more crime events will occur whether 

those events are crime productions (origins) or crime attractions (destinations). Similarly, relative 

income equality appears in four of the six crime-specific models with the coefficient always being 

negative. In general, zones with relatively lower incomes will have more robberies, burglaries, and 

vehicle thefts. The only model for which income equality did not appear was as an origin variable 

for burglaries; apparently, burglars come from zones with various income levels, at least in 

Baltimore. 

The other general variables have more limited applicability. Retail employment predicts 

both total crime origins and total crime destinations, but only predicts specifically robbery 

destinations and vehicle theft destinations; the latter tend to occur more in commercial areas than 

not. On the other hand, non-retail employment appears to be important only as a crime origin 

variable; zones with less non-retail employment tend to produce more offender trips. Distance 

from the CBD only appears as a destination variable; the closer a zone is to the metropolitan 

center, the higher the number of crimes being attracted to that zone; this variable was not important 

in the origin model. 

In other words, these models are measuring general conditions associated with crime, not 

causes per se. They capture the general contextual relationships associated with crime productions 

and attractions. But, they do not necessarily predict individual behavior. Nevertheless, the models 

can be used for prediction since the conditions appear to be quite general. 

27.55 
Adding External Trips to the Origin Model 

After an origin and destination model has been developed, the next step is to add any crime 

trips that came from outside the modeling area (external trips). In this case, these would be trips 

that came from areas that were not in either Baltimore County or the City of Baltimore (the 

modeling area). 

A simple estimate of external trips is obtained by taking the difference between the total 

number of crimes occurring in the study area (Baltimore County destinations) and the total number 

of crimes originating in the modeling area (Table 27.8). 

The difference between the number of crime enumerated within Baltimore County and that 

originating from both Baltimore County and the City of Baltimore is 1,627. This is 3.9% of the 

total Baltimore County crimes. In general, it is important that the external trips be as small as 

possible. Ortuzar and Willumsen (2001) suggest that this percentage be no greater than 5% in 

order to minimize potential bias from not including those cases in the origin model. It is not an 

absolute percentage, but more like a rule of thumb; in theory, any external trips could bias the 

origin model. But, in practice, the error will be small if external crime trips are a small percentage 

of the total number enumerated in the destination county. 

In this case, the condition holds. For the three types of crime modeled, the percentage of 

external trips was also less than 5%: robbery (4.0%), burglary (4.5%), and vehicle theft (1.4%). On 

the other hand, if the percentage of external trips is greater than approximately 5%, a user would 

be advised to widen the origin study area to include more zones in the model. 

Predicting External Trips 

If a model is being applied to another data set from which it was initially estimated, a 

problem emerges about how to estimate the number of external trips. It is one thing to apply 

simple arithmetic in order to determine how many trips originated outside the modeling area (as in 

Table 27.8). It is another to know how to calculate external trips when the model is being applied 

to other data. For the modeled zones, the coefficients are applied to the variables of the model (see 

‘Make Prediction’ below). But, the external trips have to be estimated independently. 

There is not a simple way to estimate external crime trips. Unlike regular trips that can be 

estimated through cordon counts, crime trips are not detectable while they are occurring (i.e., one 

cannot stand by a road and count offenders traveling by). Thus, they have to be estimated. 

27.56 
 

 

 

Table 27.8: 

Estimating External Crime Trips in Baltimore County 

Number of

crimes ending 

in 325 Baltimore 

County zones: 41,969 

Number of 

crimes originating 

in 532 Baltimore 

County/City zones: 40,342 

Crimes from 

outside the 

modeling area: 1,627 

Note: external trips are only added to the origin model since they are 

crime trips that originate outside the modeling area. They are not 

relevant for the destination model. 

A simple method is to calculate the number of external trips for two time periods. For 

example, external trips could be calculated from a 2010 data set by subtracting the total number of 

crimes occurring in the modeling region from the total number of crimes occurring in the study 

area (e.g., as in Table 27.8 above). If a similar calculation was made for, say, 2012, then the 

difference (the ‘trend’) could be extrapolated. To take our example, between 1993 and 1996, there 

were 1,627 external trips. If the number of external trips turned out to be 1,850 for 1997-2000, 

then the difference (1,850 - 1,627 = 223) could be applied for future years. Essentially, a slope is 

being calculated and applied as a linear equation: 

(27.32) 

where Yi is the number of crime origins during a four year period, I, and Xi is an integer for a four 

year period starting with the next period (i.e., the base year, 1997-2000, has integer value of 0). In 

other words, a linear trend is being extrapolated. 

How realistic is this? For short time periods, linear extrapolation is probably as good a 

method as any. But for longer time periods, it can lead to spurious conclusions (e.g., crime trips 

from outside the region will always increase). Short of developing a sophisticated model that 

27.57 
 

 

relates crime trips to the growth of the metropolitan area and to other metropolitan areas within, 

say, 500 miles, a linear extrapolation is one of the few methods that one can apply.10 

Make Prediction 

In CrimeStat, external trips are added on the second page of the trip generation - Make 

prediction. This is a page where the modeled coefficients and any external trips are applied to a 

data set. There are two reasons why this is a separate page from the ‘Calibrate model’ page where 

the model was calibrated. First, the coefficients might be applied to another data than that from 

which it was calibrated. For example, one might calibrate the model with a data set from 2008-

2010 and then apply to a data set covering 2011-2013. Similarly, one might take future year 

forecasts (e.g., 2025) and apply the model. In effect, the model would be predicting the number of 

future crimes if the same conditions hold over the time frame. 

A second reason for separating the calibration and application pages is to add external trips 

to the origin zones. As mentioned above, external trips are, by definition, those that were not 

modeled in the calibration. They have to be calculated independently of the model and then added 

to the estimates. 

Thus, the ‘Make prediction’ page allows these operations to occur. Figure 27.10 shows the 

page. There are several steps that have to be implemented for this page to be operative. 

1. The data file has to be input as either the primary or secondary file (not shown in

the image). In this example, the same data set is being used as was used for the

calibration. But, if it is a different data set, that will need to be input in the Data

Setup section. Whether the input data set is a primary file (the usual occurrence) or

a secondary file needs to be specified. Also, indicate whether the applied model is

to be an origin or destination model. In Figure 27.10, it is defined as an origin file.

2. A trip generation coefficients file needs to be input. These were the estimated

coefficients from the calibration stage. Inputting this file brings in the coefficients

in the order in which they were saved. They are listed in the ‘Matching parameters’

dialogue box on the right side of the page.

An alternative might be to use cordon counts from major highways coming into the region and assume that 

crime trips represent a constant proportion of those trips. Thus, if the total number of estimated external 

highway trips increases by 5%, one could assume that the external trips also increase by 5%. While this is 

plausible, it is not necessarily an accurate estimate. Talk to your Metropolitan Planning Organization or the 

State Department of Transportation if you are interested in developing this type of model as you will need 

their estimates of external trips. 

27.58 

10 
Figure 27.10: 

“Make Prediction” Setup Page 
 

3. On the left side of the page are listed all the variables in the input data set (primary

or secondary file). In the middle box, the variables are added in the same order as

in the matching parameters box. That is, each independent variable needs to be

matched to the variable from the coefficients file, one for one. This is very

important. The names do not have to be the same (e.g., if the model was calibrated

with data set and applied to another, the variable names may not be identical). But

the content and order of the variables needs to be the same. In the example, the first

variable in the coefficients file is INCEQUAL. The selected variable in the middle

box has to be the income equality variable (whatever its name). In the example, the

same data set is being used so the names are identical. This is repeated for each of

the independent variables in the coefficients file.

4. Next, any missing value codes are specified in the missing values box. Any records

with a missing value for any of the selected independent variables will be dropped

from the calculation. In the example, there are no missing value codes applied

other than the default blank field.

5. If external trips are to be added, the external trips box must be checked. External

trips could be applied in an origin model, but not in a destination model. If they are

to be added, the number of trips should be specified in the ‘Number of external

trips’ box and the zone ID field for the file indicated; in the example, 1627 is added

as external trips and the TAZ field is specified as the ID variable (TZ98).

6. The type of model to be applied is indicated in the AType of regression model@ box.

There are only two choices: Poisson (the default) and Normal (OLS). Since the

coefficients are being applied to the data, no over-dispersion correction is necessary

(since it was probably used in calibrating the model).

7. Finally, the output file name is defined in the ‘Save predicted values’ box.

For each zone, the routine will then take the appropriate variable from the input data set 

and apply the matching coefficient from trip generation coefficients file to produce a predicted 

estimate of the number of trips. To calculate this value, for the OLS model, the routine will use 

equation 27.2 above while for the Poisson model, the routine will use equation 27.6 above. For the 

latter, it will then raise the predicted log value to the power, e, to produce a prediction for the 

expected number of crime trips: 

(27.33) 

27.60 
If external trips are added, a new zone is created called EXTERNAL in the ID field that 

was indicated on the page. Then, the specified number of external trips is simply placed in that 

field with zeros being placed for the values of all the remaining variables in the file. By default, 

the output name for the predicted number of crimes will be called PREDORIG for an origin model 

and PREDDEST for a destination model. An example data set is available on the CrimeStat 

download page. 

Note: for a destination model, this ‘Make prediction’ operation is not 

necessarily needed if the same data set is used for calibration and 

prediction. This step is primarily for the origin file 

Balancing Predicted Origins and Destinations 

After the origin model and destination model are calibrated and applied to a data set, the 

final step in trip generation is to ensure that the number of predicted origins equals the number of 

predicted destinations. This is necessary for the next stage of crime travel demand modeling - trip 

distribution. Since a trip has both an origin and a destination, the total number of origins must 

equal the total number of destinations. This is an absolute requirement for the trip distribution 

model to work. The routine will return an error message if the number of origins does not equal 

the number of destinations. 

If the Poisson model is used for calibration, the routine ensures that the number of 

predicted trips equals the number of input trips. Further, if the calculation of external trips has 

been obtained by subtracting the total number of predicted origins from the total number of 

predicted destinations, and if the external trips are then added to the predicted origins, then most 

likely the total number of origins will equal the total number of destinations. However, because of 

rounding-off errors and inconsistent external trip estimates, it is possible that the sums are not 

equal. 

Consequently, it is important to balance the predicted origins and destinations to ensure 

that no problems will occur in the trip distribution model. There are two ways to do this in 

CrimeStat. First, the number of predicted destinations is held constant and the number of predicted 

origins is adjusted to match this number. This is the default choice. Second, the number of 

predicted origins is held constant and the number of predicted destinations is adjusted to match this 

number. 

The calculation is essentially a multiplier that is applied to each zone. If destinations are to 

be held constant, the multiplier is defined as the ratio of total destinations to total origins: 

27.61 
	 	

	 	 	 

	 	 	

	 	 

(27.34)

The predicted number of origins is multiplied by Mj. If, on the other hand, the origins are to be 

held constant, the multiplier is defined as the ratio of total origins to total destinations: 

(27.35)

The predicted number of destinations is multiplied by Mi. The multiplication simply ensures that 

the sums of the predicted origins and predicted destinations are equal. 

The third page in the trip generation model is the ‘Balance predicted origins & 

destinations’ page. Figure 27.11 shows the setup for this page. The steps are as follows: 

1. The box is checked indicating that it is a balancing operation.

2. The predicted origin file is input and the predicted origin variable is identified. In

the example, the predicted origin file is called ‘PredictedOrigins.dbf’ and the field

with the predicted numbers was called PREDORIG.

3. The predicted destination file is input and the predicted destination variable is

identified. In the example, the predicted destination file is called

‘PredictedDestinations.dbf’ and the field with the predicted numbers was called

PREDDEST.

Note that these files are input on this page and not on the primary or secondary file pages. 

4. Next, the type of balancing is specified - Holding destinations constant (the default)

or holding origins constant. In the example, the destinations are to be held constant.

5. Finally, the output file is specified. If the origins are to be adjusted, then only the

origin file is saved. If the destinations are to be adjusted, then only the destination

file is saved. In other words, the adjustment is applied to only one of the two

predicted crime files. In the example, the file was named

‘AdjustedPredictedOrigins.dbf’ (not shown) since the origin file was adjusted.

The output produces a new column with the adjusted values. Table 27.9 shows the origin 

output for the Baltimore data of the first 11 records. Once the balancing has been completed, the 

trip generation model is finished and the user can go on to the trip distribution model. In other 

27.62 
Figure 27.11: 

Balance Predicted Origins and Destinations Setu
words, the output file ensures that both the predicted origin file (crime productions) and predicted 

destination file (crime attractions) are balanced. 

Table 27.9: 

Adjusted Data Should Have These Fields 

Zone PREDICTED ADJORIGIN 

0001 225.818482 225.850955 

0002 187.527819 187.554785 

0003 320.877458 320.923600 

0004 75.096631 75.107430 

0005 44.981775 44.988243 

0006 32.574758 32.579442 

0007 107.334835 107.350270 

0008 74.683931 74.694671 

0009 76.425236 76.436226 

0010 34.183846 34.188762 

0011 66.975803 66.985434 

etc etc etc 

Strengths and Weaknesses of Regression Modeling of Trips 

As mentioned earlier, the use of regression for producing the trip generation model has its 

strengths and weaknesses. The advantages are that, first, the approach is applicable to crime 

incidents. Unlike regular travel behavior, crime trips have to be inferred from police reports. 

Thus, starting with counts of the number of crimes occurring in each zone and the number of 

crimes that originate from each zone, a model can be constructed. 

Second, the use of a non-linear model, such as the Poisson, allows more complex fitting of 

crime counts. In the early 1970s when trip generation models were starting to be implemented in 

Metropolitan Planning Organizations around the U.S., the major type of regression modeling 

available was OLS. At that time, researchers could not demonstrate that this method was reliable 

in terms of predicting travel. However, with the availability of software for conducting Poisson 

and other non-linear models, that criticism is no longer applicable. The Poisson model is very well 

behaved with respect to count data. It does not produce negative estimates. It requires high levels 

of an independent variable to produce a slight effect in the dependent variable, but that the level 

increases as the values of the independent variable increase. It maintains constancy between the 

sum of the input counts and the sum of the predicted counts. Non-linear models are much more 

realistic for modeling trips than OLS. 

27.64 
Third, the use of a multivariate regression model allows multiple variables to be included. 

In our example, there were four independent variables in the reduced origin and destination 

models. Trip tables, on the other hand, typically only have three or four independent predictors; it 

becomes too complicated to keep track of multiple conditions of predictor variables. Thus, a more 

complex and sophisticated model can be produced with a regression framework. 

Fourth, and finally, a regression framework allows for complex interactions to be 

estimated. For example, the log of an independent variable can be defined. An interaction 

between two of the independent variables can be examined (e.g., median household income for 

those zones having a sizeable amount of retail employment). In the trip table approach, these 

interactions are implicit in the cell means. Thus, overall, the regression framework allows for a 

more complex model than is available with a trip table approach. 

On the other hand, there are potential problems associated with a regression framework. 

First, the regression coefficients can be influenced by zone size. Since the model is estimating 

differences between zones (i.e., differences in the number of crimes as a function of differences in 

the values of the independent variables), zone size affects the level of those differences. With 

small zone sizes, there will be substantial differences between zones in both the independent and 

dependent variables. Conversely, large zone sizes will minimize within-zone differences, but will 

usually increase the estimate of the between-zone differences. The result could be an exaggeration 

of the effect of a variable that would not be seen with small zone geography. As was argued in 

Chapter 25, one should choose the smallest zone geography that is practical in order to minimize 

this problem. 

Second, a point that has been repeated again and again, these models are not behavioral 

explanations. They represent ecological correlations with crime trips. It is important to not try to 

convert these models into explanations of offender behavior. Too often, researchers have jumped 

to conclusions about individuals based on the relationships with environments and neighborhoods. 

It is important to not do this. This criticism, incidentally, applies both to the trip table as well as 

the regression approach to trip generation modeling. 

The new generation of travel demand models are specifically behavioral and involve 

modeling the behavior of specific individuals. Probabilities are calculated based on individual 

choice and a micro-simulation routine can apply these probabilities to a large metropolitan area 

(Shifton et al, 2003; Recker, 2000). While this approach offers some definite theoretical 

advantages and is the subject of much current research, to date there has not been a demonstration 

that this approach is more accurate at predicting trips than the tradition trip-based travel demand 

model. For crime, such an approach would have to be simulated. 

27.65 
 

Conclusion 

In summary, the trip generation model is a valuable tool for predicting the number of 

crimes that originate in each zone and the number of crimes that end in each zone. Even if the 

model is not behavioral, the model can be stable and useful for many years in the future. It is best 

thought of as a proxy model in which the variables in the models are proxies for conditions that are 

generating crimes, either in terms of environments that produce offenders or in terms of locations 

that attract them. 

In the next chapter, we will examine the second stage in the travel demand model - trip 

distribution. In that stage, the predicted crime origins and the predicted crime destinations are 

linked to produce crime trips. 

27.66 
References 

Boswell, M. T. & Patil, G. P. (1970). “Chance mechanisms generating negative binomial 

distributions”. In Random Counts in Scientific Work, Vol. 1, G. P. Patil, ed., Pennsylvania State 

University Press:University Park, PA, 3-22. 

Bowers, K. & Hirschfield, A. (1999). Exploring links between crime and disadvantage in NorthWest England: An analysis using Geographic Information Systems. International Journal of 

Geogrpahical Information Science, 13,b 159-184. 

Bursik, R. J., Jr. & Grasmick, H. G. (1993). Economic deprivation and neighborhood crime rates, 

1960-1980. Law and Society Review, 27, 263-268. 

Cameron, A. C. & Trivedi, P.K. (1998). Regression Analysis of Count Data. Cambridge 

University Press: Cambridge, U.K. 

Cameron, A. C. & Windmeijer, F. A. G. (1996). R-squared measures for count data regression 

models with applications to health-care utilization. Journal of Business & Economic Statistics, 

14(2), 209-20. 

Chiricos, T. (1987). Rates of Crime and Unemployment Social Problems, 34, 187-211 

Cohen, L.E. & Felson, M. (1979) Social change and crime rate trends: a routine activity approach, 

American Sociological Review, 44: 588-608. 

Culp, M. & Lee, E. J. (2005). Improving travel models through peer review. Public Roads, 68 (6), 

FHWA-HRT-05-005. Federal Highway Administration, U.S. Department of Transportation: 

Washington, DC. http://www.fhwa.dot.gov/publications/publicroads/05may/07.cfm. Accessed 

April 28, 2012. 

Der, G. & Everitt, B. S. (2002). A Handbook of Statistical Analyses using SAS. Chapman & 

Hall/CRC: London. 

Draper, N. & Smith, H. (1981). Applied Regression Analysis, Second Edition. John Wiley & Sons: 

New York. 

Ehrlich, I. (1975). On the relation between education and crime. In F. T. Juster (ed), Education, 

Youth and Human Behavior. McGraw-Hill: New York, 313-337. 

Fotheringham, A. S., Brunsdon, C. & Charlton, M. (2002). Geographically Weighted Regression: 

The Analysis of Spatially Varying Relationships. John Wiley & Sons: New York. 

27.67 
 

References (continued) 

Fowles, R. & Merva, M.. (1996). Wage Inequality and Criminal Activity, Criminology, 34, 163-

82. 

Freedman, David A. (1999). Ecological inference and ecological fallacy. International 

Encyclopedia of the Social and Behavioral Sciences, Technical Report No. 549, October. 

http://www.stanford.edu/class/ed260/freedman549.pdf. Accessed March 26, 2012. 

Hagan, J. & Peterson, R. (1994). Inequality and Crime. Stanford University Press: Palo Alto, CA. 

Hall, D. B. (2000). Zero-inflated Poisson and binomial regression with random effects: a case 

study. Biometrics, 56, 1030-1039. 

Hensher, D. A. & Button, K. J. (2002). Handbook of Transport Modeling. Elsevier Science: 

Cambridge, UK. 

ITE (2003). Trip Generation (7th edition). Institute of Transportation Engineers: Washington, DC. 

Kohfeld, C. W. & Sprague, J. (1988). Urban unemployment drives crime. Urban Affairs 

Quarterly, 24, 215-241. 

Langbein, L. I. & Lichtman, A. J. (1978). Ecological Inference. Sage University Paper series on 

Quantitative Applications in the Social Sciences, series no. 07-010. Beverly Hills and London: 

Sage Publications. 

Messner, S. (1986). Economic inequality and levels of urban homicide, Criminology, 23, 297-317. 

Miaou, S.P (1996). Measuring the Goodness-of-Fit of Accident Prediction Models. FHWA-RD96-040. Federal Highway Administration, U.S. Department of Transportation: Washington, DC. 

Microsoft (2012). SKEW - skewness function, Microsoft Office Excel 2010, Microsoft: Redmond, 

WA. http://office.microsoft.com/en-us/excel-help/skew-HP005209261.aspx. Accessed May 21, 

2012. 

Nannen, V. (2003). The Paradox of Overfitting. Artificial Intelligence, Rijksuniversitat: 

Groningen, Netherlands. http://volker.nannen.com/pdf/the_paradox_of_overfitting.pdf. Accessed 

March 11, 2010. 

27.68 
 

References (continued) 

NCHRP (1998). Integration of Land Use Planning with Multimodal Transportation Planning. 

Project 8-32(3). Prepared by Parsons Brinkerhoff Quade & Douglas, Inc. for the National 

Cooperative Highway Research Program, Transportation Research Board, National Research 

Council: Washington DC. October. 

NIST (2004). Gallery of distributions. Engineering Statistics Handbook. National Institute of 

Standards and Technology: Washington, DC. 

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm. Accessed May 21, 2012. 

Newman, O. (1972). Defensible Space: Crime Prevention Through Urban Design. Macmillan: 

New York. 

Ortuzar, J. D. & Willumsen, L. G. (2001). Modeling Transport (3rd edition). J. Wiley & Sons: 

New York. 

Park, R. & Burgess, E. (1924). Introduction to the Science of Sociology. Chicago University Press: 

Chicago. 

Pribyl, O & Goulias, K. G. (2005). Simulation of daily activity patterns incorporating interactions 

within households: Algorithm overview and performance. Transportation Research Record, 1926 

(January), 135-141. http://trb.metapress.com/content/r7u36h005758h304/. Accessed May 9, 2012. 

Radford, N. (2006). The problem of overfitting with maximum likliehood . CSC 411: Machine 

Learning and Data Mining, University of Toronto: Toronto, CA. 

http://www.cs.utoronto.ca/~radford/csc411.F06/10-nn-early-nup.pdf Accessed March 11, 2010. 

Ratcliffe, J.H. (2008). The magnitude of the crime challenge (Chapter 3). Intelligence-Led 

Policing, Willan Publishing: Cullompton. 

Recker, W. (2000). A bridge between travel demand modeling and activity-based travel analysis. 

Center for Activity Systems Analysis. Paper UCI-ITS-AS-WP-00-11. 

http://repositories.cdlib.org/itsirvine/casa/UCI-ITS-AS-WP-00-11/. Accessed May 23, 2012. 

Shaw, C. R. & McKay, H. D. (1942). Juvenile Delinquency in Urban Areas. Chicago: University 

of Chicago Press. 

27.69 
References (continued) 

Shifton, Y., Ben-Akiva, M., Proussaloglu, K., de Jong, G., Popuri, Y., Kasturirangan, K., & 

Bekhor, S. (2003). Activity-based modeling as a tool for better understanding travel behaviour. 

Conference Proceedings. 10th International Conference on Travel Behaviour Research, Lucerne, 

Switzerland. August. http://www.ivt.ethz.ch/news/archive/20030810_IATBR/shiftan.pdf. 

Accessed May 23, 2012. 

Shoup, D. (2002). Roughly right vs. precisely wrong. Access, No. 20, Spring. 20-25. 

Stack, S. (1984). Income inequality and property crime, Criminology, 22, 229-257. 

Thrasher, F. M. (1927). The Gang, University of Chicago Press: Chicago. 

Venables, W.N. & Ripley, B. D. (1997). Modern Applied Statistics with S-Plus (second edition). 

Springer-Verlag: New York. 

Wilson, J.Q. & Kelling, G. (1982) Broken Windows: The Police and Neighborhood Safety. 

Atlantic Monthly, March. 29-38. 

27.70