CrimeStat IV 

Part V: Spatial Modeling II 
 

 

 

Chapter 15: 

OLS Regression Modeling

 Ned Levine 

Ned Levine & Associates

Houston, TX 

Dominique Lord

Zachry Dept. of 

Civil Engineering 

Texas A & M University 

College Station, TX 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents 

Functional Relationships 15.1 

Normal Linear Relationships 15.1 

 Ordinary Least Squares 15.2 

 Maximum Likelihood Estimation 15.3 

 Assumptions of Normal Linear Regression 15.5 

 Normal Distribution of Dependent Variable 15.5 

 Errors are Independent, Constant, and Normally-distributed 15.5 

 Independence of Independent Variables 15.6 

 Adequate Model Specification 15.6 

 Example of Modeling Burglaries by Zones 15.7 

Example of Normal Linear Model 15.7 

Summary Statistics for the Goodness-of-Fit 15.11 

 Statistics on Individual Coefficients 15.12 

Estimated Error in the Model for Individual Coefficients 15.14 

Violations of Assumptions for Normal Linear Regression 15.16 

 Non-constant Summation 15.16 

 Non-linear Effects 15.18 

Greater Residual Errors 15.18 

Corrections to Violated Assumptions in Normal Linear Regression 15.19 

Eliminating Unimportant Variables 15.19 

 Eliminating Multicollinearity 15.19 

Transforming the Dependent Variable 15.21 

Example of Transforming Dependent Variable on Houston Burglaries 15.21 

Example of Modeling Skewed Variable with OLS 15.22 

Diagnostic Tests and OLS 15.30 

Minimum and Maximum Values for the Variables 15.30 

Skewness Tests 15.30 

Tests for Spatial Autocorrelation in the Dependent Variable 15.32 

Multicollinearity Tests 15.32 

MCMC Version of Normal (OLS) 15.32 

References 15.33 
 

 

                                                            

 

1 

Chapter 15: 

OLS Regression Modeling1 

The Regression I and Regression II modules are a series of routines for regression 

modeling and prediction. This chapter will lay out the basics of regression modeling and 

prediction and will discuss the Ordinary Least Squares (OLS) model in CrimeStat. 

Functional Relationships 

The aim of a regression model is to estimate a functional relationship between a 

dependent variable (call it yi ) and one or more independent variables (call them x1i , xKi ). In an 

actual database, these variables have unique names (e.g., ROBBERIES, POPULATION), but we 

will use general symbols to describe these variables. The functional relationship can be specified 

by an equation (15.1): 

yi  f (x1i ,, xKi )  i (15.1) 

where Y is the dependent variable, x1i , xKi are the independent variables, f ( ) is a functional 

relationship between the dependent variable and the independent variables, and  i is an error 

term (essentially, the difference between the actual value of the dependent variable and that 

predicted by the relationship). 

Normal Linear Relationships 

The simplest relationship between the dependent variable and the independent variables 

is linear with the dependent variable being normally distributed, 

y     x   x  (15.2) i 0 1 1i K Ki i 

The regression chapters are the result of the effort of many persons. The maximum likelihood routines were produced 

by Ian Cahill of Cahill Software in Edmonton, Alberta as part of his MLE++ software package. We are grateful to him 

for providing these routines and for conducting quality control tests on them. The basic MCMC algorithm in CrimeStat 

for the Poisson-Gamma and Poisson-Gamma-CAR models was designed by Dr. Shaw-Pin Miaou of College Station, 

TX. We are grateful for Dr. Miaou for this effort. Improvements to the algorithm were made by us, including the 

block sampling strategy and the calculation of summary statistics. Dr. Dominique Lord of Texas A & M University 

provided technical advice on the Poisson-based models. Dr. Byung-Jung Park of the Korea Transport Institute 

expanded the MCMC algorithms to include various dispersion functions and a Simultaneous Autoregressive function. 

Dr. Ned Levine developed the block sampling methodology and provided overall project management. The 

programmer for the routines was Ms. Haiyan Teng of Houston, TX. We are also grateful to Dr. Richard Block of 

Loyola University in Chicago (IL) for testing the MCMC and MLE routines. 

15.1
 

 

 

 

 

 

 

 

 

 

 

This equation can be written in a simple matrix notation: yi  x T

i β   i where 

T T T xi  (1, x1i , , xKi ) and β  ( 0 ,  1 , ,  K ) . The number one in the first element of xi 

represents an intercept while T denotes that the matrix x T

i is transposed. 

This function says that a unit change in each independent variable, xki , for every 

observation, is associated with a unit change in the dependent variable, yi . The coefficient of 

each variable,  k , specifies the amount of change in yi associated with that independent variable 

while keeping all other independent variables in the equation constant. The first term,  0 , is the 

intercept, a constant that is added to all observations. The error term,  i , is assumed to be 

identically and independently distributed (iid) across all observations, normally distributed with 

an expected mean of 0 and a constant standard deviation. If each of the independent variables 

has been standardized by 

x  x 

zk  k k (15.3) std (xk ) 

then the standard deviation of the error term will be 1.0 and the coefficients will be standardized, 

b1, b2, b3, and so forth. 

The equation is estimated by one of two methods, ordinary least squares (OLS) and 

maximum likelihood estimation (MLE). Both solutions produce the same results. The OLS 

method minimizes the sum of the squares of the residual errors while the maximum likelihood 

approach maximizes a joint probability density function. 

Ordinary Least Squares 

Appendix B by Luc Anselin discusses the method in more depth. Briefly, the intercept 

and coefficients are estimated by choosing a function that minimizes the residual errors by 

setting: 



N  K   yi   0  k xki xki  0 (15.4) 

i1  k 1 

for k=1 to K independent variables or, in matrix notation: 

XT (y  Xβ)  0 (15.5) 

15.2
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

XT Xβ  XT y (15.6) 

where X  (x1 , x 2 , , x N )

T and y  ( y1 , y 2 ,, y N )T . 

The solution to this system of equations yields the familiar matrix expression for 

b OLS  (b0 , b1 , , bK )

T 

T  T b OLS  (X X) 1 X y (15.7) 

An estimate for the error variance follows as 

2 N K 

2  

s   y  b -b x  /(N  K 1) (15.8) OLS i 0 k ki 

i1  k1 

or, in matrix notation, 

2 T sOLS  e e /( N  K 1) (15.9) 

Maximum Likelihood Estimation 

For the maximum likelihood method, the likelihood of a function is the joint probability 

density of a series of observations (Wikipedia, 2010; Myers, 1990). Suppose there is a sample of 

n independent observations (x1, x2 ,, xN ) that are drawn from an unknown probability density 

distribution but from a known family of distributions, for example the single-parameter 

exponential family. This is specified as f ( | θ) where θ is the parameter (or parameters if 

there are more than one) that define the uniqueness of the family. The joint density function 

will be: 

f (x , x ,, x | θ)  f (x | θ) f (x | θ) f (x | θ) (15.10) 1 2 N 1 2 N 

and is called the likelihood function: 

N 

L(θ | x1 , x2 ,, xN )  f (x1 , x2 ,, xN | θ)   f (xi |θ) (15.11) i1 

where L is the likelihood and ∏ is the product term. 

Typically, the likelihood function is interpreted in term of natural logarithms since the 

logarithm of a product is a sum of the logarithms of the individual terms. That is, 

15.3
 

 

 

 

 

 

 

 

 

 

 

 

 N 

ln f (x | θ)



  lnf (x | θ) lnf (x | θ) lnf (x | θ) (15.12) i 1 2 n

 i1 

This is called the Log likelihood function and is written as: 

N 

ln L(θ | x1 , x2 ,, xN )  lnf (xi | θ) (15.13) i1 

For the OLS model, the log likelihood is: 

N N 2 1 N

T 2 ln L   ln(2 )  ln( )  2 ( yi  xi β) (15.14) 2 2 2 i1 

where N is the sample size and σ 2 is the variance. As a comparison, in Chapter 16 we discuss 

the Poisson model in which the log likelihood is: 

N 

ln L   i  yi ln(i )  ln yi ! (15.15) i1 

where i  exp( x T

i β) is the conditional mean for zone i, and yi is the observed number of events 

for zone i. As mentioned, Anselin provides a more detailed discussion of these models in 

Appendix B. 

The MLE approach estimates the value of θ that maximizes the log likelihood of the data 

coming from this family. Because they are all part of the same mathematical family and are 

distributed as a concave function, the maximum of a joint probability density distribution can be 

easily estimated. The approach is to, first, define a probability function from this family, second, 

create a joint probability density function for each of the observations (the Likelihood function); 

third, convert the likelihood function to a log likelihood; and, fourth, estimate the value of 

parameters that maximize the joint probability through an approximation method (e.g., NewtonRaphson or Fisher scores). Because the function is regular and known, the solution is relatively 

easy. Anselin discusses the approach in detail in Appendix B of the CrimeStat manual. More 

detail can be found in Hilbe (2008) or in Train (2009). 

In CrimeStat, we use the MLE method. Because the OLS method is the most commonly 

used, a normal linear model is sometimes called an Ordinary Least Squares (OLS) regression. If 

the equation is correctly specified (i.e., all relevant variables are included), the error term,  , will 

be normally distributed with a mean of 0 and a constant variance, σ 2 . 

The OLS normal estimate is sometimes known as a Best Linear Unbiased Estimate 

(BLUE) since it minimizes the sum of squares of the residuals errors (the difference between the 

15.4
 

 

observed and predicted values of y ). In other words, the overall fit of the normal model 

estimated through OLS or maximum likelihood will produce the best overall fit for a linear 

model. However, keep in mind that because a normal function has the best overall fit does not 

mean that it fits any particular section of the dependent variable better. In particular, for count 

data, the normal model usually does a poor job of modeling the observations with the greatest 

number of events. We will demonstrate this with an example below. 

Assumptions of Normal Linear Regression 

The normal linear model has some assumptions. When these assumptions are violated, 

problems can emerge in the model, sometimes easily correctable and other times introducing 

substantial bias. 

Normal Distribution of Dependent Variable 

First, the normal linear model assumes that the dependent variable is normally 

distributed. If the dependent variable is not exactly normally distributed, it has to have its peak 

somewhere in the middle of the data range and be somewhat symmetrical (e.g., a quartic 

distribution; see Chapter 10). 

For some variables, this assumption is reasonable (e.g., with height or weight of 

individuals). However, for most variables that crime researchers work with (e.g., number of 

robberies, number of homicides, journey-to-crime distances), this assumption is usually violated. 

Most variables that are counts (i.e., number of discrete events) are highly skewed. Consequently, 

when it comes to counts and other extremely skewed variables, the normal (OLS) model will 

produce distorted results. 

Errors are Independent, Constant, and Normally-distributed 

Second, the errors in the model, the ε in equation 15.2, must be independent of each 

other, constant, and normally distributed. This fits the iid assumption mentioned above. 

Independence means that the estimation error for any one observation cannot be related to the 

error for any other observation. Constancy means that the amount of error should be more or 

less the same for every observation; there will be natural variability in the errors, but this 

variability should be distributed normally with the mean error being the expected value. 

Unfortunately, for most variables that crime researchers and analysts work with, this 

assumption is usually violated. With count variables, the errors increase with the count and are 

much higher for observations with large counts than for observation with few counts. Thus, the 

assumption of constancy is violated. In other words, the variance of the error term is a function 

15.5
 

 

of the count. The shape of the error distribution is also sometimes not normal either but may be 

more skewed. Also, if there is spatial autocorrelation among the error terms (which would be 

expected in a spatial distribution), then the error term may be quite irregular in shape; in this 

latter case, the assumption of independent observations would also be violated. 

Independence of Independent Variables 

Third, an assumption of the normal model (and any model, for that matter) is that the 

independent variables are truly independent. In theory, there should be zero correlation between 

any of the independent variables. In practice, however, many variables are related, sometimes 

quite highly. This condition, which is called multicollinearity, can produce distorted coefficients 

and overall model effects. The higher the degree of multicollinearity among the independent 

variables, the greater the distortion in the coefficients. This problem affects all types of models, 

not just the normal, and it is important to minimize the effects. We will discuss diagnostic 

methods for identifying multicollinearity later in the chapter. 

Adequate Model Specification 

Fourth, the normal model assumes that the independent variables have been correctly 

specified. That is, the independent variables are the correct ones to include in the equation and 

that they have been measured adequately. By ‘correct ones’, we mean that the independent 

variable chosen should be a true predictor of the dependent variable, not an extraneous one. 

With any model, the more independent variables that are added to the equation, in general the 

greater will be the overall fit. This will be true even if the independent variables are highly 

correlated with independent variables already in the equation or are mostly irrelevant (but may 

be slightly correlated due to sampling error). When too many variables are added to an equation, 

strange effects can occur. Overfitting of a model is a serious problem that must be seriously 

evaluated. Including too many variables will also artificially increase the model’s variance 

(Myers, 1990). 

Conversely, a correct specification implies that all the important variables have been 

included and that none have been left out. When important variables are not included, this is 

called underfitting a model. Also, not including important variables lead to a biased model 

(known as the omitted variables bias). A large bias means that the model is unreliable for 

prediction (Myers, 1990). Also, the left out variables can be shown to have irregular effects on 

the error terms. For example, if there is spatial autocorrelation in the dependent variable (which 

there usually is), then the error terms will be correlated. Without modeling the spatial 

autocorrelation (either through a proxy variable that captures much of its effect or through a 

parameter adjustment), the error can be biased and even the coefficients can be biased. 

15.6
In other words, adequate specification involves choosing the correct number of 

independent variables that are appropriate, neither overfitting nor underfitting of the model. 

Also, it is assumed that the variables have been correctly measured and that the amount of 

measurement error is very small. 

Unfortunately, we often do not know whether a model is correctly specified or not, nor 

whether the variables have been properly measured. Consequently, there are a number of 

diagnostics tests that can be brought to bear to reveal whether the specification is adequate. For 

overfitting, there are tolerance statistics and adjusted summary values. For underfitting, we 

analyze the error distribution to see if there is a pattern that might indicate lurking variables that 

are not included in the model. In other words, examining violations of the assumptions of a 

model is an important task in assessing whether there are too many variables included or whether 

there are variables that should be included but are not, or whether the specification of the model 

is correct or not. 

Example of Modeling Burglaries by Zones 

For many problems, normal regression is an appropriate tool. However, for many others, 

it is not. Let us illustrate this point. A note of caution is warranted here. This example is used to 

illustrate the application of the normal model in CrimeStat and, as discussed further below, the 

normal model with a normal error distribution is not appropriate for this kind of dataset. For 

example, figure 15.1 shows the number of residential burglaries that occurred in 2006 within 

1,179 Traffic Analysis Zones (TAZ) inside the City of Houston. The data on burglaries came 

from the Houston Police Department. There were 26,480 burglaries that occurred in 2006. They 

were then allocated to the 1,179 TAZ’s within the City of Houston. As can be seen, there is a 

large concentration of residential burglaries in southwest Houston with small concentrations in 

southeast Houston and in parts of north Houston. 

The distribution of burglaries by zones is quite skewed. Figure 15.2 shows a graph of the 

number of burglaries per zone. Of the 1,179 traffic analysis zones, 250 had no burglaries occur 

within them in 2006. On the other hand, one zone had 284 burglaries occur within it. The graph 

shows the number of burglaries up to 59; there were 107 zones with 60 or more burglaries that 

occurred in them. About 58% of the burglaries occurred in 10% of the zones. In general, a small 

percentage of the zones have the majority of the burglaries. 

Example of Normal Linear Model 

We can set up a normal linear model to try to predict the number of burglaries that 

occurred in each zone in 2006. We obtained estimates of population, employment and income 

from the transportation modeling group within the Houston-Galveston Area Council, the 

15.7
Figure 15.1:
   

250300

Houston Burglaries in 2006:

Number of Burglaries Per Zone

Figure 15.2:

150200

er of zones

50

100

Numbe

0

50

0 5 10 15 20 25 30 35 40 45 50 5

Number of burglaries per zone
 

 

 

 

 

 

 

 

 

 

 

 

 

---------------------------------------------------------------------------------------------------------------------

--------------------------------------------------------------------------------------------------------------------- 

Metropolitan Planning Organization for the area (H-GAC, 2010). Specifically, the model relates 

the number of 2006 burglaries to the number of households, number of jobs (employment), and 

median income of each zone. The estimates for the number of households and jobs were for 

2006 while the median income was that measured by the 2000 census. Table 15.1 present the 

results of the normal (OLS) model. 

Table 15.1: 

Predicting Burglaries in the City of Houston: 2006 

Ordinary Least Squares: Full Model 

(N= 1,179 Traffic Analysis Zones) 

DepVar: 2006 BURGLARIES 

N: 1,179 

Df: 1,174 

Type of regression model: Ordinary Least Squares 

F-test of model: 357.2 p≤.0001 

R-square: 0.48 

Adjusted r-square: 0.48 

Mean absolute deviation: 13.5 

1st (highest) quartile: 26.4 

2nd quartile: 10.6 

3rd quartile: 8.3 

4th (lowest) quartile: 8.8 

Mean squared predictive error: 505.1 

1st (highest) quartile: 1,497.5 

2nd quartile: 270.4 

3rd quartile: 134.3 

4th (lowest) quartile: 120.9 

Predictor DF Coefficient Stand Error Tolerance VIF t-value p 

INTERCEPT 1 12.9320 1.269 - - 10.19 0.001 

HOUSEHOLDS 1 0.0256 0.0008 0.923 1.083 31.37 0.001 

JOBS 1 -0.0002 0.0005 0.903 1.107 -0.453 n.s. 

MEDIAN 

HOUSEHOLD 

INCOME 1 -0.0002 0.00003 0.970 1.031 -6.88 0.001 

15.10
 

 

 

 

 

 

Summary Statistics for the Goodness-of-Fit 

The table presents two types of results. First, there is summary information. Information 

on the size of the sample (in this case, 1,179) and the degrees of freedom (the sample size less 

one for each parameter estimated including the intercept and one for the mean of the dependent 

variable); in the example, there are 1,174 degrees of freedom (1,179 – 1 for the intercept, 1 for 

HOUSEHOLDS, 1 for JOBS, 1 for MEDIAN HOUSEHOLD INCOME, and 1 for the mean of 

the dependent variable, 2006 BURGLARIES). 

The F-test presents an Analysis of Variance test of the ratio of the mean square error 

(MSE) of the model compared to the total mean square error (Kanji, 1993, 131; Abraham & 

Ledolter, 2006, 41-51). Next, there is the R-square (or R2

) statistic, which is the most common 

type of overall fit test. This is the percent of the total variance of the dependent variable 

accounted for by the model. More formally, it is defined as: 

R2 ( yi  yˆi )

2 

 1  (15.16) ( yi  y)

2 

where yi is the observed number of events for a zone, i, yˆi is the predicted number of events 

given a set of K independent variables, and Mean y is the mean number of events across zones. 

The R-square value is a number from 0 to 1; 0 indicates no predictability while 1 indicates 

perfect predictability. 

For a normal (OLS) model, R-square is a very consistent estimate. It increases in a linear 

manner with predictability and is a good indicator of how effective a model has fit the data. As 

with all diagnostic statistics, the value of the R-square increases with more independent 

variables. Consequently, an R-square adjusted for degrees of freedom is also calculated - the 

adjusted r-square in the table. This is defined as: 

2 1 (yi  yˆi )

2 /(N  K 1) Ra 2 (15.17) (yi  y) /(N 1) 

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

The R2

 value is sometimes called the coefficient of determination. It is an indicator of the 

extent to which the independent variables in the model predict (or explain) the dependent 

variable. One interpretation of the R2

 is the percent of the variance of Y accounted for by the 

variance of the independent variables (plus the intercept and any other constraints added to the 

model). The unexplained variance is 1 - R2

 or the extent to which the model does not explain the 

15.11
variance of the dependent variable. For a normal linear model, the R2

 is relatively 

straightforward. In the example, both the F-test is highly significant and the R2 is substantial 

(48% of the variance of the dependent variable is explained by the independent variables). 

However, for non-linear models, it is not at all an intuitive measure and has been shown to be 

unreliable (Miaou, 1996). 

The final two summary measures are Mean Squared Predictive Error (MSPE), which is 

the average of the squared residual errors, and the Mean Absolute Deviation (MAD), which is the 

average of the absolute value of the residual errors (Oh, Lyon, Washington, Persaud, & Bared, 

2003). The lower the values of these measures, the better the model fits the data. 

These measures are also calculated for specific quartiles. The 1st quartile represents the 

error associated with the 25% of the observations that have the highest values of the dependent 

variable while the 4th quartile represents the error associated with the 25% of the observations 

with the lowest value of the dependent variable. These percentiles are useful for examining how 

well a model fits the data and whether the fit is better for any particular section of the dependent 

variable. In the example, the fit is better for the low end of the distribution (the zones with zero 

or few burglaries) and less good for the higher end. We will use these values in comparing the 

normal model to other models. 

It is important to point out that the summary measures are more useful when several 

models with a different number of variables are compared with each other than for evaluating a 

single model. 

Statistics on Individual Coefficients 

The second type of information presented is about each of the coefficients. The table 

lists the independent variables plus the intercept. For each coefficient, the degrees of freedom 

associated are presented (one per variable) plus the estimated linear coefficient. For each 

coefficient, there is an estimated standard error, a t-test of the coefficient (the coefficient divided 

by the standard error), and the approximate two-tailed probability level associated with the t-test 

(essentially, an estimate of the probability that the null hypothesis of zero coefficient is correct). 

Usually, if the probability level is smaller than 5% (.05), then we reject the null hypothesis of a 

zero coefficient though frequently 1% (.01) or even 0.1% (0.001) have been used to reduce the 

likelihood that a false alternative hypothesis has been selected (called a Type I error). 

The last two parameters included in the table are the tolerance of the coefficient and the 

VIF (or Variance Inflation Factor). They are measures of multicollinearity (or one type of 

overfitting). Basically, they measure the extent to which each independent variable correlates 

with the other dependent variables in the equation. The traditional tolerance test is a normal 

15.12
 

 

 

model relating each independent variable to the other independent variables (StatSoft, 2010; 

Berk, 1977). It is defined as: 

Tol 1 R2 

i ji (15.18) 

where R2 

ji is the R-square associated with the prediction of one independent variable with the 

remaining independent variables in the model using an OLS model. The VIF is simply the 

reciprocal of tolerance: 

VIFi 1/Toli (15.19) 

In other words, the tolerance of each independent variable is the unexplained variance of 

a model that relates the variable to the other independent variables. If an independent variable is 

highly related (correlated with) to the other independent variables in the equation, then it will 

have a low tolerance. Conversely, if an independent variable is independent of the other 

independent variables in the equation, then it will have a high tolerance. In theory, the higher the 

tolerance, the better since each independent variable should be unrelated to the other independent 

variables. In practice, there is always some degree of overlap between the independent variables 

so that a tolerance of 1.0 is rarely, if ever, achieved. However, if the tolerance is low (e.g., 0.70 

or below), this suggests that there is too much overlap in the independent variables and that the 

interpretation will be unclear. In Chapter 17, we will discuss multicollinearity and the general 

problem of overfitting in more detail. 

Note that the statistic is labeled as pseudo-tolerance in the CrimeStat output. The reason 

is that this statistic is only approximate when the independent variable is skewed, a situation that 

we will discuss shortly. For a normally-distributed independent variable (or approximately 

normally-distributed), however, the tolerance test is exact. 

Looking at the output in Table 15.1, we see that the number of burglaries is positively 

associated with the intercept and the number of households and negatively associated with the 

median household income. The relationship to the number of jobs is also negative, but not 

significant. Essentially, zones with larger numbers of households but lower household incomes 

are associated with more residential burglaries. Because the model is linear, each of the 

coefficients contributes to the prediction in an additive manner. The intercept is 12.93 and 

indicates that, on average, each zone had 12.93 burglaries. For every household in the zone, 

there was a contribution of 0.0256 burglaries. For every job in the zone, there was a contribution 

of -0.0002 burglaries. For every dollar increase in median household income, there is a decrease 

of -0.0002 burglaries. Thus, to predict the number of burglaries with the full model in any one 

zone, i, we would take the intercept – 12.93, and add in each of these components: 

15.13
 

 

 

 

 

 

 

 

(BURGLARIES)  12.93  0.0256(HOUSEHOLDS)  0.0002(JOBS) i i i (15.20)  0.0002(MEDIAN HOUSEHOLD INCOME)i 

 To illustrate, TAZ 833 had 1762 households in 2006, 2,698 jobs in 2006, and had a median 

household income of $27,500 in 2000. The model’s prediction for the number of burglaries in 

TAZ 833 is: 

Number of burglaries (TAZ833) = 12.93 + 0.0256*1762 – 0.0002*2,698 

 – 0.0002*27,500 

= 52.0 

The actual number of burglaries that occurred in TAZ 833 was 78. 

Estimated Error in the Model for Individual Coefficients

 In CrimeStat, and in most statistical packages, there is additional information that can be 

output as a file. There is the predicted value for each observation. Essentially, this is the linear 

prediction from the model. There is also the residual error, which is the difference between the 

actual (observed) value for each observation, i, and that predicted by the model. It is defined as: 

 Residual errori = Observed Valuei - Predicted valuei (15.21) 

Table 15.2 below gives predicted values and residual errors for five of the observations 

from the Houston burglary data set. Analysis of the residual errors is one of the best tools for 

diagnosing problems with the model. A plot of residual errors against predicted values indicate 

whether the prediction is consistent across all values of the dependent variable and whether the 

underlying assumptions of the normal model are valid (see below). Figure 15.3 show a graph of 

the residual errors of the full model against the predicted values for the model estimated in table 

1. As can be seen, the model fits quite well for zones with few burglaries, up to about 12 

burglaries per zone. 

Table 15.2: 

Predicted Values and Residual Error for Houston Burglaries: 2006 

(5 Traffic Analysis Zones) 

Zone (TAZ) Actual value Predicted value Residual error 

833 78 52.0 26.0 

831 46 35.9 10.1 

911 89 67.6 21.4 

2173 30 42.3 -12.3 

2940 3 10.2 -7.2 

15.14
Figure 15.3:
However, for the zones with many predicted burglaries (the ones that we are most likely 

interested in), the model does quite poorly. First, the errors increase the greater the number of 

predicted burglaries. Sometimes the errors are positive, meaning that the actual number of 

burglaries is much higher than predicted and sometimes the errors are negative, meaning that we 

are predicting more burglaries than actually occurred. More importantly, the residual errors 

indicate that the model has violated one of the basic assumptions of the normal model, namely 

that the errors are independent, constant, and identically-distributed. It is clear that they are not. 

Because there are errors in predicting the zones with the highest number of burglaries and 

because the zones with the highest number of burglaries were somewhat concentrated, there are 

spatial distortions from the prediction. Figure 15.4 show a map of the residual errors of the 

normal model. As can be seen by comparing this map with the map of burglaries (figure 15.1), 

typically the zones with the highest number of burglaries (mostly in southwest Houston) were 

under-estimated by the normal model (shown in red) whereas some zones with few burglaries 

ended up being over-estimated by the normal model (e.g., in far southeast Houston). 

In other words, the normal linear model is not necessarily good for predicting Houston 

burglaries. It tends to underestimate zones with a large number of burglaries but overestimates 

zones with few. 

Violations of Assumptions for Normal Linear Regression 

There are several deficiencies with the normal (OLS) model. First, normal models are 

not good at describing skewed dependent variables, as we have shown. Since crime distributions 

are usually skewed, this is a serious deficiency for multivariate crime analysis. Second, a normal 

model can have negative predictions. With a count variable, such as the number of burglaries 

committed 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 normal 

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 highly 

skewed. If most records have values close to zero, it is very possible for a normal model to 

predict a negative value. 

Non-consistent Summation 

A third problem with the normal model is that the sum of the observed values does not 

necessarily equal the sum of the predicted values. Since the estimates of the intercept and 

coefficients are obtained by minimizing the sum of the squared residual errors (or maximizing 

the joint probability distribution, which leads to the same result), there is no balancing 

mechanism to require that they add up to the same as the input values. In calibrating the model, 

15.16
Figure 15.4:
                                                            

 

 

 

adjustments can be made to the intercept term to force the sum of the predicted values to be 

equal to the sum of the input values. But in applying that intercept and coefficients to another 

data set, there is no guarantee that the consistency of summation will hold. In other words, the 

normal method cannot guarantee a consistent set of predicted values. 

Non-linear Effects 

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

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

model would 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.2 It is far better to use a nonlinear model for a highly skewed dependent variable. 

Greater Residual Errors 

The final problem with a normal 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. As we saw with the example above, typically a normal equation produces non-constant 

residual errors with skewed data. In theory, errors in prediction 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 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 and Smith, 1981, 147). 

A highly skewed distribution tends to exacerbate 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 

nor 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). 

For example, to account for the 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 is not intuitive nor easily explained to nonspecialists. 

15.18

2 
 

Corrections to Violated Assumptions in Normal Linear Regression 

Some of the violations in the assumptions of an OLS normal model can be corrected. 

Eliminating Unimportant Variables 

One good way to improve a normal model is to eliminate variables that are not important. 

Including variables in the equation that do not contribute very much adds ‘noise’ (variability) to 

the estimate. In the above example, the variable, JOBS, was not statistically significant and, 

hence, did not contribute any real effect to the final prediction. This is an example of overfitting 

a model. Whether we use the criteria of statistical significance to eliminate non-essential 

variables or simply drop those with a very small effect is less important than the need to reduce 

the model to only those variables that truly predict the dependent variable. We will discuss the 

‘pros’ and ‘cons’ of dropping variables in Chapter 17, but for now we argue that a good model - 

one that will be good not just for description but for prediction, is usually a simple model with 

only the strongest variables included. 

To illustrate, we reduce the burglary model further by dropping the non-significant 

variable (JOBS). Table 15.3 show the results. Comparing the results with those from Table 

15.1, we can see that the overall fit of the model is actually slightly better (an F-value of 536.0 

compared to 357.2). The R2

 values are the same while the mean squared predictive error is 

slightly worse while the mean absolute deviation is slightly better. The coefficients for the two 

common independent variables are almost identical while that for the intercept is slightly less 

(which is good since it contributes less to the overall result). 

In other words, dropping the non-significant variable has led to a slightly better fit. One 

will usually find that dropping non-significant or unimportant variables makes models more 

stable without much loss of predictability, and conceptually they become simpler to understand. 

Eliminating Multicollinearity 

Another way to improve the stability of a normal model is to eliminate variables that are 

substantially correlated with other independent variables in the equation. This is the 

multicollinearity problem that we discussed above. Even if a variable is statistically significant 

in a model, if it is also correlated with one or more of the other variables in the equation, then it 

is capturing some of the variance associated with those other variables. The results are 

ambiguity in the interpretation of the coefficients as well as error in trying to use the model for 

15.19
 

 

 

 

 

 

 

 

 

 

 

 

 

---------------------------------------------------------------------------------------------------------------------

--------------------------------------------------------------------------------------------------------------------- 

Table 15.3: 

Predicting Burglaries in the City of Houston: 2006 

Ordinary Least Squares: Reduced Model 

(N= 1,179 Traffic Analysis Zones) 

DepVar: 2006 BURGLARIES 

N: 1,179 

Df: 1,175 

Type of regression model: Ordinary Least Squares 

F-test of model: 536.0 p≤.0001 

R-square: 0.48 

Adjusted r-square: 0.48 

Mean absolute deviation: 13.5 

1st (highest) quartile: 26.5 

2nd quartile: 10.6 

3rd quartile: 8.3 

4th (lowest) quartile: 8.8 

Mean squared predictive error: 505.1 

1st (highest) quartile: 1498.8 

2nd quartile: 269.5 

3rd quartile: 135.1 

4th (lowest) quartile: 120.2 

Predictor DF Coefficient Stand Error Tolerance VIF t-value p 

INTERCEPT 1 12.8099 1.240 - - 10.33 0.001 

HOUSEHOLDS 1 0.0255 0.0008 0.994 1.006 33.44 0.001 

MEDIAN 

HOUSEHOLD 

INCOME 1 -0.0002 0.00003 0.994 1.006 -7.03 0.001 

prediction. Multicollinearity means that essentially there is overlap in the independent variables; 

they are measuring the same thing. It is better to drop a multicollinear variable even if it results 

in a loss in fit since it will usually result in a simpler and more stable model. 

For the Houston burglary example, the two remaining independent variables in Table 

15.3 are relatively independent; their tolerances are 0.994 respectively, which points to little 

overlap in the variance that they account for in the dependent variable. Therefore, we will keep 

15.20
 

 

 

 

 

 

 

 

 

these variables. However, in the next chapter, we will present an example of how 

multicollinearity can lead to ambiguous coefficients. 

Transforming the Dependent Variable 

It may be possible to correct the normal model by transforming the dependent variable (in 

another program since CrimeStat does not currently do this). Typically, with a skewed dependent 

variable and one that has a large range in values, a natural log transformation of the dependent 

variable can be used to reduce the amount of skewness. The problem will occur for zones with 0 

since the natural log of 0 cannot be calculated. Consequently, one takes: 

ln y  log ( y 1) (15.22) i e i

where e is the base of the natural logarithm (2.718...) and regresses the transformed dependent 

variable against the linear predictors, 

ln y     x   x  (15.23) i 0 1 1i K Ki i

This is equivalent to the equation 

  x   x  0 1 1i K Ki i yi  e (15.24) 

with, again, e being the base of the natural logarithm. 

In doing this, it is assumed that the log transformed dependent variable is consistent with 

the assumptions of the normal model, namely that it is normally distributed with an independent 

and constant error term, ε, that is also normally distributed. 

One must be careful about transforming values that are zero since the natural log of 0 is 

unsolvable. Usually researchers will set the value of the log-transformed dependent variable to 0 

or the value of the dependent variable to a small number (e.g., 1) for cases where the raw 

dependent variable actually has a value of 0 (e.g., equation 15.22 above). But, one must be 

careful that it does not distort relationships if there are many zeros in the data. For example, in 

the burglary data, there were 250 zones (out of 1,179, or 21%) that had zero burglaries! 

Example of Transforming Dependent Variable on Houston Burglaries 

Using the Houston burglary example from above, we transformed the dependent 

variable– number of 2006 burglaries per TAZ, by taking the natural logarithm of it. All zones 

with zero burglaries were automatically given the value of 0 for the transformed variable. 

15.21
 

                                                            

  

 

The transformed variable was then regressed against the two independent variables in the 

reduced form model (from Table 15.3 above). Table 15.4 present the results: The coefficients 

are similar in sign. The R2

 value is smaller than the untransformed model (0.42 compared to 

0.48). Further, the mean squared predictive error is now much lower than the original raw values 

(1.47 compared to 505.14) and the mean absolute deviation is also much lower (1.05 compared 

to 13.50).3

 In other words, transforming the dependent variable into a logarithm has improved 

the fit of the estimate substantially. 

Another type of transformation that is sometimes used is to convert the independent 

variables and, occasionally, the dependent variable into Z-scores. The Z-score of a variable is 

defined as: 

xk  xk z  (15.25) k std(xk ) 

But all this will do is to standardize the scale of the variable as standard deviations 

around an expected value of zero, but not alter the shape. If the dependent variable is skewed, 

taking the Z-score of it will not alter its skewness. 

A third type of transformation takes the square root of the dependent variable and regress 

it in an OLS model. When we did this with the Houston burglary data, however, the fit was not 

as good as the log transformation (model not shown). The mean absolute deviation was more 

than 50% higher and the mean squared predictive error was three times higher. Again, the basic 

reason is that a count, such as the number of burglaries, is typically Poisson-distributed, meaning 

that it is exponential in form. Essentially, skewness is a fundamental property of a distribution 

and the normal model is poorly suited for modeling it. 

Example of Modeling Skewed Variable with OLS 

A simple example can illustrate this theoretically. Figure 15.5 shows an exponential 

distribution that relates a dependent variable, Y, to an independent variable, X. Think of these as 

any two variables that are positively related (e.g., crime & poverty; crime & unemployment). 

The data were created in a spreadsheet by the function Yi = eX with a random error added to 

simulate randomness. However, the underlying curve is still exponential. In Figure 15.6, we fit 

a linear model to the data using the CrimeStat module. The result show that the model tended 

The errors were calculated by, first, transforming the dependent variable by taking its natural log; second, the natural 

log was then regressed against the independent variables; third, the predicted values were then calculated; and, fourth, 

the predicted values were then converted back into raw scores by taking them as the exponents of e, the base of the 

natural logarithm. The residual errors were calculated from the re-transformed predicted values. 

15.22

3
 

 

 

 

 

 

 

 

 

 

 

 

 

---------------------------------------------------------------------------------------------------------------------

--------------------------------------------------------------------------------------------------------------------- 

Table 15.4: 

Predicting Burglaries in the City of Houston: 2006 

Log Transformed Dependent Variable 

(N= 1,179 Traffic Analysis Zones) 

DepVar: Natural log of 2006 BURGLARIES 

N: 1,179 

Df: 1,175 

Type of regression model: Ordinary Least Squares 

F-test of model: 417.4 p≤.0001 

R-square: 0.42 

Adjusted r-square: 0.42 

Mean absolute deviation: 1.05 

1st (highest) quartile: 1.23 

2nd quartile: 0.94 

3rd quartile: 0.56 

4th (lowest) quartile: 1.46 

Mean squared predictive error: 1.47 

1st (highest) quartile: 2.02 

2nd quartile: 1.14 

3rd quartile: 0.47 

4th (lowest) quartile: 2.24 

Predictor DF Coefficient Stand Error Tolerance VIF t-value p 

INTERCEPT 1 1.5674 0.067 - - 23.44 0.001 

HOUSEHOLDS 1 0.0012 0.00004 0.994 1.006 28.84 0.001 

MEDIAN 

HOUSEHOLD 

INCOME 1 -0.000006 0.000001 0.994 1.006 -4.09 0.001 

to underestimate both the upper- and lower-ends of the distribution of X, especially the high end 

while overestimating the middle range. 

Transforming the dependent variable into a natural log (i.e., Ln[X]) creates a better fit 

(Figure 15.6). Similarly, transforming the dependent variable into a square root (i.e., Sqrt[X]) is 

better than the linear though not as good as the log transformation (Figure 15.7). However, 

neither transformation are as good as fitting a true Poisson function (Figure 15.8). This can be 

15.23
24 Modeling Skewed Phenomenon: I ‐ Data Points

Y = eX

Figure 15.5:

162012

4840

0 5 10 15 20 25 30 35 40 45 50 

‐4 X
24 Modeling Skewed Phenomenon: II ‐ OLS Model

Y = eX

Figure 15.6:

162012

4840

0 5 10 15 20 25 30 35 40 45 50 

‐4 X
 

 

 

 

 

 

 

 

 

seen by comparing the Mean Square Predictive Error (MSPE) and the Mean Absolute Deviation 

(MAD) statistics including the quartiles for the MAD (Table 15.5). 

Table 15.5: 

Comparing Errors for Models Estimating Exponential Function 

Mean Squared Predictive Error and Mean Absolute Deviation 

Model

 Error OLS w. OLS w.

 Statistic OLS Ln(Y) Sqrt(Y) Poisson 

MSPE: 4.96 1.94 2.57 1.80 

MAD: 1.79 1.19 1.31 1.15 

1st quartile: 

2nd quartile: 

3rd quartile: 

2.15 

2.15 

2.16 

1.15 

1.35 

1.01 

1.48 

1.28 

1.50 

0.96 

1.36 

1.06 

4th quartile: 1.50 1.21 0.93 1.21 

As seen, the Poisson provides the best overall fit with both the MSPE and the MAD. 

While the OLS using the log-transformed dependent variable produces a reasonably good fit, 

certainly better than the OLS on the untransformed dependent variable, it still provides a poorer 

fit than a non-linear Poisson function, which is an exponential function. Further, the MAD for 

the first quartile (i.e., the data points with the highest actual values) is much worse for the OLS 

of the transformed dependent variable compared to the Poisson. Where the transformed 

dependent variable does as well if not better than the Poisson is in the last two quartiles, the low 

end of the X distribution. 

With either the log transformation or the square root transformation, the fit is better for 

the low end of the dependent variable (i.e., those observations with fewer counts of the 

dependent variable) than for the high end. The reason is because the OLS minimizes the sum of 

the squared deviations of the predictions from the dependent variable. Since it assumes 

homoscedasticity in the residual errors across the ranges of independent variables, it cannot 

adjust the errors at the high end. In other words, no matter what transformation is used with an 

OLS, the result will always be worse than a Poisson-based model. Since we are usually 

interested in the high end of the dependent variable (i.e., those observations with many counts), 

that is a substantial deficiency of the OLS model. 

15.26
24 Modeling Skewed Phenomenon: III ‐ OLS Model with LogY

Y = eX

Figure 15.7:

162012

4840

0 5 10 15 20 25 30 35 40 45 50 

‐4 X
24 Modeling Skewed Phenomenon: IV ‐ OLS Model with Square Root

Y = eX

Figure 15.8:

162012

4840

0 5 10 15 20 25 30 35 40 45 50 

‐4 X
24 Modeling Skewed Phenomenon: V ‐ Poisson Model

Y = eX

Figure 15.9:

162012

4840

0 5 10 15 20 25 30 35 40 45 50 

‐4 X
 

                                                            

 

 

Keep in mind that this was a created distribution where the data points were distributed 

equally across the X spectrum and where the errors were constant throughout (homosceadstic). 

With real data, a count variable (e.g., number of crimes, income) will usually be highly skewed 

with most observations having low values with a small percentage having high values (or vice 

versa such as with distance traveled) and the errors will typically increase with the value of the 

dependent variable. 

Diagnostic Tests and OLS 

To evaluate skewness and other violations of assumptions of a linear model, it is essential 

to examine various diagnostics about the dependent variable. The regression module has a set of 

diagnostic tests for evaluating the characteristics of the data and the most appropriate model to 

use. There is a diagnostics box on the Regression I page (see Figure 20.1 in chapter 20). 

Diagnostics are provided on: 

1. The minimum and maximum values for the dependent and independent variables 

2. Skewness in the dependent variable 

3. Spatial autocorrelation in the dependent variable 

4. Estimated values for the distance decay parameter – alpha, for use in the PoissonGamma-CAR model 

5. Multicolinarity among the independent variables 

Minimum and Maximum Values for the Variables 

The minimum and maximum values of both the dependent and independent variables are 

listed. A user should look for ineligible values (e.g., -1) as well as variables that have a very 

high range. The MLE routines are sensitive to variables with very large ranges. To minimize 

the effect, variables are internally scaled when being run (by dividing by their mean) and then rescaled for output. Nevertheless, variables with extreme ranges in values and especially variables 

where there are a few observations with extreme values can distort the results for models.4 A 

user would be better choosing a more balanced variable than using one where one or two 

observations determines the relationship with the dependent variable. 

For example, in Excel, two columns of random numbers from 1 to 10 were listed in 99 rows to represent two variables 

X1 and X2. The correlation between these two variables over the 99 rows (observations) was -0.03. An additional row 

was added and the two variables given a value of 100 each for this row. Now, the correlation between these two 

variables increased to 0.89! The point is, one or two extreme values can distort a statistical relationship. 

15.30

4 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Skewness Tests 

As we have discussed, skewness in a variable can distort a normal model by allowing 

high values to be underestimated while allowing low or middle-range values to be overestimated. 

For this reason, a Poisson-type model is preferred over the normal for highly skewed variables. 

The diagnostics utility tests for skewness using two different measures. First, the utility 

outputs the “g” statistic (Microsoft, 2003): 

 n

g  [( X  X ) / s]

3 (15.26) (n 1)(n  2) i

i



where n is the sample size, Xi is observation i, X is the mean of X, and s is the sample standard 

deviation (corrected for degrees of freedom). The sample standard deviation is defined as: 



(X  X )

2 

s   i

i (n 1) (15.27) 

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

1996): 

6 SES  (15.28)

n

An approximate Z-test can be obtained from: 

g Z(g)  (15.29) SES

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

p≤.05 level. 

An example is the number of crimes originating in each traffic analysis zone within 

Baltimore County in 1996. The summary statistics were: 



X = 75.108 

s = 96.017 

n = 325 



[( X i  X ) / s]

3  898.391 i

15.31
 

 

 

 

Therefore, 

325 

g  *898.391  2.79 

324 * 323 

6 SES   0.136 

325 

2.79 Z(g)   20.51 

0.136 

The Z of the g value shows the data are highly skewed. 

The second skewness measure is a ratio of the simple variance to the simple mean. 

While this ratio had not been adjusted for any predictor variables, it is usually a good indicator of 

skewness. Ratios greater than about 2:1 should make the user cautious about using a normal 

model. 

If either measure indicates skewness, CrimeStat prints out a message indicating the 

dependent variable appears to be skewed and that a Poisson-type model should be used. 

Testing for Spatial Autocorrelation in the Dependent Variable 

A fourth test that is available is a test for spatial autocorrelation in the dependent variable. 

It will be discussed in the spatial regression section (Chapter 19). 

Multicollinearity Tests 

The fifth type of diagnostic test is for multicollinearity among the independent predictors. 

As we have discussed in this chapter, one of the major problems with many regression models, 

whether MLE or MCMC, is multicollinearity among the independent variables. 

To assess multicollinearity, the pseudo-tolerance test is presented for each independent 

variable. This was discussed above in the chapter (see equation 15.18). 

MCMC Version of Normal (OLS) 

There is also a Markov Chain Monte Carlo (MCMC) version of the OLS model which 

assumes the dependent variable is normally distributed. This will be discussed in chapter 17 on 

Markov Chain Monte Carlo estimation and in chapter 19 on spatial regression modeling. 

15.32
 

References 

Abraham, B. & Ledolter, J. (2006). Introduction to Regression Modeling. Thompson 

Brooks/Cole: Belmont, CA. 

Berk, K. N. (1977). “Tolerance and condition in regression computations”, Journal of the 

American Statistical Association, 72 (360), 863-866. 

Draper, N. & Smith, H. (1981). Applied Regression Analysis, Second Edition. John Wiley & 

Sons: New York. 

H-GAC (2010). Transportation and air quality program, Houston-Galveston Area Council. 

http://www.h-gac.com/taq/. 

Hilbe, J. M. (2008). Negative Binomial Regression (with corrections). Cambridge University 

Press: Cambridge. 

Kanji, G. K. (1993). 100 Statistical Tests. Sage Publications: Thousand Oaks, CA. 

Miaou, S. P. (1996). Measuring the Goodness-of-Fit of Accident Prediction Models. FHWARD-96-040. Federal Highway Administration, U.S. Department of Transportation: Washington, 

DC. 

Microsoft (2003). “SKEW - skewness function”, Microsoft Office Excel 2003, Microsoft: 

Redmond, WA. 

Myers, R. H. (1990) Classical and Modern Regression with Applications, 2nd edition, Duxbury 

Press, Belmont, CA. 

Oh, J., Lyon, C., Washington, S., Persaud, B., & Bared, J. (2003). “Validation of FHWA crash 

models for rural intersections: lessons learned”. Transportation Research Record 1840, 41-49. 

StatSoft (2010). “Tolerance”, StatSoft Electronic StatisticsTextbook, StatSoft:Tulsa, OK. 

http://www.statsoft.com/textbook/statistics-glossary/t/button/t/ 

Tabachnick, B. G. & Fidell, L. S. (1996). Using Multivariate Statistics (3rd ed). Harper Collins: 

New York. 

Train, K. (2009). Discrete Choice Methods with Simulation (2nd edition). Cambridge University 

Press: Cambridge. 

15.33
References (continued) 

Venables, W.N. & Ripley, B. D. (1997). Modern Applied Statistics with S-Plus (second edition). 

Springer-Verlag: New York. 

Wikipedia (2010b). “Maximum likelihood”, Wikipedia. 

http://en.wikipedia.org/wiki/Maximum_likelihood. Accessed March 12, 2010. 

15.34