Chapter 20:

The CrimeStat Regression Module1

Ned Levine Dominique Lord Byung-Jung Park

Ned Levine & Zachry Dept. of Korea Transport 

Associates Civil Engineering Institute 

Houston, TX Texas A & M Goyang, South Korea

University

College Station, TX

Srinivas Geedipally Haiyan Teng Li Sheng

Texas Transportation Houston, TX Houston, TX

Institute

Arlington, TX

Ian Cahill

Cahill Software

Edmonton, AB

 1 The regression chapters were 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. Dr. Shaw-pin Miaou of 

College Station, TX designed the MCMC algorithm for the Poisson-Gamma-CAR model. Dr. Byung-Jung 

Park modified the algorithm to incorporate Poisson-Gamma-SAR, Poisson-Lognormal-CAR/SAR, and the 

MCMC binomial-CAR/SAR models. Dr. Srinivas Geeidpally added the MCMC Normal-CAR/SAR 

models. Dr. Dominique Lord of Texas A & M University provided technical consulting on the dispersion 

parameters in these models. Dr. Ned Levine developed the block sampling scheme and provided overall 

project management. Ms. Haiyan Teng and Dr. Li Sheng programmed the routines and added numerous 

technical improvements to the algorithms. We are also grateful to Dr. Richard Block of Loyola University in 

Chicago (IL) for testing the MCMC and MLE routines.
Table of Contents

The CrimeStat Regression Module 20.1

Regression I Module 20.1

Types of Regression Models 20.1

Input Data Set 20.3

Dependent Variable 20.3

Independent Variables 20.3

Type of Dependent Variable 20.3

Type of Dispersion Estimate 20.4

Type of Estimation Method 20.4

Spatial Autocorrelation Estimate 20.5

Type of Test Procedure 20.5

MCMC Choices 20.5

Number of Iterations 20.5

‘Burn in’ Iterations 20.6

Block Sampling Threshold 20.6

Average Block Size 20.6

Number of Samples Drawn 20.6

Calculate Intercept 20.7

Spatial Autocorrelation Estimate 20.7

Calculate Exposure Offset 20.7

Advanced Options 20.7

Initial parameter values for Phi (φ) 20.7

Rho (ρ) and Tauphi (τφ) 20.9

Alpha (α) 20.9

Diagnostic test for reasonable alpha (α) value 20.10

Value for 0 distance between records 20.12

Output 20.12

Maximum Likelihood (MLE) Model Output 20.12

MLE Summary Statistics 20.12

Information About the Model 20.12

Likelihood Statistics 20.12

Model Error Estimates 20.13

Dispersion Tests 20.13

MLE Individual Coefficient Statistics 20.14

Markov Chain Monte Carlo (MCMC) Model Output 20.16

MCMC Summary Statistics 20.16

Information About the Model 20.16
20.2

Table of Contents (continued)

Likelihood Statistics 20.17

Model Error Estimates 20.17

Dispersion Tests 20.18

MCMC Individual Coefficient Statistics 20.18

Expanded Output (MCMC Only) 20.19

Output Phi Values (CAR/SAR Models Only) 20.20

Save Output 20.20

Save Estimated Coefficients 20.22

Diagnostics Relevant for Spatial Regression 20.22

Testing for Spatial Autocorrelation in the Dependent Variable 20.22

Estimating the Value of Alpha (α) for the Poisson-CAR/SAR Models 20.23

Regression II Module 20.23

Conclusion 20.26
20.1

Chapter 20:

The CrimeStat Regression Module

We now describe the CrimeStat regression module. There are two pages in the module. 

Regression I allows the testing of a model while Regression II allows a prediction to be made 

based on an already-estimated model. Figure 20.1 displays the Regression I page.

Regression I Module

Types of Regression Models

In the current version, 18 possible regression models are available with several options 

for each of these: 

MLE Normal (OLS)

MCMC Normal

MCMC Normal-CAR

MCMC Normal-SAR

MLE Poisson

MLE Poisson with linear dispersion correction (NB1)

MLE Poisson-Gamma (NB2)

MCMC Poisson-Gamma (NB2)

MCMC Poisson-Gamma-CAR

MCMC Poisson-Gamma-SAR

MCMC Poisson-Lognormal

MCMC Poisson-Lognormal-CAR

MCMC Poisson-Lognormal-SAR

MLE Binomial Logit

MLE Binomial Probit

MCMC Binomial Logit

MCMC Binomial Logit-CAR

MCMC Binomial Logit-SAR

In addition, each of the 12 MCMC models can be run with an exposure (offset) variable 

used to define the population ‘at risk’ allowing a total of 30 possible regression models to be run.

There are two pages in the module. The Regression I page allows the testing of a model 

while the Regression II page allows a prediction to be made based on an already-estimated 
Figure 20.1:

Regression Modeling I Setup Screen
20.3

model. Also, since the Regression I module and Trip Generation module in the Crime Travel 

Demand Model duplicate regression functions, only one of these can be run at a time.

Input Data Set

The data set for the regression module is the Primary File data set. The coordinate 

system and distance units are also the same. The routine will not work unless the Primary File 

has X/Y coordinates.

Dependent Variable

To start loading the module, click on the ‘Calibrate model’ tab. A list of variables from 

the Primary File is displayed. There is a box for defining the dependent variable. The user must 

choose one dependent variable. A keystroke trick is to click on the first letter of the variable that 

will be the dependent variable and the routine will go to the first variable with that letter.

Independent Variables

There is another box for defining the independent variables. The user must choose one or 

more independent variables. In the routine, there is no limit to the number. Keep in mind that 

the variables are output in the same order as specified in the dialogue so a user might want to 

think how these should be displayed.

Type of Dependent Variable

There are five options that must be defined. The first is the type of dependent variable: 

Normal (OLS), Skewed (Poisson), Binomial logit (logistic), or Binomial probit.. The default is a 

Poisson. 

Keep in mind that the selected variable is defined by the type of link function, briefly 

discussed in Chapter 18. For a normal (OLS) model, the link (η) is the variable itself.

η = μ (18.2 repeat)

and the model relates the dependent variable to a linear combination of the independent 

variables. 

However, in the Poisson form, the link function is the natural log of the mean,

η = LL(μ) (18.3 repeat)
20.4

and the model relates the logarithm of the selected variable to a linear combination of the 

independent variables. The user selects a dependent variable, but the routine transforms it to the 

natural logarithm of the variable.

In the logistic (or logit) model, the link function is the log of the odds:

η = LL[(μ/(1 − μ)] (18.4 repeat)

and the model relates the log of the odds to a linear combination of the independent variables. 

The user selects a dependent variable, but the routine transforms it to the log of the odds.

Finally, in the probit model the link function is the inverse of the standard normal 

cumulative distribution,

η = Φ−1(μ) (18.5 repeat)

and the model relates the inverse of the standard normal cumulative distribution to a linear 

combination of the independent variables. The user selects an independent variable, but the 

routine transforms it to the inverse of the standard normal cumulative distribution.

In all cases, the output will simply list the dependent variable selected. But, the user 

must remember that the variable has been transformed by the link function (with the exception of 

the normal/OLS model where there is no transformation).

Type of Dispersion Estimate

The second model decision is the type of dispersion estimate to be used. The choices are 

Gamma, Poisson, Lognormal, and Poisson with linear correction. For the MLE models, only 

Gamma, Poisson and Poisson with linear correction are available while for the MCMC models, 

only Gamma and Lognormal are available. The default is Gamma. For the MLE Normal (OLS) 

and MCMC Normal-CAR/SAR models, the dispersion is automatically normal. For the 

binomial logit or binomial probit, the dispersion is automatically binomial.

Type of Estimation Method

The third option is the type of estimation method to be used: Maximum Likelihood 

(MLE) or Markov Chain Monte Carlo (MCMC). The default is MLE. These methods were 

discussed in Chapters 15 and 17 and in appendices B and C. 
20.5

Spatial Autocorrelation Estimate

Fourth, if the user accepts an MCMC algorithm, then a fourth decision is whether to run a 

spatial autocorrelation estimate along with it (a Conditional Autoregressive function - CAR, or a 

Simultaneous Autoregressive function - SAR). The MCMC Poisson-Gamma, MCMC PoissonLognormal, and MCMC Logit functions can be run with a spatial autocorrelation parameter. 

Type of Test Procedure

The fifth, and last model decision, is whether to run a fixed model or a backward 

elimination stepwise procedure (only with the normal or MLE models). A fixed model includes 

all selected independent variables in the regression whereas a backward elimination model starts 

with all selected variables in the model but 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.

If the fixed model is chosen, then all independent variables will be regressed 

simultaneously. However, if the stepwise backward elimination procedure is selected, the user 

must define a p-to-remove value. The choices are: 0.1, 0.05, 0.01, and 0.001. The default is 0.01. 

Traditionally, 0.05 is used as a minimal threshold for significance. We put in 0.01 as the default 

to make the model stricter; with the large datasets that typically occur in police departments, the 

less strict 0.05 criterion would not exclude many independent variables. But, the user can 

certainly use 0.05 instead.

MCMC Choices

If the user chooses the MCMC algorithm, then nine additional decisions have to be made.

Number of Iterations

The first MCMC decision is the number of iterations to be run. The default is 25,000. 

The number should be sufficient to produce reliable estimates of the parameters. Check the MC 

Error/Standard deviation ratio and the G-R statistic to be sure these are below 1.05 and 1.20 

respectively.

Note that the CAR model runs quite quickly whereas the SAR model runs very 

slowly. Unless the data set is small or a SAR model is absolutely essential, we 

recommend using a CAR function for the spatial regression models.
20.6

‘Burn in’ Iterations

The second MCMC decision is the number of initial iterations that will be dropped from 

the final distribution (the ‘burn in’ period). The default is 5,000. The number of ‘burn in’ 

iterations should be sufficient for the algorithm to reach an equilibrium state and produce reliable 

estimates of the parameters. Check the MC Error/Standard deviation ratio and the G-R statistic 

to be sure these are below 1.05 and 1.20 respectively.

Block Sampling Threshold

The third MCMC decision is whether to run all the records through the MCMC algorithm

or whether to draw block samples. This is called the Block Sampling Threshold. The algorithm

will be run on all cases unless the number of records exceeds the number specified in the block 

sampling threshold. The default threshold is 6,000 cases. If the number of cases exceeds the 

threshold, then the block sampling method is used (see below).

Note that if you raise the run the block sampling threshold for more cases, calculating 

time will increase substantially. For the non-spatial Poisson-Gamma model, the increase is 

linear. However, for the spatial Poisson-Gamma model, the increase is exponential. Further, we 

have found that we cannot calculate the spatial model for more than about 6,000 cases. In short, 

the block sampling method must be used for spatial models with a large number of cases.

Average Block Size

The fourth MCMC decision is the number of cases to be drawn in each block sample if 

the total number of records is greater than the block sampling threshold. The default is 400 

cases. Note that this is an average. Actual samples will vary in size. The output will display the 

expected sample size and the average sample size that was drawn.

Number of Samples Drawn

The fifth MCMC decision is the number of samples to be drawn if the total number of 

records is greater than the block sampling threshold. The default is 25. We have found that 

reliable estimates can be obtained from 20 to 30 samples especially if the sequence converges 

quickly and even 10 samples can produce meaningful results. Obviously, the more samples that 

are drawn, the more reliable will be the final results. But, having more samples will not 

necessarily increase the precision beyond 30.
20.7

Calculate Intercept

The sixth MCMC decision is whether to run the model with or without an intercept 

(constant). The default is with an intercept estimated. To run the model without the intercept, 

uncheck the ‘Calculate intercept’ box.

Spatial Autocorrelation Estimate

The seventh MCMC decision is whether to run a spatial autocorrelation model. There are 

two alternative spatial autocorrelation functions that can be used, a Conditional Autoregressive

(or CAR) or a Simultaneous Autoregressive (or SAR). These were defined in Chapter 19. The 

default is no spatial autocorrelation. Note that estimating the SAR function takes a long time, 

much longer than for the CAR model. Unless there is a reason for using the SAR, we 

recommend using the CAR for any spatial autocorrelation component.

Calculate Exposure/Offset

The eighth MCMC decision is whether to run a risk model. If the model is a risk or rate 

model, then an exposure (offset) variable needs to be defined. Check the ‘Calculate 

exposure/offset’ box and identify the variable that will be used as the exposure variable. The 

coefficient for this variable will automatically be 1.0.

Advanced Options

There is also a set of advanced options for the MCMC algorithm. Figure 20.2 displays 

the advanced options dialogue. We would suggest keeping the default values initially until you 

become very familiar with the routine.

Initial parameters values for Phi (φ)

The ninth, and last, MCMC decision is the prior values used for the different parameters 

being estimated. The MCMC algorithm requires an initial estimate for each parameter. There 

are default values that are used. For the beta coefficients (including the intercept), the default 

values are 0. This assumes that the coefficient is ‘not significant’ and has a large variance. It is 

frequently called a ‘non-informative’ prior. These are displayed as a blank screen for the Beta 

box. However, estimates of the beta coefficients can be substituted for the assumed 0 

coefficients. To do this, all independent variable coefficients plus the intercept (if used) must be 

listed in the order in which they appear in the model and must be separated by commas. Do not

include the beta coefficients for the spatial autocorrelation, Φi, term (if used).
Figure 20.2:

Advanced Options for MCMC Poisson-Gamma-CAR 
20.9

For example, suppose there are three independent variables. Thus, the model will have 

four coefficients (the intercept and the coefficients for each of three independent variables). 

Suppose a prior study had been done in which a Poisson-Gamma model was estimated as:

Yi = e4.5+0.3X1i−2.1X2i+3.4X3i (20.1)

The researcher wants to repeat this model but with a different data set and assumes that 

the model using the new data set will have coefficients similar to the earlier research. Thus, the 

following would be specified in the box for the betas under the advanced options:

4.5, 0.3, -2.1, 3.4 (20.2)

The routine will use these values for the initial estimates of the parameters before starting 

the MCMC process (with or without the block sampling method). The advantage is that the 

distribution will converge more quickly (assuming the model is appropriate for the new data set).

Rho (ρ) and Tauphi (τφ)

The spatial autocorrelation component, Φ, is made up of three separate sub-components, 

called Rho (ρ), Tauphi (τφ), and Alpha (α, see formula 19.5 in chapter 19). These are additive. 

Rho is roughly a global component that applies to the entire data set. Tauphi is roughly a 

neighborhood component that applies to a sub-set of the data. Alpha is essentially a localized

effect. The routine works by estimating values for Rho and Tauphi but uses a pre-defined value 

for Alpha. The default initial values for Rho and Tauphi are 0.5 and 1 respectively. The user 

can substitute alternative values for these parameters.

Alpha (α)

Alpha (α) is the exponent for the distance decay function in the spatial model. 

Essentially, the distance decay function defines the weight to be applied to the values of nearby 

records. The weight can be defined by one of three mathematical functions. First, the weight can 

be defined by a negative exponential function where:

W WWht = e−αdii (20.3)

where diiis the distance between observations and α is the value for alpha. It is automatically 

assumed that alpha will be negative whether the user puts in a minus sign or not. The user inputs 

the alpha value in this box.
20.10

Second, the weight can be defined by a restricted negative exponential whereby the 

negative exponential operates up to the specified search distance, whereupon the weight becomes 

0 for greater distances:

Up to Search distance: W WWht = e−αdii for dii ≥ 0, dii ≤ dp (20.4)

Beyond search distance: 0 for dii > dp (20.5)

where dp is the search distance. The coefficient for the linear component is assumed to be 1.0. 

Third, the weight can be defined as a uniform value for all other observations within a 

specified search distance. This is a contiguity (or adjacency) measure. Essentially, all other 

observations have an equal weight within the search distance and 0 if they are greater than the 

search distance. The user inputs the search distance and units in this box.

For the negative exponential and restricted negative exponential functions, substitute the 

selected value for α in the alpha box. 

Diagnostic test for reasonable alpha (α) value

The default function for the weight is a negative exponential with a default alpha value of 

-1 in miles. For many data sets, this will be a reasonable value. However, for other data sets, it 

will not. 

Reasonable values for alpha with the negative exponential function are obtained with the 

following procedure:

1. Decide on the measurement units to be used to calculate alpha (miles, kilometers, 

feet, etc). The default is miles. CrimeStat will convert from the units defined for the 

Primary File input dataset to those specified by the user.

2. Calculate the nearest neighbor distance from the Nna routine on the Distance 

Analysis I page. These may have to be converted into units that were selected in step 

1 above. For example, if the Nearest Neighbor distance is listed as 2000 feet, but the 

desired units for alpha are miles, convert 2000 feet to miles by dividing the 2000 by 

5280.

3. Input the dependent variable as the Z (intensity) variable on the Primary File page.

4. Run the Moran Correlogram routine on this variable on the Spatial Autocorrelation 

page (under Spatial Description). By looking at the values and the graph, decide 
20.11

whether the distance decay in this variable is very ‘sharp’ (drops off quickly) or very 

‘shallow’ (drops off slowly).

5. Define the appropriate weight for the nearest neighbor distance:

a. Assume that the weight for an observation with itself (i.e., distance = 0) is 1.0.

b. If the distance decay drops off sharply, then a low weight for nearby values 

should be given. Assume that any observations at the nearest neighbor 

distance will only have a weight of 0.5 with observations further away being 

even lower.

c. If the distance decay drops off more slowly, then a higher weight for nearby 

values should be given. Assume that any observations at the nearest neighbor 

distance will have a weight of 0.9 with observations further away being lower 

but only slightly so.

d. An intermediate value for the weight is to assume it to be 0.75.

6. A range of alpha values can be solved using these scenarios:

a. For a sharp decay, alpha is given by:

α = LL(0.5)

NNd dddddd

 (20.6)

b. For a shallow distance decay, alpha is given by:

α = LL(0.9)

NNd dddddd

 (20.7)

c. For an intermediate decay, alpha is given by:

α = LL(0.75)

NNd dddddd

 (20.8)

In all three equations, NNdistance is the nearest neighbor distance. 

These calculations will provide a range of appropriate values for α. The diagnostics 

routine automatically estimates these values as part of its output.
20.12

Value for 0 distance between records

The advanced options dialogue has a parameter for the minimum distance to be assumed 

between different records. If two records have the same X and Y coordinates (which could 

happen if the data are individual events, for example), then the distance between these records 

will be 0. This could cause unusual calculations in estimating spatial effects. Instead, it is more 

reliable to assume a slight difference in distance between all records. The default is 0.005 miles

but the user can modify this (including substituting 0 for the minimal distance).

Output

The output depends on whether an MLE or an MCMC model has been run.

Maximum Likelihood (MLE) Model Output

The MLE routines (Normal, Poisson, Poisson with linear correction, MLE PoissonGamma, Binomial Probit, and MLE Binomial Logit/Logistic) produce a standard output which 

includes summary statistics and estimates for the individual coefficients. 

MLE Summary Statistics

The summary statistics include:

Information About the Model

1. The data file

2. The dependent variable

3. The number of cases

4. The degrees of freedom (N – number of parameters estimated)

5. The type of regression model (Normal/OLS, Poisson, Poisson with linear 

correction, Poisson-Gamma, Binomial Logit)

6. The method of estimation (MLE)

Likelihood Statistics

7. Log-likelihood estimate, which is a negative number. For a set number of 

independent variables, the more negative the log-likelihood the better.

8. Log-likelihood per case. This divides the log-likelihood by the sample size (N). 

This indicates the average contribution to the log-likelihood of each observation. 

The more negative, the better.
20.13

9. Akaike Information Criterion (AIC) adjusts the log-likelihood for the degrees of 

freedom. The smaller the AIC, the better.

10. AIC per case. This divides the AIC statistic by the sample size (N). This 

indicates the average contribution to the AIC of each observation. The smaller, 

the better.

11. Bayesian Information Criterion (BIC), sometimes known as the Schwartz 

Criterion (SC), adjusts the log-likelihood for the degrees of freedom. The smaller 

the BIC, the better.

12. BIC per case. This divides the BIC/SC statistic by the sample size (N). This 

indicates the average contribution to the BIC/SC of each observation. The 

smaller, the better.

13. Deviance compares the log-likelihood of the model to the log-likelihood of a 

model that fits the data perfectly. A smaller deviance is better.

14. The probability value of the deviance based on a Chi-square test with N-K-1

degrees of freedom where K is the number of independent variables.

15. Pearson Chi-square is a test of how closely the predicted model fits the data. A 

smaller Chi-square is better since it indicates the model fits the data well.

16. The probability value of the Pearson Chi-square based on a Chi-square test with 

N-K-1 degrees of freedom where K is the number of independent variables.

Model Error Estimates

17. Mean Absolute Deviation (MAD). For a set number of independent variables, a 

smaller MAD is better.

18. Quartiles for the Mean Absolute Deviation. For any one quartile, smaller is 

better.

19. Mean Squared Predictive Error (MSPE). For a set number of independent 

variables, a smaller MSPE is better.

20. Quartiles for the Mean Squared Predictive Error. For any one quartile, smaller is 

better.

21. Squared multiple R (for linear model only). This is the percentage of the 

dependent variable accounted for by the independent variables. 

22. Adjusted squared multiple R (for linear model only). This is the squared multiple 

R adjusted for degrees of freedom.

Dispersion Tests

23. Adjusted deviance. This is a measure of the difference between the observed and 

predicted values (the residual error) adjusted for degrees of freedom. The smaller 
20.14

the adjusted deviance, the better. A value greater than 1 indicates overdispersion.

24. Probability of adjusted deviance. This is the Pearson Chi-square test with 1 degree 

of freedom.

25. Adjusted Pearson Chi-square. This is the Pearson Chi-square adjusted for degrees 

of freedom. The smaller the Pearson Chi-square, the better. A value greater than 

1 indicates over-dispersion.

26. Probability of adjusted Pearson Chi-square. This is the Pearson Chi-square test 

with 1 degree of freedom.

27. Dispersion multiplier. This is the ratio of the expected variance to the expected 

mean. For a set number of independent variables, the smaller the dispersion 

multiplier, the better. For example, in a pure Poisson distribution, the dispersion 

should be 1.0. In practice, a ratio greater than 10 indicates that there is too much 

variation that is unaccounted for in the model. Either add more variables or 

change the functional form of the model.

28. Z-test for dispersion multiplier (Poisson models only). This is a test for whether 

the dispersion parameter is significantly greater than that assumed by the Poisson 

model. It is a test of over-dispersion.

29. P-value for Z-test of dispersion parameter (Poisson models only). This is the onetail probability level associated with the Z-test.

30. Inverse dispersion multiplier. For a set number of independent variables, a larger 

inverse dispersion multiplier is better. A ratio close to 1.0 is considered good.

MLE Individual Coefficient Statistics

For the individual coefficients, the following are output:

31. The coefficient. This is the estimated value of the coefficient from the maximum 

likelihood estimate.

32. Standard Error. This is the estimated standard error from the maximum 

likelihood estimate.

33. Pseudo-tolerance. This is the tolerance value based on a normal prediction of the 

variable by the other independent variables.

34. Z-value. This is asymptotic Z-test that is defined based on the coefficient and 

standard error. It is defined as Coefficient/Standard Error.

35. p-value. This is the two-tail probability level associated with the Z-test.

Table 20.1 show the output for an MLE Poisson-Gamma model that relates the number of 

Houston 2007-09 burglaries to the number of 2008 households and the 2000 median household 

income of Traffic Analysis Zones.
20.15

Table 20.1:

Maximum Likelihood Output for Poisson-Gamma Model

Model result:

Data file: Burglaries_within_City_of_Houston.dbf

DepVar: BURG2006

N: 1179

Df: 1175

Type of regression model: Poisson-Gamma-no spatial autocorrelation

Method of estimation: MLE

 Likelihood statistics

Log-likelihood: -4430.800180

AIC: 8869.600361

BIC/SC: 8889.890048

Deviance: 1390.149554 P-value of Deviance: 0.0001

Pearson Chi-Square: 1112.717355 P-value of Chi-Square: 0.0001

 Model error estimates

Mean absolute deviation: 39.580568

 1st (highest) quartile: 124.121350

 2nd quartile: 19.377810

 3rd quartile: 6.195620

 4th (lowest) quartile: 8.940150

Mean squared predicted error: 62031.156586

 1st (highest) quartile: 242037.095867

 2nd quartile: 6445.778853

 3rd quartile: 118.261739

 4th (lowest) quartile: 154.880457

 Dispersion tests

Adjusted deviance: 1.183106 P-value of Deviance: n.s.

Adjusted Pearson Chi-Square: 0.946993 P-value of Chi-Square: n.s.

Dispersion multiplier: 1.534057 Z= 910.799548 P-value: 0.0001

Inverse dispersion multiplier: 0.651866

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

 PseudoPredictor DF Coefficient Stand Error Tolerance z-value p-value

INTERCEPT 1 2.321019 0.083077 . 27.938042 0.001

 HH2006 1 0.001160 0.000066 0.993563 17.661356 0.001

MEDHHINC00 1 -0.000008 0.000002 0.993563 -5.129752 0.001

-----------------------------------------------------------------------------
20.16

Markov Chain Monte Carlo (MCMC) Model Output

The MCMC routines (Normal-CAR/SAR, Poisson-Gamma, Poisson-Gamma-CAR/SAR, 

Poisson-Lognormal, Poisson-Lognormal-CAR/SAR, Binomial Logit, Binomial LogitCAR/SAR) produce a standard output and an optional expanded output. The standard output 

includes summary statistics and estimates for the individual coefficients. Background 

information on these models is found in chapters 16, 17, 18, and 19. 

MCMC Summary Statistics

The summary statistics include:

Information About the Model

1. The dependent variable

2. The number of records

The sample number. This is only output when the block sampling method is used.

3. The number of cases for the sample. This is only output when the block sampling 

method is used.

4. Date and time for sample. This is only output when the block sampling method is 

used

5. The degrees of freedom (N – number of parameters estimated)

6. The type of regression model (Normal/OLS, Poisson, Poisson with linear 

correction, Poisson-Gamma, Poisson-Gamma-CAR/SAR, Poisson-Lognormal, 

Poisson-Lognormal-CAR/SAR, Binomial Logit, Binomial Logit-CAR/SAR)

7. The method of estimation

8. The number of iterations

9. The ‘burn in’ period

10. The block size is the expected number of records selected for each block sample. 

The actual number may vary.

11. The number of samples drawn. This is output when the block sampling method 

used.

12. The average block size. This is output when the block sampling method used.

13. The type of distance decay function used. This is output for models that use CAR 

or SAR spatial autocorrelation functions.

14. Condition number for the distance matrix. If the condition number is large, then 

the model may not have properly converged. This is output for the PoissonGamma-CAR model only. 

15. Condition number for the inverse distance matrix. If the condition number is 

large, then the model may not have properly converged. This is output for the 

Poisson-Gamma-CAR/SAR, or Poisson-Lognormal-CAR/SAR models only.
20.17

Likelihood Statistics

16. Log-likelihood estimate, which is a negative number. For a set number of 

independent variables, the smaller the log-likelihood (i.e., the most negative) the 

better.

17. Log-likelihood per case. This divides the log-likelihood by the sample size (N). 

This indicates the average contribution to the log-likelihood of each observation. 

The more negative, the better.

18. Deviance Information Criterion (DIC) for models only. This adjusts the loglikelihood for the effective degrees of freedom. The smaller the DIC, the better.

19. Akaike Information Criterion (AIC) adjusts the log-likelihood for the degrees of 

freedom. The smaller the AIC, the better.

20. AIC per case. This divides the AIC statistic by the sample size (N). This 

indicates the average contribution to the AIC of each observation. The smaller, 

the better.

21. Bayesian Information Criterion (BIC), sometimes known as the Schwartz 

Criterion (SC), adjusts the log-likelihood for the degrees of freedom. The smaller 

the BIC, the better.

22. BIC per case. This divides the BIC/SC statistic by the sample size (N). This 

indicates the average contribution to the BIC/SC of each observation. The 

smaller, the better.

23. Deviance compares the log-likelihood of the model to the log-likelihood of a 

model that fits the data perfectly. A smaller deviance is better.

24. The probability value of the deviance based on a Chi-square test with N-K-1

degrees of freedom where K is the number of independent variables. 

25. Pearson Chi-square is a test of how closely the predicted model fits the data. A 

smaller Chi-square is better since it indicates the model fits the data well.

26. The probability value of the Pearson Chi-square based on a Chi-square test with 

N-K-1 degrees of freedom where K is the number of independent variables.

Model Error Estimates

27. Mean Absolute Deviation (MAD). For a set number of independent variables, a 

smaller MAD is better.

28. Quartiles for the Mean Absolute Deviation. For any one quartile, smaller is 

better.

29. Mean Squared Predictive Error (MSPE). For a set number of independent 

variables, a smaller MSPE is better.

30. Quartiles for the Mean Squared Predictive Error. For any one quartile, smaller is 

better.
20.18

Dispersion Tests

31. Adjusted deviance. This is a measure of the difference between the observed and 

predicted values (the residual error) adjusted for degrees of freedom. The smaller 

the adjusted deviance, the better. A value greater than 1 indicates overdispersion.

32. The probability value of the adjusted deviance based on a Chi-square test with 1 

degree of freedom.

33. Adjusted Pearson Chi-square. This is the Pearson Chi-square adjusted for degrees 

of freedom. The smaller the Pearson Chi-square, the better. A value greater than 

1 indicates over-dispersion.

34. The probability value of the adjusted Pearson Chi-square based on a Chi-square 

test with 1 degree of freedom.

35. Dispersion multiplier. This is the ratio of the expected variance to the expected 

mean. For a set number of independent variables, the smaller the dispersion 

multiplier, the better. In a pure Poisson distribution, the dispersion should be 1.0. 

In practice, a ratio greater than 10 indicates that there is too much variation that is 

unaccounted for in the model. Either add more variables or change the functional 

form of the model.

36. Inverse dispersion multiplier. For a set number of independent variables, a larger 

inverse dispersion multiplier is better. A ratio close to 1.0 is considered good.

MCMC Individual Coefficient Statistics

For the individual coefficients, the following are output:

37. The mean coefficient. This is the mean parameter value for the N-K iterations 

where k is the ‘burn in’ samples that are discarded. With the MCMC block 

sampling method, this is the mean of the mean coefficients for all block samples.

38. The standard deviation of the coefficient. This is an estimate of the standard error

of the parameter for the N-K iterations where k is the ‘burn in’ samples that are 

discarded. With the MCMC block sampling method, this is the mean of the 

standard deviations for all block samples.

39. t-value. This is the t-value based on the mean coefficient and the standard 

deviation. It is defined by Mean/Std.

40. p-value. This is the two-tail probability level associated with the t-test.

41. Adjusted standard error (Adj. Std). The block sampling method will produce 

substantial variation in the mean standard deviation, which is used to estimate the 
20.19

standard error. Consequently, the standard error will be too large. An 

approximation is made by multiplying the estimated standard deviation by

N

n

−

where � is the average sample size of the block samples and N is the number of 

records. If no block samples are taken, then this statistic is not calculated.

42. Adjusted t-value. This is the t-value based on the mean coefficient and the 

adjusted standard deviation. It is defined by Mean/Adj_Std. If no block samples 

are taken, then this statistic is not calculated.

43. Adjusted p-value. This is the two-tail probability level associated with the 

adjusted t-value. If no block samples are taken, then this statistic is not calculated.

44. MC error is a Monte Carlo simulation error. It is a comparison of the means of m

individual chains relative to the mean of the entire chain. By itself, it has little 

meaning.

45. MC error/Std is the MC error divided by the standard deviation. If this ratio is 

less than .05, then it is a good indicator that the posterior distribution has 

converged.

46. G-R stat is the Gelman-Rubin statistic which compares the variance of m

individual chains relative to the variance of the entire chain. If the G-R statistic is 

under 1.2, then the posterior distribution is commonly considered to have 

converged.

47. Spatial autocorrelation term (Phi, φ) for CAR/SAR models only. This is the 

estimate of the fixed effect spatial autocorrelation effect. It is made up of three 

components: a global component (Rho, ρ); a local component (Tauphi, τφ); and a 

local neighborhood component (Alpha, α, which is defined by the user).

48. The log of the error in the model (Taupsi). This is an estimate of the unexplained 

variance remaining. Taupsi is the exponent of the dispersion multiplier, eτψ. For 

any fixed number of independent variables, the smaller the Taupsi, the better.

Expanded Output (MCMC Only)

If the expanded output box is checked, additional information on the percentiles from the 

MCMC sample are displayed. If the block sampling method is used, the percentiles are the 

means of all block samples. The percentiles are:

49. 2.5th percentile

50. 5th percentile

51. 10th percentile

52. 25th percentile

53. 50th percentile (median)

54. 75th percentile
20.20

55. 90th percentile

56. 95th percentile

57. 97.5th percentile

The percentiles can be used to construct confidence intervals around the mean estimates 

or to provide a non-parametric estimate of significance as an alternative to the estimated t-value 

in the standard output. For example, the 2.5th and 97.5th percentiles provide approximate 95 

percent confidence intervals around the mean coefficient while the 0.5th and 99.5th percentiles 

provide approximate 99 percent confidence intervals.

The percentiles will be output for all estimated parameters including the intercept, each 

individual predictor variable, the spatial effects variable (Phi), the estimated components of the 

spatial effects (Rho and Tauphi), and the overall error term (Taupsi).

Table 20.2 show selective output from an MCMC Poisson-Lognormal-CAR spatial 

model that relates the number of Houston 2007-09 burglaries to the number of 2008 households 

and the 2000 median household income of Traffic Analysis Zones. The percentiles have been 

reduced to 0.5th, 2.5th, 97.5th, and 99.5th to fit the table.

Output Phi Values (CAR/SAR Models Only)

For the CAR and SAR models only, the individual Phi values can be output. This will 

occur if the sample size is smaller than the block sampling threshold. Check the ‘Output Phi 

value if sample size smaller than block sampling threshold’ box. An ID variable must be 

identified and a DBF output file defined. 

Save Output

The predicted values and the residual errors can be output to a ‘dbf’ file with a 

REGOUT<root name> file name where rootname is the name specified by the user. The output 

is saved 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 field 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 field 

name RESIDUAL). The file can be imported into a spreadsheet or graphics program and the 

errors plotted against the predicted dependent variable (similar to Figure 15.3 in chapter 15).
20.21

Table 20.2:

MCMC Output for Poisson-Lognormal-CAR Model

DepVar: BURG2006

N: 1179

Df: 1174

Number of iterations: 25000

Type of regression model: Poisson-Lognormal-CAR

Method of estimation: MCMC

Distance decay function: Negative exponential

 Likelihood statistics

Log-likelihood: -6087.822981

 Per case: -5.163548

DIC: 30510.458212

AIC: 12185.645963

 Per case: 6.246823

BIC/SC: 7390.366951

 Per case: 6.268335

Deviance: 414.787381 P-value of Deviance: 0.0001

Pearson Chi-Square: 422.236291 P-value of Chi-Square: 0.0001

 Model error estimates

Mean absolute deviation: 5.387914

 1st (highest) quartile: 14.262519

 2nd quartile: 5.504652

 3rd quartile: 1.340483

 4th (lowest) quartile: 0.493941

Mean squared predicted error: 149.000118

 1st (highest) quartile: 542.211088

 2nd quartile: 51.172821

 3rd quartile: 3.835512

 4th (lowest) quartile: 0.298416

 Dispersion tests

Adjusted deviance: 4.456926 P-value of Deviance: 0.0001

Adjusted Pearson Chi-Square: 20.611149 P-value of Chi-Square: 0.0001

Dispersion multiplier: 0.904852 Z = 133.050700 P-value of Z: 0.0001

Inverse dispersion multiplier: 1.105154

 MC error/

 Mean Std t-value p-value MC error std G-R stat

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

Intercept: 0.057768 0.086334 0.669124 n.s. 0.001960 0.022698 1.004705

HH2006: 0.000156 0.000064 2.448304 0.02 2.7333e-006 0.042825 1.018906

MEDHHINC00:-5.7411e-008 1.5194e-006 -0.037785 n.s. 2.7607e-008 0.018169 1.001817

Spatial autocorrelation

(Phi): 1.660699 0.063369 26.206660 0.001 0.003377 0.053283 1.026494

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

Global component

(Rho) 0.178264 0.142500 1.250969 n.s. 0.001356 0.009515 1.000174

Local component

 (Tauphi): 0.003762 0.000404 9.317862 0.001 0.000018 0.043832 1.019646

Neighborhood component

(Alpha: defined) -0.636652 Miles

-------------------------------------------------------------------------------------------------
20.22

Table 20.2 (continued)

Percentiles 0.5th 2.5th 97.5th 99.5th

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

Intercept: -0.153012 -0.103269 0.236579 0.300385

HH2006: 0.000008 0.000041 0.000289 0.000339

MEDHHINC00: -0.000004 -0.000003 0.000003 0.000004

Spatial component 

(Phi): 1.486406 1.530011 1.776679 1.804173

Global component 

(Rho): 0.001125 0.005925 0.525452 0.657165

Local component 

(Tauphi): 0.002892 0.003060 0.004649 0.005008

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

Save Estimated Coefficients

The individual coefficients can be output to a DBF file with a REGCOEFF<root name>

file name where rootname is the name specified by the user. This file can be used in the ‘Make 

Prediction’ routine under Regression II.

Diagnostics Relevant for Spatial Regression

In chapter 15, the diagnostic tests for the regression module were described. Among the 

statistics produced by the routine are two relevant for spatial regression.

Testing for Spatial Autocorrelation in the Dependent Variable

First, there is the Moran’s “I” test for spatial autocorrelation. The statistic was discussed 

extensively in Chapter 5. If the “I” is significant, CrimeStat outputs a message indicating that 

there is definite spatial autocorrelation in the dependent variable and that it needs to be 

accounted for, either by a proxy variable or by estimating a CAR or SAR model. 

A proxy variable would be one that can capture a substantial amount of the primary 

reason for the spatial autocorrelation. One such variable that we have found to be very useful is 

the distance of the location from the metropolitan center (e.g., downtown). Almost always, 

population densities are much higher in the central city than in the suburbs, and this differential 

in density applies to most phenomena including crime (e.g., population density, employment 

density, traffic density, events of all types). It represents a first-order spatial effect, which was 

discussed in Chapters 4 and 5, and is the result of other processes. Another proxy variable that 

can be used is income (e.g., median household income, median individual income) which tends 

to account for much clustering in an urban area. The problem with income as a proxy variable is 

that it is both causative (income determines spatial location) as well as a by-product of 
20.23

population densities. The combination of both income and distance from the metropolitan center 

can capture most of the effect of spatial autocorrelation.

An alternative is to use the Poisson-Gamma-CAR model to filter out some of the spatial 

autocorrelation. As we discussed above, this is useful only when all obvious spatial effects have 

already been incorporated into the model. A significant spatial effect only means that the model 

cannot explain the additional clustering of the dependent variable.

Estimating the Value of Alpha (α) for CAR/SAR Models

Second, there is an estimate of a plausible value for the distance decay function alpha, α, 

in the CAR or SAR models. The way the estimate is produced was discussed above and is 

based on assigning a proportional weight for the distance associated with the nearest neighbor 

distance, the average distance from each observation to its nearest ‘neighbor’ (see chapter 6). 

Three values of α are given in different distance units, one associated with a weight of 0.9 

( a very steep distance decay, one associated with a weight of 0.75 (a moderate distance decay), 

and one associated with a weight of 0.5 (a shallow distance decay). Users should run the Moran 

Correlogram and examine the graph of the drop off in spatial autocorrelation to assess what type 

of decay function most likely exists. The user should choose an α value that best represents the 

distance decay and should define the distance units for it. 

Regression II Module

The Regression II module allows the user to apply a model to another dataset and make a 

prediction. Figure 20.3 show the Regression II setup page. The ‘Make prediction’ routine 

allows the application of coefficients to a dataset. 

Note that, in this case, the coefficients are being applied to a different Primary File than 

that from which they were calculated. For example, a model might be calculated that predicts 

robberies for 2006. The saved coefficient file then is applied to another dataset, for example 

robberies for 2007.

There are four types of models that are fitted – normal, Poisson, binomial logit, and 

binomial probit. For the normal model, the routine fits the equation:

Yi = β0 + β1X1i + ⋯ + βkXk (20.9)
Figure 20.3:

Regression Modeling II Setup Screen
20.25

For the Poisson model, the routine fits the equation:

Yi = eβ0+β1X1i+⋯+βKXk +[Φi] (20.10)

with β0 being the intercept (if calculated), β1.... β k being the saved coefficients and Φi is the 

saved Phi values (if a CAR or SAR model was estimated). Notice that there is no error in each 

equation. Error was part of the estimation model. What were saved were only the coefficients. 

For the binomial logit model, the routine fits the equation:

P(Y = 1) = eβ0+∑ βKXK+[Φi] K

1

1+eβ0+∑ βKXK+[Φi] K

1

= 1

1+e−(β0+∑ βIXK+[Φi] K

1 ) (20.11)

with β0 being the intercept (if calculated), β1.... β k being the saved coefficients and Φi is the 

saved Phi values (if a CAR or SAR model was estimated). 

For the binomial probit model, the routine fits the equation:

p(Y = 1) = Φ−1(pi) = β0 + ∑ βKXK

K

1 (20.12)

 

with β0 being the intercept (if calculated), β1.... β k being the saved coefficients and Φi is the 

saved Phi values (if a CAR or SAR model was estimated), and Φ is the cumulative standard 

normal distribution,

For all four types of model, the coefficients file must include information on the intercept 

and each of the coefficients. The user reads in the saved coefficient file and matches the 

variables to those in the new dataset based on the order of the coefficients file. 

If the model had estimated a general spatial effect from a CAR or SAR model, then the 

general Φi will have been saved with the coefficient files. If the model had estimated specific 

spatial effects from a CAR or SAR model, then the specific Φi values will have been saved in a 

separate Phi coefficients file. In the latter case, the user must read in the Phi (Φi) coefficients 

file along with the general coefficient file.

Table 20.3 shows the output for the first 20 cases from a prediction of the number of 

burglaries per zone based on the estimation model shown in Table 20.2 (Poisson-LognormalCAR). The output will include all variables in the input data set plus the Phi coefficient and the 

predicted values. The user can then calculate residuals by subtracting the predicted from the 

actual (observed) values of the dependent variable.
20.26

Table 20.3:

File Output from Poisson-Lognormal-CAR Prediction of Houston Burglaries

TAZ03 BURG2006 PHI PREDICTED

532 19 0.633593 7.922792

534 2 -0.163279 7.030844

536 2 -0.223977 11.555803

530 107 1.323602 21.462356

537 19 0.259453 15.658255

522 55 1.537228 5.987060

538 11 0.335330 7.432503

516 10 0.364732 9.598958

481 0 -0.350902 8.693915

474 1 -0.161788 8.348133

482 7 0.009940 12.178501

496 2 -0.245535 17.342402

548 0 -1.199179 13.717904

475 4 -0.037407 8.166218

435 3 -0.425498 8.307698

476 1 -0.056756 8.897897

484 8 0.014615 16.133065

483 2 -0.066611 7.888521

477 1 -0.076599 8.166218

478 0 -0.050627 9.293352

Conclusion

This chapter has summarized the structure of the Regression I and Regression II modules 

and most of the options that are available. The help menu on the program will provide contextspecific help on individual items. Note that if you are using Windows Vista, Windows 7 or 

Windows 8, you must download a utility from Microsoft that allows the help menu to be viewed 

from the program. See Chapter 1 (p. 1.17) for details.