Chapter 21: 

Discrete Choice Modeling

 Wim Bernasco 

Netherlands Institute 

For The Study of Crime 

and Law Enforcement 

Amsterdam 

& 

Department of Spatial Economics 

VU University Amsterdam, 

Netherlands 

 Richard Block

Loyola University

Chicago, IL 

 
 

 

 

Table of Contents 

Introduction 21.1 

Discrete Choice Framework 21.3 

Multinomial and Conditional Logit 21.5 

 Multinomial Logit Model 21.5 

 Conditional Logit Model 21.6 

Probabilities in the Multinomial and Conditional Logit Models 21.6 

Data Structures 21.7 

The Multinomial Logit Model 21.7 

Example 1: Modeling Choice of Premises in Chicago Non-street Robberies 

with the Multinomial Logit Model 21.10 

 Adding another variable to the 1997 model 21.15 

 Predicting Non-street Robberies in 1998 based on the 1997 model 21.15 

 Example 1 Conclusion 21.19 

The Conditional Logit Model 21.19

 Destination Choice 21.20 

 Crime Type Choice 21.22 

Example 2: Modeling Choice of Neighborhood for Residential Burglaries 

 in The Hague with the Conditional Logit Model 21.23 

Conclusion 21.26 

References 21.28 

Attachments 21.30 

A. Modeling Correlates of Weapon Use in Houston Robberies 

with the Multinomial Logit Model 

By Ned Levine, Alan Robertson, & Barry Fosberg 21.30 
Chapter 21: 

Discrete Choice Modeling 

Introduction 

This chapter describes the discrete choice framework and the two most well-known 

models that are part of it: the Multinomial Logit (MNL) and the Conditional Logit (CL). These 

techniques require a solid background in statistics and especially regression modeling. A 

background in economics will also be beneficial, though not necessary. Analysts wishing to use 

these techniques, in particular the conditional logit model, would be advised to find an expert to 

work with in developing applications. 

The MNL and CL are two closely related statistical regression models that can be used to 

analyze a discrete outcome variable as a function of a set of independent variables. Discrete 

variables are also known as nominal or categorical variables. They can take on a finite number of 

unordered, mutually exclusive values. Both the MNL and the CL are generalizations of the logit 

model, which is used to analyze binomial (two category) outcome variables and which was 

discussed in Chapter 18. 

Gender is an example of a binomial variable (it is either male or female). The weapon 

used in a robbery (gun, knife, strong arm, or other weapon) is a multinomial variable. Other 

examples are the mode of transport used by a rapist (car, scooter, train, bus, bike, walking) or the 

neighborhood in which a burglary was committed (any one of the city’s neighborhoods). 

Although the MNL and CL models can be used for all analytical problems where the 

outcome variable is discrete (nominal, categorical), in a number of disciplines the models are 

used to study the way that people or organizations make choices. Many research questions in the 

social and behavioral sciences, including criminology, deal with understanding and predicting 

discrete choices (Bernasco & Block, 2009). Political scientists aim to understand why people 

vote and what makes them choose a particular party (Palfrey & Poole, 1987). The party vote is a 

discrete variable. Sociologists want to understand what makes people decide in favor of a 

particular education, occupation, or marriage partner (Jepsen & Jepsen, 2002). Schools, 

occupations and partners are discrete choices. In marketing research, understanding and 

predicting consumer choice is a central concern (McFadden, 1980). Most consumer choices are 

discrete, such as which brand and model of car to purchase, or in which restaurant to have lunch. 

Transportation modelers predict why commuters choose to travel by bus, train, car or bicycle 

(Train, 1980). Behavioral ecological models try to find out what influences an animal’s choice of 

where to forage, rest, or reproduce (Krebs & Davies, 1993). 

21.1 
 

 

 

 

 

 

 

 

Choice is also a central concern in crime analysis. What criteria does a police officer use 

to arrest or not arrest a juvenile? How does a robber choose a specific victim or a particular 

location to commit a robbery (Bernasco & Block, 2009)? This question addresses criminal 

location choice, which formed the major impetus to include these models in CrimeStat. 

Although the MNL and CL models are both discrete choice models and share the same 

underlying likelihood function, they are quite different in practice. The main difference between 

the MNL and the CL model lies in the assumed sources of variation in choice outcomes. The 

MNL model assumes that variation in the characteristics of decision makers (e.g., age) 

determines variation in choice outcomes, whereas the CL model assumes that variation in the 

characteristics of the alternatives themselves (e.g., presence of a bar) determines variation in the 

choice outcomes. 

Aggregated spatial interaction or ‘gravity’ models had been applied to criminal location 

choice and crime trips by Smith (1976) and Rengert (1981). These models bear a strong 

similarity in form and function to the discrete spatial choice models discussed in this chapter, but 

they are aggregated models of the volume of crime trips between areas. The discrete spatial 

choice approach was introduced in the criminological literature by Bernasco and Nieuwbeerta 

(2005) and has subsequently been applied in other studies (Bernasco, 2006, 2010a, 2010b; 

Bernasco & Block, 2009; Bernasco & Kooistra, 2010; Clare, Fernandez, & Morgan, 2009). 

Bernasco (2007) demonstrates how the discrete choice model can be reversed to form a tool in 

geographic offender profiling. 

Neither the MNL nor the CL models require that the outcome variable be interpreted as a 

choice. In fact, the models can be used to model the outcomes of any process that results a finite 

number of unordered possible outcomes. For example, one study proposed a five-category 

typology of homicides in terms of the geographical relation between victim residence, offender 

residence and homicide location (Tita & Griffiths, 2005). It then used the MNL model to study 

the effects of various interactional, motivational and situational characteristics of the homicides 

on the type of the homicide. In this study it would be difficult to interpret the outcome as a 

decision, but the multinomial model is nevertheless useful to describe the effects of the variables 

on the different outcomes. Besides spatial choice, the conditional logit model has not been used 

very often in research on crime. An exception is a study that investigated the causes of criminal 

vengeance in conflicts (Phillips, 2003). 

In the remainder of this chapter, the MNL and CL models are discussed in detail. First 

we demonstrate how the discrete choice model (encompassing both MNL and CL) is derived 

from random utility theory, and show the differences between the MNL and the CL models. Next 

we illustrate the structure of the data necessary to estimate MNL and CL models and give 

examples of both models. 

21.2 
 

 

 

 

 

 

Discrete Choice Framework 

The discrete choice framework was developed in the 1970’s by McFadden (1973) and 

others working in the field of travel demand, and the first applications of discrete choice were in 

the study of travel mode choice (i.e., the choice between train, bus, car, or airplane). Later the 

model was also applied to the choice of a travel routes and travel destinations (Ben-Akiva & 

Lerman 1985). This book is probably the most accessible and complete reference work on 

discrete choice that focuses on the conditional logit and multinomial logit model. A more 

advanced and more technical reference work is Train (2009), which is freely available 

(http://elsa.berkeley.edu/~train/). 

The discrete choice framework consists of a set of assumptions regarding four elements 

of a choice situation (Ben-Akiva & Bierlaire 1999): 

1. Decision makers. The decision maker is the person or agent that makes a choice. 

2. Alternatives. The decision maker must choose one alternative from the choice set, 

i.e. the set of available alternatives that are mutually exclusive and collectively 

include all possible choices. 

3. Attributes. Alternatives have attributes that make them attractive to the decision 

maker. The decision maker evaluates the attractiveness of all alternatives. The 

decision makers themselves can also have attributes. 

4. Decision rule. According to economic theory, the decision maker chooses the 

alternative that maximizes his/her (expected) utility (net gain, profits, 

satisfaction). 

The discussion that follows is mathematically advanced. Readers who prefer to skip the 

mathematical description of the models may want to continue reading at the ”Data structures” 

section on page 21.7. We follow the notation of Train (2009). 

A decision maker, labeled n, must make a choice among J alternatives. Note that the 

word ‘alternatives’ is used for the items, actions or locations that can be chosen, and the word 

‘choice’ is used for the decision of the decision maker in selecting one of these alternatives. By 

convention the complete set of available alternatives is referred to as the ‘choice set’, although 

‘set of alternatives’ might better describe it. 

Decision maker n obtains a level of utility (profits, satisfaction), Uni, from alternative i if 

that alterative is chosen. The principle of utility maximization asserts that the decision maker 

21.3 
 

 

 

 

 

 

 

 

 

 

 

 

 

decides in favor of the alternative i if and only if the individual expects to derive more utility 

from alternative i than from any other available alternative. Thus, if the decision maker decides 

in favor of alternative i, then that person must expect to derive less utility from each of the other 

alternatives (the expression ∀j ≠ i means ‘for all values of j such that j not equals i). 

Uni >Unj∀j ≠ i . (21.1) 

The utilities are known by the decision maker, but not by the analyst. The analyst only 

observes the J alternatives, some attributes ani of the alternatives, some attributes dn of the 

decision maker, and can specify a function V, often called representative utility or systematic 

utility, that links these observed attributes to the decision maker’s utility: 

Vni = V (ani , dn )∀i (21.2) 

The analyst incompletely observes utility, so that generally ≠ Vni U . The utility can be ni 

written as the sum of representative utility Vni and a term μni that captures the factors that 

determine utility but are not observed by the analyst, and that is treated as random. 

U ni = Vni + ε ni (21.3) 

The probability that decision maker n chooses alternative i is the probability that the 

utility associated with choosing i is greater than the utility associated with any other alternative 

in the choice set: 

Pni = Pr( U ni > U nj ∀j ≠ i) (21.4) 

Pni = Pr( Vni + ε ni > Vnj + ε nj ∀j ≠ i) (21.5) 

Pni = Pr( ε nj − ε ni < Vni −Vnj ∀j ≠ i) (21.6) 

This is the most general formulation of the discrete choice model, and any specific choice 

model that is consistent with random utility maximization can be derived from specific 

assumptions on the joint distribution of the unobserved utility term μni. CrimeStat can estimate 

the two most basic models of this family, the multinomial logit model and the conditional logit 

model. There are many others, including for example nested logit, mixed logit, and multinomial 

probit. These are described in Train (2009). 

21.4 
 

 

 

 

 

 

 

 

 

 

 

 

Multinomial and Conditional Logit 

If the unobserved random utility components μni are independent and identically 

distributed according to an extreme value distribution (also referred to as a Gumbel distribution), 

the MNL model and the CL can be derived. Originally, the general form of both was labeled the 

conditional logit model (McFadden 1973). Today both models are usually simply referred to as 

‘multinomial logit model’ or even ‘logit model’ in the discrete choice literature. CrimeStat 

distinguishes between the MNL and the CL models because despite their mathematical 

equivalence, they require a different organization of the data. In the general model that 

encompasses both the CL and the MNL, the choice probability, Pni, the probability that decision 

maker n chooses alternative i, is given by: 

e 

V ni 

Pni =

∑

J 

e 

Vnj

j =1 (21.7) 

For computational convenience, and because any function can be closely approximated 

by a linear function, representative utility Vni is usually assumed to be linear in the parameters. 

The specification of observed utility Vni is different in the MNL and the CL models. In the MNL 

model, Vni depends on the characteristics of the decision maker while in the CL model, it 

depends on the characteristics of the alternatives. 

Multinomial Logit Model 

In the MNL model, 

V ∑

K 

ni = 2 '

i X n = β ki X kn

k =1 (21.8) 

In equation 21.8, K is the number of predictor variables in the model, Xkn is the value of 

the kth predictor variable for observational unit (e.g. decision maker) n, and 2 ki is a parameter 

associated with the kth predictor variable and alternative i. Thus, as can be seen from the k, i and 

n indexes, in the MNL model, there is a separate parameter 2 i for every alternative i in the choice 

set per predictor variable (including a constant). Note that the variables Xkn vary only across the 

decision makers n, but not across the alternatives (they have no i subscript). Characteristics of 

the alternatives do not explicitly play a role in this model (implicitly they do, as we would expect 

the 2 ki to depend on characteristics of the alternatives). 

21.5 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Conditional Logit Model 

In the CL model, 

∑

K 

Vni = 2 ' Xni = βk Xkni 

k =1 (21.9) 

where K is again the number of predictor variables in the model, Xkni is the value of the kth 

predictor variable for observational unit (e.g. decision maker) n and alternative i, and a 2 k is a 

parameter associated with the kth predictor variable. Thus, as can be seen from the k, i and n 

indexes, in the CL model, there is only a single parameter for all alternatives in the choice set per 

predictor variable. 

Note that the variables Xkni vary across the decision makers n and alternatives i. 

Essentially, going from equation 21.8 to equation 21.9, the i index (that references alternatives) 

moves from the parameter 2 to the predictor variable X, a manifestation of the fact that in the 

MNL model characteristics of alternatives are implicitly included in the estimated alternativespecific parameters, while in the CL model they are explicitly measured and their effects 

estimated in generic parameters. 

Probabilities in the Multinomial and Conditional Logit Models 

Substituting equation 21.8 into equation 21.7, the multinomial logit probability that 

decision maker n chooses alternative i is: 

K 

∑βki X

e kn k =1

Pni =

∑

J K 

∑ βkj X

e kn k =1

j=1 (21.10) 

Note that in equation 21.10, the predictor variables vary across decision makers n but not 

across alternatives i. Analogously, substituting equation 21.9 into equation 21.7, the conditional 

logit model asserts that the probability that decision maker n chooses alternative i is: 

K 

∑ β k X

e kni k =1

Pni =

∑

J K 

∑ β k X

e knj k =1

j=1 (21.11) 

21.6 
 

 

 

 

 

 

 

 

 

Note that in equation 21.11, it is impossible to estimate effects of attributes of the 

decision maker that do not vary across alternatives (such as age or gender), because such 

variables (and their parameters) automatically cancel out of the equation; the characteristic of the 

decision maker cannot affect which alternative is chosen because those characteristics do not 

vary across the alternatives. This feature differentiates the conditional logit model from the 

multinomial logit model. 

It is possible, though, to estimate the interaction of characteristics of decision makers and 

characteristics of alternatives by creating or measuring variables that vary across both 

alternatives and decision makers. Such variables must have the n and the i subscript so that they 

do not cancel out in the equation. 

An example of a measured interaction is the experience that the decision maker has with 

each of the alternatives. A given decision maker has more experience with one alternative than 

others, and, therefore, is more or less likely to choose the alternative. Repeat and near repeat 

victimization may be examples of this. Another example of a measured interaction is the 

distance between decision makers and alternatives. 

An example of a created interaction is the multiplication of a characteristic of decision 

makers (e.g. gender Sn) with a characteristic of alternatives (e.g. location Li) resulting in SLni . 

The resulting variable varies across decision makers (as for a given alternative i its value is 

different for males and females) and across alternatives (because for a given decision maker n it 

varies across locations). 

Data Structures 

Although the same mathematical model underlies the MNL model and the CL model, the 

estimation of the CL model requires the data to be organized differently than the estimation of 

the MNL model. This section considers the data structures that hold the information that is 

required to estimate either model. 

The MNL model applies to a n × k matrix (where n refers to cases and k refers to 

variables that vary across cases), while the CL model applies to a n × i × k matrix, where n 

refers to cases, i refers to alternatives, and k refers to variables that vary across cases and across 

alternatives. The distinctions between these two data structures are explained below. 

The Multinomial Logit Model 

The MNL model is estimated on a data set that is similar to the data structure of most 

other regression models and many incident spreadsheets. Each row (record) represents an 

21.7 
 

 

 

 

 

 

 

 

observational unit n (a case, sometimes a decision maker) and each column represents a variable 

(a characteristic of the unit). The dependent variable is nominal and indicates which alternative 

from a set of alternatives was chosen. The variation in outcomes is explained by variation in the 

characteristics of the observational units (decision makers). Table 21.1 shows a simple example 

that describes the first 5 incidents of a larger data file. For each case we know the area where the 

offender lived (Origin), the area where the crime was committed (Destination), the offender’s 

age (Age), the type of crime committed (CrimeType), and the time of day it was committed 

(Time). There is also a variable that uniquely identifies cases (ID). 

The first record indicates that at 3AM (Time) a burglary (CrimeType) was committed in 

zone P (Destination) by an 18 year old (Age) offender who lived in zone P (Origin). Case 2 is a 

robbery committed at 7 PM in zone P by an offender aged 23 living in zone Q. The third record 

shows that someone aged 42 living in zone R purchased an illicit drug in zone S at 2pm. 

Table 21.1: 

Case file Describing 5 Incidents 

ID Origin Destination Age CrimeType Time 

1 P P 18 Burglary 3am 

2 Q P 23 Robbery 7pm 

3 R S 42 Illicit drug 2pm 

4 R Q 32 Robbery 1pm 

5 S R 19 Burglary 6am 

In principle, any variable (except ID) in the case file can be analyzed as representing a 

choice outcome (an alternative being chosen) although for some variables a choice interpretation 

is more natural than for others. Destination represents the choice outcome of the decision of 

where to commit the crime, CrimeType would be the choice outcome of the decision which type 

of crime to commit, and Time would be the choice outcome of the decision of when to commit 

the crime. Origin could also be a choice outcome, the outcome of the decision of where to live. 

Age can be a seen as the outcome of the choice at what age to commit the offence. 

If the decision to be analyzed is where to commit the offence, the first record in Table 

21.1 indicates that the offender offended in zone P rather than in zones Q, R or S. If the decision 

to be analyzed is which type of crime to commit, the first record in Table 21.1 shows that the 

offender decided to commit a burglary rather than commit a robbery or purchase an illicit drug. 

If the decision to be analyzed is when to commit the offence, the first record in Table 21.1 

indicates that the offender offended at 3am rather than at 1pm, 3pm or any other time of the day. 

21.8 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Let us assume that in Table 21.1, the outcome variable is Destination (i.e. the area in 

which the offender committed the crime). Note that we assume that each offender was able to 

choose any of the alternatives (zones P=1, Q=2, R=3, and S=4 for four alternatives), and also 

note that the data do not contain attributes of the alternatives (e.g. whether the areas are affluent, 

have mixed land use, etc.). A MNL model could be used to assess the relation between linear 

combinations of Time (T) and Age (A) with the choice of a Destination (D) zone. In this case, 

equation 21.12 becomes: 

eβ i +β TiT n +β Ai An

Pn (D = i) =

∑

4 

e 

β j +β TjT n +β Aj An

j =1 (21.12) 

where Pn(D=i) is the probability that in the in n

th case the Destination chosen is i, Tn is the Time 

of the n

th case, An is the Age of the nth case, and 2 Ti is the parameter that represents the effect of 

Time on the probability that Destination i is chosen, 2 Ai is the effect of Age on the probability 

that Destination i is chosen, and 2 i is an alternative-specific constant, representing the average 

attractiveness of alternative i in the sample. Note that if Destination has four categories, the 

multinomial logit model involves the following four categorical equations. 

eβ1 +β T 1 T n +β A1 An

Pn (D =1) =

∑

4 

e 

β j +β TjT n +β Aj An

j =1 (21.13) 

eβ 2 +β T 2 Tn +β A2 An

Pn (D = 2) =

∑

4 

e 

β j +β TjT n +β Aj An

j =1 (21.14) 

eβ 3 +β T 3 T n +β A3 An

Pn (D = 3) =

∑

4 

e 

β j +β TjT n +β Aj An

j =1 (21.15) 

eβ 4 +β T 4 Tn +β A4 An

Pn (D = 4) =

∑

4 

e 

β j +β TjT n +β Aj An

j =1 (21.16) 

All four equations are linked by having the same denominator and by 

P n (D =1) + P n (D = 2) + P n (D = 3) + P n (D = 4) =1 (21.17) 

Altogether, 12 parameters are estimated, 4 alternative-specific constants (2 1, 2 2, 2 3, 2 4), 4 

for the Time predictor variable (2 T1, 2 T2, 2 T3, 2 T4), and 4 for the Age predictor variable (2 A1, 2 A2, 

21.9 
 

 

 

 

 

 

 

 

 

 

 

 

2 A3, 2 A4). However, because the effects apply to the differences between the alternatives, the 

parameters for one of the J alternatives must be fixed, and the remaining effects are expressed in 

relation to this fixed ‘reference’ alternative. Like other programs, CrimeStat fixes these 

parameters of the reference alternative to 0 (the user can choose which alternative to use as the 

reference alternative. By default the most frequent alternative is the reference alternative. 

Example 1: Modeling Choice of Premises in Chicago Non-street Robberies with the 

Multinomial Logit Model 

In 1997, there were 1,587 robbery incidents in Chicago that did not occur on the street, in 

which a specific type of premises was robbed, and for which at least one offender was arrested. 

In 1998, there were 1,441 such incidents. In this example, characteristics of offenders and 

incidents will be used to describe differences in the type of premises victimized. The statistics 

used to differentiate models will be explained and the premise pattern of robberies in 1998 will 

be predicted using the robbery patterns of 1997. 

Figure 21.1 maps the premises type of non street robberies in 1997. In 1997, 48.6% of 

these robberies were residential, 11.3% were in parking lots and garages, 23.8% were 

commercial, 2.1% were at banks or currency exchanges, 5.5% were in schools and school yards, 

5.1% were in parks, and 3.5% were in public transit or stations. Although parks are amenities, 

not premises, they will be subsumed under ‘premises ’ here. 

Some areas of the city are nearly free of non-street robberies. Unsurprisingly, 

commercial, and bank robberies are concentrated on main streets. Residential robberies are 

widespread over large sections of the west and south sides. The remainder of this brief will look 

at crime and offender characteristics that differentiate residential robberies from each of the other 

premises types using the multinomial logit model in CrimeStat IV. 

In Table 21.2, 1,587 non-street robberies in 1997 are analyzed using the multinomial logit 

model. Residential robberies are compared to 6 other robbery premises. In the summary section 

of the table, the log likelihood ratio (LLR), Akaike Information Criterion (AIC) and Bayesian 

Information Criterion (BIC/SC) are measures of the differences between a model that includes 

offender and crime characteristics and the naïve (null) hypothesis that only includes the 

frequency of the various premises types. 

The best use for these statistics is in comparing models. The most negative log likelihood 

ratio, and the smallest positive AIC or BIC are best. Unlike the LLR, the AIC and BIC correct 

for the number of explanatory (independent) variables. This is important because a model with 

21.10 
Figure 21.1:

Distribution of Chicago Non Street R bb o eries in 1997
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

______________________________________________________________________________ 

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Table 21.2: 

Multinomial Logit Model of Crime Premises: 

Non-Street Robbery 1997 

Model result: 

Data file: 1997 CHICAGO NON-STREET ROBBERIES 

DepVar: TYPE OF PREMISES 

N: 1,587 

Df: 1,580 

Type of choice model: Multinomial logit model 

Number of Alternatives: 7 

Method of estimation: MLE 

Likelihood statistics 

Log Likelihood: -1,963.1 

Per case: -1.2 

AIC: 3,996.3 

Per case: 2.5 

BIC/SC: 4,184.2 

Per case: 2.6 

Model error estimates 

Mean absolute deviation: 0.2 

Mean squared predicted error: 0.1 

REFERENCE CHOICE: 2 RESIDENTIAL 

Predictor Coefficient Stand Error t-value p-value Odds Ratio 

3 GARAGES AND PARKING LOTS 

Alternative N=180 

Constant -1.3156 0.250 -5.27 0.001 0.27 

GUNCRIME 0.2768 0.187 1.48 n.s. 1.32 

EVENING 0.4431 0.193 2.30 0.05 1.56 

LATENIGHT -0.3754 0.235 -1.60 n.s. 0.69 

TRAVEL DIST 0.0059 0.001 4.43 0.001 1.01 

OFFAGE -0.0016 0.001 -1.88 n.s. 1.00 

OFFBLACK -0.4792 0.237 -2.02 0.05 0.62 

4 COMMERCIAL 

Alternative N=378 

Constant -0.6512 0.197 -3.31 0.001 0.52 

GUNCRIME 1.3900 0.137 10.12 0.001 4.01 

EVENING -0.0614 0.163 -0.38 n.s. 0.94 

LATENIGHT -0.5360 0.180 -2.97 0.01 0.59 

TRAVEL DIST 0.0049 0.001 4.23 0.001 1.00 

OFFAGE -0.0018 0.001 -2.66 0.01 1.00 

OFFBLACK -0.6909 0.186 -3.71 0.001 0.50 

21.12 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

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Table 21.2: (continued) 

Predictor 

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

Coefficient 

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

Stand Error 

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

t-value 

-----------

p-value 

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

Odds Ratio 

---------

5 BANKS AND CURRENCY EXCHANGES 

Alternative N=34 

Constant -2.2444 0.411 -5.46 0.001 0.11 

GUNCRIME 1.3043 0.362 3.60 0.001 3.69 

EVENING -1.5034 0.620 -2.43 0.05 0.22 

LATENIGHT -1.9124 0.742 -2.58 0.01 0.15 

TRAVEL DIST 0.0083 0.002 3.51 0.001 1.01 

OFFAGE -0.0022 0.002 -1.09 n.s. 1.00 

OFFBLACK -1.4572 0.395 -3.69 0.001 0.24 

----------

6 

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

SCHOOLS 

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

Alternative N=88 

Constant 6.6012 0.783 8.43 0.001 735.99 

GUNCRIME -2.0686 0.612 -3.38 0.001 0.13 

EVENING -2.1280 0.480 -4.44 0.001 0.12 

LATENIGHT -2.2797 0.750 -3.09 0.01 0.10 

TRAVEL DIST 0.0065 0.003 2.30 0.05 1.01 

OFFAGE -0.3692 0.040 -9.17 0.001 0.69 

OFFBLACK -1.1614 0.382 -3.04 0.01 0.31 

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

7 

-------

PARKS 

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

Alternative N=81 

Constant 3.1511 0.573 5.50 0.001 23.36 

GUNCRIME -0.6429 0.328 -1.96 0.05 0.53 

EVENING 0.0558 0.282 0.20 n.s. 1.06 

LATENIGHT -1.1378 0.459 -2.48 0.05 0.32 

TRAVEL DIST 0.0070 0.002 3.14 0.01 1.01 

OFFAGE -0.1966 0.025 -7.90 0.001 0.82 

OFFBLACK -1.2570 0.311 -4.05 0.001 0.28 

----------

8 

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

PUBLIC TRANSPORT 

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

Alternative N=55 

Constant -3.4423 0.617 -5.58 0.001 0.03 

GUNCRIME -0.6316 0.395 -1.60 n.s. 0.53 

EVENING -0.4296 0.409 -1.05 n.s. 0.65 

LATENIGHT -0.0134 0.337 -0.04 n.s. 0.99 

TRAVEL DIST 0.0091 0.002 5.21 0.001 1.01 

OFFAGE -0.0010 0.001 -0.88 n.s. 1.00 

OFFBLACK 0.6914 0.608 1.14 n.s. 2.00 

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

Reference Alternative: 

----

2 

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

Residential 

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

Multicollinearity statistics 

Predictor Pseudo-Tolerance 

GUNCRIME 0.98 

EVENING 0.93 

LATENIGHT 0.93 

TRAVEL DIST 0.99 

OFFAGE 1.00 

OFFBLACK 1.00 

21.13 
 

 

 

 

 

 

 

 

 

 

 

 

 

many explanatory variables is likely to have the most negative log likelihood ratio (LLR), but 

part of the size of the LLR results from the large number of variables used in the explanation. 

In the second section of Table 21.2, each of the other types of premises is compared to 

the reference type (residential units), which is the type that is chosen most frequently. Note that 

the coefficients will differ for each of the alternatives. This is because the variables predicting 

each alternative are unique and will differ in their weights. For some alternatives, an 

independent variable may have a significant positive effect while for other alternatives it may 

have a significant negative effect. 

For even other alternatives, the variable may not have a significant effect. For example, 

the use of a gun in a robbery (GUNCRIME) is positively associated with bank robberies but 

negatively associated with school robberies. For robberies in parks, the use of a gun is not related 

to the type of robbery. Note, also, that these are relative to the reference alternative, which in 

this case are residential robberies. 

The numbers in the far right column, the Odds Ratios, are useful for substantive 

interpretation of the model. They indicate the odds increase or decrease associated with the 

variable that the robbery took place on the specific premise compared to the reference alternative 

(residential premises). They are measured as the relative change when the corresponding 

predictor variable increases by one unit. Odds ratios above 1 indicate that the odds increase as 

the predictor variable increases, odds ratios between 0 and 1 indicate they decrease as the 

predictor variable increases. 

The p value indicates whether the odds ratio were likely to have occurred by chance if 

there was no relationship in the unit of the explanatory variable. For example, in the top panel 

on ‘Garages and parking lots’, the value of 1.56 indicates that-- if a robbery occurs in the 

evening (1) it is 1.56 times more likely (or, in other words, 56% more likely) to be in a garage or 

parking lot than at a residence. The p value indicates whether the odds ratio could have occurred 

by chance if there was no relationship. Thus, the p-value of 0.05 in the output demonstrates that 

the above odds ratio of 1.56 could have occurred by chance 5% of the time if there was no timeof-day difference in the probability that robberies take place in residences or in parking lots. 

Commercial robberies are 4 times as likely to be committed with a gun than residential 

robberies, and a difference this large could occur only .1% of the time. Robberies occurring in 

and around schools are significantly different from residential robberies on all six explanatory 

variables. They are much less likely to involve guns or be committed by black offenders and are 

slightly further away from the offender’s home. Unsurprisingly, they are all less likely to occur 

in the evening or late night and the offenders are younger than in residential robberies. 

21.14 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

The coefficients column in section two are similar to those in any regression equation. 

Coefficients are created for each choice (here premises type) including the reference category. 

They are particularly useful for prediction of choice with a new data set(see below). 

The third section indicates to what extent the explanatory variables vary together 

(multicollinearity). A pseudo tolerance below .90 indicates that this may be a problem in the 

model. If this is so, delete the variable with the lowest pseudo tolerance and run the model 

again. In this model all pseudo tolerances are above .9. Multicollinearity is not a problem. 

Adding another variable to the 1997 model 

Using the Log Likelihood, AIC, and BIC/SC statistics, it is possible to compare one 

decision making model to another. The decision making model in table 21.2 included six 

explanatory variables. Table 21.3 below adds the variable, number of offenders, to the model. 

Perhaps residential robbers are more likely to solo offenders than school yard robbers? 

However, adding the number of offenders to the model has little effect. 

The more explanatory variables, the fewer degrees of freedom (df) and the more complex 

the model. The log likelihood decreases from -1963 to -1957 (more negative is better). The AIC 

declines slightly from 3996 to 3994, but the more comprehensive BIC/SC increases from 4184 to 

4209 (closer to 0 is better for both). In other words, adding number of offenders to the model 

does not improve the differentiation of residential premises from other premises. For every 

premises type, the number of offenders is not significantly differentiated from residential 

robberies. 

Predicting non-street Robberies in 1998 based on the 1997 model 

Once a multinomial logit model is estimated, the parameter estimates can be used to 

predict a dependent variable in other data. The model developed in predicting the premises of 

robberies in 1997 can be used to predict the premises of robberies in 1998. This is done by 

saving the coefficients and applying them to the 1998 robbery data. The results can show how 

well the 1997 model estimated predicted 1998 robberies. 

21.15 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 21.3: 

Multinomial Logit Model of Crime Premises: 

Non-Street Robbery 1997 

Number of Offenders Added 

Model result: 

Data file: 1997 CHICAGO NON-STREET 

ROBBERIES 

DepVar: TYPE OF PREMISES 

N: 1,587 

Df: 1,579 

Type of choice model: Multinomial logit model

Number of Alternatives: 7 

Method of estimation: MLE 

Likelihood statistics 

Log Likelihood: -1957.2 

AIC: 3,994.3 

BIC/SC: 4,209.1 

Model error estimates 

Mean absolute deviation: 0.2 

Mean squared predicted error: 0.1 

REFERENCE CHOICE: 2 RESIDENTIAL 

Predictor 

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

Coefficient 

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

Stand Error 

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

t-value 

-----------

p-value 

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

Odds Ratio 

---------

3 GARAGES AND PARKING LOTS 

Alternative N=180 

Constant -1.2044 0.302 -3.98 0.001 0.30 

GUNCRIME 0.3039 0.191 1.59 n.s. 1.36 

EVENING 0.4481 0.193 2.32 0.05 1.57 

LATENIGHT -0.3608 0.236 -1.53 n.s. 0.70 

TRAVEL DIST 0.0059 0.001 4.46 0.001 1.01 

OFFAGE -0.0016 0.001 -1.90 n.s. 1.00 

OFFBLACK -0.4807 0.237 -2.02 0.050 0.62 

NUM OFF -0.1011 0.155 -0.65 n.s. 0.90 

4 COMMERCIAL 

Alternative N=378 

Constant -0.7871 0.221 -3.55 0.001 0.46 

GUNCRIME 1.3501 0.140 9.61 0.001 3.86 

EVENING -0.0709 0.163 -0.43 n.s. 0.93 

LATENIGHT -0.5702 0.183 -3.12 0.01 0.57 

TRAVEL DIST 0.0048 0.001 4.13 0.001 1.00 

OFFAGE -0.0017 0.001 -2.62 0.01 1.00 

OFFBLACK -0.6908 0.186 -3.71 0.001 0.50 

NUM OFF 0.1261 0.093 1.36 n.s. 1.13 

21.16 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 21.3: (continued) 

Predictor Coefficient Stand Error t-value p-value Odds Ratio 

5 BANKS AND CURRENCY EXCHANGES 

Alternative N=34 

Constant -1.2864 0.715 -1.80 n.s. 0.28 

GUNCRIME 1.4098 0.365 3.86 0.001 4.09 

EVENING -1.4180 0.621 -2.28 0.05 0.24 

LATENIGHT -1.7920 0.743 -2.41 0.05 0.17 

TRAVEL DIST 0.0085 0.002 3.56 0.001 1.01 

OFFAGE -0.0023 0.002 -1.12 n.s. 1.00 

OFFBLACK -1.4442 0.396 -3.64 0.001 0.24 

NUM OFF -0.9035 0.560 -1.61 n.s. 0.41 

6 SCHOOLS 

Alternative N=88 

Constant 6.8782 0.873 7.87 0.001 970.84 

GUNCRIME -2.0414 0.614 -3.32 0.001 0.13 

EVENING -2.1276 0.479 -4.44 0.001 0.12 

LATENIGHT -2.2976 0.752 -3.06 0.01 0.10 

TRAVEL DIST 0.0066 0.003 2.35 0.05 1.01 

OFFAGE -0.3716 0.040 -9.19 0.001 0.70 

OFFBLACK -1.1790 0.381 -3.09 0.01 0.31 

NUM OFF -0.1896 0.279 -0.68 n.s. 0.83 

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

7 PARKS 

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

Alternative N=81 

Constant 2.7911 0.618 4.52 0.001 16.30 

GUNCRIME -0.7293 0.337 -2.17 0.05 0.48 

EVENING 0.0657 0.283 0.23 n.s. 1.07 

LATENIGHT -1.1658 0.461 -2.53 0.05 0.31 

TRAVEL DIST 0.0068 0.002 3.05 0.01 1.01 

OFFAGE -0.1946 0.025 -7.83 0.001 0.82 

OFFBLACK -1.2409 0.312 -3.98 0.001 0.30 

NUM OFF 0.2586 0.177 1.46 n.s. 1.30 

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

8 PUBLIC TRANSPORT 

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

Alternative N=55 

Constant -3.0490 0.721 -4.23 0.001 0.05 

GUNCRIME -0.5504 0.400 -1.38 n.s. 0.58 

EVENING -0.4107 0.409 -1.00 n.s. 0.66 

LATENIGHT 0.0074 0.338 0.02 n.s. 1.01 

TRAVEL DIST 0.0092 0.002 5.28 0.001 1.01 

OFFAGE -0.0010 0.001 -0.90 n.s. 1.00 

OFFBLACK 0.6919 0.608 1.14 n.s. 2.00 

NUM OFF -0.3629 0.348 -1.04 n.s. 0.70 

Reference Alternative: 2 RESIDENTIAL 

21.17 
 

 

 

 

 

 

 

 

 Percent 

Correctly 

Predicted 

for 1998 

 

Type of Premises 1997 1998 

Residential 48.6% 47.8% 53.0% 

Garages/Parking 11.3% 10.5% 12.6% 

Commercial 23.8% 25.4% 33.2% 

Banks/CurrEx 2.1% 3.7% 4.8% 

Schools 5.5% 5.3% 39.2% 

Parks 5.1% 4.0% 11.7% 

Public Transit 3.5% 3.3% 5.0% 

 

 

 

 

 

 

 

 

 

 

 

 1587 1441 

 

 

 

 

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Table 21.3: (continued) 

Multicollinearity statistics

Predictor Pseudo-Tolerance 

GUNCRIME 0.94 

EVENING 0.92 

LATENIGHT 0.92 

TRAVEL DIST 0.98 

OFFAGE 0.99 

OFFBLACK 1.00 

NUM OFF 0.94 

In Table 21.4, the percentage distribution of the 7 premises types is compared for 1997 

and 1998 with the 1998 percentage correctly predicted for each type of premises using the MNL 

equation developed for 1997 robberies. Overall, not much has changed between the two years. 

Table 21.4: 

Non-Street Robberies in 1997 & 1998 

1998 Predicted by the 1997 Multinomial Logit Model 

Number of Robberies 

In order to be an improvement on the naïve assumption that the percentage of incidents at 

each premises type is no better than the overall distribution of premises in 1998, the multinomial 

logit model based on 1997 (Column 3) should predict the premises of incidents better than the 

marginal percentage distribution of incidents in 1998 (Column 2). It does for all premise types. 

A few examples: 

1. 47.8% of incidents were residential with the model correctly predicting 53.0% 

percent of them. 

2. 25.4% of incidents were commercial with the model correctly predicting 33.2% 

percent. 

21.18 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. 5.3% of incidents were in schools or school yards and the model correctly predicted 

39.2% percent. 

4. Garages and parking lots were only slightly better predicted by the model than by the 

1998 percentage distribution. 10.5% of incidents were in garages or parking lots, and 

the model correctly predicted 12.6% of these. 

Example 1 Conclusion 

When an offender chooses a type of premise to commit a robbery, the choice is not random. 

Personal characteristics such as age and racial group make a difference, but so do decisions that 

the offenders make when coming into the incident such as gun availability, distance from home, 

and time of day. This example demonstrates how Multinomial Logit models can be used to 

clarify the offender’s choice by the type of premise. The example also demonstrates that a 

model based on robbery choices made in one year can be useful in prediction of robberies in 

another year. 

Another example of the Multinomial Logit model is presented in the attachment where 

Levine, Robertson and Fosberg analyze the type of weapon used in Houston robberies. 

The Conditional Logit Model 

The CL model is estimated on a different data structure. It is a matrix where each row 

(record) represents a combination of an observational unit n (a case, often a decision maker) with 

an alternative in the choice set i, and where each column represents a variable (a characteristic of 

the observational unit and/or the alternative). In this case, each record represents a possible 

alternative that the case (or decision maker) is presented with. The dependent variable is a 

binomial variable and indicates which alternative i from a set of alternatives was chosen by 

observational unit n. 

For example, a community is divided into twenty neighborhoods (alternatives). Each of 

these is classified according to number of businesses, wealth, racial makeup and population size 

(5 variables). For each case, an offender ‘selects’ a neighborhood where the crime is committed 

(choice).1 For 100 cases and twenty alternatives, a matrix of 2,000 records and a minimum of 

six variables would be necessary. The sixth variable identifies the chosen alternative. The 

number of records can grow quickly. The following is a simplified example. The variation in 

outcomes is explained by variation in the characteristics of the alternatives. CrimeStat is able to 

construct such a file by combining a ‘case file’ and an ‘alternatives file’. Below we present a 

simplified description of the process. 

The offender may not do this rationally, of course, and may simply be at that location (a routine activity). 

Nevertheless, the model assumes that the offender has made a utility calculation to commit the crime at the 

location. To that extent, it is a decision among many alternatives. 

21.19 

1 
 

 

 

 

 

 

 

 

 

 

 

 

 

Destination Choice 

We start with the case file shown in Table 21.1, that is the file that is used for estimating 

a multinomial logit model, and by a model of the Destination (i.e.. in which area, P, Q, R or S, 

did the offender commit the crime?). Whereas a MNL model is used to assess whether linear 

combinations of characteristics associated with the cases (e.g.,Origin, Age, Time and CrimeType) 

predict which zones the offender selects to commit the crime, a CL model is used to assess 

whether characteristics of the alternatives predict the zone chosen. The alternatives are the zones 

themselves and, obviously, additional information is needed on the alternatives. 

Table 21.5 shows part of an example file, containing the four alternative destination areas 

P, Q, R and S for the decision on where to offend. The variables include an identifier (Zone), the 

percentage of the household below a poverty threshold (Poverty) and the percentage of the nonresidential land use (Non-Residential) in the zone. 

Table 21.5: 

Zone File Describing 4 Alternative Zones 

Zone Poverty Non-Residential 

P 2 40 

Q 2 16 

R 4 23 

S 5 12 

The data structure required for estimation of the CL model represents all possible 

combinations of the rows in the case file and the rows in the zone file, including the variables in 

both files. It also includes for each decision maker a binomial variable indicating the alternative 

that was chosen by the decision maker. For example, if there are 200 cases (decision makers) 

and 7 alternatives that are available, there will be 1,400 records (200 x 7) in the data set. Each 

decision maker will be represented 7 times, representing each of the 7 alternatives that the 

decision maker is confronted with. However, the decision maker will have selected only one of 

these alternatives. For that record, the value of the binomial choice variable will be 1; for the 

other six records, the value of the binomial choice variable will be 0. 

To go back to the example, Table 21.6 displays the combination of Tables 21.1 and 21.5. 

Note that the columns 1-6 of Table 21.6 are a copy of Table 21.1 with each row repeated four 

times (the first original row in rows 1-4). Also verify that columns 7-9 are copies of Table 21.5, 

with each row repeated five times (the first original row in rows 1, 5, 9, 13 and 17, the second 

original row in rows 2, 6, 10, 14 and 18, etc.). Finally note that the indicator variable, 

21.20 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 21.6: 

Case-alternative File Describing 20 Case-Alternative Combinations 

ID Org Dest Age CrimeTyp Time Zone Pov NonRes Chosen Home 

1 P P 18 Burglary 3am P 2 40 1 1 

1 P P 18 Burglary 3am Q 2 16 0 0 

1 P P 18 Burglary 3am R 4 23 0 0 

1 P P 18 Burglary 3am S 5 62 0 0 

2 

2 

2 

2 

Q 

Q 

Q 

Q 

P 

P 

P 

P 

23 

23 

23 

23 

Robbery 

Robbery 

Robbery 

Robbery 

7pm 

7pm 

7pm 

7pm 

P 

Q 

R 

S 

2 

2 

4 

5 

40 

16 

23 

62 

1 

0 

0 

0 

0 

1 

0 

0 

3 

3 

3 

3 

R 

R 

R 

R 

S 

S 

S 

S 

42 

42 

42 

42 

Illicit drug 

Illicit drug 

Illicit drug 

Illicit drug 

2pm 

2pm 

2pm 

2pm 

P 

Q 

R 

S 

2 

2 

4 

5 

40 

16 

23 

62 

0 

0 

0 

1 

0 

0 

1 

0 

4 

4 

4 

4 

R 

R 

R 

R 

Q 

Q 

Q 

Q 

32 

32 

32 

32 

Robbery 

Robbery 

Robbery 

Robbery 

1pm 

1pm 

1pm 

1pm 

P 

Q 

R 

S 

2 

2 

4 

5 

40 

16 

23 

62 

0 

1 

0 

0 

0 

0 

1 

0 

5 

5 

5 

5 

S 

S 

S 

S 

R 

R 

R 

R 

19 

19 

19 

19 

Burglary 

Burglary 

Burglary 

Burglary 

6am 

6am 

6am 

6am 

P 

Q 

R 

S 

2 

2 

4 

5 

40 

16 

23 

62 

0 

0 

1 

0 

0 

0 

0 

1 

Chosen, is set to 1 if the value in variable Destination matches the value in variable Zone. The 

variable Home will be discussed below. 

Note that in Table 21.6, the zone characteristics Pov and Nonres only vary across 

alternatives but not across cases (decision makers): the values of these two variables are just 

repeated in every case. Quite often, however, the model includes variables that vary across 

alternatives and across cases as well. The last column in Table 21.6 contains a binomial 

variable, Home, that indicated whether an alternative zone is the zone of residence of the 

offender. Thus, it has value 1 if Origin=Zone, and 0 otherwise. This variable varies both across 

alternatives (e.g. for a given ID, one alternative equals 1 and the other equal 0) and across cases 

(for a given Zone, say A, it equals 1 for case 1, but 0 for cases 2-5). In a similar fashion (but 

more difficult to verify by just inspecting the table), we could define a new variable that 

represents the distance between the alternative zone and the zone of the offender’s residence. 

21.21 
 

 

 

 

 

 

 

 

 

 

Note also that the indexing in the independent variables in the MNL and CL models 

reflects the data structures used for estimation. In the MLN model, Vin = 2 iXn, where the index n 

in the term Xn indicates that the variables only vary between cases (decision makers), so that we 

only need one row per case. In the CL model, Vin = 2 Xni, where the indices n and i in the term Xni 

reflect that the variables can vary between cases (decision makers) and between alternatives, so 

that we need multiple rows case (equation 21.11 demonstrated that in the CL model, the 

variables must vary across alternatives and may vary across decision makers, but cannot be 

estimated when they vary across decision makers only). 

Crime Type Choice 

Now let us consider another type of choice: the choice of a crime type. An offender 

urgently needing money may have to choose a criminal activity that generates the required 

amount as easily and with as little risk as possible. If we assume that burglary, robbery and illicit 

drug dealing are the available alternatives, an alternatives file could look like Table 21.7. The 

variables in this file represent attributes that may differentiate between the crime types: Expected 

Profits, Detection Risk and Time needed to search and attack a target and that may affect the 

attractiveness of these offences to the offenders. 

Table 21.7: 

Crime Type File 

(Alternative Crime Types) 

Crime type Expected 

Profits 

Detection 

Risk 

Sanction 

Severity Time Needed 

Burglary 200 .07 3 60 

Robbery 50 .15 5 20 

Illicit drug 20 .02 2 40 

Analogously to the case of destination zone choice, the data structure required for 

estimation of the CL model represents all possible combinations of the rows in the case file 

(Table 21.1) and the rows in the alternatives file (Table 21.6), including the variables in either 

file, and also including for each decision maker a binomial variable indicating the alternative 

(which crime type) that was chosen by the decision maker. 

Table 21.8 displays the combination of Tables 21.1 and 21.7. The first six columns of 

Table 21.8 are a copy of Table 21.1 with each row repeated three times. Also verify that column 

7-9 are copies of Table 21.3, with each row repeated four times (the first original row in rows 

1,4,7, 10, and 13, the second original row in rows 2, 5, 8, 11 and 14, etc.). Finally note that the 

indicator variable Chosen is set to 1 if the value in variable Type matches the value in variable 

CrimeType. 

21.22 
 

 

 

 

 

 

 

 

 

 

 

Table 21.8: 

Case-alternative File Describing 15 Case-alternative Combinations 

ID Org Dest Age CrimeType Time Type Profit Risk Sanc Time Chosen 

1 P P 18 burglary 3am burglary 200 .07 3 60 1 

1 P P 18 burglary 3am robbery 50 .15 5 20 0 

1 P P 18 burglary 3am il. drug 20 .02 2 40 0 

2 Q P 23 robbery 7pm burglary 200 .07 3 60 0 

2 Q P 23 robbery 7pm robbery 50 .15 5 20 1 

2 Q P 23 robbery 7pm il. drug 20 .02 2 40 0 

3 R S 42 il. drug 2pm burglary 200 .07 3 60 0 

3 R S 42 il. drug 2pm robbery 50 .15 5 20 0 

3 R S 42 il. drug 2pm il. drug 20 .02 2 40 1 

4 R Q 32 robbery 1pm burglary 200 .07 3 60 0 

4 R Q 32 robbery 1pm robbery 50 .15 5 20 1 

4 R Q 32 robbery 1pm il. drug 20 .02 2 40 0 

5 S R 19 burglary 6am burglary 200 .07 3 60 1 

5 S R 19 burglary 6am robbery 50 .15 5 20 0 

5 S R 19 burglary 6am il. drug 20 .02 2 40 0 

Example 2: Modeling Choice of Neighborhood for Residential Burglaries in The 

Hague with the Conditional Logit Model 

The discrete spatial choice approach was first applied to criminal location choices by 

Bernasco & Nieuwbeerta (2005). This example uses CrimeStat to replicate their analysis of 548 

cleared burglaries committed in the years 1996-2001 in the city of The Hague, the Netherlands, 

by solitary offenders ( i.e., offenders who perpetrated the burglary without known accomplishes). 

The discrete spatial choice model of burglary integrated journey-to-crime research (that 

focuses on distance traveled without considering other aspects of criminal location choice) and 

ecological research (that addresses variation in opportunities and target attractiveness, but 

ignores the distance offenders have to travel to reach the targets). 

Bernasco & Nieuwbeerta distinguished 89 neighborhoods in The Hague, which served as 

the spatial units of analysis. They argued that neighborhoods would be attractive for burglary if 

they (1) were affluent, (2) had a large proportion of single-family dwellings, (3) had high 

population turnover (4) had high ethnic heterogeneity, (5) had large numbers of households, (6) 

were situated relatively close to the city center and (7) were located relatively close to the 

offender’s residence. Note that the first six criteria are the attributes of the 89 alternative 

neighborhoods (independently of any attributes of the burglar), while the last criterion (proximity 

to offender’s home) depends on the locations of both the offender and the potential target 

neighborhoods. 

21.23 
 

 

 

 In Table 21.9, the 548 The Hague burglaries are analyzed with the conditional logit model, 

using the following 7 variables as predictors of the burglars’ selection of a target neighborhood: 

1. PROPVAL. Average value of residential properties, in 100,000 euro 

2. SINGFAM. Percentage of units that are single-family dwellings, in 10% units 

3. RESMOBIL. Percentage of residents that moved during past year, in 10% units 

4. ETNHETERO. (Ethnic Heterogeneity). Blau / Herfindahl index (x 10) 

5. PROXIMITY. Negative distance between offender neighborhood and potential target 

neighborhood, in kilometers. The authors used negative distance instead of distance because 

this yielded a model in which all expected parameters were positive. 

6. PROXCITY.Negative distance between city center and potential target neighborhood, in 

kilometers 

7. RESUNITS. Number of residential properties in the neighborhood, in 1,000 properties 

The results in Table 21.9 replicate the findings reported by Bernasco & Nieuwbeerta 

(2005, p. 308).2

 The summary section of the output reports general information about the model 

and the estimation procedure, including the names of the data file and the dependent variable. 

The output shows that the number of records is 48,772, which is 548 × 89 (i.e. the number of 

offenders multiplied by the number of The Hague neighborhoods). The number of degrees of 

freedom is 541 (the number of offenders -548, minus the number of estimated parameters -7). As 

discussed in the multinomial logit example, the likelihood statistics (Log Likelihood, Akaike 

Information Criterion, AIC ; and Bayesian Information Criterion, BIC) indicate how well the 

model fits the data (lower values indicate better fit). These statistics are only used to compare 

different models, and have no useful interpretation for a single model. 

The coefficient section reports the results for each predictor variable and include the 

estimated coefficients, their standard errors, t-values, and p-values. The odds ratios column is the 

most useful statistic for substantive interpretation of the outcome. The odds ratio (which equals 

e

β

) represents the factor by which the odds that a neighborhood is chosen for a burglary increases 

or decreases when the value of the predictor increases by one unit. An odds ratio greater than 1 

indicates that the odds increase while an odds ratio between 0 and 1 indicate that the odds 

decrease. 

For example, the odds ratio of 1.05 for variable PROPVAL indicates that as the average 

value of properties in the neighborhood increases by 100,000 euro, the odds that it is selected by 

a burglar increase by a factor 1.05 (i.e. by approximately 4.5 percent). Another example: the 

The standard errors reported here are slightly smaller than those reported by Bernasco & Nieuwbeerta 

(2005), a difference due to their correction of the standard errors for the possible interdependence among 

the burglaries (the 548 burglaries were committed by 290 unique persons; thus, some of them committed 

multiple burglaries). 

21.24 

2 
 

 

 

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estimated value of 1.67 for proximity means that if a neighborhood is located one kilometer 

closer to the offender’s home, the odds that it will be selected by this burglar increase by a factor 

1.67 (i.e. by approximately 67 percent). 

Table 21.9: 

Conditional Logit Model of Burglary Neighborhood Choice

 Model result: 

Data file: TheHagueBurglary.dbf

DepVar: CHOSEN 

N: 48,772 

Df: 541 

Type of choice model: Conditional logit model

Number of Alternatives: 89 

Method of estimation: MLE 

Likelihood statistics 

Log Likelihood: -2,203.3 

AIC: 4,420.6 

BIC/SC: 4,450.7 

Model error estimates 

Mean absolute deviation: 0.02 

Mean squared predicted error: 0.01 

Pseudo- Odds 

Predictor 

PROPVAL 

Coefficient Stand Error 

0.0445 0.112 

Tolerance 

0.33 

t-value 

0.40 

p-value 

n.s. 

Ratio 

1.05 

SINGFAM 0.1239 0.042 0.43 2.96 0.01 1.13 

RESMOBIL -0.0285 0.046 0.48 -0.62 n.s. 0.97 

ETNHETERO 0.1380 0.032 0.37 4.37 0.001 1.15 

PROXIMITY 0.5140 0.034 0.74 15.22 0.001 1.67 

PROXCITY -0.0812 0.049 0.37 -1.66 n.s. 0.92 

RESUNITS 0.3039 0.029 0.80 10.61 0.001 1.36 

Average Predicted Probability:

Mean 

CHOSEN Probability N StdDev 

0 0.011 48224 0.012 

1 0.028 548 0.024 

Total 0.011 48772 0.013 

The section also includes the pseudo-tolerances of the indicator variable (see Chapters 15 

and 17 for discussion of this statistic). If the tolerance of a variable is low, this indicates that the 

variable is strongly correlated with linear combinations of the other predictor variables in the 

equation, and that it therefore does not add much unique variability to the prediction of the 

21.25 
 

 

 

 

dependent variable. This situation is called multicollinearity and is usually solved by removing 

the variable with the lowest tolerance from the equation. Note that three of the variables are not 

significant and several have low tolerances and that a simplified model can be produced by 

dropping them without much loss of generality (not shown). 

The last section also lists average predicted probabilities for neighborhoods that were 

chosen (.028) and those that were not chosen (.011). Note that the average predicted probability 

multiplied by the total number of records yields the total number of burglary cases in the file.3 

Conclusion 

In discrete choice modeling, the dependent variable is made up of mutually exclusive and 

exhaustive categories. The category that is chosen is based upon characteristics of the decision 

maker (in the multinomial logit model), the characteristics of the alternatives (in the conditional 

logit model), or the interaction of the two (also in the conditional logit model). Interpretations of 

discrete choice models can be closely linked to the economic theory of utility maximization. Of 

all possible alternatives, the alternative is selected that maximizes gain and minimizes cost. 

The CrimeStat discrete choice module is designed for regression when the dependent 

variable consists of unordered categories such as type of weapon or neighborhood where a crime 

is committed. This is in contrast to more traditional regression that is mainly concerned with 

dependent variables that are continuous or quasi-continuous, such as rates or counts. The 

Discrete Choice module is a multinomial extension of binomial logistic regression, discussed in 

Chapter 18, which allows for only two categories of the dependent variable. 

The Discrete Choice module provides for two different models, the multinomial and the 

conditional logit model. Which one is used must be based upon the availability and relevance of 

data that reflect attributes of the categories and attributes of the cases (usually offenders, or 

crimes). To some extent it also depends upon the number of categories of the dependent variable 

since the tractability of the multinomial model decreases as the number of categories grows. 

The conditional logit model is most appropriate if the outcome is assumed to be based on 

characteristics of the alternatives or their interaction with characteristics of the situation or the 

decision-maker. The CL data structure duplicates every possible alternative for each case and 

designates one as chosen. The results summarize the difference between the chosen selection 

and all others. For example, Chicago has seventy-seven neighborhoods that vary in terms of 

wealth, number of businesses, level of drug crime, and population. They also vary in distance 

from an offender’s home. Each offender’s decision about in which neighborhood to commit the 

To do this accurately, one needs more than 3 decimal places. The CrimeStat output includes six decimal 

places. We have reduced the number of decimal places in the table to make it clearer. 

21.26 

3 
 

 

 

 

crime is based upon a comparison of the characteristics of the 77 neighborhoods. The data file 

has one record for each alternative that the decision maker faces. If 1,000 offenders are 

analyzed, the resultant file would have 77,000 records. 

The multinomial model may be appropriate if the choice has fewer categories and is 

dependent mainly on characteristics of the offender and the particular incident. A separate 

equation is constructed that compares a reference category with every other category of the 

dependent variable. For example, if weapon choice is dependent upon the victim’s age and 

gender, type of target, and time of day, then a separate equation is constructed comparing gun 

incidents, the most frequent category, to knives, other weapons and strong armed. The data file 

contains one record for each offender. 

In Chapter 22, we discuss the use of the CrimeStat discrete choice module routines to 

estimate these two models. Two additional routines are included in the discrete choice module. 

First, as discussed above the Conditional logit requires data organization that combines 

characteristics of the incident and all possible choices. CrimeStat will build this file for you. 

Second, both discrete and conditional models allow for prediction of dependent variables in one 

data set from the relationships found in another. Thus, in the example of Chicago robberies 

(above) 1998 robbery locations are predicted based on MNL coefficients of 1997 robberies. 

If an analyst wants to consider the ‘who’, ‘where’, or ‘why’ of choice among multiple 

mutually exclusive possibilities and has a model in which criminals maximize the utility of their 

choices, then the either Conditional logit or Multinomial logit in the discrete choice module are 

appropriate techniques. 

21.27 
 

References 

Ben-Akiva, M. E., & Bierlaire, M. (1999). Discrete Choice Methods and their Applications to 

Short Term Travel Decisions. In R. W. Hall (Ed.), Handbook of Transportation Science (pp. 5-

34). Norwell, MA: Kluwer. 

Ben-Akiva, M. E., & Lerman, S. R. (1985). Discrete Choice Analysis: Theory and Application to 

Travel Demand. Cambridge, MA: MIT Press. 

Bernasco, W. (2010a). Modeling Micro-Level Crime Location Choice: Application of the 

Discrete Choice Framework to Crime at Places. Journal of Quantitative Criminology, 26(1), 

113-138. 

Bernasco, W. (2010b). A Sentimental Journey to Crime; Effects of Residential History on Crime 

Location Choice. Criminology, 48, 389-416. 

Bernasco, W. (2007). The usefulness of measuring spatial opportunity structures for tracking 

down offenders: A theoretical analysis of geographic offender profiling using simulation studies. 

Psychology, Crime & Law, 13, 155-171. 

Bernasco, W. (2006). Co-Offending and the Choice of Target Areas in Burglary. Journal of 

Investigative Psychology and Offender Profiling, 3, 139-155. 

Bernasco, W., & Block, R. (2009). Where Offenders Choose to Attack: A Discrete Choice 

Model of Robberies in Chicago. Criminology, 47(1), 93-130. 

Bernasco, W., & Kooistra, T. (2010). Effects of Residential History on Commercial Robbers’ 

Crime Location Choices. European Journal of Criminology, 7(4), 251-265. 

Bernasco, W., & Nieuwbeerta, P. (2005). How Do Residential Burglars Select Target Areas? A 

New Approach to the Analysis of Criminal Location Choice. British Journal of Criminology, 45, 

296-315. 

Clare, J., Fernandez, J., & Morgan, F. (2009). Formal Evaluation of the Impact of Barriers and 

Connectors on Residential Burglars' Macro-Level Offending Location Choices. Australian and 

New Zealand Journal of Criminology, 42, 139-158. 

Jepsen, L., & Jepsen, C. (2002). An empirical analysis of the matching patterns of same-sex and 

opposite-sex couples. Demography, 39(3), 435-453. 

21.28 
 

References (continued) 

Krebs, J. R., & Davies, N. B. (1993). An Introduction to Behavioural Ecology (3th ed.). Oxford: 

Blackwell. 

McFadden, D. (1980). Econometric Models for Probabilistic Choice Among Products. The 

Journal of Business, 53(3), S13-S29. 

McFadden, D. (1973). Conditional Logit Analysis of Qualitative Choice Behavior. In P. 

Zarembka (Ed.), Frontiers in Econometrics (pp. 105-142). New York: Academic Press. 

Palfrey, T. R., & Poole, K. T. (1987). The Relationship between Information, Ideology, and 

Voting Behavior. American Journal of Political Science, 31(3), 511-530. 

Phillips, S. (2003). The Social Structure of Vengeance: A Test of Black's Model. Criminology, 

41(3), 673-708. 

Rengert, G. F. (1981). Burglary in Philadelphia: A Critique of an Opportunity Structure Model. 

In P. J. Brantingham & P. L. Brantingham (Eds.), Environmental Criminology (pp. 189-202). 

Beverly Hills, CA: Sage. 

Smith, T. S. (1976). Inverse Distance Variations for the Flow of Crime in Urban Areas. Social 

Forces, 54(4), 802-815. 

Tita, G., & Griffiths, E. (2005). Traveling to Violence: The Case for a Mobility-Based Spatial 

Typology of Homicide. Journal of Research in Crime and Delinquency, 42, 275-308. 

Train, K. E. (2009). Discrete Choice Methods with Simulation (2nd ed.). New York: Cambridge 

University Press. 

Train, K. E. (1980). A Structured Logit Model of Auto Ownership and Mode Choice. The 

Review of Economic Studies, 47(2), 357-370. 

21.29 
Attachment A 

Modeling Correlates of Weapon Use in Houston Robberies 

With the Multinomial Logit Model 

Ned Levine 

Ned Levine & Associates 

Houston, TX 

Alan Robertson 

Houston Police Department 

Houston, TX 

Barry Fosberg 

Houston Police Department 

Houston, TX 

Introduction 

We made an analysis of weapon use in Houston robberies. Because the type of weapon 

used rarely changes, the Multinomial Logit model was an appropriate modeling tool. Between 

2007 and 2009, there were 33,419 robberies that occurred within the City of Houston. Of these, 

suspect information was obtained for 3,709 of these offenses. Using the suspect information for 

these 3,709 offenses, we modeled predictors of weapon use. 

Figure 21A.1 shows the distribution of weapons within the area covered by the Houston 

Police Department. Of the weapons used, 1,744 (or 47%) involved firearms, 272 (or 7%) 

involved knives, 1,184 (or 32%) involved bodily force, 192 (or 5%) involved threat, and 317 (or 

9%) involved another weapon. Using the ‘other weapon’ as the reference category, we related 

weapon choice to 11 variables grouped into five categories: 1) Offender characteristics (age, 

gender, being of Hispanic ethnicity, being of African-American ethnicity); 2) Presence of cooffenders (the number of suspects); 3) Whether the robbery occurred on a commercial premise or 

not; 4) Time period (night, afternoon, morning), and 5) Crime location characteristics (median 

household income of the block group at the crime location, distance from the offenders residence 

to the crime location). 

Method 

The multinomial logit model was used to estimate the effect of the coefficients on 

weapon choice. The choice probability, Pni, that the offender, i, chooses a particular weapon, j, is 

estimated by an exponentiated linear combination of independent predictors associated with the 

offender, k, proportional to the choice probabilities for all weapons: 

Attachment A: 21.30 
Figure 21A.1:
 

 

 

 

 

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That is, the probability of the offender choosing any one weapon is estimated by an 

exponentiated linear combination of observed variables associated with the offenders divided by 

the sum of the exponentiated linear combination for all weapon choices. The coefficients are 

estimated across offenders but the probabilities are calculated for each offender separately. 

Table 21A.1 presents the results of the model and Table 21A.2 summarizes the initial 

frequencies and the average predicted probabilities. Compared to the use of another weapon (the 

reference group), firearm use was associated with younger Hispanic or African-American males, 

with more accomplices, and was more likely to be committed on commercial premises in higher 

income locations at night or in the early morning. Crime travel distance was farther. 

Table 21A.1: 

Multinomial Logit Predictors of Weapon Use in Houston Robberies: 2007-09 

Model result: 

DepVar: WEAPON 

N: 3709 

Df: 3696 

Likelihood statistics 

Log Likelihood: -4432.1 

AIC: 8936.3 

BIC/SC: 9160.2 

Model error statistics 

Mean absolute deviation: 0.27 

Mean squared predicted error: 0.14 

Weapon: Firearm 

Predictor Coefficient Stand Error t-value p-value Odds ratio 

Constant 0.4959 0.005 91.43 0.001 1.64 

Offender characteristics 

AGE -0.0279 0.003 -9.79 0.001 0.97 

FEMALE -0.9308 0.005 -171.60 0.001 0.39 

HISPANIC 1.0317 0.005 190.54 0.001 2.81 

AFRICANAMERICAN 1.3980 0.005 258.29 0.001 4.05 

Co-offenders 

NUMSUSPCTS 0.1774 0.005 33.08 0.001 1.19 

Type of premise 

COMMERCIAL 0.6431 0.005 118.73 0.001 1.90 

Time period 

NIGHT 0.3927 0.005 72.47 0.001 1.48 

AFTERNOON -0.2119 0.005 -39.11 0.001 0.81 

MORNING 0.1672 0.005 30.84 0.001 1.18 

Crime location 

MED HH INC 0.00001 0.000003 1.97 0.05 1.00 

TRAVEL 

DISTANCE 0.0289 0.004 6.65 0.001 1.03 

Attachment A: 21.32 
 

 

 

 

 

 

 

 

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Table 21A.1: (continued) 

Weapon: Knife 

Predictor Coefficient Stand Error t-value p-value Odds ratio 

Constant -0.6883 0.005 -126.82 0.001 0.50 

Offender characteristics 

AGE 0.0205 0.004 5.82 0.001 1.02 

FEMALE -0.1706 0.005 -31.43 0.001 0.84 

HISPANIC 0.4801 0.005 88.52 0.001 1.62 

AFRICANAMERICAN -0.1024 0.005 -18.88 0.001 0.90 

Co-offenders 

NUMSUSPCTS -0.1842 0.005 -34.03 0.001 0.83 

Type of premise 

COMMERCIAL 0.0914 0.005 16.84 0.001 1.10 

 Time period 

NIGHT 0.2574 0.005 47.45 0.001 1.29 

AFTERNOON -0.0165 0.005 -3.05 0.01 0.98 

MORNING 0.1056 0.005 19.46 0.001 1.11 

Crime location 

MED HH INC 0.000004 0.000003 1.123 n.s. 1.00 

TRAVEL 

DISTANCE -0.0300 0.005 -5.91 0.001 0.97 

Weapon: Bodily force 

Predictor Coefficient Stand Error t-value p-value Odds ratio 

Constant 0.5384 0.005 99.27 0.001 1.71 

Offender characteristics 

AGE -0.0012 0.003 -0.42 n.s. 1.00 

FEMALE 0.0542 0.005 9.99 0.001 1.06 

HISPANIC 0.2362 0.005 43.61 0.001 1.27 

AFRICANAMERICAN 0.5802 0.005 107.17 0.001 1.79 

Co-offenders 

NUMSUSPCTS -0.1342 0.005 -24.96 0.001 0.87 

Type of premise 

COMMERCIAL 0.2963 0.005 54.70 0.001 1.34 

Time period 

NIGHT 0.0861 0.005 15.88 0.001 1.09 

AFTERNOON 0.4638 0.005 85.64 0.001 1.59 

MORNING 0.3381 0.005 62.35 0.001 1.40 

Crime location 

MED HH INC 0.00001 0.000003 4.14 0.001 1.00 

TRAVEL 

DISTANCE -0.0227 0.005 -5.02 0.001 0.98 

Attachment A: 21.33 
 

 

 

 

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Table 21A.1: (continued) 

Weapon: Threat 

Predictor Coefficient Stand Error t-value p-value Odds ratio 

Constant -1.9169 0.005 -353.16 0.001 0.15 

Offender characteristics 

AGE 0.0187 0.004 5.17 0.001 1.02 

FEMALE -1.2088 0.005 -222.67 0.001 0.30 

HISPANIC 0.2279 0.005 42.00 0.001 1.26 

AFRICANAMERICAN 0.6623 0.005 122.08 0.001 1.94 

Co-offenders 

NUMSUSPCTS -0.3061 0.005 -56.47 0.001 0.74 

Type of premise 

COMMERCIAL 0.7707 0.005 142.04 0.001 2.16 

Time period 

NIGHT -0.1765 0.005 -32.52 0.001 0.84 

AFTERNOON -0.0113 0.005 -2.08 0.05 0.99 

MORNING 0.4941 0.005 91.049 0.001 1.64 

Crime location 

MED HH INC 0.00002 0.000003 4.25 0.001 1.00 

TRAVEL 

DISTANCE 0.0193 0.005 3.86 0.001 1.02 

Reference choice: Other weapon 

On the other hand, knife use was associated with older, Hispanic males with few 

accomplices. The robberies were more likely to be committed on commercial premises at night 

or early morning. Crime travel distance was shorter. 

Bodily force was associated with Hispanic or African-American females and with few 

accomplices. The robberies were more likely to be committed in higher income locations on 

commercial premises in the afternoon, morning or, to a lesser extent, late at night. The crime 

travel distance was shorter. 

Finally, threats were associated with older Hispanic or African-American males with no 

or few accomplices. The robberies were more likely to be committed on commercial premises in 

the morning in higher income locations. The crime travel distance was farther. 

Table 21A.2 shows that the average predicted probabilities for weapon use across all 

robbers exactly predicted the actual distribution of weapon use. 

Conclusion 

The most distinguishing variable is the number of suspects. More co-offenders lead to a 

greater use of firearms, suggesting the involvement of gangs. Other consistent predictors are 

Attachment A: 21.34 
 

Weapon Use (%) Probability

Firearm 1,744 (47%) 0.47 

Knife 272 (7%) 0.07 

Bodily force 1,184 (32%) 0.32 

Threat 192 (5%) 0.05 

Other weapon 317 (9%) 0.09 

TOTAL 3,709 (100%) 1.00 

 

 

 

 

 

 

 

ethnicity - Hispanic or African-Americans are more likely to use weapons than non-Hispanic 

White or Asian suspects, and gender - males are more likely to use firearms, knives or threats 

than females, who in turn are more likely to use bodily force. Commercial properties tend to be 

Table 21A.2: 

Summary of Predictions

 Average

 Frequency of Predicted

Weapon 

disproportionately associated with weapons of all sorts primarily because they are the most 

common location for robberies in general. There are subtle differences in the time period and in 

the travel distance in predicting the type of weapon used. 

Attachment A: 21.35