logistic regression and discriminant analysis...logistic regression and discriminant analysis are...

Logistic Regression and DiscriminantAnalysis

Sebastian Tillmanns and Manfred Krafft

AbstractQuestions like whether a customer is going to buy a product (purchase vs. non-purchase) or whether a borrower is creditworthy (pay off debt vs. credit default)are typical in business practice and research. From a statistical perspective, thesequestions are characterized by a dichotomous dependent variable. Traditionalregression analyses are not suitable for analyzing these types of problems,because the results that such models produce are generally not dichotomous.Logistic regression and discriminant analysis are approaches using a number offactors to investigate the function of a nominally (e.g., dichotomous) scaledvariable. This chapter covers the basic objectives, theoretical model consider-ations, and assumptions of discriminant analysis and logistic regression. Further,both approaches are applied in an example examining the drivers of sales contestsin companies. The chapter ends with a brief comparison of discriminant analysisand logistic regression.

KeywordsDichotomous Dependent Variables •Discriminant Analysis • Logistic Regression

Dr. Sebastian Tillmanns is an Assistant Professor at the Institute of Marketing at the WestfälischeWilhelms-University Münster. Prof. Dr. Manfred Krafft is Director of the Institute of Marketing atthe Westfälische Wilhelms-University Münster. The book chapter is adapted from Frenzen andKrafft (2008). We thank Linda Hollebeek for her copy-editing.

S. Tillmanns (*)Westfälische Wilhelms-Universität Münster, Muenster, Germanye-mail: [email protected]

M. KrafftInstitute of Marketing, Westfälische Wilhelms-Universität Münster, Muenster, Germanye-mail: [email protected]

# Springer International Publishing AG 2017C. Homburg et al. (eds), Handbook of Market Research,DOI 10.1007/978-3-319-05542-8_20-1

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mailto:[email protected]

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ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Foundations and Assumptions of Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Discriminant Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Foundations and Assumptions of Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Logistic Regression Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Applied Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Research Question and Sample, Model, Estimation, and Model Assessment . . . . . . . . . . . . . . . 29Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Introduction

Decision makers and researchers regularly need to explain or predict outcomes thathave only one of two values. For example, practitioners might aim to predict, whichcustomers will respond to their next mailing campaign. In this case, the outcome iseither response or no response. Variables with such a nominal scaling can bedescribed as dichotomous or binary. Table 1 lists several marketing-based examples.When approaching these issues systematically, it becomes obvious that traditionalordinary least squares (OLS) regression analyses are not suitable for analyzing thesetypes of problems, because the results that such models produce are generally notdichotomous. For example, if product purchase (coding of the dependent variableY = 1) and non-purchase (Y = 0) are considered, an OLS regression could predictpurchase probabilities of less than zero or above one, which cannot be interpreted. Inaddition, a binary dependent variable would violate the assumption of normallydistributed residuals and would render inferential statistical statements more difficult(Aldrich and Nelson 1984, p. 13). Discriminant analysis and logistic regression aresuitable methodologies for the analysis of problems in which the dependent variableis nominally scaled.

Table 1 Potential applications of discriminant analysis and logistic regression in marketing

Subject of investigation Grouping

Direct marketing/mail order business Order (yes – no)

Nonprofit marketing Donation (yes – no)

Retailing Purchase/non-purchase (choice)

Staff selection (salespeople, franchisees) Successful/unsuccessful employee

Customer win-back Defected customers return (yes – no)

Branding Repurchase (yes – no)

Product policy New product success versus failure

Innovation adoption Adopter versus non-adopter

2 S. Tillmanns and M. Krafft

Although both logistic regression and discriminant analysis are suitable forexplaining or predicting dichotomous outcomes, discriminant analysis is optimalfor classifying observations when the independent variables are normally distributed(conditional on the dependent variable Y ). In this case, the discriminant analysis ismore efficient, because it utilizes information about the distribution of the indepen-dent variables, which the logistic regression does not (Agresti 2013, p. 569).However, if the independent variables are not normally distributed (e.g., becausethey are categorically scaled) logistic regression is more appropriate, because itmakes no assumptions about the independent variables’ distribution. Further, incomparison to discriminant analysis, extreme outliers affect logistic regressionless. Generally, logistic regression is preferred to discriminant analysis (e.g., Agresti2013; Osborne 2008). Nevertheless, one way to validate the results of a logisticregression is to see whether an alternative method can replicate the findings anddiscriminant analysis can serve as such an alternative (Cambell and Fiske 1959;Anderson 1985).

In section “Discriminant Analysis,” we discuss the basic objectives, theoreticalmodel considerations, assumptions, and different steps of discriminant analysis.Analogously, the third section addresses the logistic regression. An applied exampleis presented in section “Applied Examples,” in which both methods are used toexplain which factors drive the use of sales contests in companies. This chapter endswith a conclusion and a brief comparison of discriminant analysis and logisticregression.

Discriminant Analysis

Foundations and Assumptions of Discriminant Analysis

Discriminant analysis is a multivariate (separation) method for the analysis ofdifferences between relevant groups. It can help determine which variables explainor predict the observations’ membership of specific groups. Depending on theinvestigation objectives, this enables the approach to undertake either diagnostic orpredictive analyses.

The four key objectives of discriminant analysis can be described as follows (Hairet al. 2010, p. 350; Aaker et al. 2011, p. 470):

• Determining the linear combinations of independent variable(s) that lead to thebest possible distinction between groups by maximizing the variance between thegroups relative to the varaince observed within the groups. This linear combina-tion is also called a discriminant function or axis. By inserting the independentvariables’ attribute values into the discriminant function, researchers can calculatea discriminant score for each observation.

• Examining whether, based on the group means obtained through the discriminantscores (centroids), the groups differ significantly from each other.

Logistic Regression and Discriminant Analysis 3

• Identifying the variables that contribute most effectively to the explanation ofintergroup differences.

• Allocating new observations for which the attribute values are known to therelevant group (either classification or prediction).

Based on the distance between the group centroids, it is possible to determine thestatistical significance of a discriminant function. Undertaking such analysesrequires a comparison of the distribution of the groups’ discriminant scores. Theless the distributions overlap, the more the groups based on the discriminant functiondiffer. Figure 1 illustrates this basic concept of discriminant analysis according to thedistributions of the two groups A and B: besides showing the separation value Y� (i.e., the critical discriminant score) through which the examined observations can beclassified, the discriminant axis is also shown next to the two group centroids YA andYB . The area of overlap (i.e., the white areas under the curves) between the twodistributions corresponds to the proportion of misclassified observations in group A(i.e., to the right-hand side of Y�) and group B (i.e., to the left-hand side of Y�).

Discriminant analysis is considered a dependency analysis, which distinguishesbetween nominally scaled dependent variables and metrically scaled independentvariables. When comparing discriminant analysis with regression and analysis ofvariance (ANOVA), the following analogies and differences are specifically found:

• Discriminant analysis is based on a model structure similar to the one deployed inmultiple regression. The main difference lies in the nominal level of the depen-dent variable’s measurement. In regression analysis, a normal distribution of theerror terms is assumed, and the independent variables are known or pre-determined. In discriminant analysis, the reverse applies: the independent vari-ables are assumed to have a multivariate normal distribution, while the dependentvariable is fixed; that is, the groups are defined a priori (Aaker et al. 2011, p. 471).

Separation value Y∗

Group A Group B

Assigned to group A Assigned to group BYA

__YB

Y_

Fig. 1 Distributions of the discriminant scores of two groups A and B


• If a reverse relationship existed, such that the variables of interest are dependenton a group membership, an approach such as a one-factorial multivariate analysisof variance (MANOVA) would be appropriate. Accordingly, discriminant analy-sis can also be regarded as the reversal of a MANOVA.

To apply discriminant analysis, certain assumptions have to be complied with.For a comprehensive overview of the assumptions of discriminant analysis, see Hairet al. 2010, pp. 354 et seq. and Tabachnick and Fidell 2013, pp. 384 et seq. As inmultiple regression analysis, this includes the absence of multicollinearity andautocorrelation. Further, the observed relationships should be of a linear nature.Finally, the independent variables should be normally distributed. Nonetheless,discriminant analysis is relatively robust regarding the violation of the normalityassumption, provided the violation is not caused by outliers, but by skewness. If thegroup sizes are very different, the sample should be sufficiently large to ensurerobustness: based on a conservative recommendation, robustness is assumed if thesmallest group includes 20 cases and only a few (up to five) explanatory variables(Hair et al. 2010, p. 375; Tabachnik and Fidell 2013, p. 384). The equality of theindependent variables’ variance-covariance matrices in the different groups is themost important assumption for the application of discriminant analysis. Again,inferential statistics are relatively robust against a violation of this assumption,given that the group sizes are not too small or too uneven. However, the classifica-tion accuracy might not be robust even if reasonable group sizes are adopted,because in such cases observations are frequently misclassified into groups withgreater variance. If classification is a key objective of the analysis, the homogeneityof the variance-covariance matrices should always be tested (Tabachnik and Fidell2013, pp. 384 et seq.). Finally, all independent variables are required to be at leastinterval-scaled, since the violation of this assumption leads to unequal variance-covariance matrices.

Discriminant Analysis Procedure

The application of discriminant analysis involves several steps, which we discuss inthe following.

Step 1: Problem and Group Definition As mentioned, discriminant analysis can beused to explain the differences between groups in a multivariate manner. However, itcan also serve as a procedure to classify observations with known attribute values,but unknown group membership. To apply discriminant analysis, the dependentvariable has to be defined first; that is, the groups to be analyzed need to bedetermined. The dependent variable can be drawn directly from the specific exam-ination context (e.g., product purchasers vs. non-purchasers) or from the findings ofpreceding analyses. For example, customer segments identified by means of clusteranalysis might be further investigated by undertaking discriminant analysis. Thecluster analysis might be used to reveal groups and a subsequent discriminant


analysis might use either the same or different variables for further analyticalpurposes. In the first case, the aim of the investigation would be to verify theclustering variables’ adequacy in terms of their discriminatory meaning. In thesecond case, the groups generated by means of cluster analysis are explored ingreater depth. For instance, an initial cluster analysis generates consumer segmentsbased on their purchase behavior. A subsequent discriminant analysis might then beused to explain the segment-specific differences in consumer purchase patterns bymeans of psychographic variables.

If, in its original form, the dependent variable is based on a metric scale, it can beclassified into two or more groups (e.g., low/medium/high), which can be analyzedby the subsequent discriminant analysis. The group definition is also related to thenumber of groups to be analyzed. The simplest case is two-group discriminantanalysis (e.g., ordering vs. non-ordering in mail order businesses). However, if theclassification involves more than two groups, a multigroup discriminant analysisshould be used. In determining the number of groups, the number of observations pergroup should be at least 20. Consequently, combining several groups into a singlecategory might be appropriate.

Step 2: Model Formulation As part of discriminant analysis a discriminant functionshould be estimated, which ensures an optimal separation between the groups and anassessment of the discrimination capability of the applied variables. The discrimi-nant function is a linear combination of the applied variables and generally adheresto the following scheme:

Yi ¼ β0 þXJj¼1

βjXji, (1)

where

Y: Discriminant scoreβ0: Constantβj: Discriminant coefficient of independent variable j

By using the attribute values Xji for each observation i, the discriminant functiongenerates an individual discriminant score Yi.

Step 3: Estimation of the Discriminant Function The parameters βj are estimatedsuch that the calculated discriminant scores provide an optimal separation betweenthe groups. This result requires a discriminant criterion that measures the intergroupdifferences. The estimation procedure is then carried out such that the discriminantcriterion is maximized.

Considering the distance between the group centroids for the purpose of evalu-ating the differences between the groups might initially seem obvious. However,also the variance of the discriminant scores within a group has to be considered.


While a larger distance between two group centroids improves the distinction of thegroups, the distinction is made more difficult when the variance of the groups’discriminant scores increases. This situation is illustrated in Fig. 2, where twopairs of groups A and B are represented as distributions on the discriminant axis.The centroids of the pairs of groups A (YA= -2) and B (YB ¼ 2

�are identical, but the

standard deviations of the discriminant scores differ (standard deviations upperdistributions = 1; standard deviations lower distributions = 2). Based on a compar-ison of the pairs shown at the top and at the bottom of Fig. 2, we can observe that,although the lower groups indicate identical centroids, they also have greatervariances – in this case, generating a lower discrimination.

Thus, assessments of discriminant analysis performance should aim at optimizingboth the distance of the group centroids and the spread of the observations. Toachieve optimal separation efficiency between the groups, we refer to the well-established regression- and ANOVA-based principle of variance decomposition:

Separation value Y*

Group A Group B

Group A Group B

Assigned to group A Assigned to group B0–2–4–6–8–10 2 4 6 8 10

0–2–4–6–8–10 2 4 6 8 10

Fig. 2 Groups with different centroids and variances


SS ¼ SSb þ SSW

Total variance ¼ Between � group varianceþWithin� group variance

¼ Explained varianceþ Unexplained variance

(2)

Thus, a discriminant function should be determined such that the group means(centroids) differ significantly from one another, if possible. For this purpose, werefer to the following discriminant criterion:

Γ ¼ Variance between the groups

Variance within the groups(3)

This criterion can be made more precise and converted into an optimizationproblem as follows:

Γ ¼

PGg¼1

Ig Yg � Y� �2

PGg¼1

PIgi¼1

Ygi � Yg

� �2 ¼SSb

SSw! Max!

bj, (4)

where

G: Number of groups studiedI: Group size

In order to account for different group sizes, the variance between the groups ismultiplied by the respective group size I. The discriminant coefficients βj ( j = 1,. . ., J ) have to be determined so that the discriminant criterion Γ is maximized. Themaximum value of the discriminant criterion

γ ¼ MAX Γf g (5)

is referred to as the eigenvalue, because it can be determined mathematically bysolving the eigenvalue problem (for further details refer to Tatsuoka 1988, pp. 210 etseq.).

In the multigroup case (i.e., with more than two groups), more than one discrim-inant function and more than one eigenvalue must be determined. The maximumnumber of discriminant functions is given by K = Min {G-1, J}. Generally, thenumber of independent variables J exceeds the number of groups, so that the numberof groups usually determines the number of discriminant functions to be estimated.The following applies to the order of the respective discriminant functions and theircorresponding eigenvalues:

γ1 � γ2 � γ3 � . . . � γK:


This way, the first discriminant function explains the majority of the independentvariables’ variance. The second discriminant function is uncorrelated (orthogonal) tothe first and explains the maximum amount of the remaining variance once the firstfunction has been determined. Since additional discriminant functions are alwayscalculated in order to explain the remaining variance’s maximum share, the func-tions’ explanatory power declines gradually.

The following measure represents a discriminant function’s relative importanceby deriving the respective eigenvalue share (ES):

ESk ¼ γkγ1 þ γ2 þ . . .þ γK

: (6)

The eigenvalue share equals the share of variance explained by the kth discrim-inant function relative to the variance explained by all discriminant functions Kcombined. The eigenvalues always add up to one.

Figure 3 illustrates the discriminant analysis’s estimation principle by means of agraphic representation. In this case, the determination is based on a two-group dis-criminant function. It is assumed that two attribute values X1 and X2 are available foreach of the subjects examined, which are shown in the scatter plot in Fig. 3, in whichfilled circles represent the members of group A and empty circles members of group B.In this example the optimal discriminant function is Y = -0.09X1 þ 0.86X2. Based onthis function four members of group A are misclassified as members of group B. Thearbitrarily selected discriminant function Y = 0.6X1 - 0.25X2 shows a worse classi-fication performance by having 19 misclassifications in total. A projection of the

8

6

4

2

0

–2

–1 0 4 2 3 4 5 6 7 8

Y = 0.6X1 – 0.25X2

Y = –0.09X1 + 0.86X2

X2

Separation value: 5.03(19 misclassifications)

Separation value: 3.38(4 misclassifications)

9 10 11 12 13X1

Fig. 3 Graphical representation of a two-group discriminant analysis


respective attribute value combinations on the discriminant axes corresponds to theexamined subjects’ discriminant values.

Step 4: Assessment the Discriminant Function Performance Researchers have twobasic approaches at their disposal to assess a discriminant function’s performance.The first approach compares the classification of observations with the observations’actual group membership. This approach is explained later in this chapter in relationto logistic regression, where a specific example is also provided. It is also specificallyapplicable in the context of the discriminant analysis.

A second fundamental way of assessing the discriminant function’s quality isbased on the discriminant criterion described above. The eigenvalue γ represents themaximum value of the discriminant criterion and, thus, forms a starting point forassessing the discriminant function’s quality or discriminant power. Since γ has thedisadvantage of not being standardized to values between zero and one, othermetrics based on the eigenvalue have been established for quality assessment,including the canonical correlation coefficient (c):

c ¼ffiffiffiffiffiffiffiffiffiffiffiγ

1þ γ

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVariance explained

Total variance

r(7)

In the two-group case, the canonical correlation equals the simple correlationbetween the estimated discriminant values and the binary grouping variables. Themost common criterion for testing the quality of the discriminant function is Wilks’slambda (Λ):

Λ ¼ 1

1þ γ¼ Unexplained variance

Total variance(8)

As can be seen from the above formula, Wilks’s lambda is an inverse measure ofgoodness; that is, lower (higher) values imply a better (worse) discriminant power ofthe discriminant function. Wilks’s lambda can be transformed into a test statistic,thereby enabling inferential statements on the diversity of groups. By applying thetransformation

V ¼ � N � J þ G

2� 1

� �lnΛ, (9)

where

N: Number of observationsJ: Number of independent variablesG: Number of groupsΛ: Wilks’s lambda


the resulting test statistic is approximately χ2-distributed with J (G-1) degrees offreedom. Thus, a statistical significance testing of the discriminant function can beperformed. Given that the test statistic increases with lower values of Λ, highervalues imply a greater diversity between the groups.

In the case of multigroup discriminant analysis, each discriminant function can beevaluated by means of the above measures on the basis of their respective eigen-value. Here, the kth eigenvalue γk measures the proportion of the explained variancewhich can be attributed to the kth discriminant function. There is a clear analogy tothe principal components analysis procedure in which the components are extractedfrom the independent variables (like the “main components”) in the form of dis-criminant functions (Tatsuoka 1988).

All discriminant functions and their eigenvalues are considered to assess theoverall differences between the groups. The multivariate Wilks’s lambda, which iscalculated by multiplying the univariate lambdas, is a measure that can capture this:

Λ ¼ ∏K

k¼1

1

1þ γk, (10)

where

K: Number of possible discriminant functionsγk: Eigenvalue of the k

th discriminant function

To statistically check whether the groups differ significantly from one another, aχ2-distributed test statistic can be generated by using transformation (9).

Other test statistics representing approximations of the F-distribution and Wilks’slambda are available and can be applied to test the significance of group-relateddifferences. These statistics cannot only be applied to a MANOVA but also todiscriminant analysis (e.g., Hotelling’s trace, Pillai’s trace, and Roy’s GCR). Othermeasures, such as Rao’s V and Mahalanobis’s D2 are particularly applied in thecontext of stepwise discriminant analyses (Tabachnik and Fidell 2013, p. 399; Hairet al. 2010, p. 435).

As with all statistical tests, a statistically significant test result does not necessarilyimply a substantial difference. In a sufficiently large sample size, even smalldifferences are likely to have statistical significance. Consequently, the absolutevalues of the mean differences between the groups, the canonical correlation coef-ficients, and Wilks’s lambda, should not be overlooked. For the sake of interpret-ability, it is advisable to limit analyses to two or three discriminant functions, even iffurther discriminant functions prove to be statistically significant.

Step 5: Examination and Interpretation of the Discriminant Coefficients If testingthe discriminant function(s) result(s) in a sufficient discriminatory power betweenthe groups in step 4, the independent variables can be examined. Assessing theimportance of individual independent variables can be used to, first, explain the keydifferences between the groups, thereby contributing to the interpretation of group-


based differences. Second, unimportant variables can be removed from the model ifthe goal is to specify a parsimonious model. As an alternative to simultaneouslyincluding all variables into the model for estimation, a stepwise estimation may beundertaken. Here, only variables are included in the discriminant function one at atime, which contribute significantly to the discriminant function’s improvement,depending on the level of significance that the researcher specifies.

Individual variables’ discriminatory relevance can be checked by using univariateand multivariate approaches. As part of a univariate assessment, each of the vari-ables can be tested separately to determine whether their mean values between thegroups differ significantly from one another. Likewise, undertaking discriminantanalyses of each of the variables based on Wilks’s lambda serves to isolate theirdiscriminant power. The F-test can also be used here. The result then corresponds toa one-factorial analysis of variance with the groups as factor levels.

A review of the univariate discriminatory relevance is insufficient if there arepotential interdependencies between the variables. For example, while a particularvariable may have little discriminatory meaning when viewed in isolation, it maysignificantly contribute to an increase of discriminant power in combination withother variables.

Multivariate assessments of individual variables’ discriminatory power and oftheir importance as part of the discriminant function can be undertaken by using thestandardized discriminant coefficients. These represent the influence of the indepen-dent variables on the discriminant variable and are defined as follows:

β�j ¼ βj � sj, (11)

where

βj: Discriminant coefficient of independent variable jsj: Standard deviation of independent variable j

Standardization allows for assessments independent of independent variables’scaling and their meaning. The higher the absolute value of a standardized coeffi-cient, the greater the discriminatory power of the associated variable. Theunstandardized discriminants are, however, required to calculate the discriminantscores (Hair et al. 2010, p. 381).

Deriving the correlation coefficients between the values of the respective inde-pendent variables and the discriminant scores is another alternative method tointerpret the independent variables’ influences. These correlation coefficients arecalled discriminant loadings, canonical loadings, or structure coefficients. Comparedto (standardized) discriminant coefficients, potential multicollinearity between theindependent variables affects them less. Therefore they often provide benefits interms of an unbiased interpretation of the independent variables. As a rule of thumb,loadings exceeding a magnitude of 0.4 indicate substantially discriminatory vari-ables (Hair et al. 2010, pp. 389 et seq.). The identification of variables withsufficiently high loadings allows for creating profiles of groups in terms of these


variables and to identify differences between the groups. The signs of discriminantweights and loadings reflect the groups’ relative average profile.

A sufficiently large sample size is required to obtain stable estimates of thestandardized discriminant coefficients and discriminant loadings. As a guidingvalue, a minimum of 20 observations per independent variable is required (Hair etal. 2010, p. 435; Aaker et al. 2011, p. 477).

Step 6: Prediction In discriminant analysis, several alternative predictionapproaches are available. Predictions can be based on classification functions, thedistance concept, or the probability concept (Backhaus et al. 2016, pp. 246 et seq.).Classification functions and the probability concept allow the consideration of apriori probabilities. Such probabilities can reflect theoretical knowledge about dif-ferent group sizes before any prediction is conducted. The consideration of a prioriprobabilities is especially useful, if the examined groups differ in their size. Further-more, the probability concept allows to allocate specific costs to the misclassificationof observations into a certain group. Specifically, this builds on the distance conceptand thus should be addressed last.

Classification FunctionsFisher’s classification functions can be used to predict an observation’s group-membership. They require the variances in the groups to be homogeneous and oneclassification function has to be determined for each group. Thus, in a two-groupcase, two functions have to be determined:

F1i ¼ β01 þXJj¼1

βj1Xji

F2i ¼ β02 þXJj¼1

βj2Xji

(12)

For the classification of an observation, the value of each function Fi has to becalculated. An observation is assigned to the group for which it yields the maximumvalue.

As mentioned above, classification functions allow the consideration of a prioriprobabilities P(g). A priori probabilities must add up to one and are implemented inthe classification function as follows:

Fg :¼ Fg þ lnPðgÞ (13)

It is also possible to determine individual probabilities Pi(g) for each observationi.

Distance ConceptAccording to the distance concept, an observation i needs to be assigned to a groupg, such that the distance to the centroid is minimized (i.e., to which it is closest on the


discriminant axis). This corresponds to determining whether an observation lieseither left or right of the critical discriminant score Y* (see Fig. 1). The squareddistance is the measure usually deployed in the K-dimensional discriminant spacebetween observation i and the centroid of group g:

D2ig ¼

XIi¼1

Yki � Ykg

� �2, (14)

where

Yki: Discriminant score of observation i according to discriminant function kYkg: Centroid of group g regarding discriminant function k

It is, however, not necessary to consider all possible K discriminant functions toexecute the classification. Usually, it is sufficient to limit the analysis to the signif-icant or relevant discriminant functions, which substantially facilitates thecalculation.

Classification based on the distance concept requires the variances in the groupsto be nearly homogeneous. This assumption can, for instance, be checked usingBox’s M as a test statistic. When the assumption of homogeneous variances in thegroups is violated, modified distance measures need to be calculated.

Probability ConceptThe probability concept, which builds on the distance concept, is the most flexibleapproach to the classification of observations. It allows the consideration of a prioriprobabilities and different misclassification costs in the examined group. Withoutthese modifications, the probability concept generates the same results as the dis-tance concept.

Regarding the classification of observations with the probability concept, a prioriprobabilities and conditional probabilities are combined in order to derive a post-eriori probabilities according to the Bayes theorem:

P gj Yið Þ ¼ P Yij gð ÞPi gð ÞPGg¼1 P Yij gð ÞPi gð Þ , (15)

where

P(g| Yi): A posteriori probability, that an observation is in group g, given a discrim-inant score Yi is observed.

P(Yi| g): Conditional probability, that a discriminant score Yi is observed, given that itappears in group g.

Pi(g): A priori probability, that an observation is in group g


An observation i is assigned to group g, for which the value of P(g| Yi) ismaximized. For example, if G = 2, observation i is assigned to group 1 if

P Yij g1ð ÞPi g1ð ÞP2g¼1 P Yij gð ÞPi gð Þ >

P Yij g2ð ÞPi g2ð ÞP2g¼1 P Yij gð ÞPi gð Þ : (16)

The a posteriori probabilities can be calculated on the basis of the observations’distances to the group centroids (see Tatsuoka 1988, pp. 358 et seq. for more details).Another advantage of the probability concept lies in the possibility of explicitlyincorporating the costs of a misclassification in the decision rule. The field ofmedical diagnostics can be used as an example: the consequences of not diagnosinga malignant disease are certainly more fatal than coming up with an erroneousdiagnosis. Hence, the expected value of the costs involved might be considered insuch calculations:

Eg Cð Þ ¼XGh¼1

CghP hj Yið Þ: (17)

The costs, which are quantified by Cgh, arise if an observation is assigned to groupg, though it belongs to group h. Thus, an observation i is assigned to the group gwiththe lowest expected costs Eg(C).

Marketing decisions in which misclassification costs could play a major role arefor example new product introductions. In this regard, high costs may arise due tomisclassifications of products as a “success” or a “flop” in terms of either theirlaunch or non-launch. Further, in the mail order business, substantial mis-classification costs may result from customers either being sent or not sent acatalogue of relevant product assortments.

Logistic Regression

Foundations and Assumptions of Logistic Regression

The application of logistic regression has become increasingly popular in recentyears. There is little difference between logistic regression and discriminant analysiswith respect to their objectives and applications. On the one hand, logistic regressionmay be used to examine the variables and the specific degree to which theycontribute to explaining group membership (diagnosis). On the other hand, it allowsfor classifying new observations into groups (prediction). The following sectiondescribes the basics underlying logistic regression’s estimation method. Initially, themeasures with which to assess the entire model are explained only generically,because an example in section “Logistic Regression” is used to clarify the qualitymeasures as well as the options for interpreting the coefficients. This example draws


on a study that Mantrala et al. (1998) conducted to address the (non-)adoption ofsales contests. The dependent variable takes two values:

Y ¼ 1, if sales contests are used,

0, if sales contests are not used:

�(18)

Compared to a linear regression, linking one or more independent variables to adependent variable that can only take one of two values (i.e., 0 and 1) is morecomplicated. However, this linkage can be established through a logistic regressionmodel, which can be derived as either a latent variable model or a probability model(see Long and Freese 2014). In the following, both approaches are explained.

In the latent variable model, a non-observed (i.e., latent) variable Y� is assumed,which is related to the observed independent variables in the following way:

Y�i ¼ β0 þ

XJj¼1

βjXji þ ϵi, (19)

where

β0: Constantβj: Coefficient of independent variable jϵi: Error term

Equation 19 is identical to a linear regression, in which the dependent variablecan range from -1 to 1, but it differs with regard to the dependent variable, whichis non-observable. In order to transform the continuous non-observable dependentvariable into a dichotomous one (i.e., one that can only take the values 0 and 1), thefollowing linkage is established:

Yi ¼ 1, if Y�i > 0,

0, otherwise:

�(20)

Thus, Yi is assigned a value of 1 ifY�i takes positive values and a value of 0 if Y

�i is

smaller or equal to 0. In our abovementioned example, Y�i can be viewed as the

propensity to adopt (as opposed to the non-adoption of) sales contests.To illustrate the latent variable model, we assume a single independent variable in

the following. For any given value of X of this variable, we can link the observabledependent variable Yi to its non-observable counterpart through the followingequation:

P Y ¼ 1jXð Þ ¼ P Y� > 0jXð Þ (21)

Equation 21 represents the probability of an event occurring (e.g., the use of asales contest), with either Y = 1 or Y� > 0 representing the occurrence of the event,


which is conditional on the value X of the independent variable. If Y� is substitutedby Eq. 19 (and restricted to a single independent variable), Eq. 21 results in:

P Y ¼ 1jXð Þ ¼ P β0 þ β1X1i þ ϵið Þ > 0jXð Þ (22)

It can be rearranged in:

P Y ¼ 1jXð Þ ¼ P ϵi > � β0 þ β1X1ið ÞjXð Þ (23)

In this equation, the probability of the event occurring depends on the distributionof the error ϵi. Depending on the assumptions about this distribution, either a probitor a logit model can be derived. Similar to logit models, probit models are capable ofmodeling a dichotomous dependent variable. While a normal distribution with avariance of 1 is assumed for the probit model, a logistic distribution with a varianceof π

2

3is assumed for the logit model. In the logistic regression model, this leads to the

following equation:

P Y ¼ 1jXð Þ ¼ e β0þβ1X1ið Þ

1þ e β0þβ1X1ið Þ (24)

In a more general form, which allows for more than one independent variable, thisbecomes:

Pi ¼ eZi

1þ eZi¼ 1

1þ e�Zi, (25)

where Zi serves as a linear predictor of the logistic model for the ith observationwith Zi = β0 + β1X1i + β2iX2i þ . . . þ βJXJi.

The probability model is an alternative to the latent variable model, which allowsthe logistic model to be derived without referring to a latent variable (see Theil 1970;Long and Freese 2014). In order to derive a model with an outcome ranging from0 to 1, the probabilities of an event occurring have to be transformed into odds:

OddsðY ¼ 1Þ ¼ PðY ¼ 1ÞPðY ¼ 0Þ ¼

PðY ¼ 1jXÞ1� PðY ¼ 1jXÞ (26)

Odds indicate how often something happens relative to how often it does nothappen (i.e., Y = 1 vs. Y = 0). The odds therefore represent the chance of an eventoccurring. The natural logarithm of the odds is called logit (logistic probability unit),which ranges from -1 to 1, and is linear:

ln Odds Y ¼ 1ð Þð Þ ¼ β0 þXJj¼1

βjXji (27)


The logistic function, which is an S-shaped curve, has the advantageous charac-teristic that even for infinitely small or large values of the logit Zi the resulting valuesof Pi are never outside the interval of [0,1] (Hosmer et al. 2013, pp. 6 et seq.).

The nonlinear characteristics of Eq. 25 are represented in Fig. 4, in which theexponent Xi is systematically varied from -9 to þ9. The figure shows the predictionof an indifferent result if the sum of the weighted factors (i.e., Zi) equals zero. Itssymmetry at the turning point of Pi = 0.5 is another essential feature of the logisticfunction. The constant β0 of the linear predictor Zi moves the function horizontally,while higher coefficients lead to a steeper slope of the logistic function. Negativesigns of the coefficient βj change the origin of the curve, which corresponds to thedotted line in Fig. 4 (Menard 2002, pp. 8 et seq.; Agresti 2013, pp. 119 et seq.).

As mentioned, in empirical research the actual (non-)entry of an event and not itsprobability of entry is observed. The logistic regression approach regarding theoccurrence of an event (Yi = 1) and its opposing event (Yi = 0) can thus beexpressed as follows for each observation i:

Pi Yð Þ ¼ 1

1þ e�Zi

� Yi

1� 1

1þ e�Zi

� 1�Yi

(28)

Thus, the information required to establish the probabilities regarding the z-values (logits) can be calculated using Eq. 25.

Fig. 4 Progression of the logistic function curves


The coefficients of the logistic model βj can now be estimated by maximizing thelikelihood (L) of obtaining the empirical observation values of all possible cases.Since the observed values Yi represent realizations of a binomial process with theprobability Pi, which vary depending on the expression of Xji, we are able to set upthe following likelihood function, which has to be maximized:

L ¼ ∏N

i¼1

1

1þ e�Zi

� Yi

1� 1

1þ e�Zi

� 1�Yi

! Max! (29)

Compliance with specific assumptions is required to apply logistic regressionusefully. Compared to discriminant analysis, logistic regression offers the advantagethat the assumptions of multivariate normality and identical variance-covariancematrices between the groups are not required. It should be ensured that the indepen-dent variables do not exhibit multicollinearity, errors are independent, and thelinearity of the logit is given (Field 2013, pp. 768 et seq. and pp. 792 et seq.).

The Variance Inflation Factor (VIF) is commonly derived to assess multi-collinearity. Empirical research often quotes critical VIF values that must not beexceeded. Nevertheless, it should be noted that multicollinearity problems mightalready arise with very small VIFs (Baguley 2012). Hence, in order to control formulticollinearity, researchers should exclude their specific independent variablesfrom the model one at a time and verify whether the remaining independent variablesexperience a substantial change regarding their level of significance or the directionof their effect.

In logistic regression, the outcome variable is categorical and, hence, there is nogiven linear relationship between independent and dependent variables. Neverthe-less, the assumption of linearity in logistic regression assumes a linear relationshipbetween any metric independent variable and the logit of the outcome variable. Asignificant interaction term between an independent variable and its log transforma-tion indicates that this assumption has been violated (Field 2013, p. 794; for furtherapproaches to test this assumption refer to Hosmer et al. 2013, pp. 94 et seq.).

In addition, one should ensure that outliers and other influential observations donot affect the estimated results. Some statistical packages, like SAS, offer a hugevariety of outlier statistics for logistic regression, which might be used to alter resultsin different directions. Hence, researchers should always report whether substantialchanges occur in their results when outliers have been removed.

Finally, a reasonable sample size must be used, since maximum likelihoodestimation is adopted with this method and requires a large number of observationsbased on its asymptotic properties. With respect to a study by Peduzzi et al. (1996) itis recommended that the ratio of the group size to the number of independentvariables should be at least 10:1 for the smallest group. For instance, in a samplewhere 25% of the observations have an outcome of Y = 1 (representing the leastfrequent outcome) and 4 independent variables are available, 160 observations areneeded (160 observations * 0.25 (share of the observations with the least frequentoutcome) = 40 observations; 40 observations / 4 variables =10 observations for


each independent variable). Even though this recommendation is widely accepted inliterature, Agresti (2013, p. 208) and Hosmer et al. (2013, p. 408) note that this ismerely one guideline and models that violate it should not necessarily be neglected.For example, Vittinghoff and McCulloch (2006) conclude in their simulation studythat at least 5-9 observations for each independent variable in the smallest groupshould be sufficient. A more stringent threshold is provided by Aldrich and Nelson(1984), who view 100 degrees of freedom as a minimum for a valid estimation.

Logistic Regression Procedure

The application of logistic regression comprises different steps, which are addressedin more detail below.

Step 1: Problem and Group Definition As stated, similar to two-group discriminantanalysis, logistic regression is suitable for the multivariate explanation of differencesbetween groups or for the classification of group-based observations for predictionpurposes. Logistic regression is therefore generally appropriate when a singlecategorical variable is used as a dependent variable. When more than two groupsare given, ordered or multinomial logistic regressions can be applied, which repre-sent generalizations of binary logistic regression. If the dependent variable has ametric scale, either those observations that are furthest from one another can becoded as 0 and 1 or the metric variables can be classified into multiple groups (e.g.,low, medium, high). This approach is called the “polar extreme approach” (Hair et al.2010, p. 352).

Step 2: Model Formulation Model formulation may be used to assess which of theindependent variables should be analyzed. In contrast to discriminant analysis, nometric scale is required of the explanatory variables. Rather, dummy or categoricalvariables can be considered. It is important to clarify whether these nonmetricvariables are indicator or effect coded. With indicator coding, the reference categoryis assigned the value of 0 and the resulting coefficient should be interpreted as arelative effect of a category compared to the reference category. Effect-coded vari-ables may be interpreted as a category’s relative influence compared to the averageeffect of all the categories. Unlike in indicator coding, one of the categories isassigned the value of -1. The coefficients of the categories of effect-coded variablesadd up to zero, so that the reference category’s coefficient can be calculated from thecoefficients of the other categories (for further details, see Hosmer et al. 2013, pp. 55et seq.).

Generally, the selection of the independent variables should be based on soundlogical or theoretical considerations. Nevertheless, especially in the context of BigData, a large number of interdependent independent variables with unknown qualityare often available. In this case, theoretical considerations might not be possible.However, a limited set of variables still needs to be selected to avoid overfitting.Overfitting often occurs when a large number of variables are included in a


prediction model. This might result in a good prediction performance, if a logisticregression model is applied to the observations used for its estimation. Nevertheless,as soon as a prediction is conducted for observations outside the original sample, theprediction performance is likely to suffer. Thus, researchers and practitioners mighttherefore reduce the dimensions of their data by applying a principal componentanalysis and only including a limited number of factors into their logistic regressionmodel. Alternatively, variable selection approaches can be used. Tillmanns et al.(2017) provide an extensive comparison of different approaches that can handle alarge number of independent variables in models predicting a binary outcome.

Finally, a sufficiently large number of observations is required (see section“Foundations and Assumptions of Logistic Regression”), which should be distrib-uted as evenly as possible between the analyzed groups. If a sufficiently large sampleis given, it also makes sense to split the observations into an estimation sample and aholdout sample. The regression function estimated on the basis of the estimationsample can then be used to cross-validate the results with those from the holdoutsample.

Step 3: Function Estimation Before the regression model can be estimated, theassumptions underlying logistic regression need to be checked (see section “Foun-dations and Assumptions of Logistic Regression”). If these assumptions are met, themodel can be estimated. As in multiple regression analysis or discriminant analysis,this can be done by using a stepwise procedure (“forward/backward elimination”) orby the simultaneous entry of each of the independent variables into the estimationequation (“enter”). If the logistic regression is used to test hypotheses, the lattermethod is required. In preliminary estimation runs, the potential presence of influ-ential observations or outliers should also always be checked. In this process, the useof Cook’s distance is recommended. In cases where the sample is very unbalanced(i.e., one outcome is very rare), researchers might consider ReLogit (Rare EventsLogistic Regression) as an alternative approach (for a detailed explanation see Kingand Zeng 2001).

The maximization of the likelihood function shown in Eq. 29 is achieved withstatistical packages, such as SAS or SPSS, using the Newton-Raphson algorithm.The principle underlying maximum likelihood estimation is to select the estimates ofparameter βj in a stepwise iterative analytical approach, such that the observation ofthe estimated value is assigned a maximum likelihood.

Figure 5 illustrates two logistic functions fitted to two different samples. On theordinate, the dependent variable Y is shown, wherein P(Y = 1) represents theprobability of occurrence of an event. On the abscissa, the values of an independentvariable Xi are represented. The logistic function reflects the predicted (estimated)probabilities for the occurrence of the event under the independent variables’different values. The actual observation values are marked by points. In the upperpart (a), the logistic function is suitably adapted to the observed data: high indepen-dent variable values correspond to the occurrence of the event, and vice versa.However, in the lower part (b), the logistic function is not suitably adapted to theobserved data, which is expressed by the large overlap between the two groups in the


central region of the abscissa. Entering a cutoff point of 0.5 for classifying observa-tions shows that, in example (a), four observations are misclassified whereas eightobservations are misclassified in example (b).

Step 4: Assessment of the Model Performance Before starting with the interpreta-tion of the individual coefficients, it is important to first investigate whether an

Fig. 5 Fit of the logistic function with different samples


estimated logistic regression model is in fact suitable. When reviewing this issue,one cannot rely on the traditional measurements and tests used in linear regressionanalysis (such as the coefficient of determination or F-values), because the coeffi-cients obtained from logistic regression are determined by using maximum likeli-hood estimation. Goodness-of-fit in maximum likelihood applications is usuallyassessed by using deviance (or -2LL), which is calculated as -2*log (likelihood).A perfect fit of the parameters is equivalent to a likelihood of 1, corresponding to adeviance of 0 (Aldrich and Nelson 1984, p. 59; Hosmer et al. 2013, p. 155).

In terms of interpretation, the likelihood is comparable to that of least squareserrors in conventional regression analysis. Further, -2LL is used, because it isasymptotically χ2-distributed with (N-p) degrees of freedom, where N is the numberof observations, and p the number of the parameters. Good models reflecting a“high” likelihood of close to 1 result in a deviance that is close to 0, while a bad fit ismirrored in high deviance values. Deviance values are unlimited in positive range.

Whether a deviance has to be considered as rather “high” or “low” depends on theparticular sample used and the analysis deployed. The value of the deviance can beused to test a hypothesis H0, which posits that the overall model shows a perfect fitwith the data. With either low deviance values or high significance, H0 cannot berejected and the model can be deemed as showing a good fit with the data (Menard2002, pp. 20 et seq.).

In addition to the deviance, likelihood ratio tests and pseudo-R2 statistics can beapplied, which provide additional indicators permitting assessments of the fullmodel fit compared to the null model, which only comprises an estimated valuefor β0, i.e., the intercept of the linear predictor Zi. In this process, the deviance isapplied, which can also be used for comparing incremental differences of differentmodels. The absolute difference between the deviance of the null model and the fullmodel provides an asymptotically distributed χ2 value, which can be tested againstthe null hypothesis that the full model coefficients are not significantly different from0. Thus, a likelihood-ratio test can be conducted, which is comparable to the F-test inlinear regression analysis. This test statistic is called the Model Chi-Square. High χ2

values and low significance levels suggest that the final model’s coefficients aresignificantly different from 0. For this test, the 5% level is usually set as the criticallevel of significance (Hosmer et al. 2013, p. 40; Menard 2002, p. 21; Tabachnik andFidell 2013, pp. 448 et seq.).

In logistic regressions, the deviances of the full and the null model can be used tocalculate McFadden’s R2. This pseudo-R2 metric is expressed in the followingequation:

McFadden0s R2 ¼ 1� LL1

LL0, (30)

where

LL1: Natural logarithm of the likelihood of the full model.LL0: Natural logarithm of the likelihood of the null model


There are additional pseudo-R2 statistics, which are also based on a comparativegoodness-of-fit of the null model and the full model. Similarly, the R2 statistics ofCox and Snell, as well as of Nagelkerke suggest that higher values correspond toenhanced model fit, with the maximum value of 1 corresponding to perfect model fit.

Assessments of the model performance can also be undertaken on the basis of theattained classification results, where the predicted values Pi are compared with theactual (observed) values Yi. As part of the Hosmer-Lemeshow test, the accuracy ofpredictions is checked against the null hypothesis, which posits that the differencebetween the predicted and observed values is equal to zero (Hosmer et al. 2013, pp.157 et seq.). These observations are divided into 10 groups of approximately equalsize based on the predicted values Pi. By means of a Chi-Square test, the extent towhich the observed and predicted frequencies differ is checked. A low χ2 value ofthe test statistic, coupled with a high significance level, implies a good model fit.

Further, to assess the predictive accuracy of logistic models, the confusion matrixor classifications matrix is considered. In this 2*2 matrix, the predicted groupmemberships on the basis of the logistic regression model are compared to theempirically observed group memberships (Menard 2002, p. 31). The correctlyclassified items are then available on the main diagonal, while the misclassifiedobservations appear off the main diagonal. The proportion of correctly classifiedelements – as facilitated by logistic regression (which is called the match quality orthe hit ratio) – should be higher than the matches obtained from a random allocation.This is a limitation, since the matches attained by using the model are inflated whenthe parameter estimation of logistic regression and the calculation of the hit rate arebased on the same sample (Morrison 1969, p. 158; Hair et al. 2010, p. 373; Afifi et al.2012, p. 265). Using the estimated coefficients from a calibration sample is thereforeexpected to generate lower hit rates for other (holdout) samples. With approximatelyequal groups represented by the dependent variable, using the maximum chancecriterion (MCC) to assess the classification performance is recommended. Byapplying the MCC metric, the classification performance is based on the share ofthe larger groups within the total sample (Morrison 1969, p. 158; Hair et al. 2010, p.384; Aaker et al. 2011, p. 478). Nevertheless, if two groups of rather unequal size areconsidered, adopting the MCC metric is inadequate.

The proportional chance criterion (PCC) is particularly recommended whenanalyzing two groups of unequal size or when seeking a classification that is equallyas good for both groups. The PCC is equivalent to a random hit rate of α2 þ (1-α)2,where α represents the proportion of a group to the total number of observations(Morrison 1969, p. 158; Hair et al. 2010, p. 384; Aaker et al. 2011, p. 478). Whetherto use the PCC, MCC, or a different classification criterion depends on the particularsubject of investigation. For example, it may be meaningful to minimize the mis-classification of only one of the two groups, when assessing credit risks or failure ofnew products.

The Receiver Operator Characteristic (ROC) is another well-established measurefor assessing a logistic regression model’s classification performance (see Hosmer etal. 2013, pp. 173 et seq. for a more detailed discussion). Within this analysis, thesensitivity (i.e., the true positive rate: the probability that a certain outcome is


predicted to occur, given that the outcome occurred) and specificity (i.e., the truenegative rate: the probability that a certain outcome is predicted not to occur, giventhat the outcome did not occur) are derived. Classification performance depends onthe choice of an appropriate cutoff point (i.e., the probability that is necessary toassign a predicted outcome of either 0 or 1 to an object). Generally, an outcome of 1is predicted for probabilities greater than 0.5. Nevertheless, the choice of an appro-priate cutoff point has implications for the classification performance in terms ofsensitivity and specificity. Hence, it is worthwhile deriving a cutoff point thatmaximizes both sensitivity and specificity. The ROC curve considers all possiblecutoff points by plotting sensitivity versus 1 - specificity and represents the likeli-hood that an object with an actual outcome of 1 has a higher probability than anobject with an actual outcome of 0. If the logistic regression model has no predictivepower, the curve has a slope of one, resulting in an area under the ROC curve of 0.5.Higher values indicate a good predictive power and typically range between 0.6 and0.9 (Hilbe 2009, p. 258). Hosmer et al. (2013, p. 177) provide a general rule forevaluating classification performance regarding the area under the ROC curve: ROC= 0.5: no discrimination; 0.7 � ROC < 0.8: acceptable discrimination; 0.8 � ROC< 0.9: excellent discrimination; ROC � 0.9: outstanding discrimination.

Table 2 provides an overview of the different criterions, which can be used toassess the performance of a logistic regression.

Step 5: Examination and Interpretation of the Coefficients If the total model fit isconsidered acceptable, one can start testing and interpreting the coefficients in termsof their significance, direction, and relative importance. It should be noted that theparameter estimates of a logistic regression are much more difficult to interpret thanthose attained with linear regression. In linear regression, the coefficient correspondsto the absolute change in the dependent variables with a one-unit increase in theindependent variables. The nonlinear nature of the logistic function makes theinterpretation more difficult. For a demonstration, refer to Fig. 4 and suppose twovariables X1 and X2. For example, Zi might increase from A to A’ or from A’ to A”because of an increase of X1 by one unit. The resulting change of the probability forY = 1 depends heavily on the starting point of the increase – A or A’. These starting

Table 2 Acceptable ranges for logistic regression performance measures

Criterion Range of (acceptable) values

Deviance (-2LL) Deviance close to 0; significance level close to 100%

Likelihood-ratio test(“Model χ2”)

Highest possible χ2 value; significance level < 5%

Hosmer-Lemeshow test Lowest possible χ2 value; significance level close to 100%

Proportional chancecriterion (PCC)

Classification should be better than the proportional chance: α2þ (1-α)2, with α = relative size of a group

Area under the ROCcurve

ROC = 0.5: no discrimination0.7 � ROC < 0.8: acceptable discrimination0.8 � ROC < 0.9: excellent discriminationROC � 0.9: outstanding discrimination


points in turn depend on the value of the other variables in the model, like X2 in ourexample. Thus, in logistic regression coefficients represent only the change in thedependent variable’s logit with a one-unit change in the independent variables(Aldrich and Nelson 1984, p. 41; Hair et al. 2010, p. 422; Agresti 2013, p. 163).The logit is the natural logarithm of the “chance of winning;” that is, the ratio of theprobability that the dependent variable is equal to 1 divided by its counter probability(see Eqs. 26 and 27). For interpretation purposes, it may be easier to use the “oddsratios,” or effect coefficients, which are obtained by means of the eβ transformation(Hosmer et al. 2013, pp. 50–51). Specifically, odds ratios show how the odds changewith a one-unit increase in the independent variables. The odds describe the ratio ofthe probability of occurrence of an event to its counter probability (i.e., chance) asdisplayed in Eq. 26. Odds range from 0 to 1, whereby values below one indicatethat the chance of an event occurring becomes lower with increasing values of theindependent variable and values above one indicate the opposite.

To verify the significance of the independent variables, either the Wald statistic orthe likelihood-ratio test can be used. The confidence interval of individual coeffi-cients can be determined based on the χ2-distributed Wald statistic, which is calcu-lated from the square of the ratio of the coefficient and the variable’s standard error.This formula applies to metric variables with one degree of freedom only. Forcategorical variables, the variable’s degrees of freedom have to be considered inaddition. With the likelihood-ratio test, the full model is tested against a reducedmodel, which is reduced by the variable under consideration. The significance test isperformed on the basis of the difference between both models’ deviances, whichagain follows a χ2-distribution.

The direction of the variables’ effects with significant coefficients can beinterpreted directly. As can be seen in Fig. 4, negative signs imply that the proba-bility Pi decreases, while positive signs imply increasing probabilities with highervalues of the variable under consideration. Statements regarding the relative impor-tance of each variable can be made based on the aforementioned “odds ratios.” Theirlevel is, however, dependent on the scaling of the variables. Furthermore, a constantchange in the odds ratios does not result in a constant change in probabilities, and themagnitude of the effect on the probabilities is not symmetric around one (Hoetker2007). Hence, alternative interpretations are discussed in the following.

In order to derive the relative importance of each predictor, different options areavailable. First, a standardized coefficient can be calculated to reflect the strength ofthe effect independent from the scaling of the independent variables. This effectstrength can be interpreted similar to the standardized coefficients used in linearregression, as this metric specifies the number of standard deviations by which thelogit changes when the independent variable increases by one standard deviation (forfurther details refer to Menard 2002, pp. 51 et seq.).

Marginal effects, i.e., the partial derivative of the logit function for an indepen-dent variable Xj, represent another alternative (Leclere 1992, pp. 771 et seq.). Themarginal effect of Xj can be derived by applying the following formula:


@Pi

@Xj¼ e

� β0þPJ

j¼1βjXj

�

1þ e� β0þ

PJ

j¼1βjXj

� !2βj (31)

Equation 31 can be reduced to:

@Pi

@Xj¼ βjPi 1� Pið Þ (32)

Regarding the relative importance of different variables, it has to be consideredthat marginal effects depend on the scale of the examined variables. Furthermore,marginal effects vary across different values of an independent variable as can beseen easily in our example in Fig. 4. Because of the different gradients of the curve inpoints A, A’, and A”, the marginal effects at these points are substantially different.The issue regarding marginal effects’ dependence on the scaling of independentvariables can be resolved by standardizing the independent variables on the onehand, while on the other hand using their elasticities to interpret the logistic regres-sion coefficients.

Elasticities are easier to interpret than the coefficients’ scale-variant partialderivatives, because they are dimensionless and quantify the percentage change inthe probability Pi under a 1% change in the respective independent variable of thelogistic model. The elasticity of the probability Pi regarding infinitesimally smallchanges of Xj is obtained by adoption of the following equation (Leclere 1992, p.772):

ej, i ¼ Xj

Pi

@Pi

@Xj¼ Xj

Pi

e� β0þ

PJ

j¼1βjXj

�

1þ e� β0þ

PJ

j¼1βjXj

� !2(33)

The equation can be simplified to

ej, i ¼ Xj

Pi

@Pi

@Xj¼ Xj 1� Pið Þ � βj: (34)

The elasticity thus results from the multiplication of the partial derivative ofprobability Pi (with respect to the independent variable Xj) with Xj divided by Pi.For Xj the mean value Xj is often applied (Leclere 1992, pp. 773 et seq.). As withpartial derivatives, elasticities depend on the initial values of Pi and the independentvariables’ values. However, the lacking dimensionality of elasticities allows directcomparisons of various independent variables’ relative influence on the probabilityPi.


From the perspective of non-econometricians, sensitivity analysis may be anappealing way of interpreting logistic regression models. In order to conduct asensitivity analysis, the probability Pi is examined for different values of theindependent variable. Usually, the value of a single independent variable is variedsystematically (e.g., þ10%, þ20%, ..., -10%, -20%), while the other variables arekept constant. The difference between the initially estimated and the resulting newprobability Pi can then be interpreted as the relative importance of each significantindependent variable for Pi. The advantage of sensitivity analysis lies in the visual-ization of the absolute effect of the independent variables’ different values on theprobability Pi (Leclere 1992, pp. 772 et seq.). In section “Interpretation of theCoefficients,” we provide an example for such a sensitivity analysis.

Several authors emphasize that the interpretation of interaction effects in logisticregression is often insufficient, as the marginal effect of an interaction between twovariables is not simply the coefficient of their interaction and the associated level ofsignificance (e.g., Hoetker 2007). Ai and Norton (2003) show that the interactioneffect of two independent variables X1 and X2 is the cross-derivative of Pi withrespect to each. For two continuous variables X1 and X2 this results in the followingequation:

@2Pi

@X1@X2

¼ β12Pi 1� Pið Þ

þ β1 þ β12X2ð Þ β2 þ β12X1ð Þ Pi 1� Pið Þ 1� 2Pið Þð Þ (35)

Based on Eq. 35, Norton et al. (2004, p. 156) emphasize four important implica-tions for the interpretation of interaction effects in logistic regression. First, eq. (35)shows that the interaction effect is not equal to the coefficient of the interaction β12and an interaction can even exist, if β12 = 0. Second, the statistical significance of aninteraction must be derived for the entire cross-derivative and not just the coefficientof the interaction. Third, the interaction effect is conditional on the independentvariables. Fourth, Eq. 35 consists of two additive terms, which can take differentsigns. Accordingly, the interaction effect may have different signs for differentvalues of the independent variables and therefore, the sign of the interaction coef-ficient does not necessarily represent the sign of the interaction effect. Thus, werecommend to derive the interaction effects for each observation and plot it againstthe predicted values of the dependent variable in order to reveal the full interactioneffect (see e.g., Norton et al. 2004 for an example).

Step 6: Prediction At the beginning of section “Logistic Regression,” we pointedout that logistic regression can also be used for prediction purposes. As in discrim-inant analysis, a holdout sample can be applied to validate the prediction perfor-mance of a logistic regression. Alternatively, cross-validation can be undertaken bymeans of the U-method or the jackknife method (Hair et al. 2010, pp. 374 et seq.;Aaker et al. 2011, p. 478). Examples for applying logistic regression in a predictioncontext include the check of creditworthiness, where financial service providers first


analyze good and poor credit agreements, in order to then be in a position to assesscurrent loan applications and their associated risks. The methods presented in thepreceding sections are illustrated below in an application to sales contests amongsales people.

Applied Examples

Research Question and Sample, Model, Estimation, and ModelAssessment

Mantrala et al. (1998) examine the use of sales contests (SCs) in the USA andGermany. Although SCs represent commonly used motivational tools in practice,they are often neglected in research. The authors therefore develop hypotheses basedon institutional economics that investigate the influence that the sales force size (x1),the ease and adequacy of output measurement (x2), the replaceability of sales people(x3), and the length of the sales cycle (x4) has on the probability that SCs are used.This might help managers to determine, whether their firm should use sales contestsaccording to market standards. In addition to 118 observations from the USA, thestudy considers 270 German cases. Of the 270 original cases, 39 have missing valuesfor at least one of the four variables. The remaining 231 observations are made up of91 sales forces that do not use SCs (39.39%) and 140 sales forces that do deploy SCsas part of their incentive system (60.61%). The latter data set is also described inKrafft et al. (2004) and forms the basis of the following example. In our example,231 observations are used to estimate four parameters, such that 226 degrees offreedom remain. The data set thus meets the most stringent criteria in terms ofsample size and the ratio of sample size to parameters to be estimated. Table 3provides an overview of the independent variables’ mean values and whether theydiffer significantly with regard to the use of sales contests. The analyses are based onone-factorial analyses of variances and provide initial evidence that the independentvariables can discriminate between the groups in our sample. This initial analysis isespecially suitable for a discriminant analysis, which requires a metric scale for theindependent variables.

Table 3 Determinants of the adoption of sales contests

Independent variable

Means

FTotal(n = 231)

No SC(n = 91)

SC(n = 140)

Sales force size (x1) 269.16 42.12 416.74 13.58 ***

Adequacy of output measures (x2) 0.51 0.44 0.55 18.99 ***

Replaceability of sales people (x3) 7.98 6.89 8.69 11.83 ***

Length of the sales cycle (x4) 12.27 19.08 7.85 24.85 ***

***p < .001


Discriminant Analysis

Model Estimation and Model AssessmentThe estimation of the discriminant function yields an eigenvalue of 0.255. Asmentioned in section “Discriminant Analysis Procedure,” higher values indicate ahigher quality of the discriminant function. Since the measure is not standardized tovalues between zero and one, Wilks’s lambda is a more appropriate measure toindicate a discriminant function’s discriminant power. Wilks’s lambda is an inversemeasure of goodness, such that lower values represent a better discriminant power.In the given application, Wilks’s lambda yields a value of 0.797 and indicates thatthe discriminant function significantly discriminates between companies that usesales contests and those that do not use them (χ2 (4), p < 0.001). A classificationmatrix can be applied to derive the predictive power of the discriminant function andis depicted in Table 4. Given the group sizes in our example, the proportional chancecriterion (PCC) is 52.25% (i.e., 0.612 þ (1 - 0.61)2). In relation to the PCC, the hitrate of 69.70% (i.e., (61 þ 100) / 231) can be considered sufficient.

Interpretation of the CoefficientsEven though Table 3 provides initial evidence about the discriminatory meaning ofthe variables in our example, particularly by highly significant F statistics, theinterdependencies between the variables are not considered and a multivariateassessment might provide more valuable insights. Examining the standardizeddiscriminant coefficients and the discriminant loadings, which are displayed inTable 5, is one such approach.

In our example, the standardized discriminant coefficients and discriminantloadings indicate that the length of the sale cycle and adequacy of output measuresare the best predictors for the use of sales contests. Notably, slight differences can beobserved with regard to the relative importance of both variables. Because discrim-inant loadings are superior to discriminant coefficients in terms of an unbiasedinterpretation (see section “Discriminant Analysis Procedure”), we recommend totake them into account for assessing variables’ relative importance.

Table 4 Classification matrix regarding the (non-)adoption of sales contests after applying adiscriminant analysis

Predicted groupmembership

Observed groupmembership No sales contests

Adoption of salescontests

Correctclassifications (%)

No sales contests 61 30 67.03


40 100 71.43

Total 69.70


Logistic Regression

Model Estimation and Model AssessmentAccording to the Cook’s distance statistic, we do not find any influential outliers inour data set. After seven iterations of the logistic regression, no further improvementin the model likelihood could be achieved (improvement < 0.01%). The full modelhas a deviance of 232.109 (-2LL of the null model: 309.761). This relatively lowvalue in combination with a significance level of 1.000 for the deviance indicates agood model fit. This is also confirmed by a high significance level of the Hosmer-Lemeshow test. The logistic model exhibits a likelihood ratio value of 77.652, whichis highly significant ( p < 0.001). Thus, compared to the null model, the inclusion ofthe four independent variables results in a significantly improved model fit. A LL1

value of -116.055 is obtained for the full model, while an LL0 value of -154.880 isobserved for the null model. The resulting McFadden’s R2 of 0.2507 for the finalmodel may be viewed as relatively good, given that only four variables are used.

Considering the proportional chance criterion (PCC) of 52.25%, an overall hit rate of75.32% of correctly classified observations ((58 þ 116) / 231) indicates a goodprediction performance within the calibration sample. In addition, the hit rate of63.74% for the (smaller) group of sales forces without SCs indicates a suitable classi-fication performance. Table 6 shows the classification table of the logistic regression inour example.

The predicted group memberships in Table 6 are based on a cutoff point of 0.5 forthe predicted probability regarding the use of SCs. Consequently, if the predictedprobabilities are larger than or equal to 50%, the observations are assigned to the SC

Table 5 Discriminant coefficients and loadings

Independent variableStandardized discriminantcoefficients

Discriminantloadings

Sales force size (x1) 0.443 0.482

Adequacy of output measures(x2)

0.531 0.570

Replaceability of sales people(x3)

0.311 0.450

Length of the sale cycle (x4) -0.526 -0.652

Table 6 Classification matrix regarding the (non-)adoption of sales contests after applying alogistic regression

Predicted group membership

Observed groupmembership

No salescontests

Adoption of salescontests Correct classifications (%)

No sales contests 58 33 63.74


24 116 82.86

Total 75.32


group. In our example, a true positive rate (i.e., sensitivity) of 0.83 (i.e., 116 /(116 þ 24)) and a true negative rate (i.e., specificity) of 0.64 (i.e., 58 / (58 þ 33))are achieved. Especially if the groups under consideration are unequal in size or if amisclassification is associated with costs, applying a different cutoff point might bemore reasonable. For example, misclassification costs might appear if firms predicthouseholds’ responses to a direct mailing campaign and seek to avoid reactance (i.e.,costs) by those who are not interested in the offering, but who would be misclassifiedas potential respondents. In this case, the firm would try to limit the false positiverate. The Receiver Operator Characteristic (ROC) curve in Fig. 6 visualizes thetrade-off between the true positive rate and the false positive rate, when differentcutoff points are used in our example. If higher cutoff points are chosen, the falsepositive rate decreases (e.g., from 0.36 (i.e., 33 / (33 þ 58)) to 0.24 (i.e., 22 /(22þ 69)) if a cutoff point of 0.6 instead of 0.5 is chosen). Choosing a cutoff point of0.49 would maximize the hit ratio to 0.77 in the given setting. Generally, the areaunder the curve is frequently used as a meaningful measure to evaluate the predictiveperformance of logistic regression models. In the given example, it yields a value of0.8164, indicating that the logistic regression provides an excellent discrimination.

1.0

0.8

0.6

0.4

0.2

0

0 0.2 0.4

Cutoff point (maximal hit ratio)

Cutoff point (varying hit ratio)

0.6

0.7

0.8

0.9

0.50.49

0.4

Area under thecurve: 0.8164

0.3 0.20.1

1-Specificity (false positive rate)

Spe

cific

ity (

true

pos

itive

rat

e)

0.6 0.8 1.0

Fig. 6 Receiver Operator Characteristic (ROC) curve


Interpretation of the CoefficientsTable 7 shows the coefficients of the four variables included in the logistic regres-sion, as well as the constant. With the exception of the significance level and thedirection of the effect of each variable, a direct interpretation of the coefficients of thelogistic regression model is not possible. As an indicator of the change in oddsresulting from a one unit change in the predictor, the odds ratios (see Table 7) arehelpful for interpreting the coefficients (see Field 2013, pp. 766 et seq. for a detailedexplanation).

In our example, the odds of the use of SCs are defined as follows:

odds ¼ probability, that SCs are used

probability, that no SCs are used(36)

The probability of using SCs (i.e., P(SCs)) is derived by applying the logisticregression coefficients in our example to formula (25), while the probability, thatSCs are not used is simply 1-P(SCs):

P SCsð Þ ¼ 1

1þ e� 2:3121þ0:008x1þ3:0574x2þ0:1110x3�0:0258x4ð Þ (37)

The odds ratio represents the odds before and after a one unit change of thepredictor variable:

odds ratio ¼ odds after a one unit change

original odds(38)

In our example, the odds ratio for sales force size is derived by e0.0080 = 1.008and indicates that an increase of the salesforce by one salesperson increases the oddsof using SCs by 1.008 or 0.8%. If the length of the sales cycle is increased byone month, the odds of using SCs are reduced by 0.974, equivalent to 2.6% (e0.0258).As mentioned in section “Logistic Regression Procedure,” odds ratios are dependenton the scaling of the variables. This can easily be seen when comparing themagnitude of the odds ratios with the mean values in Table 7. Further, a change ofodds ratios does not result in a constant change of probabilities. Thus, in order toderive insights on the relative importance of the determinants of the use of SCs, wecalculated the variables’ elasticities based on the means of the non-SC observations.If these means are used in the logistic regression function, we observe a probabilityof P = 41.07% that SCs are used.

As elasticities are dimensionless, they can be compared directly with each other atthe absolute level. However, just like the partial derivatives, they are valid only for acertain point examined in the logistic probability function (e.g., the means of thenon-SC observations in our application) and vary when other points (e.g., the meansof the SC observations in our application) are examined.

All elasticities shown in Table 7 exhibit absolute values that differ substantiallyfrom 0; that is, the influence of the independent variables on the probability that SCs


Table

7Elasticities

andsensitivity

analysisof

thesign

ificantvariablesregardingtheuseof

salescontests

Independ

entvariable

Log

it-coefficient

SE

Odd

sRatio

Means

Elasticity

(based

onno

SC

mean)

Probabilityin

%thatSCsare

used

when...

isinclud

edin

the

logisticfunctio

na

No

SC

Total

SC

NoSC

mean

Total

mean

SC

mean

Sales

forcesize

0.00

80b

0.00

21.00

842

.12

269.16

416.74

0.19

7741

.07

80.95

93.23

Adequ

acyof

output

measures

3.05

74b

0.89

721

.272

0.44

0.51

0.55

0.79

2041

.07

46.09

49.41

Replaceability

ofsales

peop

le0.1110

b0.04

31.117

6.89

7.98

8.69

0.45

0641

.07

44.03

45.98

Lengthof

thesales

cycle

0.02

58c

0.01

20.97

419

.08

12.27

7.85

-0.290

541

.07

45.38

48.22

Con

stant

2.31

21b

0.65

2

NoSC

:Non

-deploym

entof

salescontests,S

C:Deploym

entof

salescontests,S

E:Stand

arderror

a The

respectiv

emeanvalueof

theSCsampleisreplaced

only

forthevariableun

derconsideration(ceterispa

ribu

s)bsign

ificant

atthe0.1%

level

c significant

atthe5%

level


are deployed is comparatively high. However, there are substantial differencesbetween the elasticities: in terms of magnitude, the strongest influence stems fromthe variable “adequacy of output measures,” followed by “replaceability of salespeople,” “length of the sales cycle,” and “sales force size.”

In the event that P = 41.07%, it may be deduced that with a 1% increase in thesize of the sales force (i.e., from 42.12 to 42.54 sales people), the probability Pi thatSCs are used increases by 0.20% to approximately 41.15%. If the length of the19.08 week sales cycle is increased by 1% to approximately 19.27 weeks, theprobability Pi decreases by -0.29% to 40.95%. For the other two significant variablesused in our example, the effect of a 1% change in the influencing factors can becalculated analogously.

For sales mangers it might be revealing to know, how the probability that SCs areused changes when the values of the independent variables are changed by a certainlevel (e.g., “what effect is observed on the probability that SCs are used, when thelength of the sales cycle increases from 10 to 15 weeks?”). In our example of asensitivity analysis in Table 7, the mean values of the non-SC observations areselected as the baseline again. If only the mean values of the non-SC observations areused in the logistic function, we derive a probability of 41.07% that SCs are used.For our sensitivity analysis, we now replace the mean values of the non-SC obser-vations with the mean values of the SC observations one at a time for eachindependent variable. The values of the other variables are at the same time keptconstant at the means of the non-SC observations. The estimated probabilities shownin Table 7 indicate that the largest influence on the probability that SCs are usedemanates from “sales force size.” If the number of sales people increases from 42.12(i.e., mean for sales forces that do not use SCs) to 416.74 (i.e., mean value for salesforces that deploy SCs), the estimated probability to use SCs rises from 41.07% to93.23%. Notably, the variable is ranked as least important with regard to its elastic-ity. This can be explained by the substantial difference between the means of thenon-SCs and SCs observations, which exhibit a ratio of almost 1:10. While elastic-ities are good at indicating a variable’s influence for a rather limited range of values,sensitivity analyses are useful to reveal probability changes for larger changes of theindependent variable. The variable “adequacy of output measures” also exerts asubstantial, although clearly smaller influence on the change in the probability thatSCs are used. If this standardized multiple-item variable (which is normed between0 and 1) equals 0.55 (i.e., the mean of cases where SCs are used), rather thanequaling 0.44, the probability of using SCs increases from 41.07% to 49.41%.While a substantial impact is also exerted by the “length of the sales cycle” on theestimated change in the dependent variables, the variable “replaceability of salespeople” exerts the lowest influence on Pi.

Logistic regression can also be used for prediction or cross-validation purposes.As part of cross-validation, the estimated coefficients can be applied to a holdoutsample. In this case, only a part of the sample is used for calibrating the logisticfunction, which is then used to predict the actual outcomes of the remaining sample.Comparing the predicted outcomes with the actual outcomes of the holdout sampleprovides a good indication about the predictive power of the logistic function.


Especially in direct marketing, it is crucial to know which customers will respond toa direct marketing instrument (e.g., direct mailings) in order to select the mostpromising customers. In this case, choosing the logistic function with the best out-of-sample prediction performance is pivotal for the success of a marketing cam-paign. Evaluations of the predictive performance, which are based on the calibrationsample, might be particularly misleading when using models that suffer fromoverfitting (see section “Logistic Regression Procedure” and Tillmanns et al. 2017for further explanations).

However, the estimated function can also be used to facilitate managementdecisions: companies considering whether or not to use SCs might use the findingsof the logistic regression model presented in our application. Based on the attributevalues given in a certain company, managers can derive the probability whether SCsare used by companies with similar attribute values. As an illustration, we use theobserved attributes of a pharmaceutical company (see Table 8). The attributes of thiscompany are then applied to Eq. 25 with the coefficients of the logistic regressionmodel, which results in a probability of Pi = 28.05% that SCs are used. Since theobserved company currently does not use SCs, the current (non-)use of sales contestscorresponds to the typical practice as observed in the sample.

Conclusion

In the preceding sections, we addressed the fundamentals of discriminant analysisand logistic regression. Table 9 provides a compact overview of both methods interms of their essential characteristics.

In principle, both methods are suitable for research questions in which thedependent variable has a categorical scale level with two or more groups. Theymight be applied for either classification or prediction purposes.

Compared to discriminant analysis, logistic regression has a number of keybenefits, which relate particularly to the comparatively high robustness of theestimation results. For example, logistic regression allows to conduct analyseseven in cases where the assumptions of discriminant analysis are violated, such asfor the analysis of nonmetric independent variables (Hosmer et al. 2013, p. 22). As inlinear regression, categorical variables can be analyzed using dummy variables,while in discriminant analysis, such variables would violate the assumption ofhomogeneous variances in the groups (Hair et al. 2010, p. 341 and p. 426).

Table 8 Data for a pharmaceutical company regarding the use of sales contests

Independent variable Logit coefficient Observation at the sample company

Sales force size þ0.0080 18

Adequacy of output measures þ3.0574 0.39a

Replaceability of sales people þ0.1110 5a

Length of the sales cycle (in weeks) -0.0258 20aFor further information on these scales refer to Krafft et al. (2004) and the literature cited there


Further, the assumption of multivariate normally distributed independent vari-ables, which is required for the use of discriminant analysis, is frequently not met inpractical applications. In these situations, logistic regression is also preferable. Thesame holds for studies in which the analyzed groups have very different sizes(Tabachnik and Fidell 2013, p. 380). Another important advantage of logisticregression is that through the regression procedure, asymptotic t-statistics can be

Table 9 Overview of logistic regression and discriminant analysis

Logistic regression Discriminant analysis

Objectives

■ Identification of variables that contribute tothe explanation of group membership

■ Prediction of group membership for out-of-sample observations

■ Determination of linear combinations of theindependent variables that optimize theseparation between the groups and minimizethe misclassification of observations

■ Identification of variables that contribute toexplaining differences between groups

■ Predicting the group membership for out-of-sample observations

Estimation principle

Maximum likelihood approach Maximization of the variance between thegroups, relative to the variance within thegroups

Scaling of the variables

■ Dependent variable: nominal scale■ Independent variables: metric and/or

nominal scales

■ Dependent variable: nominal scale■ Independent variables: metric scales

Assessment of the significance and strength of influence of the independent variables

■ Wald test, likelihood ratio test■ Odds ratio, standardized coefficients, partial

derivatives, elasticities, sensitivity analyses

■ F-test (univariate ANOVA)■ Standardized discriminant coefficients,

discriminant loadings

Interpretation of the coefficients

Coefficients represent the effect of a one-unitchange in the independent variables on thelogit

Discriminant coefficients and weights reflectthe relative average group profile

Sample size (recommendations)

■ A minimum of 10 observations perindependent variable in the smallest group

■ Large samples sizes are recommendedbecause of the asymptotic properties of themaximum likelihood estimates in the modelparameters

■ A minimum of 20 observations per group■ A minimum of five observations per

independent variable. Better: 20observations per independent variable toattain stable estimates for the standardizedcoefficients and weights

Assumptions/recommendations

■ Nonlinear relationships■ No multicollinearity■ Errors are independent■ Linearity of the logit

■ Linear relationships■ No multicollinearity■ Multivariate normal distribution of the

independent variables■ Homogeneity of the variance-covariance

matrices of the independent variables


provided for the estimated coefficients. The confidence intervals obtained in dis-criminant analysis are, on the other hand, not interpretable (Morrison 1969, pp. 157et seq.).

Compared to discriminant analysis, logistic regression is thus an extremely robustestimation method. However, it should not be concluded that logistic regression isalways the best choice. Discriminant analysis might provide more efficient estimateswith higher statistical power if group sizes do not turn out to be too unequal and incases where the assumptions of discriminant analysis are met (Press and Wilson1978, p. 701; Tabachnik and Fidell 2013, p. 380 and p. 443). Further, because of theasymptotic properties of the maximum likelihood estimation, the use of logisticregression often requires larger sample sizes than discriminant analysis. Addition-ally, researchers should examine whether the assumed nonlinear development of theprobability Pi is suitable for the specific research context. If, for example, a linearchange of Pi is more appropriate, the logistic regression should be avoided. Instead,researchers should check whether linear approaches such as the linear probabilitymodel (LPM) or discriminant analysis are more appropriate (Aldrich and Nelson1984).

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