CFA-SEM
Chapter 6
Logistic Regression: Regression with a Binary Dependent Variable
LEARNING OBJECTIVES
Upon completing this chapter, you should be able to do the following:
State the circumstances under which logistic regression should be used instead of multiple regression.
Identify the types of dependent and independent variables used in the application of logistic regression.
Describe the method used to transform binary measures into the likelihood and probability measures used in logistic regression.
Chapter 6
Logistic Regression: Regression with a Binary Dependent Variable
LEARNING OBJECTIVES continued . . .
Upon completing this chapter, you should be able to do the following:
Interpret the results of a logistic regression analysis and assessing predictive accuracy, with comparisons to both multiple regression and discriminant analysis.
Understand the strengths and weaknesses of logistic regression compared to discriminant analysis and multiple regression.
Chapter 6
Logistic Regression: Regression with a Binary Dependent Variable
Logistic Regression . . . is a specialized form of regression that is designed to predict and explain a binary (two-group) categorical variable rather than a metric dependent measure. Its variate is similar to regular regression and made up of metric independent variables. It is less affected than discriminant analysis when the basic assumptions, particularly normality of the independent variables, are not met.
Logistic Regression Defined
Logistic Regression May Be Preferred . . .
When the dependent variable has only two groups, logistic regression may be preferred for two reasons:
Discriminant analysis assumes multivariate normality and equal variance-covariance matrices across groups, and these assumptions are often not met. Logistic regression does not face these strict assumptions and is much more robust when these assumptions are not met, making its application appropriate in many situations.
Even if the assumptions are met, some researchers prefer logistic regression because it is similar to multiple regression. It has straightforward statistical tests, similar approaches to incorporating metric and nonmetric variables and nonlinear effects, and a wide range of diagnostics.
Multiple Regression Decision Process
Stage 1: Objectives of Logistic Regression
Stage 2: Research Design for Logistic Regression
Stage 3: Assumptions of Logistic Regression
Stage 4: Estimation of the Logistic Regression Model and Assessing Overall Fit
Stage 5: Interpretation of the Results
Stage 6: Validation of the Results
Logistic regression is best suited to address two research objectives . . .
Identifying the independent variables that impact group membership in the dependent variable.
Establishing a classification system based on the logistic model for determining group membership.
Stage 1: Objectives of Logistic Regression
Stage 2: Research Design for Logistic Regression
The binary nature of the dependent variable (0 – 1) means the error term has a binomial distribution instead of a normal distribution, and it thus invalidates all testing based on the assumption of normality.
The variance of the dichotomous variable is not constant, creating instances of heteroscedasticity as well.
Neither of the above violations can be remedied through transformations of the dependent or independent variables. Logistic regression was developed to specifically deal with these issues.
Stage 3: Assumptions of Logistic Regression
The advantages of logistic regression are primarily the result of the general lack of assumptions.
Logistic regression does not require any specific distributional form for the independent variables.
Heteroscedasticity of the independent variables is not required.
Linear relationships between the dependent and independent variables are not required.
Stage 4: Estimation of Logistic Regression Model and Assessing Overall Fit
Transforming the dependent variable
Estimating the coefficients
Transforming a probability into odds and logit values
Model estimation
Assessing the goodness of fit
Estimating the Coefficients
Two basic steps . . .
Transforming a probability into odds and logit values
Model estimation using a maximum likelihood approach, not least squares as in multiple regression
The estimation process maximizes the likelihood that an event will occur – the event being a respondent is assigned to one group versus another
Transforming a Probability into Odds and Logit Values
The logistic transformation has two basic steps:
Restating a probability as odds, and
Calculating the logit values.
Instead of using ordinary least squares to estimate the model, the maximum likelihood method is used.
The basic measure of how well the maximum likelihood estimation procedure fits is the likelihood value.
Model Estimation Fit – Between Model comparisons . . .
Comparisons of the likelihood values follow three steps:
Estimate a Null Model – which acts as the “baseline” for making comparisons of improvement in model fit.
Estimate Proposed Model – the model containing the independent variables to be included in the logistic regression.
Assess – 2LL Difference.
Comparison to Multiple Regression . . .
Correspondence of Primary Elements of Model Fit
Multiple Regression Logistic Regression
Total Sum of Squares -2LL of Base Model
Error Sum of Squares -2LL of Proposed Model
Regression Sum of Squares Difference of -LL for Base and Proposed Models
F test of model fit Chi-square Test of -2LL Difference
Coefficient of determination “Pseudo” R2 measures
Stage 5: Interpretation of the Results
Testing for significance of the coefficients – based on the Wald statistic
Interpreting the coefficients
Directionality of the relationship
Magnitude of the relationship of metric independent variables
Interpreting nonmetric independent variables
Directionality of the Relationship
A positive relationship means an increase in the independent variable is associated with an increase in the predicted probability, and vice versa. But the direction of the relationship is reflected differently for the original and exponentiated logistic coefficients.
Original coefficient signs indicate the direction of the relationship.
Exponentiated coefficients are interpreted differently since they are the logarithms of the original coefficients and do not have negative values. Thus, exponentiated coefficients above 1.0 represent a positive relationship and values less than 1.0 represent negative relationships.
Magnitude of the Relationship . . .
The magnitude of metric independent variables is interpreted differently for original and exponentiated logistic coefficients:
Original logistic coefficients – are less useful in determining the magnitude of the relationship since the reflect the change in the logit (logged odds) value.
Exponentiated coefficients – directly reflect the magnitude of the change in the odds value. But their impact is multiplicative and a coefficient of 1.0 denotes no change (1.0 times the independent variable = no change).
Rules of Thumb 6–1
Logistic Regression
Logistic regression is the preferred method for two-group (binary) dependent variables due to its robustness, ease of interpretation and diagnostics.
Sample size considerations for logistic regression are primarily focused on the size of each group, which should have 10 times the number of estimated model coefficients (the number of variables).
Sample size should be met in both the analysis and holdout samples.
Model significance tests are made with a chi-square test on the differences in the log likelihood values (-2LL) between two models.
Rules of Thumb 6–1 continued . . .
Logistic Regression
Coefficients are expressed in two forms: original and exponentiated to assist in interpretation.
Interpretation of the coefficients for direction and magnitude is:
Direction can be directly assessed in the original coefficients (positive or negative signs) or indirectly in the exponentiated coefficients (less than 1 are negative, greater than 1 are positive).
Magnitude is best assessed by the exponentiated coefficient, with the percentage change in the dependent variable shown by: Percentage change = (Exponentiated Coefficient – 1.0) * 100
Stage 6: Validation of the Results
Involves ensuring both the internal and external validity of the results.
The most common form of estimating external validity is creation of a holdout or validation sample and calculating the hit ratio.
A second approach is cross-validation, typically achieved with a jackknife or “leave-one-out” process of calculating the hit ratio.
Variable Description Variable Type
Data Warehouse Classification Variables
X1 Customer Type nonmetric
X2 Industry Type nonmetric
X3 Firm Size nonmetric
X4 Region nonmetric
X5 Distribution System nonmetric
Performance Perceptions Variables
X6 Product Quality metric
X7 E-Commerce Activities/Website metric
X8 Technical Support metric
X9 Complaint Resolution metric
X10 Advertising metric
X11 Product Line metric
X12 Salesforce Image metric
X13 Competitive Pricing metric
X14 Warranty & Claims metric
X15 New Products metric
X16 Ordering & Billing metric
X17 Price Flexibility metric
X18 Delivery Speed metric
Outcome/Relationship Measures
X19 Satisfaction metric
X20 Likelihood of Recommendation metric
X21 Likelihood of Future Purchase metric
X22 Current Purchase/Usage Level metric
X23 Consider Strategic Alliance/Partnership in Future nonmetric
Description of HBAT Primary Database Variables