Stepwise Binomial Logistic Regression
Prepared by Dr Wayne Bovey
Objective
According to IBM, logistic regression (also known as logit model) is often used for classification and predictive analytics. Logistic regression estimates the probability of an event occurring based on a given data set of independent variables. Since the outcome is a probability, the dependent variable is bounded between 0 and 1.
The vignette conducts a stepwise logistic regression on a dataset of 208 respondents experiencing significant organisational change. Respondents reported self-efficacy, irrational ideas, maladaptive defence mechanisms, emotion, behavioural intentions and reaction towards change in their organisation.
This vignette has two objectives. First, model and identify statistically significant relationships between the outcome and explanatory variables. Second, predict outcomes and evaluate the accuracy of those predictions.
Workflow
The raw data set was wrangled and tidied before processing. Since this was a logistic regression, the outcome variable, a seven-point Likert scale, was replaced with a binary variable. Next, a brief exploratory analysis comprising a statistical summary, correlation, and comparative analysis was conducted to understand the variables.
Proceeded to conduct a stepwise binomial logistic regression, identifying statistically significant explanatory variables. Reviewed fit statistics for the stepwise model.
In the final section of this vignette, predicted outcomes using the model. Evaluated the model’s prediction performance with a confusion matrix heatmap, model fit statistics and ROC curve.
Results
1. Explore variables
Before building the logit model, the outcome variable (reaction to organisational change) was transformed from an interval variable to a binary variable. The following tables show the conversion of the outcome variable from a seven-point Likert scale (Table 1) to a binary scale (Table 2) with corresponding frequencies. Scale measures opposing change were coded as “0”, and measures supporting change were coded as “1”. The neutral measure on the Likert scale was dropped from the binary scale.
| Table 1 Original seven-point Likert scale | |
| Reaction to change | Freq |
|---|---|
| Totally Oppose | 73 |
| Oppose | 71 |
| Partially Oppose | 90 |
| Neutral | 124 |
| Partially Support | 189 |
| Support | 23 |
| Totally Support | 46 |
| Table 2 New binary scale for logistic regression | |
| Reaction to change | Freq |
|---|---|
| 0 | 184 |
| 1 | 359 |
The data set was then filtered to only those respondents who reported experiencing significant organisational change. Table 3 provides a statistical summary of the proposed explanatory variables.
| Table 3 Statistical summary of explanatory variables | ||||||||||||
| variable | n | mean | sd | median | trimmed | mad | min | max | range | skew | kurtosis | se |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| self_efficacy | 208 | 5.55 | 0.81 | 5.65 | 5.63 | 0.78 | 1.71 | 6.94 | 5.24 | −1.19 | 2.71 | 0.06 |
| needs_approval | 208 | 4.19 | 1.40 | 4.00 | 4.22 | 1.48 | 1.00 | 7.00 | 6.00 | −0.14 | −0.73 | 0.10 |
| fears_failure | 208 | 4.06 | 1.51 | 4.00 | 4.07 | 1.48 | 1.00 | 7.00 | 6.00 | 0.00 | −0.83 | 0.10 |
| labelling_blame | 208 | 3.19 | 1.32 | 3.00 | 3.11 | 1.48 | 1.00 | 7.00 | 6.00 | 0.55 | −0.31 | 0.09 |
| catastrophising | 208 | 3.70 | 1.32 | 3.50 | 3.66 | 1.48 | 1.00 | 7.00 | 6.00 | 0.15 | −0.89 | 0.09 |
| managing_feelings | 208 | 3.88 | 1.31 | 4.00 | 3.88 | 1.48 | 1.00 | 6.50 | 5.50 | −0.05 | −0.72 | 0.09 |
| anxious_thoughts | 208 | 4.10 | 1.16 | 4.00 | 4.13 | 0.74 | 1.50 | 7.00 | 5.50 | −0.17 | −0.38 | 0.08 |
| avoidance | 208 | 2.25 | 0.99 | 2.00 | 2.14 | 0.74 | 1.00 | 5.50 | 4.50 | 1.05 | 0.90 | 0.07 |
| past_influences | 208 | 2.79 | 1.30 | 2.50 | 2.71 | 1.48 | 1.00 | 6.00 | 5.00 | 0.56 | −0.76 | 0.09 |
| facing_reality | 208 | 4.54 | 1.44 | 5.00 | 4.60 | 1.48 | 1.00 | 7.00 | 6.00 | −0.35 | −0.80 | 0.10 |
| passive_existence | 208 | 4.23 | 1.24 | 4.00 | 4.26 | 1.48 | 1.00 | 7.00 | 6.00 | −0.18 | −0.46 | 0.09 |
| dissociation | 208 | 2.74 | 1.21 | 2.50 | 2.61 | 0.74 | 1.00 | 6.50 | 5.50 | 0.90 | 0.24 | 0.08 |
| displacement | 208 | 3.04 | 1.28 | 3.00 | 2.99 | 1.48 | 1.00 | 7.00 | 6.00 | 0.34 | −0.50 | 0.09 |
| isolation_of_affect | 208 | 3.41 | 1.49 | 3.50 | 3.38 | 2.22 | 1.00 | 7.00 | 6.00 | 0.16 | −0.92 | 0.10 |
| reaction_formation | 208 | 4.17 | 1.25 | 4.00 | 4.17 | 1.48 | 1.50 | 7.00 | 5.50 | 0.07 | −0.65 | 0.09 |
| denial | 208 | 2.51 | 1.03 | 2.00 | 2.44 | 0.74 | 1.00 | 6.50 | 5.50 | 0.79 | 0.40 | 0.07 |
| projection | 208 | 2.49 | 1.11 | 2.00 | 2.39 | 0.74 | 1.00 | 6.00 | 5.00 | 0.88 | 0.09 | 0.08 |
| passive_aggression | 208 | 2.62 | 1.08 | 2.50 | 2.52 | 0.74 | 1.00 | 6.50 | 5.50 | 0.88 | 0.55 | 0.07 |
| acting_out | 208 | 3.55 | 1.34 | 3.50 | 3.54 | 1.48 | 1.00 | 6.50 | 5.50 | 0.04 | −0.88 | 0.09 |
| emotion | 208 | 3.80 | 1.15 | 3.80 | 3.79 | 1.19 | 1.00 | 6.90 | 5.90 | 0.09 | −0.32 | 0.08 |
| behavioural_intentions | 208 | 5.08 | 1.13 | 5.25 | 5.15 | 1.26 | 1.75 | 7.00 | 5.25 | −0.57 | −0.21 | 0.08 |
Chart 1 supports Table 1, showing the correlation coefficients between each explanatory variable.
Because of the number of explanatory variables under consideration,
prepared two separate pairs plots. Chart 2 compares the relationship
between irrational ideas and the outcome variable, reaction to
organisational change. Chart 3 compares the relationship between
maladaptive defence mechanisms and the outcome variable, reaction to
organisational change.
2. Stepwise logistic regression
Conducted a binomial stepwise logistic regression implementing forward selection and backward elimination. Both models derived the same statistically significant explanatory variables.
Table 4 summarises logistic regression fit statistics using the stepwise approach.
| Table 4 Stepwise logistic regression model fit statistics (arranged by p.value) | ||||
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | −27.3799 | 4.6754 | −5.8562 | 0.0000 |
| behavioural_intentions | 3.0873 | 0.5860 | 5.2685 | 0.0000 |
| emotion | 2.3993 | 0.5055 | 4.7461 | 0.0000 |
| reaction_formation | −0.7021 | 0.2517 | −2.7898 | 0.0053 |
| past_influences | 0.8061 | 0.2986 | 2.6996 | 0.0069 |
| anxious_thoughts | 0.6389 | 0.2959 | 2.1593 | 0.0308 |
| avoidance | 0.6689 | 0.3334 | 2.0062 | 0.0448 |
In place of the coefficient of determination (R2) as a measure of fit, a pseudo-R2 value is adopted when the outcome variable is nominal or ordinal. There are several variants of pseudo-R2. Table 5 shows pseudo-R2 ranging from 0.6908 to 0.8996 for the selected variants.
| Table 5 Pseudo-R2 variants for stepwise model | |
| variant | pseudo-R2 |
|---|---|
| McFadden | 0.6908 |
| Nagelkerke | 0.8199 |
| VeallZimmermann | 0.8409 |
| McKelveyZavoina | 0.8996 |
3. Prediction and evaluation
3.1 Model prediction
The next step involved predicting each respondent’s reaction to organisational change. Table 6 shows the predicted reaction compared to the actual reaction in a small sample of cases, along with significant explanatory variables.
| Table 6 Sample of model predictions | |||||||
| anxious_thoughts | avoidance | past_influences | reaction_formation | emotion | behavioural_intentions | reaction | prediction |
|---|---|---|---|---|---|---|---|
| 3.00 | 1.00 | 4.50 | 3.00 | 2.95 | 4.40 | 0 | 0.0684 |
| 5.50 | 2.00 | 5.50 | 4.00 | 2.45 | 4.40 | 0 | 0.1914 |
| 4.50 | 2.50 | 4.00 | 4.00 | 2.85 | 3.20 | 0 | 0.0033 |
| 5.50 | 1.50 | 3.50 | 2.00 | 2.05 | 5.05 | 0 | 0.2816 |
| 4.00 | 1.00 | 1.50 | 4.00 | 2.55 | 4.90 | 0 | 0.0109 |
| 6.00 | 3.00 | 5.50 | 4.00 | 4.50 | 6.25 | 1 | 1.0000 |
| 3.50 | 2.00 | 3.50 | 4.00 | 1.90 | 1.85 | 0 | 0.0000 |
| 5.50 | 1.50 | 2.00 | 4.50 | 4.80 | 5.85 | 1 | 0.9943 |
| 5.00 | 1.00 | 3.50 | 4.00 | 4.70 | 4.65 | 0 | 0.8936 |
| 6.00 | 1.50 | 2.00 | 3.50 | 2.70 | 4.90 | 0 | 0.1439 |
| 2.00 | 2.00 | 2.00 | 5.50 | 2.35 | 4.35 | 0 | 0.0004 |
| 4.00 | 1.00 | 2.00 | 3.00 | 3.80 | 5.05 | 1 | 0.5145 |
| 6.50 | 4.00 | 1.00 | 4.00 | 3.95 | 5.80 | 1 | 0.9921 |
| 4.00 | 1.50 | 4.00 | 2.50 | 3.60 | 5.10 | 1 | 0.8839 |
| 5.50 | 3.50 | 2.00 | 3.50 | 3.85 | 5.70 | 1 | 0.9886 |
3.2 Model evaluation
The predictive performance of the stepwise model was evaluated with a confusion matrix, model statistics and ROC curve.
Chart 4 confusion matrix summarises predictions by categorising and comparing predicted against the actual response for reaction to change. The confusion matrix shows good performance for the stepwise model, recording 92.8 per cent accuracy (true positive and true negative). False positive (top left) and false negative (bottom right) predictions account for the remaining 7.2 per cent.
Table 7 summarises the stepwise model prediction performance.
| Table 7 Summary of model prediction metrics | ||
| .metric | .estimator | .estimate |
|---|---|---|
| accuracy | binary | 0.9279 |
| kap | binary | 0.8529 |
| sens | binary | 0.9328 |
| spec | binary | 0.9213 |
| ppv | binary | 0.9407 |
| npv | binary | 0.9111 |
| mcc | binary | 0.8530 |
| j_index | binary | 0.8541 |
| bal_accuracy | binary | 0.9271 |
| detection_prevalence | binary | 0.5673 |
| precision | binary | 0.9407 |
| recall | binary | 0.9328 |
| f_meas | binary | 0.9367 |
The ROC curve (receiver operating characteristic curve) plots the true positive rate (sensitivity) against the false positive rate (specificity) at all classification thresholds. AUC (area under the curve) measures the entire two-dimensional area underneath the ROC curve. It evaluates how well a logistic regression model classifies positive and negative outcomes at every possible threshold. An AUC from 0.9 to 1 is regarded as “A” grade in classification performance. Chart 5 illustrates the ROC curve for the stepwise model with AUC of 0.972.
References:
Self-efficacy was measured using the ‘Self-efficacy scale:
Construction and validation’ by Sherer, Maddux, Mercandante,
Prentice-Dunn and Rogers, published in Psychological
Reports.
Irrational ideas were measured using the ‘Irrational belief scale’
developed by Malouff and Schutte, published in the Sourcebook of
Adult Assessment Strategies, based on Ellis and Harper’s work,
published in A New Guide to Rational Living.
Maladaptive defence mechanisms were measured using selected items from
‘The Defense Style Questionnaire’ by Andrews, Singh and Bond, published
in The Journal of Nervous and Mental Disease.
Emotion was measured using ‘A semantic differential mood scale’ by Lorr
and Wunderlich, published in the Journal of Clinical
Psychology.
Session information and package update
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