Machine Learning Multivariate 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.
This vignette conducts a machine learning multivariate binomial 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 trains a multivariate binomial logistic regression model. The model is then tested and evaluated for accuracy in predicting the target or outcome variable.
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. Conducted a brief exploratory analysis comprising a statistical summary, correlation and comparative analysis to understand the variables.
Commenced building the logit model by splitting the data into training and testing sets at a percentage ratio of 75:25 or 3:1 split. Implemented stratified sampling to ensure that both levels of the response variable in the test set had roughly equivalent proportions of values to the training set.
Model specifications and feature engineering were formulated before fitting the model on the training data and reviewing the results.
The model was then applied to the unseen test set to predict the response variable, reaction to organisational change. 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. Because the focus is on predicting either support or opposition to change, the neutral measure on the Likert scale was removed from the binary outcome variable.
| 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 |
|---|---|
| oppose | 184 |
| support | 359 |
The data set was filtered to analyse only those respondents who reported experiencing significant organisational change. Table 3 is a statistical summary of the proposed explanatory variables for the logistic regression.
| 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 coefficient between each 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. Build and fit model
The logistic regression model was specified with additional feature engineering to normalise predictor variables. The model was then fitted to the training data. Table 4 shows model fit statistics for the training set arranged by p-value.
| Table 4 Model fit statistics (arranged by p.value) | ||||
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| behavioural_intentions | 5.5018 | 1.5519 | 3.5452 | 0.0004 |
| emotion | 3.7447 | 1.1378 | 3.2912 | 0.0010 |
| reaction_formation | −1.4276 | 0.5780 | −2.4698 | 0.0135 |
| past_influences | 1.4822 | 0.7784 | 1.9043 | 0.0569 |
| passive_aggression | 1.2442 | 0.7388 | 1.6842 | 0.0921 |
| isolation_of_affect | 1.0488 | 0.6621 | 1.5842 | 0.1132 |
| passive_existence | −1.0412 | 0.7086 | −1.4693 | 0.1417 |
| projection | −1.0643 | 0.8141 | −1.3074 | 0.1911 |
| displacement | 0.9718 | 0.7580 | 1.2820 | 0.1998 |
| anxious_thoughts | 0.7126 | 0.5741 | 1.2413 | 0.2145 |
| avoidance | 0.7410 | 0.6106 | 1.2136 | 0.2249 |
| facing_reality | −0.8787 | 0.7411 | −1.1857 | 0.2357 |
| denial | −0.9507 | 0.8341 | −1.1398 | 0.2544 |
| dissociation | −0.8328 | 0.7811 | −1.0662 | 0.2863 |
| acting_out | −0.6300 | 0.6772 | −0.9303 | 0.3522 |
| labelling_blame | 0.6776 | 0.7691 | 0.8810 | 0.3783 |
| self_efficacy | 0.4152 | 0.6488 | 0.6399 | 0.5222 |
| managing_feelings | 0.4103 | 0.6813 | 0.6023 | 0.5470 |
| needs_approval | 0.3831 | 0.7651 | 0.5007 | 0.6166 |
| catastrophising | 0.2974 | 0.6966 | 0.4269 | 0.6695 |
| fears_failure | 0.2394 | 0.6650 | 0.3601 | 0.7188 |
It is noted in Table 4 and later in Chart 5 that the order of less significant variables may differ depending on the random split of observations in the training and test sets. Chart 4 illustrates the training set’s most significant explanatory variables, with a p-value less than 0.05.
Chart 5 visualises estimates for each explanatory variable in this random split.
3. Model prediction and evaluation
The model fitted on the training data was applied to the unseen test data to predict the response variable. Table 5 compares actual reaction and predicted reaction (.pred) to organisational change in a small sample of cases extracted from the test set (scroll to far right of table for a sample of predictions.)
| Table 5 A sample of predictions from the test set | |||||||||||||||||||||||
| self_efficacy | needs_approval | fears_failure | labelling_blame | catastrophising | managing_feelings | anxious_thoughts | avoidance | past_influences | facing_reality | passive_existence | dissociation | displacement | isolation_of_affect | reaction_formation | denial | projection | passive_aggression | acting_out | emotion | behavioural_intentions | reaction | .pred_oppose | .pred_support |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.35 | 5.50 | 4.00 | 4.00 | 5.50 | 3.50 | 3.00 | 1.00 | 4.50 | 6.50 | 5.00 | 4.50 | 5.50 | 3.00 | 3.00 | 2.50 | 2.00 | 2.50 | 4.50 | 2.95 | 4.40 | oppose | 0.9626 | 0.0374 |
| 6.00 | 3.50 | 4.50 | 2.50 | 4.00 | 5.00 | 4.50 | 2.50 | 4.00 | 5.50 | 4.00 | 4.00 | 4.00 | 5.50 | 4.00 | 2.50 | 4.00 | 3.50 | 3.50 | 2.85 | 3.20 | oppose | 0.9995 | 0.0005 |
| 6.53 | 4.50 | 2.50 | 2.00 | 2.50 | 2.00 | 5.50 | 1.50 | 2.00 | 5.00 | 5.00 | 2.50 | 1.00 | 3.00 | 4.50 | 2.00 | 1.50 | 1.00 | 5.50 | 4.80 | 5.85 | support | 0.1995 | 0.8005 |
| 5.71 | 6.00 | 5.00 | 3.00 | 4.00 | 4.50 | 5.50 | 3.50 | 2.00 | 4.50 | 3.50 | 3.00 | 3.50 | 4.50 | 3.50 | 2.50 | 2.00 | 3.00 | 3.00 | 3.85 | 5.70 | support | 0.0002 | 0.9998 |
| 4.41 | 6.00 | 3.00 | 5.50 | 5.00 | 6.00 | 5.50 | 3.00 | 4.50 | 6.00 | 5.00 | 2.00 | 2.50 | 4.00 | 5.50 | 2.00 | 4.00 | 4.50 | 3.50 | 2.55 | 3.35 | oppose | 0.9991 | 0.0009 |
| 5.59 | 6.50 | 6.00 | 2.50 | 6.00 | 5.50 | 6.00 | 1.50 | 5.00 | 6.00 | 5.00 | 1.50 | 4.00 | 1.00 | 3.50 | 3.00 | 2.50 | 2.50 | 2.50 | 2.25 | 4.80 | support | 0.9119 | 0.0881 |
| 5.59 | 4.00 | 3.50 | 3.50 | 2.50 | 2.50 | 3.50 | 2.00 | 2.50 | 6.00 | 3.00 | 2.50 | 4.00 | 4.50 | 4.00 | 2.00 | 2.50 | 3.00 | 5.00 | 3.95 | 5.50 | support | 0.0257 | 0.9743 |
| 6.18 | 4.50 | 6.50 | 2.00 | 5.00 | 4.00 | 5.50 | 3.50 | 2.50 | 5.00 | 5.50 | 2.00 | 3.50 | 4.00 | 5.00 | 3.00 | 2.00 | 2.50 | 5.00 | 5.10 | 6.20 | support | 0.0000 | 1.0000 |
| 5.71 | 2.00 | 2.50 | 2.00 | 2.00 | 2.50 | 2.00 | 2.00 | 3.00 | 2.50 | 3.00 | 2.00 | 2.00 | 3.00 | 3.50 | 2.00 | 2.00 | 2.00 | 2.00 | 3.75 | 5.75 | support | 0.0291 | 0.9709 |
| 5.24 | 6.00 | 6.00 | 4.00 | 4.00 | 5.00 | 3.50 | 3.00 | 4.50 | 5.50 | 5.00 | 4.50 | 4.50 | 1.50 | 3.00 | 3.50 | 4.00 | 4.50 | 5.50 | 2.75 | 3.55 | oppose | 0.9994 | 0.0006 |
The predictive performance of the machine learning logistic regression model was evaluated with a confusion matrix, model statistics and ROC curve.
Chart 6 confusion matrix summarises predictions by categorising and comparing predicted against the actual response for reaction to change. The confusion matrix calculates 83 per cent accuracy (true positive and true negative). False positive (top left) and false negative (bottom right) predictions account for the remaining 17 per cent.
Table 6 summarises the binomial logistic regression prediction performance.
| Table 6 Summary of logistic regression prediction metrics | ||
| .metric | .estimator | .estimate |
|---|---|---|
| accuracy | binary | 0.8302 |
| kap | binary | 0.6595 |
| sens | binary | 0.8696 |
| spec | binary | 0.8000 |
| ppv | binary | 0.7692 |
| npv | binary | 0.8889 |
| mcc | binary | 0.6638 |
| j_index | binary | 0.6696 |
| bal_accuracy | binary | 0.8348 |
| detection_prevalence | binary | 0.4906 |
| precision | binary | 0.7692 |
| recall | binary | 0.8696 |
| f_meas | binary | 0.8163 |
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 7 illustrates the ROC curve with an AUC of 0.8986 for this model.
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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## tzdb 0.4.0 2023-05-12 [1] CRAN (R 4.4.0)
## urlchecker 1.0.1 2021-11-30 [1] CRAN (R 4.4.0)
## usethis 2.2.3 2024-02-19 [1] CRAN (R 4.4.0)
## utf8 1.2.4 2023-10-22 [1] CRAN (R 4.4.0)
## vctrs 0.6.5 2023-12-01 [1] CRAN (R 4.4.0)
## vip * 0.4.1 2023-08-21 [1] CRAN (R 4.4.0)
## withr 3.0.0 2024-01-16 [1] CRAN (R 4.4.0)
## workflows * 1.1.4 2024-02-19 [1] CRAN (R 4.4.0)
## workflowsets * 1.1.0 2024-03-21 [1] CRAN (R 4.4.0)
## xfun 0.46 2024-07-18 [1] CRAN (R 4.4.1)
## xml2 1.3.6 2023-12-04 [1] CRAN (R 4.4.0)
## xtable 1.8-4 2019-04-21 [1] CRAN (R 4.4.0)
## yaml 2.3.9 2024-07-05 [1] CRAN (R 4.4.1)
## yardstick * 1.3.1 2024-03-21 [1] CRAN (R 4.4.0)
##
## [1] C:/Users/wayne/AppData/Local/R/win-library/4.4
## [2] C:/Program Files/R/R-4.4.0/library
##
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