Stepwise Multiple Regression
Prepared by Dr Wayne Bovey
Objective
According to Wikipedia, stepwise regression is a method of fitting regression models whereby an automatic procedure chooses the predictive variables. The procedure is a step-by-step iterative process that considers a variable for addition to or subtraction from the set of explanatory variables based on a pre-specified criterion. Usually, stepwise regression takes the form of forward selection, backward elimination, or bidirectional elimination.
The vignette conducts a stepwise regression on a dataset of 218 respondents experiencing significant organisational change. Respondents reported their self-efficacy, irrational ideas, maladaptive defence mechanisms, emotion, behavioural intentions and reaction towards change in their organisation.
This vignette compares and evaluates the goodness of fit for each model derived from the stepwise regression.
Workflow
The raw data set was wrangled and tidied before processing. Conducted a brief exploratory analysis comprising a statistical summary and correlation analysis to understand the variables.
Proceeded to implement forward selection, backward elimination and bidirectional elimination, which resulted in two final models. The preferred model was selected for further investigation. This included:
- a review of model assumptions
- model correlation and network analysis
- visualising the relationship between the most significant explanatory variables and the response variable
- visualising the relationship between the most significant explanatory variable and the response variable segmented by categorical variables.
Results
1. Explore variables
The data set was filtered to those respondents who reported experiencing significant organisational change. Table 1 is a statistical summary of the proposed explanatory variables and the response variable (reaction).
| vars | n | mean | sd | median | trimmed | mad | min | max | range | skew | kurtosis | se | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| self_efficacy | 1 | 218 | 5.56 | 0.80 | 5.65 | 5.64 | 0.76 | 1.71 | 6.94 | 5.24 | -1.21 | 2.80 | 0.05 |
| needs_approval | 2 | 218 | 4.20 | 1.39 | 4.00 | 4.23 | 1.48 | 1.00 | 7.00 | 6.00 | -0.14 | -0.70 | 0.09 |
| fears_failure | 3 | 218 | 4.08 | 1.51 | 4.00 | 4.09 | 1.48 | 1.00 | 7.00 | 6.00 | -0.01 | -0.85 | 0.10 |
| labelling_blame | 4 | 218 | 3.19 | 1.32 | 3.00 | 3.11 | 1.48 | 1.00 | 7.00 | 6.00 | 0.54 | -0.32 | 0.09 |
| catastrophising | 5 | 218 | 3.73 | 1.32 | 3.50 | 3.69 | 1.48 | 1.00 | 7.00 | 6.00 | 0.14 | -0.89 | 0.09 |
| managing_feelings | 6 | 218 | 3.91 | 1.30 | 4.00 | 3.91 | 1.48 | 1.00 | 6.50 | 5.50 | -0.08 | -0.68 | 0.09 |
| anxious_thoughts | 7 | 218 | 4.12 | 1.16 | 4.00 | 4.15 | 0.74 | 1.50 | 7.00 | 5.50 | -0.18 | -0.39 | 0.08 |
| avoidance | 8 | 218 | 2.25 | 0.98 | 2.00 | 2.14 | 0.74 | 1.00 | 5.50 | 4.50 | 1.04 | 0.91 | 0.07 |
| past_influences | 9 | 218 | 2.79 | 1.31 | 2.50 | 2.71 | 1.48 | 1.00 | 6.00 | 5.00 | 0.58 | -0.72 | 0.09 |
| facing_reality | 10 | 218 | 4.56 | 1.43 | 5.00 | 4.62 | 1.48 | 1.00 | 7.00 | 6.00 | -0.34 | -0.78 | 0.10 |
| passive_existence | 11 | 218 | 4.23 | 1.23 | 4.00 | 4.26 | 1.48 | 1.00 | 7.00 | 6.00 | -0.15 | -0.47 | 0.08 |
| dissociation | 12 | 218 | 2.73 | 1.20 | 2.50 | 2.61 | 0.74 | 1.00 | 6.50 | 5.50 | 0.89 | 0.27 | 0.08 |
| displacement | 13 | 218 | 3.06 | 1.28 | 3.00 | 3.01 | 1.48 | 1.00 | 7.00 | 6.00 | 0.34 | -0.52 | 0.09 |
| isolation_of_affect | 14 | 218 | 3.40 | 1.50 | 3.50 | 3.37 | 2.22 | 1.00 | 7.00 | 6.00 | 0.16 | -0.94 | 0.10 |
| reaction_formation | 15 | 218 | 4.14 | 1.23 | 4.00 | 4.14 | 1.48 | 1.50 | 7.00 | 5.50 | 0.11 | -0.59 | 0.08 |
| denial | 16 | 218 | 2.51 | 1.04 | 2.00 | 2.44 | 0.74 | 1.00 | 6.50 | 5.50 | 0.77 | 0.30 | 0.07 |
| projection | 17 | 218 | 2.50 | 1.11 | 2.00 | 2.40 | 0.74 | 1.00 | 6.00 | 5.00 | 0.86 | 0.07 | 0.07 |
| passive_aggression | 18 | 218 | 2.59 | 1.06 | 2.50 | 2.49 | 0.74 | 1.00 | 6.50 | 5.50 | 0.92 | 0.67 | 0.07 |
| acting_out | 19 | 218 | 3.56 | 1.34 | 3.50 | 3.55 | 1.48 | 1.00 | 6.50 | 5.50 | 0.04 | -0.87 | 0.09 |
| emotion | 20 | 218 | 3.80 | 1.13 | 3.80 | 3.79 | 1.11 | 1.00 | 6.90 | 5.90 | 0.09 | -0.22 | 0.08 |
| behavioural_intentions | 21 | 218 | 5.08 | 1.11 | 5.25 | 5.15 | 1.19 | 1.75 | 7.00 | 5.25 | -0.58 | -0.17 | 0.08 |
| reaction | 22 | 218 | 4.23 | 1.91 | 5.00 | 4.31 | 1.48 | 1.00 | 7.00 | 6.00 | -0.27 | -1.37 | 0.13 |
Chart 1 supports Table 1, showing correlation coefficients between each variable.
2. Stepwise regression
Prepared and implemented three methods of stepwise regression (forward selection, backward elimination and bidirectional elimination). The three methods produced two different but similar models. Reviewed the parameters for both models and compared model fit.
2.1 Model 1
Table 2 summarises the parameters for Model 1, derived from forward selection. The number of explanatory variables reduced from 21 in Table 1 to seven, shown in Table 2. Table 2 p-values show that the explanatory variables isolation of affect, avoidance, emotion and behavioural intentions are statistically significant. The coefficient of determination is also significant, with an adjusted R2 of 0.718.
| Table 2 Parameters for Model 1 | |||||
| Parameter | Coefficient | SE | 95% CI | t(210) | p |
|---|---|---|---|---|---|
| (Intercept) | -3.82 | 0.51 | (-4.82, -2.82) | -7.53 | < .001 |
| behavioural intentions | 1.05 | 0.09 | (0.87, 1.23) | 11.66 | < .001 |
| emotion | 0.57 | 0.09 | (0.39, 0.74) | 6.41 | < .001 |
| avoidance | 0.26 | 0.08 | (0.10, 0.42) | 3.14 | 0.002 |
| isolation of affect | 0.11 | 0.05 | (0.01, 0.21) | 2.20 | 0.029 |
| denial | -0.15 | 0.08 | (-0.30, 1.55e-03) | -1.95 | 0.052 |
| fears failure | 0.08 | 0.05 | (-8.75e-03, 0.17) | 1.78 | 0.076 |
| reaction formation | -0.09 | 0.06 | (-0.20, 0.02) | -1.56 | 0.120 |
| Model: reaction ~ behavioural_intentions + emotion + avoidance + isolation_of_affect + denial + fears_failure + reaction_formation (218 Observations) Residual standard deviation: 1.014 (df = 210) R2: 0.727; adjusted R2: 0.718 |
|||||
Chart 2 plots important explanatory variables for Model 1.
The equation derived for Model 1 follows.
\[ \small \begin{aligned} \operatorname{\widehat{reaction}} &= -3.82 + 1.05(\operatorname{behavioural\_intentions})\ + \\ &\quad 0.57(\operatorname{emotion}) + 0.26(\operatorname{avoidance})\ + \\ &\quad 0.11(\operatorname{isolation\_of\_affect}) - 0.15(\operatorname{denial})\ + \\ &\quad 0.08(\operatorname{fears\_failure}) - 0.09(\operatorname{reaction\_formation}) \end{aligned} \]
2.2 Model 2
Table 3 summarises the parameters for Model 2, derived from backward elimination and bidirectional elimination. The number of explanatory variables reduced from 21 in Table 1 to 11 in Table 3. Table 3 p-values show that the explanatory variables reaction formation, anxious thoughts, isolation of affect, avoidance, emotion and behavioural intentions are statistically significant. The coefficient of determination is also significant, with an adjusted R2 of 0.725.
| Table 3 Parameters for Model 2 | |||||
| Parameter | Coefficient | SE | 95% CI | t(206) | p |
|---|---|---|---|---|---|
| (Intercept) | -3.85 | 0.62 | (-5.08, -2.63) | -6.23 | < .001 |
| fears failure | 0.08 | 0.05 | (-0.02, 0.17) | 1.50 | 0.134 |
| catastrophising | -0.10 | 0.07 | (-0.23, 0.03) | -1.47 | 0.142 |
| managing feelings | 0.12 | 0.07 | (-0.02, 0.26) | 1.71 | 0.089 |
| anxious thoughts | 0.14 | 0.07 | (1.95e-03, 0.27) | 2.00 | 0.047 |
| avoidance | 0.23 | 0.09 | (0.06, 0.39) | 2.65 | 0.009 |
| facing reality | -0.10 | 0.06 | (-0.22, 0.02) | -1.69 | 0.092 |
| isolation of affect | 0.11 | 0.05 | (0.02, 0.21) | 2.28 | 0.023 |
| reaction formation | -0.11 | 0.06 | (-0.23, -7.19e-04) | -1.98 | 0.049 |
| denial | -0.12 | 0.08 | (-0.28, 0.03) | -1.57 | 0.118 |
| emotion | 0.58 | 0.10 | (0.39, 0.77) | 6.10 | < .001 |
| behavioural intentions | 1.04 | 0.09 | (0.86, 1.21) | 11.50 | < .001 |
| Model: reaction ~ fears_failure + catastrophising + managing_feelings + anxious_thoughts + avoidance + facing_reality + isolation_of_affect + reaction_formation + denial + emotion + behavioural_intentions (218 Observations) Residual standard deviation: 1.002 (df = 206) R2: 0.739; adjusted R2: 0.725 |
|||||
Chart 3 plots important explanatory variables for Model 2.
The equation derived for Model 2 follows.
\[ \small \begin{aligned} \operatorname{\widehat{reaction}} &= -3.85 + 0.08(\operatorname{fears\_failure})\ - \\ &\quad 0.1(\operatorname{catastrophising}) + 0.12(\operatorname{managing\_feelings})\ + \\ &\quad 0.14(\operatorname{anxious\_thoughts}) + 0.23(\operatorname{avoidance})\ - \\ &\quad 0.1(\operatorname{facing\_reality}) + 0.11(\operatorname{isolation\_of\_affect})\ - \\ &\quad 0.11(\operatorname{reaction\_formation}) - 0.12(\operatorname{denial})\ + \\ &\quad 0.58(\operatorname{emotion}) + 1.04(\operatorname{behavioural\_intentions}) \end{aligned} \]
3. Compare models
Table 4 compares the results of both models side-by-side.
| Model 1 (Response: reaction) | Model 2 (Response: reaction) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Predictors | Estimates | std. Beta | CI | standardized CI | p | Estimates | std. Beta | CI | standardized CI | p |
| (Intercept) | -3.82 *** | 0.00 | -4.82 – -2.82 | -0.07 – 0.07 | <0.001 | -3.85 *** | 0.00 | -5.08 – -2.63 | -0.07 – 0.07 | <0.001 |
| behavioural intentions | 1.05 *** | 0.61 | 0.87 – 1.23 | 0.51 – 0.72 | <0.001 | 1.04 *** | 0.60 | 0.86 – 1.21 | 0.50 – 0.71 | <0.001 |
| emotion | 0.57 *** | 0.33 | 0.39 – 0.74 | 0.23 – 0.44 | <0.001 | 0.58 *** | 0.34 | 0.39 – 0.77 | 0.23 – 0.45 | <0.001 |
| avoidance | 0.26 ** | 0.13 | 0.10 – 0.42 | 0.05 – 0.22 | 0.002 | 0.23 ** | 0.12 | 0.06 – 0.39 | 0.03 – 0.20 | 0.009 |
| isolation of affect | 0.11 * | 0.09 | 0.01 – 0.21 | 0.01 – 0.16 | 0.029 | 0.11 * | 0.09 | 0.02 – 0.21 | 0.01 – 0.17 | 0.023 |
| denial | -0.15 | -0.08 | -0.30 – 0.00 | -0.17 – 0.00 | 0.052 | -0.12 | -0.07 | -0.28 – 0.03 | -0.15 – 0.02 | 0.118 |
| fears failure | 0.08 | 0.07 | -0.01 – 0.17 | -0.01 – 0.14 | 0.076 | 0.08 | 0.06 | -0.02 – 0.17 | -0.02 – 0.14 | 0.134 |
| reaction formation | -0.09 | -0.06 | -0.20 – 0.02 | -0.13 – 0.01 | 0.120 | -0.11 * | -0.07 | -0.23 – -0.00 | -0.15 – -0.00 | 0.049 |
| catastrophising | -0.10 | -0.07 | -0.23 – 0.03 | -0.16 – 0.02 | 0.142 | |||||
| managing feelings | 0.12 | 0.08 | -0.02 – 0.26 | -0.01 – 0.18 | 0.089 | |||||
| anxious thoughts | 0.14 * | 0.08 | 0.00 – 0.27 | 0.00 – 0.16 | 0.047 | |||||
| facing reality | -0.10 | -0.08 | -0.22 – 0.02 | -0.16 – 0.01 | 0.092 | |||||
| Observations | 218 | 218 | ||||||||
| R2 / R2 adjusted | 0.727 / 0.718 | 0.739 / 0.725 | ||||||||
| AIC | 634.648 | 633.144 | ||||||||
|
||||||||||
Output 1 compares the fit statistics for Models 1 and 2. Both models produced very similar results. Therefore, either model could be selected depending on requirements. It could be argued that Model 1 is preferred because it achieves this close result with fewer explanatory variables and has the lower BIC. It could be argued Model 2 is preferred because it has a lower AIC and higher adjusted R2.
Output 1 Model fit statistics
$Models
Formula
1 "reaction ~ behavioural_intentions + emotion + avoidance + isolation_of_affect + denial + fears_failure + reaction_formation"
2 "reaction ~ fears_failure + catastrophising + managing_feelings + anxious_thoughts + avoidance + facing_reality + isolation_of_affect + reaction_formation + denial + emotion + behavioural_intentions"
$Fit.criteria
Rank Df.res AIC AICc BIC R.squared Adj.R.sq p.value Shapiro.W
1 8 210 634.6 635.5 665.1 0.7270 0.7179 1.028e-55 0.9925
2 12 206 633.1 634.9 677.1 0.7386 0.7246 6.128e-54 0.9947
Shapiro.p
1 0.3354
2 0.6461Another test for comparing regression models is the analysis of variance (ANOVA). ANOVA determines which model has the best parsimonious fit. Output 2 is an ANOVA comparison of both models. With four additional explanatory variables, the p-value for Model 2 is not less than 0.05. Therefore it is inferred that Model 1, derived from forward selection, has a better fit for this data set.
Output 2 ANOVA model comparison
Analysis of Variance Table
Model 1: reaction ~ behavioural_intentions + emotion + avoidance + isolation_of_affect +
denial + fears_failure + reaction_formation
Model 2: reaction ~ fears_failure + catastrophising + managing_feelings +
anxious_thoughts + avoidance + facing_reality + isolation_of_affect +
reaction_formation + denial + emotion + behavioural_intentions
Res.Df RSS Df Sum of Sq F Pr(>F)
1 210 216.00
2 206 206.79 4 9.2144 2.2948 0.0605 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 14. Supplementary analysis
Model 1 was assessed as the preferred model. The remainder of this vignette provides supplementary analysis.
4.1 Model assumptions
Chart 4 checks assumptions for Model 1. This includes the model’s linearity, homogeneity, outliers, multicollinearity and residuals. Despite variable labels overlapping in the collinearity chart, collinearity levels are low in this model.
Chart 4 Model 1 assumptions
4.2 Model correlations
Chart 5 summarises the correlation coefficients for Model 1.
Chart 6 visualises correlated relationships between the variables. Highly correlated variables, either positively or negatively, appear closer together and are joined by a stronger path.
4.3. Visualise relationships
Chart 7 visualises the relationship between the two most significant explanatory variables and the response variable.
4.4. Visualise relationship with categorical variables
Finally, Chart 8 visualises the relationship between behavioural intentions and reaction segmented by categorical variables.
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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## RColorBrewer 1.1-3 2022-04-03 [1] CRAN (R 4.4.0)
## rcompanion * 2.4.36 2024-05-27 [1] CRAN (R 4.4.0)
## Rcpp 1.0.13 2024-07-17 [1] CRAN (R 4.4.1)
## readr * 2.1.5 2024-01-10 [1] CRAN (R 4.4.0)
## readxl 1.4.3 2023-07-06 [1] CRAN (R 4.4.0)
## remotes 2.5.0 2024-03-17 [1] CRAN (R 4.4.0)
## rlang 1.1.4 2024-06-04 [1] CRAN (R 4.4.0)
## rmarkdown 2.27 2024-05-17 [1] CRAN (R 4.4.0)
## rootSolve 1.8.2.4 2023-09-21 [1] CRAN (R 4.4.0)
## rprojroot 2.0.4 2023-11-05 [1] CRAN (R 4.4.0)
## rstatix 0.7.2 2023-02-01 [1] CRAN (R 4.4.0)
## rstudioapi 0.16.0 2024-03-24 [1] CRAN (R 4.4.0)
## sandwich 3.1-0 2023-12-11 [1] CRAN (R 4.4.0)
## sass 0.4.9 2024-03-15 [1] CRAN (R 4.4.0)
## scales 1.3.0 2023-11-28 [1] CRAN (R 4.4.0)
## see * 0.8.5 2024-07-17 [1] CRAN (R 4.4.1)
## sessioninfo 1.2.2 2021-12-06 [1] CRAN (R 4.4.0)
## shiny 1.8.1.1 2024-04-02 [1] CRAN (R 4.4.0)
## sjlabelled 1.2.0 2022-04-10 [1] CRAN (R 4.4.0)
## sjmisc 2.8.10 2024-05-13 [1] CRAN (R 4.4.0)
## sjPlot * 2.8.16 2024-05-13 [1] CRAN (R 4.4.0)
## sjstats 0.19.0 2024-05-14 [1] CRAN (R 4.4.0)
## stringi 1.8.4 2024-05-06 [1] CRAN (R 4.4.0)
## stringr * 1.5.1 2023-11-14 [1] CRAN (R 4.4.0)
## survival 3.5-8 2024-02-14 [2] CRAN (R 4.4.0)
## svglite 2.1.3 2023-12-08 [1] CRAN (R 4.4.0)
## systemfonts 1.1.0 2024-05-15 [1] CRAN (R 4.4.0)
## TH.data 1.1-2 2023-04-17 [1] CRAN (R 4.4.0)
## tibble * 3.2.1 2023-03-20 [1] CRAN (R 4.4.0)
## tidyr * 1.3.1 2024-01-24 [1] CRAN (R 4.4.0)
## tidyselect 1.2.1 2024-03-11 [1] CRAN (R 4.4.0)
## tidyverse * 2.0.0 2023-02-22 [1] CRAN (R 4.4.0)
## timechange 0.3.0 2024-01-18 [1] CRAN (R 4.4.0)
## 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)
## viridisLite 0.4.2 2023-05-02 [1] CRAN (R 4.4.0)
## withr 3.0.0 2024-01-16 [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)
## zoo 1.8-12 2023-04-13 [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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