# How to interpret the result of a bootstrapped multiple regression?

My data didn't have normally distributed residuals, so I have used bootstrapping to run a multiple regression analysis.

My model is the following: independent variables A and B predict the dependent variable C.

Variable A and B are the average scores on multiple Likert items used to assess personality traits, variable C is the average score on multiple Likert items used to assess life satisfaction.

The results of my analysis in JASP are the following:

Model Summary - C
Model R Adjusted R² RMSE R² Change F Change df1 df2 p
H_0 0.000 0.000 0.000 1.636 0.000 0 414
H_1 0.145 0.021 0.016 1.622 0.021 4.438 2 412 0.012
Coefficients
95% CI
Model Unstandardized Standard Error Standardized t p Lower Upper
H_0 (Intercept) 6.158x10^-16 0.080 7.669×10-15 1.000 -0.158 0.158
H_1 (Intercept) 5.594x10^-16 0.080 7.024x10^-15 1.000 -0.157 1.57
B 0.023 0.093 0.013 0.246 0.806 -0.160 0.205
A -0.283 0.098 -0.149 -2.900 0.004 -0.475 -0.091
Bootstrap Coefficients
95% CI
Model Unstandardized Bias Standard Error p* Lower Upper
H_0 (Intercept) 0.002 0.002 0.080 0.974 -0.169 0.147
H_1 (Intercept) 0.002 6.162x10^-4 0.080 0.981 -0.166 0.148
B 0.023 -0.001 0.084 0.780 -0.147 0.185
A -0.278 0.005 0.096 0.002 -0.476 -0.099

I am confused on how to interpret these results and how to report them. If there is any part of the results missing that I should also provide, let me know.

What I gather is the following:

Model Summary:

• Without Bootstrapping, my model explains 2% of variance (R²) with a statistically significant p-value of 0.012.

Coefficients:

• The impact of B is negligible (confidence interval includes 0, p-value is >.05)
• A's impact is significant (confidence interval does not include 0, p-value is <.05)

Bootstrap Coefficients:

• Essentially the same as in Coefficients, but with slightly different values.

Do I understand it correctly that the Bootstrap Coefficients table essentially just replaces the Coefficients table, that my bootstrapped model explains 2% of the variance, and that I can go ahead and report as I would report a normal multiple regression analysis using the values in Model Summary - C and Bootstrap Coefficients, with the added information that it's the result of a 5000 replicates bootstrapping?

• Welcome to Cross Validated! There's much positive to be said for bootstrapping as a way to validate a model-building scheme. I was wondering, however, just what you meant by saying "My data was not normally distributed, so I have used bootstrapping to run a multiple regression analysis." Was it that the outcome values themselves weren't normally distributed, or that the residuals around the model predictions weren't normally distributed? Normality of the outcome values isn't necessary at all; see this page among others.
– EdM
Commented Apr 28 at 18:06
• You are right that bootstrapping didn't change much. But see @EdM's comment. Also, "negligible" is not synonymous with "not significant". Nor is Since you haven't told us what A, B, C, $H_0$ or $H_1$ are, or how they are measured, it's hard to say more. However, the fact that $R^2$ is only 2% and that nothing changed much in the bootstrap model is some sign that neither A nor B did much. A is significant, but it appears to be small (although, again, we'd need to know what it is and how it's measured to really say). Commented Apr 28 at 18:37
• Thank you. I have updated the post (I was indeed talking about the residuals, I ran the Shapiro Wilk test which was highly significant, and the QQ-Plot had a significant s-shape) and added some information about the variables. I did not define H_0 and H_1 in JASP. I know I can do a hierarchical regression by telling it to add model terms to the null model, but I didn't do that. So I unfortunately don't know how to say more about H_0/H_1.
– sam
Commented Apr 28 at 20:02

All that said, Peter From summarized the results pretty well in his comment. There might be a "statistically significant" association between A and outcome, but assessing its practical significance depends on applying your knowledge of the subject matter: what your measurements of predictors and outcomes represent in practice.
Finally, with this number of observations you might consider a more complex model. First, your model assumes that each of A and B has a strictly linear association with outcome. It can be helpful to relax that assumption by fitting those predictors flexibly, as with a regression spline. Second, you might include an interaction term between predictors A and B, to relax the implicit assumption of your model that the association of each predictor with outcome is independent of the level of the other.