Extension to Multiple Regression

1. Extension to Multiple Regression

2. Simple regression • With simple regression, we have a single predictor and outcome, and in general things are straightforward though issues may arise regarding outliers and violations of assumptions • The basic model is Y = b0 + b1X + e

3. Multiple Regression • Adding more predictors sounds simple enough, and, true enough, the basic model doesn’t change much Y = b0 + b1X + b2X + b3X … + bnX + e • The key idea here is that we are getting a linear combination of the predictors1and assessing the correlation of that combination with the outcome • Squaring that correlation provides our model R2 • However there is much more to deal with

4. Prediction and Explanation • Focus of the analysis can be seen as falling somewhere on two continuums of prediction and explanation • Low Prediction-----------------------------------High Prediction • Low Explanation-----------------------------------High Explanation • Example of high prediction/low explanation • Model R2 must be very strong, exclusive focus on raw coefficients, little concern about variable importance, and actual prediction on a new set of data • Example of high explanation/low prediction • Model R2 can even be weak but at least statistically significant, focus on standardized coefficients and variable importance, little if any validation • Most of psych research tends to fall in low prediction, high explanation • This is not a good thing as it leads to satisfaction with models that may be marginally useful at best

5. Macro Analysis • Model fit • Statistical significance • Amount of variance accounted for in the DV (R2) • Standard error of estimate • The interpretation is no different with the addition of predictors • However, do know that as we have noted, there is never a zero correlation between variables unless forced (e.g. via experimental design) • As such, in practice adding a variable will always increase R2 • A bias-adjusted R2 becomes even more important to report as model complexity increases.

6. Micro Analysis • Raw coefficients • Standardized coefficients • As if we ran the model after standardizing our predictors first • This puts them and their subsequent coefficients on the same scale for easier interpretation • However, just because one is larger than another doesn’t mean it is statistically or meaningfully so • Interval estimates for coefficients • Measures of variable importance • E.g. semi-partial correlation

7. Problems • Issues with outliers, violations of the assumptions, and overfitting remain and the integrity of the model must be thoroughly examined • Furthermore, one must be on the look out for things like collinearity (high correlations among predictors) and suppression (unusual coefficients due to predictor-DV relationships)

8. Multiple Approaches • Sequential (hierarchical) regression • Exploratory (stepwise) regression • Tests of interactions (moderation) • Mediating effects • Model comparison

9. Summary • Most of the regressions you see will exhibit these types of problems: • Focus on explanation at the expense of prediction • Lack of or inadequately tested assumptions • Lack of bias-adjusted model fit indices • Lack of comparison of results to robust regression nor validation of their own model (much less comparison it to other theoretically motivated possible models) • Inadequately examine differences among predictors • Poorly perform exploratory approaches when doing them • Are a product of someone not doing more appropriate and complex analysis (e.g. path analysis) • In short, while commonly used, MR is also usually poorly performed • Try and do better!