Multiple Linear Regression Introduction

Multiple Regression One continuous Y, two or more X variables. X variables may be continuous or dichotomous k groups may be represented by k-1 dichotomous dummy variables

Obtain Weights for the X Variables Create a weighted combination of the Xs Such that the correlation between Y and is as large as possible. That is, a is predicted Y when all Xs are zero b i is number of points Y changes for each one point change in X i, above and beyond the effect of all other predictors.

Standardized (Beta) Weights  i is the number of standard deviations that Y i changes for each standard deviation change in X i, above and beyond the effect of all other predictors.

Sequential Analysis The predictors may be entered into the model all at once (simultaneous), or In sets of one or more (sequential) Order of entry may be determined by –Temporal relationships among predictors –A causal model –Economic considerations –Other considerations

Economic Considerations Want to predict college GPA. Enter inexpensive predictors first –High school GPA –Verbal and quantitative SAT –Evaluation of an essay submitted by student –Ratings from a panel of professors who interviewed the student on campus.

Stepwise Selection A statistical algorithm is used to determine order of entry. The goal is to create a model that has fewer predictors but does nearly as well as a model with all predictors. Stepwise selection is among the most misunderstood analyses known to man. It commonly leads to inappropriate conclusions.

Who Will Fail College Physics? McCammon, S., Golden, J., & Wuensch, K. L. (1988) Predict grades in physics classes from –Critical Thinking test scores (CT) –Thurstone’s Primary Mental Abilities Test (IQ) –Arithmetic skills test scores (ARI) –Algebra skills test scores (ALG) –Math anxiety scale scores (ANX)

Simultaneous Analysis R is the correlation between the weighted predictors and Y R =.40 and was statistically significant. Model explained 16% of the variance in grades. Every predictor was sig. correlated with grades (zero-order r). But in the model only ALG and CT had significant unique effects.

Stepwise Analysis Tried both Forwards Selection and Backwards Selection Both led to a model with only ALG and CT. We recommended that Physics use just the ALG and CT tests to predict who is at risk of failing. The motivation for using stepwise was economic – why use 5 predictors when 2 will do as well?

Does Sex Matter? McCammon insisted that I address this issue. Means and variances differed little between the sexes. Just to please McCammon, I did the analysis separately for men and women.

Sex Matters Among the men, not a single predictor was significantly related to grades. Among the women, every predictor was significantly related to grades. Women’s performance is class is well related to their abilities. There must be some other more important factor for predicting men’s performance.

Expert Reviewers Those at the physics journal to which we submitted the manuscript rejected it. They argued that it was not appropriate to publish an unexpected finding (the sex difference). Such “hypothesis-induced blindness” is not all that uncommon, unfortunately.

Political Correctness We submitted the manuscript to a Science Education journal. One reviewer insisted that it not be published as it is “sexist” to compare the sexes. We convinced the editor otherwise.