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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.
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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 andis as large as possible. • That is, • a is predicted Y when all Xs are zero • bi is number of points Y changes for each one point change in Xi, above and beyond the effect of all other predictors.
Standardized (Beta) Weights • i is the number of standard deviations that Yi changes for each standard deviation change in Xi, 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.
