Prediction Confidence Intervals, Cross- validation, and Predictor Selection

Skill Set Why is the confidence interval for an individual point larger than for the regression line? Describe the steps in forward (backward, stepwise, blockwise, all possible regressions) predictor selection. What is cross- validation? Why is it important? What are the main problems as far as R- square and prediction are concerned with forward (backward, stepwise, blockwise, all possible regressions)

Prediction v. Explanation Prediction is important for practice –WWII pilot training Ability tests, e.g., eye-hand coordination Built an airplane that flew Fear of heights Favorite flavor ice cream –Age and driving accidents Explanation is crucial for theory. Highly correlated vbls may not help predict, but may help explain. Team outcomes as function of team resources and team backup.

Confidence Intervals CI for the line, i.e., the mean score: =MSR. N= sample size. The df are for MSR (variance of residuals). CI for a single person’s score: Note shape.

Computing Confidence Intervals Suppose: Find CI for line (mean) at X=1. df = N-k-1 = = 18. CI =.29 to For an individual at X=1, what is the CI? CI = 3.81 to 7.79

Review Why is the confidence interval for the individual wider than a similar interval for the regression line? Why are the confidence intervals regression curved instead of being straight lines?

Shrinkage R 2 is biased (sample value is too large) because of capitalizing on chance to minimize SSe in sample. If the population value of R 2 is zero, the expected value in the sample is R 2 =k/(N-1) where k is the number of predictors and N is the number of people in the sample. If you have many predictors, you can make R 2 as large as you want. What is the expected value of R-square if N = 101 and k =10? Ethical issue here. Common adjustment or shrinkage formula: This is reported by SAS (PROC REG) under ‘Adj R-Sq.’ Adjusts for both k and N and size of initial R 2.

Shrinkage Examples Suppose R 2 is.6405 with k = 4 predictors and a sample size of 30. Then R 2 =.6405 N= Adj R R 2 =.30 N= Adj R Note small N means lots of shrinkage but also smaller initial R 2 shrinks more.

Cross-Validation Compute a and b(s) (can have one or more IVs) on initial sample. Find new sample, do not estimate a and b, but use a and b to find Y’. Compute correlation between Y and Y’ in new sample; square. Ta da! Cross- validation R 2. Cross-validation R 2 does not capitalize on chance and estimates operational R 2.

Cross-validation (2) Double cross-validation Data splitting Expert judgment weights (don’t try this at home) Math Estimates Fixed: Random:

Review What is shrinkage in the context of multiple regression? What are the things that affect the expected amount of shrinkage? What is cross-validation? Why is it important?

Predictor Selection Widely misunderstood and widely misused. Algorithms labeled forward, backward, stepwise, etc. NEVER use for work involving theory or explanation (hint: this clearly means your thesis and dissertation). NEVER use for estimating importance of variables. Use SOLELY for economy (toss predictors).

All Possible Regressions GPA (Y) GREQGREVMATAR GPA (Y) 1 GREQ.6111 GREV MAT AR Mean S.D Data from Pedhazur example. GPA is grade point average. GREQ is Graduate Record Exam, Quantitative. GREV is GRE Verbal. MAT is Miller Analogies Test. AR is Arithmetic Reasoning test.

All Possible Regressions (2) kR2R2 Variables in Model 1.385AR 1.384GREQ 1.365MAT 1.338GREV 2.583GREQ MAT 2.515GREV AR 2.503GREQ AR 2.493GREV MAT 2.492MAT AR 2.485GREQ GREV 3.617GREQ GREV MAT 3.610GREQ MAT AR 3.572GREV MAT AR 3.572GREQ GREV AR 4.640GREQ GREV MAT AR Note how easy it is to choose the model with the highest R 2 for any given number of predictors. In predictor selection, you also need to worry about cost. You get both V and Q GRE in one test. Also consider what change in R 2 means. Accuracy in prediction of dropout.

Predictor Selection Algorithms Forward – build up from start with p value. End when no variables meet PIN. May include duds. Backward – Start with all vbls and pull out with POUT. May lose gems. Stepwise – Start forward, check backward at each step. Not guaranteed to give best R 2. Blockwise – not used much. Forward by blocks, then any method (eg stepwise) within block to choose best predictors.

Things to Consider in PS Algorithms consider statistical significance, but you have to consider practical significance and cost, i.e., algorithms don’t work well. Surviving variables are often there by chance. Do the analysis again and you would choose a different set. OK for prediction. The value of correlated variables is quite different when considered in path analysis and SEM.

Hierarchical Regression Alternative to predictor selection algorithms Theory based (a priori) tests of increments to R-square

Example of Hierarchical Reg Does personality increase prediction of med school success beyond that afforded by cognitive ability? Collect data on 250 med students for first two years. Model 1: Model 2 R 2 =.10, p<.05 R 2 =.13, p<.05 Model test: F(2,245)=4.22, p <.05

Review Describe the steps in forward (backward, stepwise, blockwise, all possible regressions) predictor selection. What are the main problems as far as R- square and prediction are concerned with forward (backward, stepwise, blockwise, all possible regressions) Why avoid predictor selection algorithms when doing substantive research (when you want to explain variance in the DV)?