Psychology 202b Advanced Psychological Statistics, II

1. Psychology 202bAdvanced Psychological Statistics, II February 15, 2011

2. Overview • Confidence intervals for individual predictions • Transformations • An introduction to the method of maximum likelihood estimation • The Box-Cox procedure for selecting an optimal transformation.

3. Predicting for individuals • Recall that the MSe in a regression represents the variance of deviations from the conditional mean. • Individuals deviate from the conditional mean.

4. Predicting for individuals (cont.) • Therefore, to quantify uncertainty about individuals, we will need two components: • Uncertainty about the conditional mean itself • Uncertainty about how far from the conditional mean the individual falls.

5. Predicting for individuals (cont.) • Hence the variance of the sampling distribution of predicted individual values will be the variance of the conditional mean PLUS the MSe. • Illustration in R. • Confidence intervals for individual prediction in SAS.

6. Transformations • Controversial • Guidelines for when to use: • When the data set is very large and a transformation seems necessary • When there is a theoretical reason to expect that a transformation will be necessary (e.g., proportions, response times).

7. Transformations (cont.) • One common purpose for transformations is to correct heteroscedasticity. • Transformations that are known to correct particular forms of heteroscedasticity are called “variance stabilizing transformations.”

8. Variance stabilizing transformations • If error variance is proportional to the conditional mean of Y, the square root of Y will stabilize the variance. • If error variance is proportional to the square of the conditional mean, the log of Y will stabilize the variance. • If error variance is proportional to the conditional mean to the fourth power, the negative reciprocal will stabilize the variance.

9. Variance stabilizing transformations (cont.) • If the variance is proportional to the conditional mean  (1 – the conditional mean), the arcsin of the square root of Y will stabilize the variance. • Note that if Y is a proportion, the error variance is known to be proportional to the conditional mean  (1 – the conditional mean). • (There are better ways to approach modeling proportions.)

10. Variance stabilizing transformations (cont.) • Illustration of variance stabilizing transformations in R. • An example of a transformation in SAS.