1. Psychology 202b Advanced 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.