Simultaneous Inferences and Other Regression Topics KNNL – Chapter 4

Bonferroni Inequality Application: We want simultaneous Confidence Intervals for b0 and b1 such that we can be (1-a)100% confident that both intervals contain true parameter: A1 ≡ Event that CI for b0 does not cover b0 A2 ≡ Event that CI for b1 does not cover b1 Then: The probability that both intervals are correct is ≥ 1-2a Thus, if we construct (1-(a/2))100% CIs individually, Pr{Both Correct} ≥ 1-2(a/2) = 1-a

Joint Confidence Intervals for b0 and b1

Simultaneous Estimation of Mean Responses Working-Hotelling Method: Confidence Band for Entire Regression Line. Can be used for any number of Confidence Intervals for means, simultaneously Bonferroni Method: Can be used for any g Confidence Intervals for means by creating (1-a/g)100% CIs at each of g specified X levels

Bonferroni t-table (a = 0.05, 2-sided)

Simultaneous Predictions of New Responses Scheffe’s Method: Widely used method for making simultaneous tests and confidence intervals. Like W-H, based on F-distribution, but does increase with g, the number of simultaneous predictions Bonferroni Method: Can be used for any g Confidence Intervals for means by creating (1-a/g)100% CIs at each of g specified X levels

Regression Through the Origin In some applications, it is believed that the regression line goes through the origin This implies that E{Y|X} = b1X (proportional relation) Note, that if we imply that all Y=0 when X=0, then the variance of Y is 0 when X=0 (not consistent with the regression models we have fit so far) Should only be used if there is a strong theoretical reason Analysis of Variance and R2 interpretation are changed. Should only use t-test for slope

Regression Through the Origin

Measurement Errors Measurement Error in the Dependent Variable (Y): As long as there is not a bias (consistently recording too high or low), no problem (Measurement Error is absorbed into e). Measurement Error in the Independent Variable (X): Causes problems in estimating b1 (biases downward) when the observed (recorded) value is random. See next slide for description. Measurement Error in the Independent Variable (X): Not a problem when the observed (recorded) value is fixed and actual value is random (e.g. temperature on oven is set at 400⁰ but actual temperature is not)

Measurement Error in X (Random)

Inverse Prediction/Calibration

Choice of X Levels Note that all variances and standard errors depend on SSXX which depends on the spacing of the X levels, and the sample size. Depending on the goal of research, when planning a controlled experiments, and selecting X levels, choose: 2 levels if only interested in whether there is an effect and its direction 3 levels if goal is describing relation and any possible curvature 4 or more levels for further description of response curve and any potential non-linearity such as an asymptote