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Some assumptions are needed on the ANOVA F test. Assume that both the response and residual plots look good. It is crucial that there are no outliers. Then a rule of thumb is that if n − p is large, then the ANOVA F test p–value is approximately correct. An analogy can be made with the central limit theorem, Y is a good estimator for µ if the Yi are iid N(µ, σ 2 ) and also a good estimator for µ if the data are iid with mean µ and variance σ 2 if n is large enough. More on the robustness and lack of robustness of the ANOVA F test can be found in Wilcox (2005).

19) can be used to motivate the test for whether the reduced model can be used instead of the full model. Similarly, the sufficient predictor can be used to unify the interpretation of coefficients and to explain models that contain interactions and factors. , xp are held fixed. Denote a model by SP = α + β T x = α + β1x1 + · · · + βpxp . , p. ∂xi CHAPTER 1. INTRODUCTION 26 Of course, holding all other variables fixed while changing xi may not be possible. For example, if x1 = x, x2 = x2 and SP = α + β1x + β2x2 , then x2 can not be held fixed when x1 increases by one unit, but d SP = β1 + 2β2x.

Then as a 1D regression model, log(Y )|SP = α + SP + e. The parameters are again estimated by maximum likelihood and the survival function is Sx (t) ≡ SY |x (t) = S0 t exp(β TA x) , and Sˆx (t) = Sˆ0 t ˆ T x) exp(β A ˆ and σˆ . 6 Variable Selection A standard problem in 1D regression is variable selection, also called subset or model selection. , xp−1)T are the p − 1 nontrivial predictors and that (1, x)T has full rank. Then variable selection is a search for a subset of predictor variables that can be deleted without important loss of information.

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