Consider models each with two predictors, temperature and white blood count (WBC), for which temperature is always assumed to be linearly related to the appropriate property of the response, and WBC may or may not be linear (depending on the particular model you formulate for each question). Test:

a. H0 : WBC is not associated with the response versus Ha : WBC is linearly associated with the property of the response.

b. H0 : WBC is not associated with Y versus Ha : WBC is quadratically associated with Y . Also write down the formal test of linearity against this quadratic alternative.

c. H0 : WBC is not associated with Y versus Ha : WBC related to the property of the response through a smooth spline function; for example, for WBC the model requires the variables WBC, WBC

, and WBC where WBC and WBC represent nonlinear components (if there are four knots in a restricted cubic spline function). Also write down the formal test of linearity against this spline function alternative.

d. Test for a lack of fit (combined nonlinearity or non-additivity) in an overall model that takes the form of an interaction between temperature and WBC, allowing WBC to be modeled with a smooth spline function.