Refer to Table 5.9. Apply exact logistic regression to the model discussed in Section 5.7.3 for Cephalexin Use (C), Age (A), Length of Stay (L), and Diarrhea Outcome (D).

a. Obtain an exact P-value for testing the hypothesis of no C effect against the alternative of a positive effect, and construct a 95% "exact" confidence interval for the conditional C-D odds ratio.

b. Construct the four partial tables relating C to D, for the combinations of levels of (A,L). Note that three of the tables have no data for the row with C = 1. For the sole partial table having observations at both levels of C, find a 95% "exact" confidence interval for the odds ratio, and find an exact one-sided P-value. Compare to results using the entire data set. Comment about the contribution to inference of tables having only a single positive row total or a single positive column total.

c. Obtain the ordinary ML fit of the logistic regression model. To investigate the sensitivity of the estimated C effect, find the change in the estimate and standard error after adding one observation to the data set, a case with no diarrhea when (C, A,L) = (1, 1, 1).