The following data are from a study of the colony-forming activity of six bacterial strains (only strain 3 reported here) under exposure to three pH levels (4.5, 6.5, 8.5) and three concentrations of...

The following data are from a study of the colony-forming activity of six bacterial strains (only strain 3 reported here) under exposure to three pH levels (4.5, 6.5, 8.5) and three concentrations of chlorine dioxide CLO₂
in phosphate buffer (20, 50, 80 ppm). (Chlorine dioxide is important in sanitation for controlling bacterial growth.) After suspension of bacteria in the solutions, colony counts were taken on samples from the solutions at recorded time intervals. Use Y = ln(count) in all analyses. (The data from Vipa Hemstapat, North Carolina State University, used with permission.)

(a) Characterize the response of the bacterial strain to CLO₂
for each of the nine pH × CLO₂
combinations by fitting the Weibull model using Y = ln(count) as the dependent variable and time as the independent variable. You should get convergence in all cases with reasonable starting values; try α = 20, δ = 20, and γ = 2. Summarize your results with a 3 × 3 table of the estimates of the parameters.

(b) Verify algebraically that the time to 50% decline in the colony is estimated by

Use your fitted Weibull response curves to estimate t50 in each case. Do an analysis of variance of the 3 × 3 table of “times to 50% count.” (You do not have an estimate of error with which to test the main effects of concentration and pH, but the analysis will show the major patterns.) Summarize the results.

(c) The nonlinear function of interest in Part (b) is t50. Use the Wald procedure to find the approximate standard error and 95% confidence interval estimate of t50 for the middle cell of your 3 × 3 table. You will have to obtain the partial derivatives of t50 with respect to the three parameters and recover the variance–covariance matrix for θ, and then use these results.