Chromosome aberration assays are used to determine whether or not a substance induces structural changes in chromosomes

Week 5 Homework

Homework Problem 5.1 (25 points) Chromosome aberration assays are used to determine whether or not a substance induces structural changes in chromosomes. One study compared the results of two substances at various doses. A large number of cells were sampled at each dose to see how many were aberrant. 

(1)     Fit a binomial GLM to determine if there is evidence of a difference between the two substances.

(2)     Use the dose and the logarithm of dose as an explanatory variable in separate GLMs, and compare. Which is better, and why?

(3)     Compute the 95% confidence interval for the dose regression parameter, and interpret.

(4)     Why would estimation of the ED50 be inappropriate?

Homework Problem 5.2 (30 points) A study of the habitats of the noisy miner (a small but aggressive native Australian bird; data set: nminer) recorded whether noisy miners were present in various two hectare transects in buloke woodland patches (Miners), and considered the following potential explanatory variables: the number of eucalypt trees (Eucs); the number of buloke trees (Bulokes); the area of contiguous remnant patch vegetation in which each site was located (Area); whether the area was grazed (Grazed: 1 means yes); whether shrubs were present in the transect (Shrubs: 1 means yes); and the number of pieces of fallen timber (Timber). Fit a suitable logistic regression model for predicting the presence of noisy miners in two hectare transects in buloke woodland patches. You do not need to present the detailed calculations.

(1)     Find 𝛽̂ , 𝑠𝑒(𝛽̂ ) and 95% Wald confidence interval of 𝛽 . Interpret your finding about 𝛽 .

(2)     Perform the diagnostic analysis. You should include: plot of the quantile residuals against the fitted values transformed to the constant-information scale; plot of the working responses against the linear predictors; the Q-Q plot of the quantile residuals; plot of the Cook’s distance 𝐷 and determine if there are any outliers and/or influential observations.

Homework Problem 5.3 (25 points) A study of seed germination used two types of seeds and two types of root stocks. Use the proportion of seeds germinating as 𝑦 and Seeds and Extract as the explanatory variables. Fit a logistic regression model using the data germ. Then fit a Bernoulli GLM with the logit link using data germBin which contains record whether or not each individual seed germinates.

(1)     Show that both the Bernoulli and binomial glms produce the same values for the parameter estimates and standard errors.

(2)     Show that the two models produce different values for the residual deviance, but the same values for the null deviance.

(3)     Show that the two models produce similar results from the sequential likelihood-ratio tests.

(4)     Compare the log-likelihoods for the binomial and Bernoulli distributions. Comment.

(5)     Explain why overdispersion cannot be detected in the Bernoulli model.

 Homework Problem 5.4 (30 points) The times to death (in weeks) of two groups of leukemia patients whose white blood cell counts were measured (data set: leukwbc) were grouped according to a morphological variable called the ag factor.

(1)     Plot the survival time against white blood cell count (WBC), distinguishing between ag- positive and ag-negative patients. Comment on the relationship between WBC and survival time, and the ag factor.

(2)     Plot the survival time against 𝑙𝑜𝑔 10(𝑊𝐵𝐶), and argue that using 𝑙𝑜𝑔 10(𝑊𝐵𝐶) is likely to be a better choice as an explanatory variable.

(3)     Fit a GLM(Gamma; log) model to the data, including the interaction term between the ag factor and 𝑙𝑜𝑔 10(𝑊𝐵𝐶), and show that the interaction term is not necessary.

(4)     Refit the GLM without the interaction term, and evaluate the model using diagnostic tools.

(5)     Plot the fitted lines for each ag-factor on a plot of the observations.

(6)     The original source uses an exponential distribution, which is a gamma distribution with 𝜙 = 1. Does this seem reasonable?