FMPH 223 Longitudinal Data Analysis - Spring 2024
Final Exam
You have 3 hours to complete the test. There are ten questions for a total of 100 points. Each question is worth 10 points.
Please return the exam as a .pdf document. No particular formatting is required.
In a randomized, double-blind, parallel-group, multicenter study comparing two oral anti-fungal treatments (Itraconazole and Terbinafine) for toenail infection, patients were evaluated for the degree of onycholysis (separation of the nail plate from the nail bed) at baseline (week 0) and at weeks 4, 8, 12, 24, 36, and 48 thereafter. The onycholysis outcome variable is binary ("none or mild" versus "moderate or severe"). This variable was evaluated on 294 patients, for a total of 1908 measurements. The main objective of the analyses is to compare the effects of the two treatments on changes in the probability of moderate or severe onycholysis over the duration of the study.
The data are in the file Toenail.dat. Each row of the data set contains the following five variables: ID = patient ID; Y = binary onycholysis response, 0 = none or mild, 1 = moderate or severe; Treatment = 1 for Terbinafine (novel drug), 0 for Itraconazole (standard treatment); Month = the exact timing of measurements in months; Index = visit number for each participant, 1-7, corresponding to scheduled visits at 0, 4, 8, 12, 24, 36, and 48 weeks. (Some participants have missing visits.)
Figure 1 displays the observed proportions and log-odds of onycholysis by study week,
for each of the two treatment arms.
Note: the first 36 rows in Toenail.dat include study and data description, and they should be omitted when reading in the dataset - use read . table( . . . , skip=36).
Figure 1: Observed (i) proportions and (ii) log-odds of onycholysis by study week for the two treatment arms in the Onycholysis randomized clinical trial.
1. Consider a generalized linear mixed effects model, with randomly varying intercepts, for the patient-specific log odds of moderate or severe onycholysis. Fit a model with linear trends for the log-odds over time, with common intercept for the two treatment groups, but different slopes:
(M1) logit{E(Yij |bi )} = (β1 + bi ) + β2 Monthij + β3 Treatmenti × Monthij ,
where, given bi , Yij is assumed to have a Bernoulli distribution. Assume that bi ~ N(0,σb(2)).
In the model equation M1 above, what is the interpretation of parameters β2 and β3 ?
2. Based on the results of the M1 model fit, is Itraconazole an effective treatment of onycholysis? Summarize the efficacy of this drug treatment.
3. Based on this model, is Terbinafine effective in treating onycholysis? Summarize the effects of this drug.
4. Figure 1 suggests that the logistic GLME with linear time trend may not be correctly describing the time effect. Consider models M2 and M3, that expand model M1 in the following ways :
(M2) A model including quadratic polynomial time trends
(M3) A model including cubic polynomial time trends
All models include subject-specific random intercepts and suitable time × treatment interactions.
Show that model M2 provides a better fit to the data than the linear time trend model. Use the likelihood ratio test or the Wald test. State the statistical hypothesis being tested.
5. Show that model M3 does not provide a better fit than the quadratic time trend model M2. Use the likelihood ratio test or the Wald test. State the statistical hypothesis being tested.
6. Using M2, is there a significant difference in efficacy between the two treatment arms? State and test an appropriate statistical hypothesis.
7. Figure 1 suggests that the time trend may be suitably modeled by the following model M4, using a linear spline time trend, with a knot at 6 months (approximately week 24):
(M4) logit{E(Yij |bi )} = (β1 + bi ) + β2 Monthij + β3 (Monthij − 6)+ +β4 Treatmenti × Monthij + β5 Treatmenti × (Monthij − 6)+ ,
where x+ = max(x,0) for any real number x; As before, bi ~ N(0,σb(2)) are independent subject-specific random intercepts.
Show that model M4 provides a better fit to the data than models M1 and M2.
8. Based on M4, is there a significant difference in efficacy between the two treatment arms? State and test an appropriate statistical hypothesis.
9. Based on M4, estimate the following quantities, including 95% confidence intervals:
(a) ORI , the odds ratio of onycholysis at one year versus baseline, for a participant in the Itraconazole arm.
(b) ORT , the odds ratio of onycholysis at one year versus baseline, for a participant in the Terbinafine arm.
(c) Treatment effect at one year of treatment, defined as the ratio of odds ratios ROR = ORT /ORI .
10. Briefly summarize the results of the analysis of the data from the onycholysis ran- domized clinical trial, conducted at questions 1-3. Mention at least two strengths and at least two limitations of the conclusions regarding the efficacy of Terbinafine versus Itraconazole from this analysis.