Q40. Interpreting a treatment–time interaction
[6 marks]
The model in Question 39 produces:
Random effects:
Group Term Std.Dev.
patientID Intercept 1.20| Fixed effect | Estimate | Standard error | p-value |
|---|---|---|---|
| Intercept | -0.80 | 0.20 | 0.000063 |
treatmentB |
0.10 | 0.25 | 0.689 |
weeks |
-0.10 | 0.02 | <0.000001 |
treatmentB:weeks |
-0.08 | 0.03 | 0.0077 |
(a) Interpret the random-intercept standard deviation of 1.20.
(b) Interpret the weeks coefficient and the treatmentB:weeks coefficient.
(c) Calculate the fitted conditional odds ratio for treatment B versus treatment A at week 8, comparing patients with the same random-intercept value.
(d) Assess this conclusion:
“The treatment p-value is 0.689, so there is no evidence that the treatments differ.”
Definitions. G_i=0 or 1 codes treatment A or B; t_{ij} is the visit week and b_i the patient random intercept. The coefficients \beta_G,\beta_T,\beta_{GT} multiply treatment, time, and their interaction. The linear predictor is a conditional log odds, so subtracting two predictors and exponentiating gives a conditional odds ratio. The treatment comparison holds week and random-intercept value fixed.
The fitted linear predictor is \widehat\eta_{ij} =-0.80+0.10G_i-0.10t_{ij}-0.08G_it_{ij}+b_i.
(a) Random-intercept standard deviation. The unexplained patient-specific intercepts have estimated standard deviation 1.20 on the log-odds scale; their estimated variance is 1.20^2=1.44. This is not a standard deviation of disease probabilities.
For equal fixed covariates, increasing b_i by one standard deviation changes conditional odds by the factor \frac{\operatorname{odds}(b_i+1.20)}{\operatorname{odds}(b_i)} =e^{1.20}\approx3.32.
(b) Weeks and interaction coefficients. For treatment A (G_i=0), the fitted predictor is \widehat\eta_A(t,b)=-0.80-0.10t+b. Thus, within a patient, an additional week changes the conditional log odds by -0.10 and multiplies the odds by e^{-0.10}\approx0.905, approximately a 9.52\% reduction in odds per week.
For treatment B (G_i=1), \widehat\eta_B(t,b) =(-0.80+0.10)+(-0.10-0.08)t+b =-0.70-0.18t+b. The time coefficient for B is therefore -0.18, giving a weekly odds multiplier e^{-0.18}\approx0.835, or approximately a 16.47\% reduction.
The interaction coefficient -0.08 is the difference between the B and A slopes on the log-odds scale. Equivalently, the ratio of their weekly odds multipliers is \frac{e^{-0.18}}{e^{-0.10}}=e^{-0.08}\approx0.923. It is not an extra 8-percentage-point weekly reduction in probability.
(c) Conditional odds ratio at week 8. Comparing patients with the same random-intercept value b, \begin{aligned} \log\{\widehat{\operatorname{OR}}_{B:A}(t)\} &=\widehat\eta_B(t,b)-\widehat\eta_A(t,b)\\ &=0.10-0.08t. \end{aligned} The random intercept cancels. At week 8, \boxed{ \widehat{\operatorname{OR}}_{B:A}(8) =\exp\{0.10-0.08(8)\}=e^{-0.54}\approx0.583. } Conclusion: At week 8, treatment B has approximately 41.7\% lower fitted conditional odds of infection than treatment A, comparing patients with the same random-intercept value. This is an estimated odds ratio, not a probability ratio or a test of the week-8 contrast.
(d) Assessing the treatment p-value claim. The main treatment coefficient is the B–A difference at week zero, because the interaction term is then zero. Its p-value 0.689 tests H_0:\beta_G=0, not whether the treatment curves agree at every week. It supplies insufficient evidence of a baseline difference.
The interaction p-value 0.0077<0.05 tests H_0:\beta_{GT}=0, and provides evidence that the treatment groups have different time slopes on the conditional log-odds scale. Equality of the entire treatment-specific predictors would require both \beta_G=0 and \beta_{GT}=0, which is a joint hypothesis.
Key result. With an interaction, a main-effect test concerns the reference value of the other predictor. A nonsignificant baseline treatment effect does not rule out a treatment difference that changes over time.