Both time-dependent and age-dependent models can distinguish the three mechanisms (nonfunctionalization, neofunctionalization, and subfunctionalization) with high probabilities when the sample size is large.
These results indicate that the probabilistic models derived from the birth-death process with a time-dependent and age-dependent loss rates are useful for understanding the duplication and loss mechanisms of gene families that evolve over time in a single population with caveats discussed.
Models were fit using Cox regression, in which each case was compared with its matched risk set (with matching based on white or nonwhite race), cumulative exposure was time dependent, and the time variable was age.
We used restricted cubic splines 17 to explore the trend shapes of the baseline hazard function, 20 the logit transformed propensity score, and time dependent age at the start of each 15 day segment.
We used Cox proportional hazards regression models [hazard ratios (HRs)] with time-dependent exposures and age as the time scale to estimate associations between air pollution exposure and cause-specific mortality.
We did all analyses for these endpoints both with and without time dependent coefficients for age.
We did not introduce a time dependent covariate for age because all patients were enrolled at their 65th birthday, so that age was already accounted for in duration of follow-up.
With the addition of the period of diagnosis as 1987-92 or 1993-9 as a dichotomous time dependent variable and adjustment for age at the start of the study, the hazard ratio was 1.59 (1.07 to 2.38; P=0.022).
Methodologically rigorous conclusions would require a much larger study, with more explicit control of variables such as renal function, use of inotropes and baseline-dependent and time-dependent variables (age, gender and admission diagnosis) included in a rigorous multivariable analysis.
For age during follow-up, we estimated effect modification by adding an interaction term between exposure and a time dependent categorical variable indicating age group (<60, 60-74, >75).
Sex and the time-dependent factors current age, calendar year and HBV, HCV and HIV notification were a priori included in all multivariable models.
