GENDEP trial dataset. Results from a Xxx proportional hazards model, investigating effect of antidepressant treatment on time to remission. Variable Hazard Ratio Std. Err. 95% CI Nortriptyline* 0.680 0.117 0.485, 0.952 *Reference group Escitalopram Results from Table 2.2, indicate a statistically significant hazard ratio of 0.680 (95% CI: 0.485, 0.952) for the effect of antidepressant treatment, indicating a 32% lower remission rate in the Nortriptyline group compared to Escitalopram. The fundamental assumption of proportional hazards can be evaluated using a test of non zero slope in a generalised linear regression of the scaled Xxxxxxxxxx residuals on time (Xxxxxxxx and Xxxxxxxx 1994), with the null hypothesis of a zero slope (equating to proportional hazards). The test for non-proportional hazards in the effect of antidepressant treatment was entirely non-significant (p=0.9872) suggesting no violation of the proportional assumption. Time to depression remission was also modelled using the parametric survival comparator, the Weibull model. The Weibull model is adjusted for antidepressant treatment, alongside baseline MADRS, duration of current episode, score, gender, and baseline BMI. For the IG model, covariates may have a different impact on the initial distance (c=𝑒𝑥𝑝(𝜃1)) or velocity (μ= 𝜃2). Baseline MADRS score, duration of current depressive episode and gender (reference category male) were considered possible predictors of distance from remission, 𝑒𝑥𝑝(𝜃1). BMI and antidepressant treatment were used in the second (velocity) linear predictor, 𝜃2. BMI is modelled in 𝜃2 as previous research suggests different rates of remission in different weight categories. This mechanism of action is believed to be from a differential response to treatment rather a more (or less severe) initial depression for some categories of BMI, thus I explore BMI effects on the velocity to recovery. Later in this chapter I test for an interaction with randomised antidepressant treatment, it follows treatment and BMI must be considered in the same linear predictor. Results from the modelling are summarised in Table 2.3.
GENDEP trial dataset. Results from regression output for randomised antidepressant treatment on time to remission under the Weibull and inverse Gaussian model. Inverse Gaussian model Estimated model parameters Estimate Std. Err. 95% CI P-value (θ1) Logarithm of initial depression status Baseline MADRS 0.025 0.004 0.016, 0.034 <0.001 Duration of episode (wks) 0.006 0.002 0.002, 0.009 0.001 Gender* 0.176 0.066 0.047, 0.306 0.007 Constant (θ2) Mean depression status 0.444 0.178 0.094, 0.793 0.013 Antidepressant** -0.084 0.044 -0.171, 0.002 0.057 Baseline BMI (kg/m2) -0.009 0.004 -0.018, 0.000 0.043 Constant 0.539 0.120 0.304, 0.775 <0.001 Weibull model Estimated model parameters Estimate Std. Err. 95% CI P-value Logarithm of relative hazard Baseline MADRS -0.052 0.014 -0.080, -0.024 <0.001 Duration of episode (wks) -0.010 0.005 -0.020, 0.001 0.062 Gender* -0.210 0.173 -0.549, 0.130 0.226 Antidepressant** -0.316 0.174 -0.658, 0.026 0.070 Baseline BMI (kg/m2) -0.030 0.018 -0.064, 0.005 0.089 Constant -3.369 0.766 -4.871, -1.867 <0.001 * Reference male ** Reference Escitalopram In Table 2.3, comparing the IG model and Weibull model, I generally observe similar patterns in the direction and magnitude of effect on time to depression remission. For the distance linear predictor, representing the logarithm of initial depression status, the baseline MADRS score is significant, with p<0.001, with a positive regression coefficient signifying that initial distance tends to be higher for participants with higher baseline depression scores. The coefficient for duration of the depressive episode and gender are also positive, indicating that initial depression status was further from remission for women and those with a longer duration of depression. For the velocity linear predictor, the regression coefficients are all negative, indicating a negative effect slowing remission. Antidepressant has a non-significant negative coefficient -0.084 (95% CI -0.171, 0.002), consistent with the Weibull model in this table, where Nortriptyline delays remission. From these fully adjusted
GENDEP trial dataset. Survival curves estimated using the inverse Gaussian and Weibull models by randomised antidepressant treatment, solid lines are the reference treatment group, Escitalopram. Prior evidence suggested body mass index (BMI) was a moderator to treatment effect (Xxxx, Mors et al. 2009). Participant BMI was adjusted for in the velocity linear predictor with randomised treatment. The IG model supported previous findings of more and faster remission with Escitalopram, although no evidence of significant interaction with BMI was found (velocity regression coefficient -.005, 95% CI -0.023 to 0.012). Without pursuing the statistical inference further, it was possible to explore and visualise how BMI affected time to remission in the two
GENDEP trial dataset. Median relative dose by Escitalopram, solid line and Nortriptyline, dashed line, by trial week for those participants not in remission. Error bars are minimum and maximum quantities. Regression analysis, Table 3.1, showed that relative dose was strongly predicted by randomised treatment, with an F-statistic of 32 (Xxxxxxx and Stock 1997) and beta=-0.131 (95%CI -0.18 to -0.09), implying that the relative daily dose of Nortriptyline on average over the 12 week period was 13% lower than Escitalopram. Sex and age showed marginally significant associations, but perhaps surprisingly prior duration and age-of-onset of depression and BMI were unrelated to relative dose. The residuals from this regression are extracted for inclusion in subsequent survival analyses. Since treatment allocation was random these residuals meet the assumptions required for a TSRI estimator. These are referred to as Stage 1 residuals. Table 3.1 GENDEP trial dataset. Results from predicting relative dose using linear regression. Regression output for prediction of relative dose from randomised antidepressant treatment and baseline covariates. The association measure is the regression coefficient (standard error), with 95% confidence intervals. CoEfficient (SE) 95% CI P-value MADRS 0.003 (0.002) -0.001, 0.006 0.117 Prior duration of depression (weeks) 0.000 (0.001) -0.001, 0.001 0.853 Sex (female) 0.046 (0.024) 0.000, 0.093 0.052 Age of depression onset (years) 0.001 (0.001) -0.002, 0.004 0.399 BMI (kg/m2) 0.003 (0.002) -0.001, 0.008 0.149 Treatment (Nortriptyline) -0.131 (0.023) -0.175, -0.086 <0.001 Age (years) -0.002 (0.001) -0.005, 0.000 0.060 In total I observed 143 participants in remission over 3269 treatment weeks. In Figure 3.3, survival functions from Xxxxxx-Xxxxx, Xxx and IG models are plotted for median dichotomised high and low relative doses. These analyses were adjusted for no further covariates. The shaded area shows 95% pointwise CIs for Xxxxxx Xxxxx and Xxx models, IG model from 500 bootstrap replications of the survival function used to estimate a standard error and CI. All three suggest that a higher dose is associated with increased time to remission in all three methods, mean time to remission was 7.82 weeks (95%CI = 7.3 to 8.3) for lower doses and 9.40 weeks (95%CI = 9.0 to 9.8) for higher doses. I now explore the hypothesis that these counter-intuitive dose-response results can be explained by titration bias.
GENDEP trial dataset. Survival function for time to depression remission by relative dose, dichotomised by median split of average weekly dose. Estimated using Xxxxxx-Xxxxx; Xxx model; inverse Gaussian Models. Shaded area shows 95% confidence intervals. Table 3.2 shows results from the Cox and IG models. For each covariate two rows of estimates are shown, those in the first row are for the standard analysis while those in the second row were obtained using the TSRI estimator from models that also included the Stage 1 residuals as an additional covariate. In the standard Cox model while, as expected, baseline MADRS score was highly significant, with log HR = -0.051 (95% CI =-0.079 to -0.023) implying higher severity being associated with longer time to remission, higher relative dose was not significantly associated, and the estimated coefficient also implied a decreasing log HR = -0.180 (95% CI -0.933 to 0.573). Like the simple summary statistics suggested, this implies an increase in relative dose was associated with a decreased chance of remission and longer time to remission. Table 3.2 GENDEP trial dataset. Results from observational analysis of relative dose. Regression output for dose response on time to remission under the Cox and Inverse Gaussian model with and without the instrumental variable stage 1 residuals introduced as a control variable. Xxx Proportional Hazards Model With stage 1 residuals Time to remission Log HR 95% CI P-value Log HR 95% CI P-value MADRS -0.051 -0.079, -0.023 <0.001 -0.056 -0.088, -0.024 0.001 Duration of depressive episode (weeks) -0.009 -0.020, 0.001 0.086 -0.010 -0.020, 0.001 0.061 Sex (female) -0.153 -0.492, 0.187 0.378 -0338 -0.747, 0.072 0.106 Age of onset (years) -0.007 -0.024, 0.009 0.378 -0.007 -0.024, 0.011 0.453 BMI (kg/m2) -0.034 -0.068, -0.001 0.050 -0.039 -0.076, -0.003 0.036 Relative dose -0.180 -0.933, 0.573 0.639 3.012 0.086, 5.938 0.044 Stage1 Residuals -3.508 -6.569, -0.447 0.025 Inverse Gaussian Model With stage 1 residuals Time to remission CoEff 95% CI P-value CoEff 95% CI P-value Distance Linear Predictor MADRS 0.026 0.018, 0.035 <0.001 0.028 0.014, 0.041 <0.001 Duration of depressive episode (weeks) 0.006 0.002, 0.009 0.001 0.006 0.002, 0.010 0.003 Sex (female) 0.196 0.066, 0.325 0.003 0.240 0.063, 0.417 0.008 Velocity Linear Predictor Age of onset (years) -0.002 -0.006, 0.003 0.457 -0.002 -0.00, 0.003 0.522 BMI (kg/m2) -0.010 -0.019, -0.002 0.020 -0.012 -0.021, -0.002 0.015 Relative dose -0.029 -0.218, 0.160 0.767 0.87...
GENDEP trial dataset. Long plot showing in red those participants who have encountered the threshold for first hitting time depression remission on the Xxxxxxxxxx-Xxxxxx Depression Rating Scale over 12 study weeks. There are two points to note from Figure 4.2. First, the heterogeneity in the data and variable patterns of follow up for the study participants. Second, the distinction between the first hitting time and sustained remission as seen in this practical example. Here many participants who have passed the threshold for first hitting, and denoted to have had the event progress into depression relapses even within the time frame of the trial. In the growth curve model, the dependent variable is MADRS score at weeks 1 to 12. The median number of observations per participant (censoring observations following first hitting) was 6 of a possible 12. The growth curve model will use all available data up to first hitting and inference from all participants will contribute to the averaged treatment effects. At the end of the study period, 237participants (49.6%) of the trial population reached first-hit remission. Right censored observations for the GCT model are those participants who dropped out (n=71) or switched (n=74) treatment during the study period, while for the Weibull comparison model, all those who remitted prior to week 12 additionally contribute no further information beyond that time. The growth curve model assumes a random intercept and characterises the treatment effect as a linear interaction between treatment and time, with time being zero at baseline consistent with randomisation yielding no-group difference. The parametric survival comparator is the Weibull model adjusted for the effect of treatment on the estimated hazard of remission. Maximum likelihood was used for modelling building utilising the Likelihood ratio tests. In the following section I illustrate this methodology and explore the best model fit for a trial dataset; REML was used for the presented model results.
