Possible data-generating models of longitudinal continuous outcomes and intercurrent events to investigate estimands

Figures & data

Table

Box 1: A case study in short-term major depressive disorder trials We consider a two-arm, randomized placebo-controlled, single center, trial of six weeks duration with superiority objective. To achieve a realistic case study where we know the full data, i.e., the intercurrent events and all the outcomes before and after the intercurrent events, we fitted a longitudinal generalized least squares model to an existing dataset from a depression trial (study 003-002). We used this model to inform the final postulated model (Table 2). Based on the same depression trial, we formulated the rules for intercurrent events. The clinical outcome is Montgomery and Asberg Depression Rating Scale (MADRS10) and is planned to be measured weekly for six weeks. We simulated 500 trials (18) to allow for approximately 0.1 precision (+/-1.96*SE) in the estimate of treatment effect from the simulated trials ( supplemental material). Assumed treatment effect (δ) at end of trial at six weeks was -3.5 points on MADRS10 scale, a standard deviation (SD) of outcomes at week 6 of 7.5, to achieve 90% power (1-β) and maximum accepted type 1 error (α) of 0.025 (for a one-sided t-test) resulted in needing 95 patients per arm. We assume early separation in the outcome between the arms with the treatment effect maintained to the end of trial. Intercurrent events of interest are treatment discontinuation due to lack of efficacy (LoE) at week 3 and due to adverse events (AE) at week 2. These assumptions are based on study 003-002 (source trial for the case study). The expected proportion of LoE at trial level is 35% (21% in control, 14% in experimental)], the expected proportion of AE at trial level is 15% (5% in control, 10% in experimental). The patients not experiencing any intercurrent events are randomly allocated evenly at trial level (they are the remainder of simulated subjects that do not experience any intercurrent event). A downward time trend of outcomes is expected (improvement), based on the source trials used to extract the DGMs parameters.

Table 1. Outcomes DGM specification and notation (Selection Model)

$Y_{\mathrm{ij}}=({\beta }_{1j}+{\epsilon }_{\mathrm{ij}})+{\beta }_{2\mathrm{jTreat}}\mathrm{*Treatmen}{t}_i$

$Y_{\mathrm{ij}}$ = repeated measurements of continuous outcome for patient $i$ at baseline (j = 0) or follow-up visit $j$ $(e.g.,\mathrm{ j}=1,\dots , 6)$ ${\epsilon }_{\mathrm{ij}}\sim \hspace{0.25em}N(0,{\Sigma }_{{\epsilon }_{\mathrm{ij}}}^2)$ $\mathit{Trea}{t}_i$ = randomised treatment for patient $i$, indicator is 0 for control and 1 for treatment ${\epsilon }_{\mathrm{ij}}$= correlated residuals for patient $i$ at baseline (0) and follow-up visit $j$ (week 1 to week 6) ${\Sigma }_{{\epsilon }_{\mathrm{ij}}}^2$= unstructured covariance matrix of residuals ${\epsilon }_{\mathrm{ij}}$

Table 2. Assumed true parameters for trajectories in each arm for the target trial

Parameter name	Parameter value
beta.baseline ( ${\beta }_{10}$)	29.79
beta_week1 ( ${\beta }_{11}$)	−1
beta_week2 ( ${\beta }_{12}$)	−1.5
beta_week3 ( ${\beta }_{13}$)	−2
beta_week4 ( ${\beta }_{14}$)	−2.5
beta_week5 ( ${\beta }_{15}$)	−3
beta_week6 ( ${\beta }_{16}$)	−3.5
beta_v1_treatment ( ${\beta }_{21\mathrm{Treat}}$)	−1
beta_v2_treatment ( ${\beta }_{22\mathrm{Treat}}$)	−1.5
beta_v3_treatment ( ${\beta }_{23\mathrm{Treat}}$)	−2
beta_v4_treatment ( ${\beta }_{24\mathrm{Treat}}$)	−2.5
beta_v5_treatment ( ${\beta }_{25\mathrm{Treat}}$)	−3
beta_v6_treatment ( ${\beta }_{26\mathrm{Treat}}$)	−3.5

Table 3. Parameters extracted and used for linear predictors in the logit models for intercurrent events for the SM DGM, stochastic implementation

Logistic regression model	Parameter name	Parameter value
LoE at trial level	${\varphi }_0^{\mathrm{Lo}{E}_{\mathit{trial}}}$	1.077405
	${\varphi }_1^{\mathrm{Lo}{E}_{\mathit{trial}}}$	−0.355617
AE in experimental arm	${\varphi }_0^{A{E}_{\exp }}$	−2.370965
	${\varphi }_1^{A{E}_{\exp }}$	0.272188
AE in control arm	${\varphi }_0^{A{E}_{\mathit{ctrl}}}$	−2.78108
	${\varphi }_1^{A{E}_{\mathit{ctrl}}}$	−0.39944

Table 4. Model parameters for each pattern of the PMMM DGM

Parameters	Proportions:	0.35	0.15	0.50	Check at parameter level
	Trial level	LoE in both arms	AE in both arms	No IE in both arms
		0.14 in exp 0.21 in ctrl	0.10 in exp 0.05 in ctrl	1:1 random unconstrained
Betas	29.79	29.79	29.79	29.79	29.79
Beta1	−1	1	0.1	-2.75	−1.01
Beta2	−1.5	1.2	0.1	-3.85	−1.49
Beta3	−2	1.2	0.1	-4.85	−1.99
Beta4	−2.5	1.8	0.3	-6.35	−2.5
Beta5	−3	2.8	1	-8.25	−2.995
Beta6	−3.5	0	0	-7	−3.5
Beta7	−1	0.1	-2	-0.7	−1.006756703
Beta8	−1.5	0.8	-4	-1.15	−1.508520608
Beta9	−2	-0.2	-4	-1.5	−1.992814326
Beta10	−2.5	-0.5	-3.4	-2.25	−2.504744698
Beta11	−3	-0.5	-5	-2	−2.986739496
Beta12	−3.5	-1	-5	-4	−3.486994798

Table 5. Model parameters for SPM DGM

Linear mixed-effects model

Parameter name	Parameter value
Fixed effects
${\beta }_0$	29.374914
${\beta }_1$	−0.529752
${\beta }_2$	−0.577650
Random effects
$b_i\sim \mathrm{ N}(0, {\sigma }_b^2)$	${\sigma }_b^2=\left(\begin{array}{cc}21.140783& 1.307053\\ \mathrm{ }1.307053& 1.044141\end{array}\right)$
Random error
${\epsilon }_i \sim \mathrm{ N}(0, {\sigma }_i^2 )$	$\sigma = 3.285955\mathrm{ }$

Table 6. Model parameters for logit models of SPM DGM

Logistic regression model	Parameter name	Parameter value
LoE at trial level	${\beta }_0$	−0.0877193
	$c_1$	−0.60629
	$\alpha$	1
AE in experimental arm	${\beta }_0$	−1.353383
	$\alpha$	1
AE in control arm	${\beta }_0$	−2.155388
	$\alpha$	1

Table 7. Parameters of the survival models for intercurrent events in the JM DGM

Survival model	Parameter name	Parameter value
LoE at trial level	${\beta }_0$	$2.8789$
	$c_1$	$0.3111$
	$\alpha$	$-0.0279$
	$\lambda$	$1.7618$
	$\nu$	$1.7618$
AE at trial level	${\beta }_0$	$3.7550$
	$c_2$	$-0.5394$
	$\alpha$	$0.0076$
	$\lambda$	$0.9719$
	$\nu$	$0.9719$

Table 8a. Descriptive statistics intercurrent events

Selection Model DGM – deterministic rule

	Control		Treatment		Total
Intercurrent event	N	%	N	%	N	%
AE	10.1	5.3	18.1	9.5	28.2	14.9
LoE	39.5	20.8	26.8	14.1	66.3	34.9
Total	49.6	26.1	44.9	23.6	94.5	49.8

Averaged over 500 simulated trials. Trial sample size = 190

Table 8b. Descriptive statistics intercurrent events

Pattern-mixture mixed model DGM

	Control		Treatment		Total
Intercurrent event	N	%	N	%	N	%
AE	10	5.3	19	10.0	29	15.3
LoE	40	21.1	27	14.2	67	35.3
Total	50	26.4	46	24.2	96	50.6

Averaged over 500 simulated trials. Trial sample size = 190

Table 8c. Descriptive statistics intercurrent events

Shared parameter model DGM

	Control		Treatment		Total
Intercurrent event	N	%	N	%	N	%
AE	9.9	5.2	19.7	10.4	29.6	15.6
LoE	40.8	21.5	25.6	13.5	66.4	34.9
Total	50.7	26.7	45.3	23.9	96.0	50.5

Averaged over 500 simulated trials. Trial sample size = 190

Table 8d. Descriptive statistics intercurrent events

Joint model DGM

	Control		Treatment		Total
Intercurrent event	N	%	N	%	N	%
AE	10.0	5.2	19.3	10.1	29.2	15.4
LoE	41.7	21.9	24.6	12.9	66.2	34.9
Total	51.7	27.1	43.9	23.0	95.4	50.3

Averaged over 500 simulated trials. Trial sample size = 190

Figure 1. Individual trajectories by treatment arm/study 003-021

Graphs to visualise each pattern in the trials simulated with each DGM

Figure 2. Treatment discontinuation due to lack of efficacy patterns by each DGM and arm

Figure 3. Treatment discontinuation due to adverse event patterns by each DGM and arm

Figure 4. No intercurrent events patterns by each DGM and arm

Figure 5. All patterns stacked together as a trial simulated by each DGM and arm