Robust analyzes for longitudinal clinical trials with missing and non-normal continuous outcomes

Figures & data

Figure 1. Spaghetti plots of the change of the log-transformed CD4 count data from the baseline separated by the two treatments. The blue line represents the mean of the observed outcomes at each visit point. The black lines refer to the individual outcome profiles.

Figure 2. Normal QQ plots of the conditional residuals at each visit. The conditional residuals were obtained by fitting the sequential linear regressions on the current outcomes against the available history and calculating the residuals at each visit. The p-values for the Shapiro-Wilk test and the symmetry test are also provided.

Table 1. The outcome mean model assumption specified by common CBI scenarios.

CBI scenarios	Formations of the outcome means in Assumption 3.2
J2R	$\mathbb{E}(Y_{i,t}\mid H_{i,s-1},R_{i,s-1}=1,R_{i,s}=0,A_i=a)=\mathbb{E}(Y_{i,t}\mid H_{i,s-1},A_i=0)$ for any a and s.
CR	$\mathbb{E}(Y_{i,t}\mid R_{i,s-1}=1,R_{i,s}=0,A_i=a)=\mathbb{E}(Y_{i,t}\mid A_i=0)$ for any a and s.
CIR	$\mathbb{E}(Y_{i,t}\mid R_{i,s-1}=1,R_{i,s}=0,A_i=a)=\mathbb{E}(Y_{i,s-1}\mid A_i=a)+\mathbb{E}(Y_{i,t}\mid A_i=0)-\mathbb{E}(Y_{i,s-1}\mid A_i=0)$ for any a and s.
LMCF	$\mathbb{E}(Y_{i,t}\mid R_{i,s-1}=1,R_{i,s}=0,A_i=1)=\mathbb{E}(Y_{i,s-1}\mid A_i=1)$ and $\mathbb{E}(Y_{i,t}\mid R_{i,s-1}=1,R_{i,s}=0,A_i=0)=\mathbb{E}(Y_{i,t}\mid A_i=0)$ for any s.

Table 2. Summary of the simulation methods and results.

(a) Different estimation and imputation approaches in the seven methods used for comparison.
Method	Imputation model	Imputation method	Analysis model
Oracle	True	Mean imputation	Huber regression
Observed	/	/	Huber regression
No-intermediate	Weighted Huber regression	Mean imputation	Huber regression
	(only at the last time point)
MI	LS	MI	LS
MI Robust	LS	MI	Huber regression
LSE	Weighted Huber regression	Mean imputation	LS
Robust	Weighted Huber regression	Mean imputation	Huber regression
(b) Simulation results under the normal distributions without or with extreme outliers based on 1000 MC simulations. Here the true value $\tau =71.18\mathrm{\%}$.
				Var est		Relative bias		Coverage rate		Power
				($\times {10}^{-2}$)		($\mathrm{\%}$)		($\mathrm{\%}$)		($\mathrm{\%}$)
Case	Method	Point est ($\times {10}^{-2}$)	True var ($\times {10}^{-2}$)	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	RMSE ($\times {10}^{-2}$)
No outliers	Oracle	70.13	3.17	3.32	3.27	4.87	3.26	95.30	94.50	97.20	97.30	17.82
	Observed	56.87	4.27	3.84	4.23	−10.11	−0.76	87.70	89.00	81.30	79.50	25.12
	No-intermediate	39.63	2.76	2.67	2.75	−3.16	−0.53	51.50	52.90	68.80	68.60	35.65
	MI	70.89	3.02	5.35	3.22	77.00	6.39	98.90	95.00	92.80	98.00	17.38
	MI Robust	70.94	3.20	5.51	3.37	72.36	5.48	98.80	94.90	91.20	97.10	17.87
	LSE	70.92	3.03	3.25	3.16	7.12	4.07	95.40	94.90	97.70	97.70	17.41
	Robust	70.10	3.27	3.41	3.37	4.51	3.30	94.80	94.20	96.60	96.70	18.09
Outliers in both groups	Oracle	69.51	3.28	3.36	3.31	2.37	0.84	94.90	94.80	96.80	97.10	18.18
	Observed	56.73	4.24	3.94	4.35	−7.20	2.54	88.70	90.80	80.00	77.70	25.15
	No-intermediate	39.11	2.69	2.71	2.77	0.55	3.02	50.90	51.70	64.80	64.60	36.02
	MI	67.18	4.88	20.19	11.50	313.93	135.87	99.90	99.50	16.90	52.50	22.43
	MI Robust	99.88	4.86	11.50	4.87	136.84	0.22	96.80	74.30	94.30	99.50	36.18
	LSE	47.00	3.57	11.21	10.78	213.46	201.62	98.20	98.30	16.70	19.80	30.69
	Robust	73.20	3.55	3.57	3.49	0.72	−1.70	94.30	94.10	97.30	97.40	18.93
(c) Simulation results under the t-distribution based on 1000 MC simulations. Here the true value $\tau =68.09\mathrm{\%}$.
			Var est		Relative bias		Coverage rate		Power
			($\times {10}^{-2}$)		($\mathrm{\%}$)		($\mathrm{\%}$)		($\mathrm{\%}$)
Method	Point est ($\times {10}^{-2}$)	True var ($\times {10}^{-2}$)	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	RMSE ($\times {10}^{-2}$)
Oracle	69.87	2.72	2.87	2.77	5.26	1.57	95.10	94.20	98.50	98.70	16.59
Observed	57.10	3.46	3.25	3.58	−6.09	3.46	89.90	91.20	88.00	85.50	21.59
No-intermediate	39.76	2.22	2.28	2.32	2.61	4.28	53.50	53.50	76.50	76.30	32.01
MI	70.42	3.00	5.35	3.16	78.07	5.27	99.30	95.40	90.90	97.60	17.47
MI Robust	70.38	2.79	4.71	2.89	68.85	3.51	99.10	94.10	94.60	98.40	16.85
LSE	70.45	2.93	3.24	3.09	10.63	5.63	96.50	95.00	97.70	98.10	17.20
Robust	69.77	2.75	2.94	2.85	6.85	3.63	94.70	94.30	98.20	98.40	16.58

${\hat{V}}_1$ denotes the variance estimate obtained by Rubin's rule in MI and linearization in mean imputation-based methods; ${\hat{V}}_{\mathrm{Boot}}$ denotes the bootstrap variance estimates.

Figure 3. Simulation results under different distributions based on 1000 MC simulations for the methods MI, MI Robust, LSE, and Robust. It visualizes the numerical results in Table 2 and Tables S1–S3 in the supplementary material. The first row shows the type-1 error and power when the outcomes are normally distributed without outliers. The second row presents the results with the incorporation of the outliers in the simulated data. The third row illustrates the results under a heavy-tailed dataset.

Table 3. Simulation results under the normal distribution without or with extreme outliers, where the true value $\tau =72.26\mathrm{\%}$ and the missingness pattern is MNAR in the treated group.

				Var est		Relative bias		Coverage rate		Power
				($\times {10}^{-2}$)		($\mathrm{\%}$)		($\mathrm{\%}$)		($\mathrm{\%}$)
Case	Method	Point est ($\times {10}^{-2}$)	True var ($\times {10}^{-2}$)	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	${\hat{V}}_1$	${\hat{V}}_{\mathrm{Boot}}$	RMSE ($\times {10}^{-2}$)
No outliers	Oracle	71.83	3.21	3.42	3.26	6.62	1.42	95.20	94.30	97.40	97.50	17.91
	Observed	65.86	4.08	3.70	4.10	−9.33	0.54	93.00	93.50	91.80	89.70	21.19
	No-intermediate	54.77	3.12	3.05	3.12	−2.08	−0.00	83.00	82.50	87.80	87.70	24.85
	MI	72.69	3.19	5.09	3.29	59.33	3.11	98.40	95.40	94.20	97.90	17.87
	MI Robust	72.63	3.40	5.26	3.46	54.71	1.93	98.60	94.50	93.20	97.00	18.43
	LSE	72.58	3.18	3.39	3.26	6.68	2.42	96.30	95.10	97.90	97.90	17.83
	Robust	71.79	3.43	3.56	3.48	3.85	1.53	95.00	94.40	96.70	96.70	18.51
Outliers in both groups	Oracle	71.30	3.31	3.46	3.31	4.62	−0.09	95.10	94.60	97.10	97.60	18.21
	Observed	65.55	4.20	3.79	4.21	−9.76	0.26	93.50	94.00	92.60	89.40	21.56
	No-intermediate	54.24	3.13	3.09	3.17	−1.04	1.35	82.90	82.70	88.40	87.10	25.24
	MI	51.49	3.96	15.58	11.52	293.48	190.92	99.70	99.20	10.00	24.10	28.76
	MI Robust	78.86	3.83	7.54	3.80	97.05	−0.80	98.90	92.60	91.30	98.70	20.64
	LSE	48.23	3.70	11.65	11.36	215.02	207.18	99.00	98.50	17.40	19.20	30.77
	Robust	73.52	3.63	3.73	3.57	2.58	−1.64	95.50	94.30	97.10	97.30	19.09

${\hat{V}}_1$ denotes the variance estimate obtained by Rubin's rule in MI and linearization in mean imputation-based methods; ${\hat{V}}_{\mathrm{Boot}}$ denotes the bootstrap variance estimates.

Table 4. Analysis of the repeated CD4 count data under J2R.

Variable	Method	Point est	$95\mathrm{\%}$ CI	CI length
Mean arm 1	MI	−0.68	(−0.84, −0.53)	0.31
	MI Robust	−0.67	(−0.83, −0.52)	0.31
	LSE	−0.54	(−0.67, −0.42)	0.25
	Robust	−0.53	(−0.64, −0.41)	0.23
Mean arm 2	MI	−0.39	(−0.55, −0.24)	0.31
	MI Robust	−0.39	(−0.55, −0.23)	0.32
	LSE	−0.23	(−0.35, −0.11)	0.24
	Robust	−0.27	(−0.38, −0.16)	0.22
Difference	MI	0.29	(0.07, 0.51)	0.44
	MI Robust	0.28	(0.06, 0.50)	0.44
	LSE	0.31	(0.20, 0.41)	0.21
	Robust	0.26	(0.16, 0.35)	0.19