Comparing methods to combine functional loss and mortality in clinical trials for amyotrophic lateral sclerosis

Figures & data

Figure 1 Overview of the relationships between ALSFRS-R, mortality and treatment.

Notes: In this diagram, treatment can have either a direct effect on mortality by γ or an indirect effect on mortality by modifying the ALSFRS-R through β and subsequently affecting mortality by α. Classically, longitudinal (e.g., linear mixed) and survival (e.g., Cox) models analyze either γ or β. Joint models incorporate all relationships and simultaneously model γ, β and α.
Abbreviation: ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale.

Table 1 Null and alternative hypothesis of each analytical strategy to evaluate the combined treatment effect on survival and functional loss

Analytical strategy	Null hypothesis (H0)		Alternative hypothesis (H1)
CAFS	H0 : P(Xtreatment> Xplacebo) = 0.5	The probability that a patient on treatment has a better outcome X (either survival or functional status) than a patient on placebo is 0.5	H1 : P(Xtreatment> Xplacebo) ≠ 0.5	The probability that a patient on treatment has a better outcome X (either survival or functional status) than a patient on placebo is higher or less than 0.5
Death or 6-point loss	H0 : γcombined = 0	There is no difference between treatment arms in the probability of an event (death or 6-point loss) at any time point during follow-up	H1 : γcombined ≠ 0	There is a difference between treatment arms in the probability of an event (death or 6-point loss) at any time point during follow-up
Joint model	H0 : γadjusted = 0 ∩ βadjusted = 0	There is no difference between treatment arms in the probability of an event (death) at any time point, adjusted for functional status, and there is no difference between treatment arms in functional decline, adjusted for survival	H1 : γadjusted ≠ 0 ∪ βadjusted ≠ 0	There is a difference between treatment arms in the probability of an event (death) at any time point, adjusted for functional status, and/or there is a difference between treatment arms in functional decline, adjusted for survival
Cox or LME test	H0 : γcrude = 0 ∩ βcrude = 0	There is no difference between treatment arms in the probability of an event (death) at any time point, and there is no difference between treatment arms in functional decline	H1 : γcrude ≠ 0 ∪ βcrude ≠ 0	There is a difference between treatment arms in the probability of an event (death) at any time point, and/or there is a difference between treatment arms in functional decline

Notes: Taking the exponent of γ will provide the HR (treatment vs control). The mean difference between treatment arms in rates of decline in ALSFRS-R is given by β. Note that the joint model incorporates the relationship between survival and function and thus adjusts the treatment effect, whereas in the Cox or LME test, this adjustment does not take place.

Abbreviations: CAFS, combined assessment of function and survival; LME, linear mixed effects; HR, hazard ratio; ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale.

Figure 2 Rates of functional decline and mortality in the PRO-ACT database with a simulated treatment scenario.

Notes: (A and B) Observed rates of functional decline and death in the 1,469 selected PRO-ACT patients (Table 2). Our simulation (n = 50,000) shows a good fit with the observed dataset and exhibits a similar pattern over time. As illustration, a hypothetical treatment effect was induced, which reduced the rate of decline by 15% and the hazard by 34%. To illustrate the interaction (α) between the ALSFRS-R and survival, we divided subjects into ten equally sized groups according to their ALSFRS-R baseline score (C) or observed rate of decline during follow-up (D). Green represents the patients with the highest baseline score or the slowest rate of decline.
Abbreviations: ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale; PRO-ACT, Pooled Resource Open-Access ALS Clinical Trials.

Table 2 Baseline characteristics of the PRO-ACT database’s placebo patients

Patient characteristics	PRO-ACT database (N = 1,469)
Age, years	57 (11)
Sex, female	544 (37%)
Onset, bulbar	311 (21%)
Symptom duration at enrollment
Mean, months	17 (7)
Distribution, n (%)
<12 months	414 (28%)
12–24 months	815 (56%)
>24 months	240 (16%)
Diagnostic delay
Mean, months	10 (6)
Distribution, n (%)
<6 months	435 (30%)
6–12 months	597 (41%)
>12 months	434 (29%)
FVC, % predicted	91 (14)
ALSFRS-R total score	38 (5)
ΔFRS at baseline, points per month	−0.68 (0.5)
BMI, kg/m^2	26 (4)
Plasma creatinine, μmol/L	70 (15)

Notes: Data are in mean (SD) or n (%). ΔFRS, $\frac{\text{ALSFRS}-{\mathrm{R}}_{\text{baseline}}-48}{\text{Disease duration}}$

Abbreviations: FVC, forced vital capacity; ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale; BMI, body mass index; PRO-ACT, Pooled Resource Open-Access ALS Clinical Trials.

Table 3 Empirical power of each strategy for trials with a maximum follow-up duration of 18 months

Survival (HR, treatment vs placebo)	ALSFRS-R (slope reduction)	Cox model	LME model	Cox or LME model (omnibus test)	Joint model (omnibus test)	CAFS	Death or 6 pt.
1	0	0.050	0.052	0.046	0.051	0.047	0.048
0.66	0	0.273	0.058	0.214	0.404	0.118	0.050
0.5	0	0.586	0.053	0.495	0.804	0.212	0.050
1	15%	0.137	0.405	0.333	0.314	0.268	0.260
0.66	15%	0.566	0.406	0.563	0.616	0.526	0.274
0.5	15%	0.837	0.406	0.791	0.884	0.689	0.285
1	30%	0.378	0.921	0.888	0.871	0.755	0.729
0.66	30%	0.828	0.934	0.946	0.948	0.928	0.765
0.5	30%	0.958	0.931	0.976	0.987	0.966	0.772

Note: All simulations are based on a monthly follow-up scheme (SD = 0.16) with a maximum of 18 months of follow-up and 150 patients per treatment arm; each scenario was repeated 10,000 times (Figure S2).

Abbreviations: HR, hazard ratio; ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale; Cox, Cox proportional hazard; LME, linear mixed effects; CAFS, combined assessment of function and survival; pt., points decrease on ALSFRS-R from baseline.

Figure 3 Empirical power of the four analytical strategies for different treatment scenarios.

Notes: (A) Visual presentation of Table 3 of the empirical power after 18 months of follow-up. Panels from left to right show the effects of different treatments on functional decline (reduction in ALSFRS-R slope, β); on the x-axis are the treatment effects on survival (reduction in hazard rate, γ). (B) Direct comparison between the CAFS and the joint model for 18 months (solid lines) and 12 months (dashed lines) of follow-up.
Abbreviations: ALSFRS-R, revised amyotrophic lateral sclerosis functional rating scale; CAFS, combined assessment of function and survival; HR, hazard ratio; LME, linear mixed effects; PH, proportional hazard.