Online Appendix
Detailed Statistical Methods
A.1. Models
We utilized Bayesian hierarchical models to combine data from all trials while accounting for
the variation of survival times (or rehospitalization indications) within and between studies. One
important feature of the Bayesian hierarchical models is that, because they allow for borrowing
of information across studies/subgroups, they allow us to make inferences about subgroups that
may be under-represented in a portion of the trials. Moreover, missing values are treated as
additional variates whose values can be naturally imputed as part of the estimation procedure
(Schafer (1996), and Little and Rubin(1987)).
The Bayesian paradigm requires specification of both the likelihood for the observations as
well as priors for the model parameters. Preliminary analysis of the survival data from all trials
indicated that the Weibull parametric survival model adequately fit the survival data, in particular,
with results similar to those obtained under the Cox-Proportional Hazards Model. We thus
utilized the Weibull parametric survival model to time to death from all causes. Under this model,
we formulate the hazard function for an individual 𝑖 observed in study 𝑠 as:
ℎ𝑖 (𝑡) = ℎ0𝑠 (𝑡)𝑒𝑥𝑝(𝜏𝑠 𝑍𝑖 + 𝛾𝑠 𝑊𝑖 + 𝜈𝑍𝑖 𝑊𝑖 + 𝛽′𝑋𝑖 )
(1)
where ℎ0𝑠 (∙) is the baseline hazard function in study 𝑠. We assumed that ℎ0𝑠 (𝑡) =
𝑎𝑠 𝑡 𝑎𝑠 −1
,
𝑏𝑠 𝑎𝑠
that is, the hazard function from a Weibull distribution with shape parameter 𝑎𝑠 and scale
parameter 𝑏𝑠 . Furthermore, 𝑍𝑖 refers to treatment, 𝑊𝑖 for comorbidity count, and 𝑋𝑖 the vector
with the remaining covariates for subject 𝑖. We are interested in investigating whether the
efficacy of ICD relative to control is modified by comorbidities and thus we also include an
interaction between treatment and comorbidity. The above formulation allows for trial-specific
1
baseline hazard functions, as well as trial-specific main effects of treatment and comorbidities.
To allow for borrowing of information, we further assumed that:
[𝑎𝑠 |𝜇𝑎 , 𝜎2𝑎 ] ~ LogNormal(𝜇𝑎 , 𝜎2𝑎 ),
[𝑏𝑠 |𝜇𝑏 , 𝜎2𝑏 ] ~ LogNormal(𝜇𝑏 , 𝜎2𝑏 ),
[𝜏𝑠 |𝜇𝜏 , 𝜎2𝜏 ] ~ Normal(𝜇𝜏 , 𝜎2𝜏 ),
[𝛾𝑠 |𝜇𝛾 , 𝜎2𝛾 ] ~ Normal (𝜇𝛾 , 𝜎2𝛾 ).
(2)
We completed the model formulation with the following priors:
[𝜇𝑎 , 𝜇𝑏 , 𝜇𝜏 , 𝜇𝛾 , 𝜈, 𝛽] ∝ 1, (3)
that is, we assumed non-informative priors for the population (mean) parameters. Moreover,
we assumed that for the variance components,
[𝜎𝑎2 ] ~ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐺𝑎𝑚𝑚𝑎(𝑐, 𝑑), [𝜎𝑏2 ] ~ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐺𝑎𝑚𝑚𝑎(𝑐, 𝑑), [𝜎𝜏2 ] ~ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐺𝑎𝑚𝑚𝑎(𝑐, 𝑑),
[𝜎𝛾2 ] ~ 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝐺𝑎𝑚𝑚𝑎(𝑐, 𝑑).
In our applications, we utilized 𝑐 = 2, 𝑑 = 1. These values were chosen to be minimally
informative and such that, the preliminary analysis of simpler Bayesian hierarchical models (e.g.
only random effects for treatment effect or only trial-specific baseline hazard functions), resulted
in similar inference to those from the frequentist Cox-proportional hazards model.
We formulated a model for the indicator of at least one rehospitalization by analogy. Let 𝜋𝑖
denote the probability that individual 𝑖 had at least one rehospitalization. Then, we assumed
𝑙𝑜𝑔𝑖𝑡(𝜋𝑖 ) = (𝛼𝑠 + 𝜏𝑠 𝑍𝑖 + 𝛾𝑠 𝑊𝑖 + 𝜈𝑍𝑖 𝑊𝑖 + 𝛽′𝑋𝑖 )
(4)
with a hierarchical formulation similar to that described by equations (2)-(3).
Variations of the above models were formulated to address specific sensitivity analysis. All
models were estimated using Markov Chain Monte Carlo (MCMC) methods with WinBUGS. A
sample of code written to analyze the data is available in Appendix C.
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A.2. Missing Data
Let 𝐾𝑗𝑖 denote the indicator of comorbid condition 𝑗 in individual 𝑖. To allow for imputation of
missing values, we assume that each comorbid condition follows a Bernoulli model. Specifically,
let 𝑞𝑗 denote the probability of presence of comorbid condition 𝑗. This probability is assumed
known and based on clinical assumptions from the trial populations. That is, for trials missing
certain comorbidities, overall rates for the trial could be extrapolated from similar trials. The
MADIT I and MADIT II patients in our cohort are very similar (ischemic/CAD, EF 35-40% cutoff)
and could be used to impute rates of comorbid disease in each-other's populations. The
ischemic subgroup in SCD-HeFT (52%) could be added to this group. In contrast, patients in the
DEFINITE trial were similar to the non-ischemic group in SCD-HeFT (48%), and thus overall
rates of disease in these populations could inform one another. Therefore, we assume that
𝐾𝑗𝑖 ~𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 (𝑞𝑗 ).
Each comorbid condition is imputed via simulation prediction within each iteration of the
MCMC procedure. Thus the procedure incorporates the uncertainty on the missing value of the
comorbid condition. For sensitivity analysis, we also consider the extreme situations where the
uncertainty is removed by assuming 𝑞𝑗 = 0 (best case – subject with missing data does not have
condition) or 𝑞𝑗 = 1 (worst case – subject with missing data has condition). Finally, we note that
in model (1), the comorbidity count is 𝑊𝑖 = ∑𝑗 𝐾𝑗𝑖 . Thus, for individuals with any missing
comorbidity, the comorbidity count varies according to the imputed comorbidity values.
A3. Posterior Inference
Under the Weibull proportional hazards and logistic regression models (1)-(4) there are
natural measures of treatment effectiveness as given by the hazard ratio and odds ratio,
respectively. In addition, we consider absolute measures of treatment effectiveness. Specifically,
we calculate the risk differences (RD) and, its reciprocal, the number needed to treat (NNT). We
3
use the procedure proposed by Austin (2010) for logistic regression which naturally extends to
the proportional hazards model. We note that inferences are all within the Bayesian approach.
The advantage is that it allows us to obtain posterior credible intervals for these measures
without relying on asymptotic results or on bootstrap methods. In this Section we describe the
procedure. Specifically, under the survival model, we calculate risk differences using survival
probabilities at given times. Specifically, to compute the risk differences in the survival model, we
determine the survival probability at a given year 𝑡 post-randomization in the hypothetical
situation when all subjects have 𝑤 comorbidities and are treated with ICD. Denote the subjectspecific survival probabilities as 𝑆𝑖1 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 1). Likewise, we determine survival
probability at a given year 𝑡 post-randomization in the hypothetical situation when all subjects
have 𝑤 comorbidities but are not treated with ICD. The corresponding survival probabilities are
𝑆𝑖0 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 0). These probabilities are derived from the model specified in equation (1).
Note that these probabilities depend on covariates 𝑋𝑖 . The risk difference, based on survival
probabilities at year 𝑡 post-randomization when all subjects have 𝑤 comorbidities is computed by
integrating out those probabilities with respect to the distribution of the covariates. Using the
empirical distribution of the covariates in the study and Monte Carlo integration this leads to the
risk difference:
𝑅𝐷(𝑡|𝑊 = 𝑤) =
1
0
∑𝑁
𝑖=1 𝑆𝑖 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 1) − 𝑆𝑖 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 0).
𝑁
The inverse of the above quantity gives the number needed to treat. Note that this quantity
also depends on the number of comorbidities. We can also find a marginal risk difference by
averaging over the empirical distribution of the comorbidities and obtain:
𝑅𝐷(𝑡) =
1
0
∑𝑁
𝑖=1 𝑆𝑖 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤𝑖 , 𝑍𝑖 = 1) − 𝑆𝑖 (𝑡|𝑋𝑖 , 𝑊𝑖 = 𝑤𝑖 , 𝑍𝑖 = 0).
𝑁
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By analogy, under the logistic regression model we compute the probability of rehospitalization when
all subjects have 𝑤 comorbidities and are treated with ICD, that is, 𝜋𝑖1 (𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 1). Likewise, we
determine probability of rehospitalization when all subjects have 𝑤 comorbidities but are not treated with
ICD. The corresponding survival probabilities are 𝜋𝑖0 (𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 0). These probabilities are derived
from equation (4). Thus, in the logistic regression model we obtain the risk difference when all subjects
have 𝑤 comorbidities as:
𝑅𝐷(𝑤) =
1
0
∑𝑁
𝑖=1 𝜋𝑖 (𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 1) − 𝜋𝑖 (𝑋𝑖 , 𝑊𝑖 = 𝑤, 𝑍𝑖 = 0).
𝑁
We can also find a marginal risk difference by averaging over the empirical distribution of the
comorbidities and obtain:
1
0
∑𝑁
𝑖=1 𝜋𝑖 (𝑋𝑖 , 𝑊𝑖 = 𝑤𝑖 , 𝑍𝑖 = 1) − 𝜋𝑖 (𝑋𝑖 , 𝑊𝑖 = 𝑤𝑖 , 𝑍𝑖 = 0).
𝑅𝐷 =
𝑁
References
Little RJA and Rubin DB. (2002). Statistical Analysis with Missing Data. Wiley-Interscience; 2 edition.
Schafer (1997). Analysis of Incomplete Multivariate Data. Chapman & Hall. Chapman and Hall/CRC; 1
edition.
Austin PC (2010). Absolute risk reductions, relative risks, relative risk reductions, and numbers needed to
treat can be obtained from a logistic regression model. Journal of Clinical Epidemiology, 63: 2-6.
5
Table 1. Definitions of comorbidities by trial.
Smoking
Ischemic
heart
disease
Chronic
kidney
disease
MADIT-I
MADIT-II
DEFINITE
SCD-HeFT
Any cigarette smoking prior to
randomization
Any cigarette smoking prior to
randomization
History of current or past smoking
History of current or past smoking
Prior MI, as inclusion for trial
Prior MI, as inclusion for trial
Only non-ischemic cardiomyopathy
included (significant ischemic disease
was an exclusion)
Patients with ischemic
cardiomyopathy, prior MI, prior CABG,
or prior PCI
Calculated eGFR (<60 mL/min/1.73 m2) by the used the Chronic Kidney Disease Epidemiology Collaboration equation.*
Diabetes
History of insulin-dependent diabetes
History of diabetes
History of insulin dependent or noninsulin dependent diabetes
History of diabetes
Pulmonary
disease
Interstitial pulmonary findings on x-ray
Not available
History of COPD
History of pulmonary disease
Not available
Not available
History of atrial fibrillation and/or flutter,
either paroxysmal or chronic
History of atrial fibrillation or atrial
flutter
History of peripheral vascular disease
Not available
History of peripheral vascular disease
Not available
Atrial
fibrillation
Peripheral
vascular
disease
Unless noted, definitions are as gathered and stated on the original trial case report form. Further diagnostic details are not available.
*Levey AS, Stevens LA, Schmid CH, et al. A new equation to estimate glomerular filtration rate. Ann Intern Med. May 5
2009;150(9):604-612.
MADIT-I: Multicenter Automatic Defibrillator Implantation Trial I; MADIT-II: Multicenter Automatic Defibrillator Implantation Trial II;
DEFINITE: Defibrillators in Non-Ischemic Cardiomyopathy Treatment Evaluation; SCD-HeFT: Sudden Cardiac Death in Heart Failure
Trial; MI: myocardial infarction; CAD: coronary artery disease; CABG: coronary artery bypass grafting; PCI: percutaneous coronary
intervention; eGFR: estimated glomerular filtration rate; COPD: chronic obstructive pulmonary disease;
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Table 2. Distribution of number of comorbidities, by treatment assignment.
No. of
comorbidities
Control
(n=1577)
ICD
(n=1771)
0
69
84
1
319
358
2
569
673
3
450
459
4
149
173
5
20
23
6
1
1
7
0
0
7
Table 3. Baseline characteristics stratified by number of observed comorbidities
Trials
MADIT-I
MADIT-II
DEFINITE
SCD-HeFT
Treatment Assignment
Control
ICD
Age
Female
Race
White
Black
Other
Medical History
Hypertension
Ischemic Heart Disease
Prior CABG
Medical Therapy
Beta-Blocker
ACE Inhibitor
Antiarrhythmic Therapy
LVEF
LBBB
<2 Comorbidities ≥2 Comorbidities
P-value
(n=830)
(n=2518)
<0.001
22 (3)
157 (6)
74 (9)
1015 (40)
248 (30)
210 (8)
486 (59)
1136 (45)
0.841
388 (47)
1189 (47)
442 (53)
1329 (53)
<0.001
55 (13)
62.4 (10.6)
<0.001
254 (31)
425 (17)
<0.001
587 (71)
2078 (83)
189 (23)
318 (13)
54 (7)
122 (5)
268 (46)
132 (16)
68 (12)
1271 (55)
1918 (76)
1088 (47)
<0.001
<0.001
<0.001
624 (75)
746 (90)
14 (2)
22.7 (6.7)
172 (30)
1605 (64)
2143 (85)
92 (4)
23.6 (6.3)
390 (18)
<0.001
<0.001
0.004
0.001
<0.001
Baseline characteristics, co-morbidities, treatment assignment, and medical therapy, stratified by
number of comorbidity. Values are presented as number (%) or mean (standard deviation).
ICD: implantable cardioverter-defibrillator; MADIT-I: Multicenter Automatic Defibrillator
Implantation Trial I; MADIT-II: Multicenter Automatic Defibrillator Implantation Trial II; DEFINITE:
Defibrillators in Non-Ischemic Cardiomyopathy Treatment Evaluation; SCD-HeFT: Sudden
Cardiac Death in Heart Failure Trial; CABG: coronary artery bypass grafting; ACE: angiotensinconverting enzyme; LVEF: left-ventricular ejection fraction; LBBB: left bundle branch block.
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Table 4. Unadjusted outcomes by comorbidity.
<2 Comorbidities (n=830)
≥2 Comorbidities (n=2518)
Control (n=388)
ICD (n=442)
Control (n=1189)
ICD (n=1329)
63 (16)
42 (10)
351 (30)
270 (20)
Cardiac, arrhythmic
25 (40)
5 (12)
128 (37)
59 (22)
Cardiac, non-arrhythmic
8 (13)
13 (31)
64 (18)
70 (26)
Cardiac, NOS
7 (11)
2 (5)
46 (13)
49 (18)
Non-cardiac
9 (14)
9 (21)
61 (17)
62 (23)
Unknown
14 (22)
13 (31)
52 (15)
30 (11)
Rehospitalization
142 (54)
192 (63)
629 (62)
853 (73)
ICD complication
0 (0)
57 (18)
5 (1)
249 (21)
0 (0)
10 (3)
0 (0)
23 (2)
Death
Infection
Values are presented as number (%); cause-specific mortality presented as percentages of all deaths. ICD complications are
described in the Appendix, and only apply to ICD patients and those in the control group who crossed-over.
ICD: implantable cardioverter-defibrillator; NOS: not otherwise specified.
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Table 5. All missing comorbidities imputed to true (patient has disease).
No. of
comorbidities
Log HR for ICD vs. Control
Absolute Survival Difference at 5
years (ICD – Control)
Median
95% CI
Median
95% CI
0
-1.26
(-2.15, -0.35)
0.10
(0.05, 0.16)
1
-1.08
(-1.91, -0.27)
0.12
(0.06, 0.17)
2
-0.91
(-1.69, -0.12)
0.13
(0.07, 0.18)
3
-0.73
(-1.48, 0.01)
0.13
(0.08, 0.17)
4
-0.56
(-1.27, 0.18)
0.11
(0.07, 0.15)
5
-0.38
(-1.08, 0.38)
0.08
(0.02, 0.14)
6
-0.21
(-0.89, 0.58)
0.04
(-0.05, 0.13)
posterior probability of interaction between treatment and comorbidity, p=0.002.
HR: hazard ratio; ICD: implantable cardioverter-defibrillator; CI: confidence interval.
10
Table 6. All missing comorbidities imputed to false (patient does not have disease).
No. of
comorbidities
Log HR for ICD vs. Control
Absolute Survival Difference at 5
years (ICD – Control)
Median
95% CI
Median
95% CI
0
-0.94
(-1.79, -0.15)
0.13
(0.06, 0.19)
1
-0.77
(-1.54, -0.04)
0.13
(0.08, 0.19)
2
-0.60
(-1.31, 0.05)
0.13
(0.08, 0.17)
3
-0.43
(-1.14, 0.26)
0.10
(0.05, 0.15)
4
-0.26
(-0.99, 0.46)
0.06
(-0.03, 0.14)
5
-0.08
(-0.86, 0.7)
0.00
(-0.13, 0.12)
6
0.09
(-0.76, 0.96)
-0.05
(-0.2, 0.09)
posterior probability of interaction between treatment and comorbidity, p=0.02.
HR: hazard ratio; ICD: implantable cardioverter-defibrillator; CI: confidence interval.
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