mlma: Mixed-Level Meta-Analysis for Survival Data

Description

Mixed-level meta-analysis of a time-to-event endpoint. Combines evidence from individual-level studies and from study-level trials where aggregate evidence is in the form of survival proportions along a survival curve, e.g. surv = c(.9,.5,.2) at months = c(1,6,12).

Fixed effects that share the same name in the study-level and individual-level model formula are treated as having a common effect. This allows flexibility in separating study-level and patient-level effects by using distinct term labels.

MLMA allows for a general random effects structure; multivariate normal frailties.

Usage

mlma(
   ipd.formula,
   meta.formula,
   random.formula,
   ipd.data,
   meta.data,
   ipd.groups,
   meta.groups,
   sigma2,
   study.group.interaction,
   max.iter=5,
   min.sample=300,
   est.delta=.1,
   mc.step=2,
   df=30,
   echo = TRUE,
   max.sample = NULL,
   converge.index,
   error=1/100,
   fixed=FALSE
)

Arguments

ipd.formula

Formula for patient-level Cox model as would be supplied to coxph

meta.formula

Formula for study-level data surv~log(time)+trt*x

random.formula

Formula of random effects conditional on group factor as would be specified in coxme, i.e. ~(1+trt|study)

ipd.data

Patient-level data frame

meta.data

Study-level data frame containing survival estimates by study and treatment group

ipd.groups

Number of clusters in patient-level dataset

meta.groups

Number of clusters in study-level dataset

sigma2

Variances for aggregate survival estimates of meta.data

study.group.interaction

Factor that is the study and treatment group interaction for meta.data

max.iter

Maximum iterations of MCEM algorithm

min.sample

Starting sample size for samples at E step

est.delta

Stopping criterion threshold based on standardized relative differences for model parameters

mc.step

Determines how sample size is increased to reduce MC error

Degrees of freedom for proposal multivariate T distribution

echo

Logical, if TRUE prints maximum weight and standardized difference between iteration estimates of model parameters

max.sample

If given, then a fixed escalation is used so that if max.iter is reached, final sample size is max.sample

converge.index

Index of which parameters to include in the convergence assessment. Default is all parameters

error

MLE stopping rule; MLE considered reached when change in log-likelihood for NR maximization is less than error

fixed

Logical, should fixed effects model be used

Value

List of
iterationsNumber of MCEM iterations
est.convergeMaximal difference in parameter estimates between consecutive iterations relative to their standard error
max.weightMaximum importance weight at E step
sd.loglikStandard deviaiton of each importance-sample conditional Log-likelihood, which can be helpful for monitoring the performance of the proposal distribution
loglikImportance-sample average Conditional Log-likelihood at each iteration
coefFinal estimated fixed effects
vcovFinal estimated frailty variances
clusterMean frailty effects at final MCEM iteration
varList of variance of model parameters with components coef and vcov

Details

The estimation is based on likelihood methods for the combined likelihood contributions of the individual- and trial-level data. The individual-level model is a Cox proportional hazards mixed model (PHMM). The study-level model is a GLMM on survival proportions (KM estimates) for each treatment group within study. The link is the log-complementary-log transform. When all outcomes across studies and types follow the PHMM, the log-negative-log survival is linear in the covariates of the individual-level model.

Conditional on the frailty terms, the transformed survival estimates are assumed to have an MVN distribution where the covariance-variance matrix is regarded as known and is determined by the KM survival estimate variances sigma2 and the correlation between survival estimates within the same cluster.

Shared parameters between the individual-level and study-level models are determined by common term.labels for ipd.formula and meta.formula. So, for example, if ipd.formula was Surv(time,event)~trt*x and meta.formula was surv~log(time)+trt*x then the shared terms are trt,x and trt:x and these would be combined. If instead meta.formula was surv~log(time)+trt*x.bar, then only trt would be common, and only study level data would be used to estimate x.bar and trt:x.bar factors.

When random effects are included, an MCEM algorithm is used to obtain the full model estimates. The expectation step utilizes importance sampling based on a multivariate T distribution for the joint frailties.

The stopping rule for the EM algorithm is when the maximum of the last three standardized maximum differences are below est.delta or when max.iter has been reached.

The MC error is monitored by the CV of the maximal standardized parameter differences, based on the three most recent iterations. If the CV of the current iteration is greater than the last, the sample size is increased according to $N+\frac{N}{mc.step}$.

The stopping rule for the M step Newton-Rapshon procedure is based on the relative difference of consecutive log-likelihood, stopping when this difference is <error

Examples

Run this code

#MIXED-LEVEL META-ANALYSIS SURVIVAL DATA

data(ipd.data)
example(ipd.data)

data(meta.data)
example(meta.data)

set.seed(401)

#FIXED EFFECTS

fit <- mlma(
    Surv(time,event)~trt,surv~log(time)+trt,
    ipd.data=ipd.data,
    meta.data=meta.data,
    ipd.groups=8,meta.groups=2,
    study.group=meta.data$sub.group,
    sigma2=meta.data$sigma2,
    fixed=TRUE
    )

fit$coef				#MODEL FIT
sqrt(diag(fit$var))	          	#STANDARD ERROR


#MIXED EFFECTS, BASELINE FRAILTY BY GROUP

fit <- mlma(
    Surv(time,event)~trt,surv~log(time)+trt,~(1|group),
    ipd.data,
    meta.data,
    ipd.groups=8,meta.groups=2,
    study.group=meta.data$sub.group,
    sigma2=meta.data$sigma2,
    max.iter=3
    )

fit$coef				#MODEL FIT
sqrt(diag(fit$var$coef))		#STANDARD ERROR

#ECOLOGICAL BIAS
#MIXED-LEVEL WITH ECOLOGICAL BIAS
#MIXED EFFECTS MLMA SEPARATING STUDY-LEVEL AND INDIVIDUAL-LEVEL X EFFECTS
#SIMULATION EFFECTS TO GENERATE MIXED.BIASED

beta = array(c(0,-.5,-.2,0,-.4,0))
names(beta) <- c("int","trt","x","x.bar","trt and x","trt and x.bar")

data(mixed.biased)

fit <- mlma(
     Surv(time,event)~trt*I(x-x.bar)+trt*x.bar,surv~log(time)+trt*x.bar,
     ~(1|group),
    mixed.biased$ipd,
    mixed.biased$meta,
    ipd.groups=5,meta.groups=5,
    mixed.biased$meta$sigma2,
    mixed.biased$meta$sub.group,
    max.iter=3
    )

fit$coef				#MODEL FIT
sqrt(diag(fit$var$coef))		#STANDARD ERROR


sqrt(diag(fit$vcov))	#ESTIMATED FRAILTY STANDARD DEVIATION


#95\% CI FOR EQUIVALENCE OF STUDY- AND INDIVIDUAL-LEVEL X (MAIN EFFECT)
C <- c(0,-1,1,0,0,0,0)
ci(C,fit$coef,fit$var$coef)

#95\% FOR EQUIVALENCE OF STUDY- AND INDIVIDUAL-LEVEL X-TRT (INTERACTION)
C <- c(0,0,0,-1,1,0,0)
ci(C,fit$coef,fit$var$coef)