jumble: Stratified Randomization

Kevin W. McConeghy

2020-01-06

Source: vignettes/04-stratification.Rmd

Welcome to jumble! This program was written as a companion to academic work performed by researchers at Brown University to perform different randomization strategies with cluster-randomized nursing home trials.

Introduction

In this vignette we go over how to do a stratified randomization, both with a single category and using a distance measure.

Example dataset

The example dataset used in this package is freely available and comes from a clinical HIV therapy trial conducted in the 1990s.(Hammer SM 1996) See: ?jumble::ACTG175

Note An important limitation of most clinical trials is that enrollment is graduated with individual patients consented and enrolled one at a time. However, cluster randomized trials can often identify all potential clusters before randomization occurs which allows more finessed randomization techniques.

Load dataset

Single covariate stratification

A common method for stratification is to take 1-2 covariates which are considered to be very important clinically for the outcome. Often a cluster is used (i.e. randomized within facility, hospital, village). In a cluster randomized trial, a variable which is correlated with the cluster may be more likely to be imbalanced so if you were particularly concerned about one thing that would be a potential.

Identify a strong predictor of the primary outcome

Allow the trial design is a little more complicated, we will simply look at the cens event (A CD4 T cell drop of 50 or more, AIDs defining event, or death).

## 
## Call:
## glm(formula = cens ~ ., family = binomial, data = df[, c("cens", 
##     vars)])
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.2829  -0.8319  -0.6907   1.2494   2.1987  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)   
## (Intercept) -0.477387   1.257932  -0.380  0.70432   
## age          0.004197   0.008559   0.490  0.62384   
## race        -0.283396   0.180542  -1.570  0.11649   
## gender       0.329055   0.290614   1.132  0.25752   
## symptom      0.527787   0.175816   3.002  0.00268 **
## wtkg         0.009280   0.005528   1.679  0.09322 . 
## hemo        -0.275731   0.356463  -0.774  0.43921   
## msm         -0.124234   0.254553  -0.488  0.62552   
## drugs       -0.576851   0.249003  -2.317  0.02052 * 
## karnof      -0.018931   0.012015  -1.576  0.11513   
## oprior       0.114416   0.436392   0.262  0.79318   
## str2         0.477076   0.152291   3.133  0.00173 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1228.4  on 1053  degrees of freedom
## Residual deviance: 1184.6  on 1042  degrees of freedom
## AIC: 1208.6
## 
## Number of Fisher Scoring iterations: 4

We identify a few important predictors, str2 is an indicator for prior exposure to anti-retroviral therapy.

In general if you have no prior knowledge, you should select strata which are highly predictive of outcome.

For an example, lets examine str2.

Random assignment within symptom 

df_str <- rnd_str(df, str2, pidnum)
head(df_str)
##   str2 pidnum group
## 1    0  10896     b
## 2    0  10900     a
## 3    0  10907     b
## 4    0  10913     a
## 5    0  10920     a
## 6    0  10923     b

Join back to dataset

## Joining, by = c("str2", "pidnum")
##    
##       a   b
##   0 218 218
##   1 309 309

Nearly equal assignment by strata=0 or strata=1

How does our stratified randomization perform vs. simple randomization?
We can see the how much tighter covariate differences are if we perform a simulation of 10000 randomizations under either strategy.

## 10000 randomizations for Random Assignment - Simple 
## 
## Time difference of 14.18693 secs
## 10000 randomizations for Stratification, one var
## Time difference of 1.118263 mins

As you can see, stratification is very effective at tightening a simple strata variable.

What about a continuous measure age? You can stratify by category:

## 10000 randomizations for - continuous categorized
## Time difference of 1.181809 mins

Works pretty well!

If these variables are strongly correlated with outcome then you will see some reduction in variance, but re-randomization will work better if you have a large number of predictors with weak associations with the outcome.

Multivariate stratification

An additional option is to create multiple strata, but this can get complicated and runs into issues with finite numbers in individual cells. It is dependent on sample size, but usually is not feasible to do more than 1:3 stratifications.

The alternative to a single stratification, is to use a multivariate balance measure, create arbitrary strata (since its a continuous measure) and randomly assign within strata.

Mahalanobis computation

In this case we are computing the distance of each observation from the group mean.

\[ M \equiv ({X_i} - \bar{X}) ' cov(X)^{-1} ({X_i} - \bar{X}) \]

## 10000 randomizations for M-distance stratification
## Time difference of 1.243111 mins

Not significantly different from simple randomization!

##           age  race gender symptom  wtkg  hemo   msm drugs karnof oprior  str2
## age      1.00 -0.07   0.09    0.03  0.18 -0.26  0.19  0.07  -0.11   0.06  0.07
## race    -0.07  1.00  -0.29   -0.06 -0.06 -0.08 -0.32  0.10   0.06   0.03 -0.07
## gender   0.09 -0.29   1.00    0.06  0.22  0.11  0.61 -0.17  -0.01   0.06  0.00
## symptom  0.03 -0.06   0.06    1.00  0.01 -0.06  0.11  0.01  -0.10   0.03  0.08
## wtkg     0.18 -0.06   0.22    0.01  1.00 -0.09  0.15  0.00   0.03   0.00 -0.09
## hemo    -0.26 -0.08   0.11   -0.06 -0.09  1.00 -0.39 -0.09   0.09   0.02  0.11
## msm      0.19 -0.32   0.61    0.11  0.15 -0.39  1.00 -0.24  -0.06   0.02 -0.02
## drugs    0.07  0.10  -0.17    0.01  0.00 -0.09 -0.24  1.00  -0.06  -0.04  0.01
## karnof  -0.11  0.06  -0.01   -0.10  0.03  0.09 -0.06 -0.06   1.00  -0.06 -0.09
## oprior   0.06  0.03   0.06    0.03  0.00  0.02  0.02 -0.04  -0.06   1.00  0.13
## str2     0.07 -0.07   0.00    0.08 -0.09  0.11 -0.02  0.01  -0.09   0.13  1.00

Notice how most variables have very weak correlations? This is why the stratified analysis with a Mahalanobis distance doesn’t perform better than simple randomization. The Mahalanobis distance is an absolute measure from the covariate means, so it balances groups well when covariates are highly correlated (i.e. units grouped together by distance have similar values). But if each variable randomly varies from the mean, then using distance will not balance the groups more than simple randomization. For example, two units could be wildly different by characteristics but technically have the same distance from the means. The only variable which balances at all is MSM which is correlated a little with gender.

If we repeat the analysis using MSM, Hemo, gender, and age.

Mahalanobis distance using only 4 moderately correlated variables.

## 10000 randomizations for M-distance stratification
## Time difference of -2.589606 secs

See how the distance measure performs better when selecting a few correlated variables?

Contacting authors

The primary author of the package was Kevin W. McConeghy. See here

References

Hammer SM, et al. 1996. “A Trial Comparing Nucleoside Monotherapy with Combination Therapy in Hiv-Infected Adults with Cd4 Cell Counts from 200 to 500 Per Cubic Millimeter.” N Eng J M 335: 1081–90.