Sequential Allocation for Randomized Experiments

Description

Potential randomization schemes are prospectively evaluated when units are assigned to treatment arms upon entry into the experiment. The schemes are evaluated for balance on covariates and on predictability (i.e., how well could a site worker guess the treatment of the next unit enrolled).

Details

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

McEntegart, D. J. (2003). The Pursuit of Balance Using Stratified and Dynamic Randomization Techniques: An Overview. Drug information Journal 37: 293-308.

Rosenberger, W. F. and J. M. Lachin (2004). Randomization in Clinical Trials: Theory and Practice. New York: Wiley.

Maximum Allocation Imbalance for Covariates

Description

Evaluate maximum of the allocation imbalance for all covariates

Usage

MAICimbal(alloc, xmat)

Arguments

Value

maximum of covariate imbalances

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

See Also

WAICimbal

Examples

alloc <- rbinom(60,1,.5)
xmat <- matrix(rbinom(240,1,.4),ncol=4)

MAICimbal(alloc, xmat) 

Sequential Allocation for Prospective Experiments

Description

Simulates results of allocations using complete randomization (CR), random allocation rule (RAR), biased coin design (BCD), permuted block design (PBD), stratified permuted block design (SPBD), covariate-adaptive randomization (CAR), big stick design (BSD), and covariate-adjusted imbalance tolerance (CAIT) designs. The order of the prospective enrollees is permuted for a preset number of iterations; for each iteration, the allocations are determined for each of the methods listed above. The allocations are then evaluated for balance on the covariates and for predictability (i.e., how well an observer could guess the next treatment assignment).

Usage

SeqAlloc(xmat, carwt, strata = NULL, blksize, pbcd, pcar, bsdtol, caittol, 
         niter, seed = 12345)

Arguments

Value

List containing summary statistics (minimum, 25th percentile, median, mean, 75th percentile, 90th percentile, 95th percentile, maximum) for evaluation measures, including AI, Rsquared, MAIC, WAIC, perccorr, and perccorr_strat.

Note

Because the program allows for sequential allocation methods, it can be slow when the data set and/or number of iterations is large.

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Blackwell, David and J. L. Hodges (1957). Design for the Control of Selection Bias. Annals of Mathematical Statistics 28: 449-460.

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Rosenberger, W. F. and Lachin, J. M. (2004). Randomization in Clinical Trials: Theory and Practice. New York: Wiley.

Examples

sampsize <- 200
percent <- c(0.5,0.8,0.2,0.4)
set.seed(200)

xmat <- matrix(rbinom(sampsize*length(percent),1,rep(percent,sampsize)),
              nrow=sampsize,ncol=length(percent),byrow=TRUE)
colnames(xmat) <- c("C1","C2","C3","C4")
strat_factor <- xmat[,2]*2 + xmat[,4] + 1

SeqAlloc(xmat,carwt=c(.4,.3,.2,.1),strata=strat_factor,blksize=c(2,6),
         pbcd=.7,pcar=.8,bsdtol=2,caittol=5,niter=10, seed = 20850)

Plot the evaluation criteria for the designs

Description

Provides boxplots and scatterplots of balance and predictability measures for candidate sequential allocations.

Usage

SeqAllocplot(mysim, bporder = NULL, stratum = F, cexsize = 0.7)

Arguments

Value

Produces selected plots of predictability and balance for randomization schemes.Requires user to click window or press "enter" to progress through plots.

Note

These are example plots; the plotting code can be extracted from this function or the function is easily modified if different plots are desired.

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332

See Also

SeqAlloc

Examples

sampsize <- 200
percent <- c(0.5,0.8,0.2,0.4)
set.seed(200)

xmat <- matrix(rbinom(sampsize*length(percent),1,rep(percent,sampsize)),
              nrow=sampsize,ncol=length(percent),byrow=TRUE)
colnames(xmat) <- c("C1","C2","C3","C4")
strat_factor <- xmat[,1]*4 + xmat[,2]*2 + xmat[,4] + 1

mysim <- SeqAlloc(xmat,carwt=c(.4,.3,.2,.1),strata=strat_factor,blksize=c(2,6),
                   pbcd=.7,pcar=.67,bsdtol=2,caittol=3,niter=10, seed = 30924)

SeqAllocplot(mysim,bporder = c(3,4,7,8), stratum = FALSE, cexsize=0.6)

Weighted Average of Allocation Imbalance for Covariates

Description

Evaluate weighted average of the allocation imbalance for all covariates

Usage

WAICimbal(alloc, xmat, carwt)

Arguments

Value

Weighted average of covariate imbalances

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

See Also

MAICimbal

Examples

alloc <- rbinom(60,1,.5)
xmat <- matrix(rbinom(240,1,.4),ncol=4)
carwt <- c(.1,.3,.2,.4)

WAICimbal(alloc, xmat, carwt) 

Biased Coin Design Allocation

Description

Allocates observations sequentially using a biased coin design. If equal numbers have been allocated to treatment and control, the next observation is allocated to treatment with probability 1/2. If there are more units assigned to treatment than control, the next observation is allocated to control with probability p; if there are more units assigned to control than treatment, the next observation is allocated to treatment with probability p.

Usage

bcdalloc(n, pbcd, tol = 1e-06)

Arguments

Value

Vector of zeroes and ones indicating the allocation of each unit to treatment (1) or control (0)

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Efron, B. (1971). Forcing a sequential experiment to be balanced. Biometrika, 58(3), 403-417.

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Examples

bcdalloc(20,0.7)

Sequential Allocation for CAIT Method

Description

Performs the sequential allocation for the covariate-adjusted-imbalance-tolerance (CAIT) method of allocating observations in a randomized experiment.

Usage

caitalloc(carwt_x1, p, tol)

Arguments

Value

Vector with the allocation to treatment (denoted by 1) and control (denoted by 0)

Note

This function works for categorical as well as binary covariates. The matrix carwt_x1 can be created from an X matrix by using the function weight_x in this package.

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Pocock, S.J. and R. Simon (1975). Sequential Treatment Assignment with Balancing for Prognostic Factors in a Controlled Clinical Trial. Biometrics 31, 103-115.

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

See Also

weight_x

Examples

wtmat <- matrix(runif(10000), ncol=100) 
wtmat <- (wtmat * lower.tri(wtmat)) + t(wtmat * lower.tri(wtmat)) 
diag(wtmat) <- 1 

caitalloc(wtmat,1,3)

Sequential Allocation Using Covariate Adaptive Randomization

Description

Performs the sequential allocation for the covariate-adjusted randomization (CAR) method of allocating observations in a randomized experiment.

Usage

caralloc(xmat, carwt, p, tol)

Arguments

Value

Vector with the allocation to treatment (denoted by 1) and control (denoted by 0)

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Pocock, S. J. and R. Simon (1975). Sequential Treatment Assignment with Balancing for Prognostic Factors in A Controlled Clinical Trial. Biometrics 31: 103-115.

Examples

sampsize <- 200
percent <- c(0.5,0.8,0.2,0.4)
carwt <- c(.4,.3,.2,.1)

set.seed(5798)

xmat <- matrix(rbinom(sampsize*length(percent),1,rep(percent,sampsize)),
              nrow=sampsize,ncol=length(percent),byrow=TRUE)
colnames(xmat) = c("C1","C2","C3","C4")
strat_factor = xmat[,1]*4 + xmat[,2]*2 + xmat[,4] + 1

caralloc(xmat,carwt,1,3)

Evaluate Optimal Guess

Description

This function calculates the optimal guess for the allocation of the next observation in a sequential allocation scheme, and evaluates whether that guess is correct.

Usage

cguess(alloc)

Arguments

Details

Blackwell and Hodges (1957) proposed evaluating the predictability of a randomized allocation scheme by having an observer guess that the next unit would be allocated to the treatment arm that currently has fewer subjects. This function creates the guesses that would be generated by an observer following this strategy, then compares them with the actual allocation. If units 1 through j have more treatment than control units, then the observer guesses that observation j+1 is assigned to the control arm. If units 1 through j have more control than treatment units, then the observer guesses that observation j+1 is assigned to the treatment arm. If units 1 through j have the same number of control and treatment units, then the observer flips a coin with probability 0.5 for the allocation of the next unit. To reduce the variability in the percentage guessed correctly, this is recorded as a response of 0.5; however, the function is easily changed to allow the observer to guess treatment or control with probability 0.5 for that case.

Value

Vector of length length(alloc) containing values 0, 0.5, and 1. A value is 1 if the guess is correct, 0 if it is incorrect, and 0.5 if the allocation preceding the observation in question was equal.

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Blackwell, David and Hodges, J. L. (1957). Design for the Control of Selection Bias. Annals of Mathematical Statistics 28: 449-460.

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Examples

alloc <- rbinom(50,1,.5)

cguess(alloc)

Evaluate covariate imbalance using R-squared.

Description

Evaluate imbalance in the covariates from the allocation by finding R-squared for a centered regression through the origin.

Usage

covimbal(y, xmatc)

Arguments

Value

Value of R-squared from centered regression

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Examples

alloc <- rbinom(60,1,.5)
xmat <- matrix(rbinom(240,1,.4),ncol=4)
xmatc <- scale(xmat,center=TRUE,scale=FALSE) 

covimbal(alloc,xmatc)

Stratified permuted block allocation

Description

Allocates observations to treatment (1) or control (0) within strata

Usage

stratblockalloc(n, blksize)

Arguments

Details

Allocates observations within each block of size blksize so that half of the observations in each block are in the treatment group and the other half are in the control group.

Value

vector of allocations to treatment (1) or control (0) group

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Examples

stratblockalloc(20,2)

Find summary statistics

Description

Finds the Minimum, 25th percentile, Median, Mean, 75th percentile, 90th percentile, 95th percentile, and Maximum of the vector

Usage

summary95(x)

Arguments

Details

The code from the 'summary' function in R is modified to include the 90th and 95th percentiles

Value

vector of summary statistics for data

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr, adapted from 'summary' function in R.

Examples

summary95(rnorm(40))

Calculate weight matrix from matrix of covariates

Description

Calculate a vector of weights relating vector xrow with the rows of xmat, according to the weights in vector carwt. For each row in xmat, the positions of agreement with xrow are found. The weight for that row of xmat is then the sum of the elements of carwt corresponding to the matches.

Usage

weight_x(xrow, xmat, carwt)

Arguments

Value

Vector of weights relating vector xrow to the rows of xmat

Author(s)

Xiaoshu Zhu xiaoshuzhu@westat.com and Sharon Lohr

References

Lohr, S. and X. Zhu (2015). Randomized Sequential Individual Assignment in Social Experiments: Evaluating the Design Options Prospectively. Sociological Methods and Research. [Advance online publication: December 27, 2015] doi: 10.1177/0049124115621332.

Examples

xrow <- c(1,0,1,1)
xmat <- matrix(rbinom(60,1,.4),ncol=4)
carwt <- c(.6,.4,.2,.1)

weight_x(xrow, xmat, carwt)