Imputation for categorical variables using log linear models

Description

This function performs multiple imputation under a log-linear model as described by Schafer (1997), using his cat package, either with or without posterior draws.

Usage

catImp(
  obsData,
  M = 10,
  pd = FALSE,
  type = 1,
  margins = NULL,
  steps = 100,
  rseed
)

Arguments

Details

By default catImp will impute using a log-linear model allowing for all two-way associations, but not higher order associations. This can be modified through use of the type and margins arguments.

With pd=FALSE, all imputed datasets are generated conditional on the MLE of the model parameter, referred to as maximum likelihood multiple imputation by von Hippel and Bartlett (2021).

With pd=TRUE, regular 'proper' multiple imputation is used, where each imputation is drawn from a distinct value of the model parameter. Specifically, for each imputation, a single MCMC chain is run, iterating for steps iterations.

Imputed datasets can be analysed using withinBetween, scoreBased, or for example the bootImpute package.

Value

A list of imputed datasets, or if M=1, just the imputed data frame.

References

Schafer J.L. (1997). Analysis of incomplete multivariate data. Chapman & Hall, Boca Raton, Florida, USA.

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a partially observed categorical dataset
set.seed(1234)
n <- 100

#for simplicity we simulate completely independent variables
temp <- data.frame(x1=ceiling(3*runif(n)), x2=ceiling(2*runif(n)), x3=ceiling(2*runif(n)))

#make some data missing
for (i in 1:3) {
  temp[(runif(n)<0.25),i] <- NA
}

#impute using catImp, assuming two-way associations in the log-linear model
imps <- catImp(temp, M=10, pd=FALSE, rseed=4423)

#impute assuming a saturated log-linear model
imps <- catImp(temp, M=10, pd=FALSE, type=3, rseed=4423)

Simulated example data with continuous outcome measured repeatedly over time

Description

A dataset in the wide form containing simulated data with a repeatedly measured outcome. Some outcome values are missing. The missing data pattern is monotone. There are two baseline covariates.

Usage

ctsTrialWide

Format

A data frame with 500 rows and 7 variables:

id

ID for individual

trt

A numeric 0/1 variable indicating control or active treatment group

v

A baseline covariate

y0

Baseline measurement of the outcome variable

y1

Outcome measurement at visit 1

y2

Outcome measurement at visit 2

y3

Outcome measurement at visit 3

Imputation for a mixture of continuous and categorical variables using the general location model.

Description

This function performs multiple imputation under a general location model as described by Schafer (1997), using the mix package. Imputation can either be performed using posterior draws (pd=TRUE) or conditonal on the maximum likelihood estimate of the model parameters (pd=FALSE), referred to as maximum likelihood multiple imputation by von Hippel and Bartlett (2021).

Usage

mixImp(
  obsData,
  nCat,
  M = 10,
  pd = FALSE,
  marginsType = 1,
  margins = NULL,
  designType = 1,
  design = NULL,
  steps = 100,
  rseed
)

Arguments

Details

See the descriptions for marginsType, margins, designType, design and the documentation in ecm.mix for details about how to specify the model.

Imputed datasets can be analysed using withinBetween, scoreBased, or for example the bootImpute package.

Value

A list of imputed datasets, or if M=1, just the imputed data frame.

References

Schafer J.L. (1997). Analysis of incomplete multivariate data. Chapman & Hall, Boca Raton, Florida, USA.

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a partially observed dataset with a mixture of categorical and continuous variables
set.seed(1234)

n <- 100

#for simplicity we simulate completely independent categorical variables
x1 <- ceiling(3*runif(n))
x2 <- ceiling(2*runif(n))
x3 <- ceiling(2*runif(n))
y <- 1+0.5*(x1==2)+1.5*(x1==3)+x2+x3+rnorm(n)

temp <- data.frame(x1=x1,x2=x2,x3=x3,y=y)

#make some data missing in all variables
for (i in 1:4) {
  temp[(runif(n)<0.25),i] <- NA
}

#impute conditional on MLE, assuming two-way associations in the log-linear model
#and main effects of categorical variables on continuous one (the default)
imps <- mixImp(temp, nCat=3, M=10, pd=FALSE, rseed=4423)

Multivariate normal model imputation

Description

This function performs multiple imputation under a multivariate normal model as described by Schafer (1997), using his norm package, either with or without posterior draws.

Usage

normImp(obsData, M = 10, pd = FALSE, steps = 100, rseed)

Arguments

Details

This function imputes from a multivariate normal model with unstructured covariance matrix, as described by Schafer (1997). With pd=FALSE, all imputed datasets are generated conditional on the MLE of the model parameter, referred to as maximum likelihood multiple imputation by von Hippel and Bartlett (2021).

With pd=TRUE, regular 'proper' multiple imputation is used, where each imputation is drawn from a distinct value of the model parameter. Specifically, for each imputation, a single MCMC chain is run, iterating for steps iterations.

Imputed datasets can be analysed using withinBetween, scoreBased, or for example the bootImpute package.

Value

A list of imputed datasets, or if M=1, just the imputed data frame.

References

Schafer J.L. (1997). Analysis of incomplete multivariate data. Chapman & Hall, Boca Raton, Florida, USA.

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a partially observed dataset from multivariate normal distribution
set.seed(1234)
n <- 100
temp <- MASS::mvrnorm(n=n,mu=rep(0,4),Sigma=diag(4))

#make some values missing
for (i in 1:4) {
  temp[(runif(n)<0.25),i] <- NA
}

#impute using normImp
imps <- normImp(data.frame(temp), M=10, pd=FALSE, rseed=4423)

Normal regression imputation of a single variable

Description

Performs multiple imputation of a single continuous variable using a normal linear regression model. The covariates in the imputation model must be fully observed. By default normUniImp imputes every dataset using the maximum likelihood estimates of the imputation model parameters, which here coincides with the OLS estimates, referred to as maximum likelihood multiple imputation by von Hippel and Bartlett (2021). If pd=TRUE is specified, it instead performs posterior draw Bayesian imputation.

Usage

normUniImp(obsData, impFormula, M = 5, pd = FALSE)

Arguments

Details

Imputed datasets can be analysed using withinBetween, scoreBased, or for example the bootImpute package.

Value

A list of imputed datasets, or if M=1, just the imputed data frame.

References

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a dataset with one partially observed (conditionally) normal variable
set.seed(1234)
n <- 100
x <- rnorm(n)
y <- x+rnorm(n)
x[runif(n)<0.25] <- NA
temp <- data.frame(x=x,y=y)

#impute using normImp
imps <- normUniImp(temp, y~x, M=10, pd=FALSE)

Reference based imputation of repeated measures continuous data

Description

Performs multiple imputation of a repeatedly measured continuous endpoint in a randomised clinical trial using reference based imputation as proposed by doi:10.1080/10543406.2013.834911Carpenter et al (2013). This approach can be used for imputation of missing data in randomised clinical trials.

Usage

refBasedCts(
  obsData,
  outcomeVarStem,
  nVisits,
  trtVar,
  baselineVars = NULL,
  baselineVisitInt = TRUE,
  type = "MAR",
  M = 5
)

Arguments

Details

Unlike most implementations of reference based imputation, this implementation imputes conditional on the maximum likelihood estimates of the model parameters, rather than a posterior draw. If one is interested in frequentist valid inferences, this is ok provided the bootstrapping used, for example with using the bootImpute package.

Intermediate missing values are imputed assuming MAR, based on the mixed model fit to that patient's treatment arm. Monotone missing values are imputed using the specified imputation type.

Baseline covariates must be numeric variables. If you have factor variables you must code these into suitable dummy indicators and pass these to the function.

Value

A list of imputed datasets, or if M=1, just the imputed data frame.

References

Carpenter JR, Roger JH, Kenward MG. Analysis of Longitudinal Trials with Protocol Deviation: A Framework for Relevant, Accessible Assumptions, and Inference via Multiple Imputation. (2013) 23(6) 1352-1371

von Hippel PT & Bartlett JW (2019) Maximum likelihood multiple imputation: Faster imputations and consistent standard errors without posterior draws arXiv:1210.0870v10.

Examples

#take a look at ctsTrialWide data
head(ctsTrialWide)

#impute the missing outcome values twice assuming MAR
imps <- refBasedCts(ctsTrialWide, outcomeVarStem="y", nVisits=3, trtVar="trt",
                    baselineVars=c("v", "y0"), type="MAR", M=2)

#now impute using jump to reference method
imps <- refBasedCts(ctsTrialWide, outcomeVarStem="y", nVisits=3, trtVar="trt",
                    baselineVars=c("v", "y0"), type="J2R", M=2)

#for frequentist valid inferences we use bootstrapping from the bootImpute package
## Not run: 
  #bootstrap 10 times using 2 imputations per bootstrap. Note that to do this
  #we specify nImp=2 to bootImpute by M=1 to the refBasedCts function.
  #Also, 10 bootstraps is far too small to get reliable inferences. To do this
  #for real you would want to use a lot more (e.g. at least nBoot=1000).
  library(bootImpute)
  bootImps <- bootImpute(ctsTrialWide, refBasedCts, nBoot=10, nImp=2,
                         outcomeVarStem="y", nVisits=3, trtVar="trt",
                         baselineVars=c("v", "y0"), type="J2R", M=1)

  #write a small wrapper function to perform an ANCOVA at the final time point
  ancova <- function(inputData) {
    coef(lm(y3~v+y0+trt, data=inputData))
  }
  ests <- bootImputeAnalyse(bootImps, ancova)
  ests

## End(Not run)

Score based variance estimation for multiple imputation

Description

This function implements the score based variance estimation approach described by von Hippel and Bartlett (2021), which is based on earlier work by Wang and Robins (1998).

Usage

scoreBased(imps, analysisFun, scoreFun, pd = NULL, dfComplete = NULL, ...)

Arguments

Value

A list containing the overall parameter estimates, its corresponding covariance matrix, and degrees of freedom for each parameter.

References

Wang N., Robins J.M. (1998) Large-sample theory for parametric multiple imputation procedures. Biometrika 85(4): 935-948. doi:10.1093/biomet/85.4.935.

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a partially observed dataset
set.seed(1234)
n <- 100
x <- rnorm(n)
y <- x+rnorm(n)
y[1:50] <- NA
temp <- data.frame(x,y)
#impute using normUniImp, without posterior draws
imps <- normUniImp(temp, y~x, M=10, pd=FALSE)

#define a function which performs our desired analysis on a dataset, returning
#the parameter estimates
yonx <- function(inputData) {
  fitmod <- lm(y~x, data=inputData)
  list(est=c(fitmod$coef,sigma(fitmod)^2))
}

#define a function which when passed a dataset and parameter
#vector, calculates the likelihood score vector
myScore <- function(inputData, parm) {
  beta0 <- parm[1]
  beta1 <- parm[2]
  sigmasq <- parm[3]
  res <- inputData$y - beta0 - beta1*inputData$x
  cbind(res/sigmasq, (res*inputData$x)/sigmasq, res^2/(2*sigmasq^2)-1/(2*sigmasq))
}

#call scoreBased to perform variance estimation
scoreBased(imps, analysisFun=yonx, scoreFun=myScore)

Within between variance estimation

Description

This function implements the within-between variance estimation approach. If the imputations were generated using posterior draws, it implements the approach proposed by Barnard & Rubin (1999). If posterior draws were not used, it implements the WB approach described by von Hippel and Bartlett (2021).

Usage

withinBetween(imps, analysisFun, pd = NULL, dfComplete = NULL, ...)

Arguments

Value

A list containing the overall parameter estimates, its corresponding covariance matrix, and degrees of freedom for each parameter.

References

Barnard J, Rubin DB. Miscellanea. Small-sample degrees of freedom with multiple imputation. Biometrika 1999; 86(4): 948-955. doi:10.1093/biomet/86.4.948

von Hippel P.T. and Bartlett J.W. Maximum likelihood multiple imputation: faster, more efficient imputation without posterior draws. Statistical Science 2021; 36(3) 400-420 doi:10.1214/20-STS793.

Examples

#simulate a partially observed dataset
set.seed(1234)
n <- 100
x <- rnorm(n)
y <- x+rnorm(n)
y[1:50] <- NA
temp <- data.frame(x,y)

#impute using normImp
imps <- normImp(temp, M=100, pd=TRUE, rseed=4423)

#define a function which analyses a dataset using our desired
#analysis model, returning the estimated parameters and their
#corresponding variance covariance matrix
analysisFun <- function(inputData) {
  mod <- lm(y~x, data=inputData)
  list(est=coef(mod), var=vcov(mod))
}
withinBetween(imps,analysisFun, dfComplete=c(n-2,n-2))