smcfcs for covariate measurement error correction

Jonathan Bartlett

2019-03-30

This short vignette introduces the capabilities of smcfcs to accommodate classical covariate measurement error. We consider the cases where internal validation data and then internal replication data are available.

Validation data

We will simulate a dataset with internal validation data where the true covariate (x) is observed for 10% of the sample, while every subject has an error-prone measurement (w) observed:

set.seed(1234)
n <- 1000
x <- rnorm(n)
w <- x+rnorm(n)
y <- x+rnorm(n)
x[(n*0.1):n] <- NA
simData <- data.frame(x,w,y)

We have generated code where the error-prone measurement w is equal to the true covariate x plus some independent normally distributed measurement error. Since x is observed for some of the subjects in the case of interval validation data, this is a regular missing data problem. The error-prone measurement w serves as an auxiliary variable for the purposes of imputation of x. In particular, we will impute using `smcfcs’ such that w is not in the substantive model. This encodes the so called non-differential error assumption, that says that conditional on x, the error-prone measurement w provides no independent information about the outcome y. An initial attempt to do this is:

library(smcfcs)
imps <- smcfcs(simData, smtype="lm", smformula="y~x",
                     method=c("norm", "", ""),m=5)
## [1] "Outcome variable(s): y"
## [1] "Passive variables: "
## [1] "Partially obs. variables: x"
## [1] "Fully obs. substantive model variables: "
## [1] "Imputation  1"
## [1] "Imputing:  x  using    plus outcome"
## [1] "Imputation  2"
## [1] "Imputation  3"
## [1] "Imputation  4"
## [1] "Imputation  5"
## Warning in smcfcs.core(originaldata, smtype, smformula, method,
## predictorMatrix, : Rejection sampling failed 10 times (across all
## variables, iterations, and imputations). You may want to increase the
## rejection sampling limit.

We see from the output that smcfcs has not mentioned that it is using w anywhere. This is because w is fully observed and is not involved in the substantive model. To force w to be conditioned on when imputing x, we must pass an appropriate predictorMatrix to smcfcs:

predMat <- array(0, dim=c(3,3))
predMat[1,2] <- 1

We have specified that the first variable, x, be imputed using w. Note that we do not need to tell smcfcs to impute x using y, as this will occur automatically by virtue of y being the outcome variable in the substantive model. We can now impute again, passing predMat as the predictorMatrix:

imps <- smcfcs(simData, smtype="lm", smformula="y~x",
                     method=c("norm", "", ""),m=5,
               predictorMatrix=predMat)
## [1] "Outcome variable(s): y"
## [1] "Passive variables: "
## [1] "Partially obs. variables: x"
## [1] "Fully obs. substantive model variables: "
## [1] "Imputation  1"
## [1] "Imputing:  x  using  w  plus outcome"
## [1] "Imputation  2"
## [1] "Imputation  3"
## [1] "Imputation  4"
## [1] "Imputation  5"
## Warning in smcfcs.core(originaldata, smtype, smformula, method,
## predictorMatrix, : Rejection sampling failed 3 times (across all variables,
## iterations, and imputations). You may want to increase the rejection
## sampling limit.

Now we can fit the substantive model to each imputed dataset and use the mitools package to pool the estimates and standard errors using Rubin’s rules:

library(mitools)
impobj <- imputationList(imps$impDatasets)
models <- with(impobj, lm(y~x))
summary(MIcombine(models))
## Multiple imputation results:
##       with(impobj, lm(y ~ x))
##       MIcombine.default(models)
##                results         se      (lower    upper) missInfo
## (Intercept) 0.06561899 0.06463153 -0.08585809 0.2170961     79 %
## x           0.90668067 0.04989936  0.79561655 1.0177448     69 %

We note from the results that the fraction of missing information for the coefficient of x is high. This should not surprise us, given that x was missing for 90% of the sample and the error-prone measurement w is quite a noisy measure of x.

Replication data

We will now demonstrate how smcfcs can be used to impute a covariate x which is not observed for any subjects, but we have for at least a subset of the sample two or more error-prone replicate measurements. We first simulate the dataset:

x <- rnorm(n)
w1 <- x+rnorm(n)
w2 <- x+rnorm(n)
w2[(n*0.1):n] <- NA
y <- x+rnorm(n)
x <- rep(NA,n)
simData <- data.frame(x,w1,w2,y)

Note that now x is missing for every subject. Every subject has an error-prone measurement w1 of x, and 10% of the sample have a replicated measurement w2.

We will now impute x using smcfcs. To do this we specify that x be imputed using the latnorm method. In addition, we pass a matrix to the errorProneMatrix argument of smcfcs, whose role is to specify, for each latent normal variable to be imputed, which variables in the data frame are error-prone measurements. smcfcs then imputes the missing values in x, assuming a normal classical error model for the error-prone replicates.

errMat <- array(0, dim=c(4,4))
errMat[1,c(2,3)] <- 1
imps <- smcfcs(simData, smtype="lm", smformula="y~x",
                     method=c("latnorm", "", "",""),m=5,
               errorProneMatrix=errMat)
## [1] "Outcome variable(s): y"
## [1] "Passive variables: "
## [1] "Partially obs. variables: x"
## [1] "Fully obs. substantive model variables: "
## [1] "Imputation  1"
## [1] "Imputing:  x  using  w1,w2  plus outcome"
## [1] "Imputation  2"
## [1] "Imputation  3"
## [1] "Imputation  4"
## [1] "Imputation  5"
## Warning in smcfcs.core(originaldata, smtype, smformula, method,
## predictorMatrix, : Rejection sampling failed 16 times (across all
## variables, iterations, and imputations). You may want to increase the
## rejection sampling limit.

Analysing the imputed datasets, we obtain:

impobj <- imputationList(imps$impDatasets)
models <- with(impobj, lm(y~x))
summary(MIcombine(models))
## Multiple imputation results:
##       with(impobj, lm(y ~ x))
##       MIcombine.default(models)
##                results         se      (lower    upper) missInfo
## (Intercept) 0.03496717 0.04182397 -0.05136883 0.1213032     45 %
## x           0.98851115 0.08722297  0.76760671 1.2094156     90 %

If we summarise one of the imputed datasets (below), we will see that smcfcs has not only imputed the missing values in x, but also the ‘missing’ values in w2. We hyphenate missing here because typically a study with replicate error-prone measurements will have intentionally planned to only take a second error-prone measurement on a random subset, so the values were never intended to be measured.

summary(imps$impDatasets[[1]])
##        x                  w1                 w2          
##  Min.   :-3.53379   Min.   :-5.12005   Min.   :-5.49923  
##  1st Qu.:-0.63548   1st Qu.:-0.95665   1st Qu.:-0.95605  
##  Median : 0.06161   Median :-0.01666   Median : 0.01264  
##  Mean   : 0.01502   Mean   :-0.02141   Mean   :-0.01175  
##  3rd Qu.: 0.63233   3rd Qu.: 0.87663   3rd Qu.: 0.93432  
##  Max.   : 3.09835   Max.   : 3.93341   Max.   : 4.01630  
##        y          
##  Min.   :-5.2178  
##  1st Qu.:-0.8870  
##  Median : 0.1150  
##  Mean   : 0.0323  
##  3rd Qu.: 1.0501  
##  Max.   : 3.9721

One thing to be wary of when imputing covariates measured with error, particularly with replication data, is that convergence may take longer than in the regular missing data setting. To examine this, we re-impute one dataset using 100 iterations, and then plot the estimates against iteration number:

imps <- smcfcs(simData, smtype="lm", smformula="y~x",
                     method=c("latnorm", "", "",""),m=1,numit=100,
               errorProneMatrix=errMat)
## [1] "Outcome variable(s): y"
## [1] "Passive variables: "
## [1] "Partially obs. variables: x"
## [1] "Fully obs. substantive model variables: "
## [1] "Imputation  1"
## [1] "Imputing:  x  using  w1,w2  plus outcome"
## Warning in smcfcs.core(originaldata, smtype, smformula, method,
## predictorMatrix, : Rejection sampling failed 18 times (across all
## variables, iterations, and imputations). You may want to increase the
## rejection sampling limit.
plot(imps$smCoefIter[1,2,])

This plot suggests it would probably be safer to impute using more than 10 iterations per imputation.

Multiple covariates measured with error

smcfcs can impute multiple covariates measured with error when internal replication data are available. It allows for a separate error variance for each such covariate. The following code adds a second covariate which is itself measured by two error-prone measurements, but this time with a smaller error variance. It then defines the errorProneMatrix, imputes and analyses the imputed datasets:

x <- rnorm(n)
x1 <- x+rnorm(n)
x2 <- x+rnorm(n)
w2[(n*0.1):n] <- NA
z <- x+rnorm(n)
z1 <- z+0.1*rnorm(n)
z2 <- z+0.1*rnorm(n)
y <- x-z+rnorm(n)
x <- rep(NA,n)
z <- rep(NA,n)
simData <- data.frame(x,x1,x2,z,z1,z2,y)

errMat <- array(0, dim=c(7,7))
errMat[1,c(2,3)] <- 1
errMat[4,c(5,6)] <- 1
imps <- smcfcs(simData, smtype="lm", smformula="y~x+z",
                     method=c("latnorm", "", "","latnorm", "", "", ""),m=5,
               errorProneMatrix=errMat)
## [1] "Outcome variable(s): y"
## [1] "Passive variables: "
## [1] "Partially obs. variables: x,z"
## [1] "Fully obs. substantive model variables: "
## [1] "Imputation  1"
## [1] "Imputing:  x  using  z,x1,x2  plus outcome"
## [1] "Imputing:  z  using  x,z1,z2  plus outcome"
## [1] "Imputation  2"
## [1] "Imputation  3"
## [1] "Imputation  4"
## [1] "Imputation  5"
## Warning in smcfcs.core(originaldata, smtype, smformula, method,
## predictorMatrix, : Rejection sampling failed 11 times (across all
## variables, iterations, and imputations). You may want to increase the
## rejection sampling limit.

We now analyse the imputed datasets, remembering to add z into the substantive model:

impobj <- imputationList(imps$impDatasets)
models <- with(impobj, lm(y~x+z))
summary(MIcombine(models))
## Multiple imputation results:
##       with(impobj, lm(y ~ x + z))
##       MIcombine.default(models)
##                  results         se      (lower     upper) missInfo
## (Intercept)  0.008810966 0.03569360 -0.06193726  0.0795592     21 %
## x            0.915837749 0.09581146  0.68398545  1.1476900     84 %
## z           -0.921928399 0.05555190 -1.04849325 -0.7953635     74 %

We see that the fraction of missing information is lower for z than for x. This is a consequence of the fact that we generated the error-prone measurements of z to have smaller error variance than for the corresponding error-prone measurements of x.