Author: Jonathan Bartlett

# smcfcs for covariate measurement error correction

#### Jonathan Bartlett

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 16 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) 0.08850686 0.08171538 -0.1136864 0.2907001     87 %
    ## x           0.91596526 0.11612856  0.6100080 1.2219226     95 %

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"

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) -0.09181376 0.03709867 -0.1653430 -0.01828453     21 %
    ## x            0.86868846 0.15799531  0.4424182  1.29495872     97 %

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                 y
    ##  Min.   :-3.03077   Min.   :-4.36681   Min.   :-4.78473   Min.   :-4.43675
    ##  1st Qu.:-0.82756   1st Qu.:-0.93609   1st Qu.:-1.01688   1st Qu.:-1.02676
    ##  Median :-0.02556   Median :-0.01339   Median :-0.02595   Median :-0.13283
    ##  Mean   :-0.02281   Mean   :-0.00018   Mean   :-0.06838   Mean   :-0.08455
    ##  3rd Qu.: 0.78154   3rd Qu.: 0.94560   3rd Qu.: 0.91115   3rd Qu.: 0.87570
    ##  Max.   : 4.12664   Max.   : 5.20210   Max.   : 4.38671   Max.   : 4.25531

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 6 times (across all variables,
    ## iterations, and imputations). You may want to increase the rejection sampling
    ## limit.

    plot(imps)

This plot suggests it would probably be safer to impute using slightly 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 50 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) -0.06986784 0.03671767 -0.1424910  0.002755316     19 %
    ## x            0.87143107 0.06457764  0.7326503  1.010211874     60 %
    ## z           -0.89482480 0.04187590 -0.9809260 -0.808723639     44 %

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.