After generating a complete data set, it is possible to generate missing data. defMiss defines the parameters of missingness. genMiss generates a missing data matrix of indicators for each field. Indicators are set to 1 if the data are missing for a subject, 0 otherwise. genObs creates a data set that reflects what would have been observed had data been missing; this is a replicate of the original data set with “NAs” replacing values where missing data has been generated.
By controlling the parameters of missingness, it is possible to represent different missing data mechanisms: (1) missing completely at random (MCAR), where the probability missing data is independent of any covariates, measured or unmeasured, that are associated with the measure, (2) missing at random (MAR), where the probability of subject missing data is a function only of observed covariates that are associated with the measure, and (3) not missing at random (NMAR), where the probability of missing data is related to unmeasured covariates that are associated with the measure.
These possibilities are illustrated with an example. A data set of 1000 observations with three “outcome” measures" x1, x2, and x3 is defined. This data set also includes two independent predictors, m and u that largely determine the value of each outcome (subject to random noise).
def1 <- defData(varname = "m", dist = "binary", formula = 0.5) def1 <- defData(def1, "u", dist = "binary", formula = 0.5) def1 <- defData(def1, "x1", dist = "normal", formula = "20*m + 20*u", variance = 2) def1 <- defData(def1, "x2", dist = "normal", formula = "20*m + 20*u", variance = 2) def1 <- defData(def1, "x3", dist = "normal", formula = "20*m + 20*u", variance = 2) dtAct <- genData(1000, def1)
In this example, the missing data mechanism is different for each outcome. As defined below, missingness for x1 is MCAR, since the probability of missing is fixed. Missingness for x2 is MAR, since missingness is a function of m, a measured predictor of x2. And missingness for x3 is NMAR, since the probability of missing is dependent on u, an unmeasured predictor of x3:
defM <- defMiss(varname = "x1", formula = 0.15, logit.link = FALSE) defM <- defMiss(defM, varname = "x2", formula = ".05 + m * 0.25", logit.link = FALSE) defM <- defMiss(defM, varname = "x3", formula = ".05 + u * 0.25", logit.link = FALSE) defM <- defMiss(defM, varname = "u", formula = 1, logit.link = FALSE) # not observed set.seed(283726) missMat <- genMiss(dtAct, defM, idvars = "id") dtObs <- genObs(dtAct, missMat, idvars = "id")
missMat## id x1 x2 x3 u m
## 1: 1 0 0 0 1 0
## 2: 2 0 0 0 1 0
## 3: 3 1 0 0 1 0
## 4: 4 1 0 0 1 0
## 5: 5 1 1 0 1 0
## ---
## 996: 996 0 0 0 1 0
## 997: 997 1 0 1 1 0
## 998: 998 0 0 0 1 0
## 999: 999 0 0 0 1 0
## 1000: 1000 0 0 0 1 0
dtObs## id m u x1 x2 x3
## 1: 1 0 NA -1.0470073 3.6976385 1.1773706
## 2: 2 1 NA 37.7016446 38.6445468 40.1291761
## 3: 3 1 NA NA 42.3835492 41.4593958
## 4: 4 1 NA NA 19.1998720 21.5308830
## 5: 5 1 NA NA NA 39.9907045
## ---
## 996: 996 0 NA -0.6970442 0.4634872 -0.8588669
## 997: 997 0 NA NA 17.8357998 NA
## 998: 998 0 NA 0.4872474 0.1678724 1.9330914
## 999: 999 1 NA 19.5751706 18.9628603 18.0742193
## 1000: 1000 1 NA 19.2024422 20.2697473 22.0805539The impacts of the various data mechanisms on estimation can be seen with a simple calculation of means using both the “true” data set without missing data as a comparison for the “observed” data set. Since x1 is MCAR, the averages for both data sets are roughly equivalent. However, we can see below that estimates for x2 and x3 are biased, as the difference between observed and actual is not close to 0:
# Two functions to calculate means and compare them rmean <- function(var, digits = 1) { round(mean(var, na.rm = TRUE), digits) } showDif <- function(dt1, dt2, rowName = c("Actual", "Observed", "Difference")) { dt <- data.frame(rbind(dt1, dt2, dt1 - dt2)) rownames(dt) <- rowName return(dt) } # data.table functionality to estimate means for each data set meanAct <- dtAct[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3))] meanObs <- dtObs[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3))] showDif(meanAct, meanObs)
## x1 x2 x3
## Actual 20.1 20.2 20.2
## Observed 20.0 18.6 19.1
## Difference 0.1 1.6 1.1After adjusting for the measured covariate m, the bias for the estimate of the mean of x2 is mitigated, but not for x3, since u is not observed:
meanActm <- dtAct[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3)), keyby = m] meanObsm <- dtObs[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3)), keyby = m]
# compare observed and actual when m = 0 showDif(meanActm[m == 0, .(x1, x2, x3)], meanObsm[m == 0, .(x1, x2, x3)])
## x1 x2 x3
## Actual 10.5 10.5 10.5
## Observed 10.5 10.5 8.9
## Difference 0.0 0.0 1.6# compare observed and actual when m = 1 showDif(meanActm[m == 1, .(x1, x2, x3)], meanObsm[m == 1, .(x1, x2, x3)])
## x1 x2 x3
## Actual 29.3 29.4 29.4
## Observed 29.1 29.3 28.1
## Difference 0.2 0.1 1.3Longitudinal data with missingness
Missingness can occur, of course, in the context of longitudinal data. missDef provides two additional arguments that are relevant for these types of data: baseline and monotonic. In the case of variables that are measured at baseline only, a missing value would be reflected throughout the course of the study. In the case where a variable is time-dependent (i.e it is measured at each time point), it is possible to declare missingness to be monotonic. This means that if a value for this field is missing at time t, then values will also be missing at all times T > t as well. The call to genMiss must set repeated to TRUE.
The following two examples describe an outcome variable y that is measured over time, whose value is a function of time and an observed exposure:
# use baseline definitions from the previous example dtAct <- genData(120, def1) dtAct <- trtObserve(dtAct, formulas = 0.5, logit.link = FALSE, grpName = "rx") # add longitudinal data defLong <- defDataAdd(varname = "y", dist = "normal", formula = "10 + period*2 + 2 * rx", variance = 2) dtTime <- addPeriods(dtAct, nPeriods = 4) dtTime <- addColumns(defLong, dtTime)
In the first case, missingness is not monotonic; a subject might miss a measurement but returns for subsequent measurements:
# missingness for y is not monotonic defMlong <- defMiss(varname = "x1", formula = 0.2, baseline = TRUE) defMlong <- defMiss(defMlong, varname = "y", formula = "-1.5 - 1.5 * rx + .25*period", logit.link = TRUE, baseline = FALSE, monotonic = FALSE) missMatLong <- genMiss(dtTime, defMlong, idvars = c("id", "rx"), repeated = TRUE, periodvar = "period")
Here is a conceptual plot that shows the pattern of missingness. Each row represents an individual, and each box represents a time period. A box that is colored reflects missing data; a box colored grey reflects observed. The missingness pattern is shown for two variables x1 and y:
In the second case, missingness is monotonic; once a subject misses a measurement for y, there are no subsequent measurements:
# missingness for y is monotonic defMlong <- defMiss(varname = "x1", formula = 0.2, baseline = TRUE) defMlong <- defMiss(defMlong, varname = "y", formula = "-1.8 - 1.5 * rx + .25*period", logit.link = TRUE, baseline = FALSE, monotonic = TRUE) missMatLong <- genMiss(dtTime, defMlong, idvars = c("id", "rx"), repeated = TRUE, periodvar = "period")
