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 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
missMat <- genMiss(dtAct, defM, idvars = "id")
dtObs <- genObs(dtAct, missMat, idvars = "id")
```

```
## id x1 x2 x3 u m
## 1: 1 0 1 0 1 0
## 2: 2 0 0 1 1 0
## 3: 3 0 0 0 1 0
## 4: 4 0 0 0 1 0
## 5: 5 0 1 0 1 0
## ---
## 996: 996 1 0 0 1 0
## 997: 997 0 0 0 1 0
## 998: 998 1 0 0 1 0
## 999: 999 0 0 0 1 0
## 1000: 1000 0 0 0 1 0
```

```
## id m u x1 x2 x3
## 1: 1 1 NA 36.6365556 NA 41.0068126
## 2: 2 1 NA 39.5738913 40.0380326 NA
## 3: 3 1 NA 21.9053925 19.6687317 20.3413756
## 4: 4 0 NA 20.7216820 17.4932394 20.6237354
## 5: 5 0 NA -0.5949337 NA -0.9755929
## ---
## 996: 996 1 NA NA 41.2367090 43.1127863
## 997: 997 1 NA 40.6415501 38.3769832 40.5953556
## 998: 998 1 NA NA 19.1901652 19.2015971
## 999: 999 0 NA 19.4270147 19.1216662 19.5905361
## 1000: 1000 0 NA -0.8709804 0.9607724 -0.9439384
```

The 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 20.6 20.6 20.6
## Observed 20.3 19.1 19.7
## Difference 0.3 1.5 0.9
```

After 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 10.4 10.4 10.5
## Observed 10.3 10.3 8.8
## Difference 0.1 0.1 1.7
```

```
# compare observed and actual when m = 1
showDif(meanActm[m == 1, .(x1, x2, x3)], meanObsm[m == 1, .(x1, x2, x3)])
```

```
## x1 x2 x3
## Actual 29.9 29.9 30.0
## Observed 29.8 29.9 28.7
## Difference 0.1 0.0 1.3
```

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 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")
```