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).

```
<- 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)
def1
<- genData(1000, def1) dtAct
```

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`

:

```
<- 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
defM
set.seed(283726)
<- genMiss(dtAct, defM, idvars = "id")
missMat <- genObs(dtAct, missMat, idvars = "id") dtObs
```

` 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 1 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 2.37 -0.0938 1.79
## 2: 2 0 NA 21.00 19.0643 21.76
## 3: 3 0 NA NA -2.1260 -2.04
## 4: 4 0 NA NA 0.6345 2.91
## 5: 5 0 NA NA NA 17.11
## ---
## 996: 996 0 NA 17.71 19.6705 18.74
## 997: 997 1 NA NA 40.8860 NA
## 998: 998 0 NA 19.83 20.6475 NA
## 999: 999 1 NA 20.29 23.6448 18.81
## 1000: 1000 0 NA 20.99 20.3019 19.97
```

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
<- function(var, digits = 1) {
rmean round(mean(var, na.rm = TRUE), digits)
}
<- function(dt1, dt2, rowName = c("Actual", "Observed", "Difference")) {
showDif <- data.frame(rbind(dt1, dt2, dt1 - dt2))
dt rownames(dt) <- rowName
return(dt)
}
# data.table functionality to estimate means for each data set
<- dtAct[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3))]
meanAct <- dtObs[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3))]
meanObs
showDif(meanAct, meanObs)
```

```
## x1 x2 x3
## Actual 19.8 19.8 19.8
## Observed 19.9 18.4 18.0
## Difference -0.1 1.4 1.8
```

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:

```
<- dtAct[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3)), keyby = m]
meanActm <- dtObs[, .(x1 = rmean(x1), x2 = rmean(x2), x3 = rmean(x3)), keyby = m] meanObsm
```

```
# compare observed and actual when m = 0
showDif(meanActm[m == 0, .(x1, x2, x3)], meanObsm[m == 0, .(x1, x2, x3)])
```

```
## x1 x2 x3
## Actual 9.7 9.8 9.7
## Observed 9.8 9.9 8.4
## Difference -0.1 -0.1 1.3
```

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

```
## x1 x2 x3
## Actual 30.4 30.4 30.4
## Observed 30.7 31.0 28.6
## Difference -0.3 -0.6 1.8
```

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
<- genData(120, def1)
dtAct <- trtObserve(dtAct, formulas = 0.5, logit.link = FALSE, grpName = "rx")
dtAct
# add longitudinal data
<- defDataAdd(varname = "y", dist = "normal", formula = "10 + period*2 + 2 * rx",
defLong variance = 2)
<- addPeriods(dtAct, nPeriods = 4)
dtTime <- addColumns(defLong, dtTime) 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
<- defMiss(varname = "x1", formula = 0.2, baseline = TRUE)
defMlong <- defMiss(defMlong, varname = "y", formula = "-1.5 - 1.5 * rx + .25*period",
defMlong logit.link = TRUE, baseline = FALSE, monotonic = FALSE)
<- genMiss(dtTime, defMlong, idvars = c("id", "rx"), repeated = TRUE,
missMatLong 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
<- defMiss(varname = "x1", formula = 0.2, baseline = TRUE)
defMlong <- defMiss(defMlong, varname = "y", formula = "-1.8 - 1.5 * rx + .25*period",
defMlong logit.link = TRUE, baseline = FALSE, monotonic = TRUE)
<- genMiss(dtTime, defMlong, idvars = c("id", "rx"), repeated = TRUE,
missMatLong periodvar = "period")
```