# Simulation Arguments Details for simglm

#### 2018-06-01

The tidy simulation framework uses simulation arguments as the basis for specifying the models to be simulated. The goal of this vignette is to document more thoroughly the various arguments that are possible within each function. It is recommended that new users start with the “Tidy Simulation with simglm” vignette prior to working through this vignette.

# Fixed Arguments

Arguments associated with the fixed portion of the model are needed for each fixed variable that needs to be simulated. The fixed variables specified come from the formula argument. Interactions or the intercept are not included in the fixed arguments as these are generated automatically. Let’s start with an example.

library(simglm)

set.seed(321)

# To-DO: Add knot variable and debug

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
sample_size = list(level1 = 10, level2 = 20)
)

fixed_data <- simulate_fixed(data = NULL, sim_arguments)
head(fixed_data, n = 20)
X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id
1 0 231.1471 33 1 0 231.1471 33 Treatment 1 1
1 1 158.6388 33 1 1 158.6388 33 Treatment 2 1
1 2 171.6605 33 1 2 171.6605 33 Treatment 3 1
1 3 176.4105 33 1 3 176.4105 33 Treatment 4 1
1 4 176.2812 33 1 4 176.2812 33 Treatment 5 1
1 5 188.0455 33 1 5 188.0455 33 Treatment 6 1
1 6 201.8052 33 1 6 201.8052 33 Treatment 7 1
1 7 186.9941 33 1 7 186.9941 33 Treatment 8 1
1 8 190.1734 33 1 8 190.1734 33 Treatment 9 1
1 9 163.4426 33 1 9 163.4426 33 Treatment 10 1
1 0 190.4310 51 1 0 190.4310 51 Treatment 1 2
1 1 224.5378 51 1 1 224.5378 51 Treatment 2 2
1 2 185.6498 51 1 2 185.6498 51 Treatment 3 2
1 3 253.2978 51 1 3 253.2978 51 Treatment 4 2
1 4 145.3968 51 1 4 145.3968 51 Treatment 5 2
1 5 155.8598 51 1 5 155.8598 51 Treatment 6 2
1 6 193.6821 51 1 6 193.6821 51 Treatment 7 2
1 7 192.6100 51 1 7 192.6100 51 Treatment 8 2
1 8 197.3275 51 1 8 197.3275 51 Treatment 9 2
1 9 193.3907 51 1 9 193.3907 51 Treatment 10 2

The following example shows the five types of variables that can be generated. These types are specified with the var_type argument and can be one of the following five types:

• ‘time’: This type of variable is used for a time metric variable
• ‘continuous’: This is a continuous distribution, default is to use the rnorm distribution function for generation.
• ‘ordinal’: This generates ordinal data and is generated using the sample function.
• ‘factor’: This is a categorical or factor variable specification representing groups or categories.
• ‘knot’: This type of variable is generated based on another variable and represents a knot category variable.

Each variable type will be explored in more detail.

Finally, another common argument for all fixed variable types is the argument var_level. This defaults to var_level = 1 which would be a variable defined at the level 1 of the model (i.e. unique value for each row in the data). These can be changed to reflect the data level that is desired. For example, var_level = 2 would repeat values for each level 2 cluster found in the data and var_level = 3 would do the same for each level 3 cluster. Therefore for variables that are at level 2 or level 3, there will be fewer unique values in the data compared to level 1 variables.

## Time Variable

For time variables used in longitudinal or repeated measures designs, the default metric is 0 to level1 sample size minus 1. This can be seen above in the output. To change the time variable metric the time_levels argument can be used. A vector of values to specify for the time variable can be given directly. For example, the following could be passed to alter the metric of the time variable: time_levels = c(0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5, 6) where now the time variable would increment in 0.5 units for the first 7 measurements and 1 unit increments for the last three measurements. This could represent a case when the measurements are collected every 6 months for the first 7 measurements and yearly after that.

Below is the output including the manual time variable specification. The only requirement is that the length of the time_levels argument must match the level1 sample size.

set.seed(321)

# To-DO: Add knot variable and debug

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time',
time_levels = c(0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5, 6)),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
sample_size = list(level1 = 10, level2 = 20)
)

fixed_data <- simulate_fixed(data = NULL, sim_arguments)
head(fixed_data, n = 20)
X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id
1 0.0 231.1471 33 1 0.0 231.1471 33 Treatment 1 1
1 0.5 158.6388 33 1 0.5 158.6388 33 Treatment 2 1
1 1.0 171.6605 33 1 1.0 171.6605 33 Treatment 3 1
1 1.5 176.4105 33 1 1.5 176.4105 33 Treatment 4 1
1 2.0 176.2812 33 1 2.0 176.2812 33 Treatment 5 1
1 2.5 188.0455 33 1 2.5 188.0455 33 Treatment 6 1
1 3.0 201.8052 33 1 3.0 201.8052 33 Treatment 7 1
1 4.0 186.9941 33 1 4.0 186.9941 33 Treatment 8 1
1 5.0 190.1734 33 1 5.0 190.1734 33 Treatment 9 1
1 6.0 163.4426 33 1 6.0 163.4426 33 Treatment 10 1
1 0.0 190.4310 51 1 0.0 190.4310 51 Treatment 1 2
1 0.5 224.5378 51 1 0.5 224.5378 51 Treatment 2 2
1 1.0 185.6498 51 1 1.0 185.6498 51 Treatment 3 2
1 1.5 253.2978 51 1 1.5 253.2978 51 Treatment 4 2
1 2.0 145.3968 51 1 2.0 145.3968 51 Treatment 5 2
1 2.5 155.8598 51 1 2.5 155.8598 51 Treatment 6 2
1 3.0 193.6821 51 1 3.0 193.6821 51 Treatment 7 2
1 4.0 192.6100 51 1 4.0 192.6100 51 Treatment 8 2
1 5.0 197.3275 51 1 5.0 197.3275 51 Treatment 9 2
1 6.0 193.3907 51 1 6.0 193.3907 51 Treatment 10 2

## Continuous Variable

Continuous variables are generating using distribution functions (i.e. rnorm). Any distribution function found within R, or user written distribution functions can be used, however the default used in rnorm. To change the distribution function used, the argument dist can be specified. For example, if the Gamma distribution is desired for the weight variable, the following code would achieve this:

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time',
time_levels = c(0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5, 6)),
weight = list(var_type = 'continuous', dist = 'rgamma',
shape = 3),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
sample_size = list(level1 = 10, level2 = 20)
)

fixed_data <- simulate_fixed(data = NULL, sim_arguments)
head(fixed_data, n = 20)
X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id
1 0.0 5.922362 39 0 0.0 5.922362 39 Control 1 1
1 0.5 1.500918 39 0 0.5 1.500918 39 Control 2 1
1 1.0 1.894797 39 0 1.0 1.894797 39 Control 3 1
1 1.5 2.307843 39 0 1.5 2.307843 39 Control 4 1
1 2.0 1.975996 39 0 2.0 1.975996 39 Control 5 1
1 2.5 2.882207 39 0 2.5 2.882207 39 Control 6 1
1 3.0 3.064936 39 0 3.0 3.064936 39 Control 7 1
1 4.0 1.703499 39 0 4.0 1.703499 39 Control 8 1
1 5.0 7.518809 39 0 5.0 7.518809 39 Control 9 1
1 6.0 2.442868 39 0 6.0 2.442868 39 Control 10 1
1 0.0 7.855513 58 1 0.0 7.855513 58 Treatment 1 2
1 0.5 1.008858 58 1 0.5 1.008858 58 Treatment 2 2
1 1.0 3.452912 58 1 1.0 3.452912 58 Treatment 3 2
1 1.5 7.881209 58 1 1.5 7.881209 58 Treatment 4 2
1 2.0 2.262056 58 1 2.0 2.262056 58 Treatment 5 2
1 2.5 3.255559 58 1 2.5 3.255559 58 Treatment 6 2
1 3.0 4.160648 58 1 3.0 4.160648 58 Treatment 7 2
1 4.0 2.333586 58 1 4.0 2.333586 58 Treatment 8 2
1 5.0 1.900442 58 1 5.0 1.900442 58 Treatment 9 2
1 6.0 1.445119 58 1 6.0 1.445119 58 Treatment 10 2

This would be the resulting distribution of the weight variable generated.

library(ggplot2)

ggplot(fixed_data, aes(x = weight)) +
geom_density() +
theme_bw()

## Ordinal Variable

Ordinal variable specification uses the sample function within R to generate ordinal variables that are whole integers. The required argument for these types of variables is levels. The levels argument takes a range or vector of integer values to be passed to the sample function within R. For example, these three specifications for the levels argument are valid: 3:60, seq(4, 60, 2), c(3, 10, 18, 24, 54, 60).

An additional optional argument is replace. The replace argument specified whether the sampling is done with or without replacement. The default behavior is to do sampling with replacement. If sampling without replacement is desired set replace = FALSE. See sample for more details on this argument.

Finally, the probability of selecting each value specified to the levels argument is also able to be specified through the prob argument. If prob is specified, it takes a vector of probabilities that must be the same length as the levels argument. The default behavior is for each value specified with levels to be equally likely to be sampled.

## Factor Variable

Factor variables are generated similarly to ordinal variables with the sample function, however factor variables allow the generation of text or categorical variables in addition to numeric grouping variables. Therefore the only needed argument for factor variables is a vector of numeric or text strings representing the different groups to be generated. For example, both of these specifications are equivalent: c(1, 2, 3, 4) and c('Freshman', 'Sophomore', 'Junior', 'Senior'). Both of these specifications would generate data for these four groups, the difference is that the text labels will be used when text strings are specified.

An additional optional argument is replace. The replace argument specified whether the sampling is done with or without replacement. The default behavior is to do sampling with replacement. If sampling without replacement is desired set replace = FALSE. See sample for more details on this argument.

Finally, the probability of selecting each value specified to the levels argument is also able to be specified through the prob argument. If prob is specified, it takes a vector of probabilities that must be the same length as the levels argument. The default behavior is for each value specified with levels to be equally likely to be sampled.

To come…

# Random Error Arguments

By default, the random error is generated as random normal with a mean of 0 and standard deviation of 1. If this is the desired behavior, no additional simulation arguments need to be specified in the simulation arguments. For example, the code below generates random error using the fixed arguments already shown above.

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
sample_size = list(level1 = 10, level2 = 20)
)

error_data <- simulate_error(data = NULL, sim_arguments)
head(error_data, n = 20)
error level1_id id
1.7049032 1 1
-0.7120386 2 1
-0.2779849 3 1
-0.1196490 4 1
-0.1239606 5 1
0.2681838 6 1
0.7268415 7 1
0.2331354 8 1
0.3391139 9 1
-0.5519147 10 1
0.3477014 1 2
1.4845918 2 2
0.1883255 3 2
2.4432598 4 2
-1.1534395 5 2
-0.8046717 6 2
0.4560691 7 2
0.4203326 8 2
0.5775845 9 2
0.4463561 10 2

## Optional Arguments for Random Error

The two main optional arguments for the random error component include a dist and variance that represent the error generating function and variance of the random error respectively. For example, if a t-distribution is desired for simulation of random error, dist = 'rt' can be used to specify the t-distribution as the generating distribution. When distributions other than random normal are specified, it is common that additional arguments will need to be specified for the generating distribution function (i.e. rt). In the rt example, the df argument is needed to specify the degrees of freedom for the t-distribution. Below is an example using a t-distribution.

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
error = list(dist = 'rt', df = 4),
sample_size = list(level1 = 10, level2 = 20)
)

error_data <- simulate_error(data = NULL, sim_arguments)
head(error_data, n = 20)
error level1_id id
2.7754417 1 1
-0.6105358 2 1
0.2388096 3 1
0.2364648 4 1
-0.5580779 5 1
1.5918703 6 1
5.3320327 7 1
0.3197329 8 1
-0.1541429 9 1
-1.7929619 10 1
-1.4292669 1 2
-1.6045961 2 2
-1.0007634 3 2
-1.5413444 4 2
-0.1818689 5 2
-1.7195087 6 2
0.2629315 7 2
0.8104919 8 2
0.2150117 9 2
2.1937363 10 2

Finally, the variance of the random error can also be specified to be something other than the default 1. For example, if the variance is desired to be 10, then the argument variance = 10 will set this variance. This variance works with any distribution function. For example, the following will generate data as a t-distribution with a variance of 10.

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
error = list(dist = 'rt', df = 4, variance = 10),
sample_size = list(level1 = 10, level2 = 20)
)

error_data <- simulate_error(data = NULL, sim_arguments)
var(error_data$error) ## [1] 19.36487 Note that the variance does not actually equal 10 here. This is due to the generating distribution specified (i.e. t-distribution with 4 degrees of freedom) has a theoretical variance of 2. In cases when the random error has a variance other than 1, the variable needs to be standardized prior to converting to the desired variance value. The standardization can be done in two ways. One, an empirical mean and standard deviation of the generating distribution can be obtained by setting the argument ther_sim = TRUE. For example: set.seed(321) sim_arguments <- list( formula = y ~ 1 + time + weight + age + treat + (1 + time| id), fixed = list(time = list(var_type = 'time'), weight = list(var_type = 'continuous', mean = 180, sd = 30), age = list(var_type = 'ordinal', levels = 30:60, var_level = 2), treat = list(var_type = 'factor', levels = c('Treatment', 'Control'), var_level = 2)), error = list(dist = 'rt', df = 4, variance = 10, ther_sim = TRUE), sample_size = list(level1 = 10, level2 = 20) ) error_data <- simulate_error(data = NULL, sim_arguments) var(error_data$error)
## [1] 6.893405

This simulation takes longer to run as the generated empirical mean and standard deviation for the standardization draws a large number of random values from the specified distribution. To speed up this process, the theoretical values can be specified directly within the ther_val argument as a vector with the mean specified first followed by the standard deviation. This is shown below:

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
error = list(dist = 'rt', df = 4, variance = 10, ther_val = c(0, sqrt(2))),
sample_size = list(level1 = 10, level2 = 20)
)

error_data <- simulate_error(data = NULL, sim_arguments)
var(error_data$error) ## [1] 9.682433 ## Heterogeneity of Variance It is also possible to generate heterogeneity of variance for the random error. # Random Effect Arguments The random effect arguments are passed through a named element of the list to the simulation arguments called ‘randomeffect’. Within this element, each random effect specified in the formula must be specified as a named list. The named list allows the user to specify the names that will be used in the data simulation for the random effects. The default behavior of these random effects is to be simulated from a random normal distribution with mean 0 and variance of 1. Many of the same arguments that have been discussed with the error simulation are possible for the random effect simulation arguments. These include: dist, variance, ther_sim, ther_val and other arguments that are needed to pass on the data generating function. In addition, the argument var_level first introduced with the fixed effects are present here indicating which level of the data structure the random effect should be generated at. These arguments will not be discussed in more detail here, but readers can see examples of these arguments within the fixed effects and random error sections above. Below is an example that simulated two random effects in a two-level nested model and specifies the variance of 8 and 3 for the two random effects respectively. Finally, you will notice in the final output, the names of the random effect columns are specified by the names included in the list associated with the “randomeffect” portion of the simulation arguments. set.seed(321) sim_arguments <- list( formula = y ~ 1 + time + weight + age + treat + (1 + time| id), fixed = list(time = list(var_type = 'time'), weight = list(var_type = 'continuous', mean = 180, sd = 30), age = list(var_type = 'ordinal', levels = 30:60, var_level = 2), treat = list(var_type = 'factor', levels = c('Treatment', 'Control'), var_level = 2)), randomeffect = list(int_id = list(variance = 8, var_level = 2), time_id = list(variance = 3, var_level = 2)), sample_size = list(level1 = 10, level2 = 20) ) random_data <- simulate_randomeffect(data = NULL, sim_arguments) head(random_data, n = 20) int_id time_id level1_id id 4.822195 1.588733 1 1 4.822195 1.588733 2 1 4.822195 1.588733 3 1 4.822195 1.588733 4 1 4.822195 1.588733 5 1 4.822195 1.588733 6 1 4.822195 1.588733 7 1 4.822195 1.588733 8 1 4.822195 1.588733 9 1 4.822195 1.588733 10 1 -2.013949 -0.185436 1 2 -2.013949 -0.185436 2 2 -2.013949 -0.185436 3 2 -2.013949 -0.185436 4 2 -2.013949 -0.185436 5 2 -2.013949 -0.185436 6 2 -2.013949 -0.185436 7 2 -2.013949 -0.185436 8 2 -2.013949 -0.185436 9 2 -2.013949 -0.185436 10 2 ## Cross-Classified Random Effects If cross-classified random effects are desired, these can be specified directly within the formula syntax as you would with lme4. For example, y ~ 1 + time + weight + age + treat + (1 + time| id) + (1 | neighborhood_id). When documenting the simulation arguments for the additional cross-classified random effect (i.e. (1 | neighborhood_id)), specifying cross_class = TRUE with that random effect to identify that this is indeed a cross-classified random effect. Secondly, you can also specify directly the number of clusters that are associated with this cross-classified factor, this can be different than those specified in the sample_size argument. This can be done with the num_ids argument. Below is an example with a single cross-classified random effect representing neighborhoods individuals belong in set.seed(321) sim_arguments <- list( formula = y ~ 1 + time + weight + age + treat + (1 + time| id) + (1 | neighborhood_id), fixed = list(time = list(var_type = 'time'), weight = list(var_type = 'continuous', mean = 180, sd = 30), age = list(var_type = 'ordinal', levels = 30:60, var_level = 2), treat = list(var_type = 'factor', levels = c('Treatment', 'Control'), var_level = 2)), randomeffect = list(int_id = list(variance = 8, var_level = 2), time_id = list(variance = 3, var_level = 2), int_nid = list(variance = 5, var_level = 2, cross_class = TRUE, num_ids = 12)), sample_size = list(level1 = 10, level2 = 20) ) random_data <- simulate_randomeffect(data = NULL, sim_arguments) head(random_data, n = 20) int_id time_id int_nid neighborhood_id level1_id id 4.822195 1.588733 -0.5269349 1 1 1 4.822195 1.588733 0.8972466 2 2 1 4.822195 1.588733 -6.1054441 7 3 1 4.822195 1.588733 1.9702697 12 4 1 4.822195 1.588733 3.8171861 8 5 1 4.822195 1.588733 -0.1530702 4 6 1 4.822195 1.588733 3.8171861 8 7 1 4.822195 1.588733 -2.0036117 10 8 1 4.822195 1.588733 -2.0036117 10 9 1 4.822195 1.588733 3.8171861 8 10 1 -2.013949 -0.185436 3.8171861 8 1 2 -2.013949 -0.185436 0.8972466 2 2 2 -2.013949 -0.185436 3.8171861 8 3 2 -2.013949 -0.185436 1.9702697 12 4 2 -2.013949 -0.185436 -2.0036117 10 5 2 -2.013949 -0.185436 -6.1054441 7 6 2 -2.013949 -0.185436 -0.8219536 11 7 2 -2.013949 -0.185436 -0.8219536 11 8 2 -2.013949 -0.185436 0.8972466 2 9 2 -2.013949 -0.185436 1.3721717 6 10 2 # Missing Data Arguments Missing data is the standard rather than the exception in real world data. Therefore, generating data that include missing is important for generating data that are representative of real world data. In the power analysis context, most power analyses do not include missing data in the power computations (particularly if these are closed form solutions), but power can be negatively affected by missing data. Fortunatly, the simglm package contains many useful frameworks for generating missing data. To generate missing data, the generate_missing function can be used for this. This function is called after the complete data are generated. This means that the complete data and missing data will be generated. Currently three types of missing data are supported, dropout, random, and mar (missing at random). The specification of missing data arguments is done using a named element to the simulation argument list called “missing_data”. ## Random Missing Data First, lets explore random missing data. This structure is equivalent to the missing completely at random framework if you are familiar with Rubin’s missing data classifications. To generate random missing data the type = 'random' is specified. Two additional arguments are needed to generate the missing data, first the missing proportion needs to be specified with miss_prop and the name of the new response variable with missing data needs to be specified with new_outcome. Below is an example generating data with random missing data. set.seed(321) sim_arguments <- list( formula = y ~ 1 + time + weight + age + treat + (1 + time| id), reg_weights = c(4, 0.5, 0.75, 0, 0.33), fixed = list(time = list(var_type = 'time'), weight = list(var_type = 'continuous', mean = 180, sd = 30), age = list(var_type = 'ordinal', levels = 30:60, var_level = 2), treat = list(var_type = 'factor', levels = c('Treatment', 'Control'), var_level = 2)), randomeffect = list(int_id = list(variance = 8, var_level = 2), time_id = list(variance = 3, var_level = 2)), missing_data = list(miss_prop = .25, new_outcome = 'y_missing', type = 'random'), sample_size = list(level1 = 10, level2 = 20) ) data_w_missing <- sim_arguments %>% simulate_fixed(data = NULL, .) %>% simulate_randomeffect(sim_arguments) %>% simulate_error(sim_arguments) %>% generate_response(sim_arguments) %>% generate_missing(sim_arguments) head(data_w_missing, n = 10) X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id int_id time_id error fixed_outcome random_effects y miss_prob missing y_missing 1 0 231.1471 33 1 0 231.1471 33 Treatment 1 1 0.2244243 2.533554 2.3826970 177.6903 0.2244243 180.2974 0.646 0 180.2974 1 1 158.6388 33 1 1 158.6388 33 Treatment 2 1 0.2244243 2.533554 -0.3191794 123.8091 2.7579779 126.2479 0.088 1 NA 1 2 171.6605 33 1 2 171.6605 33 Treatment 3 1 0.2244243 2.533554 0.7167563 134.0753 5.2915315 140.0836 0.534 0 140.0836 1 3 176.4105 33 1 3 176.4105 33 Treatment 4 1 0.2244243 2.533554 -0.3144774 138.1379 7.8250850 145.6485 0.560 0 145.6485 1 4 176.2812 33 1 4 176.2812 33 Treatment 5 1 0.2244243 2.533554 -0.3251463 138.5409 10.3586386 148.5744 0.119 1 NA 1 5 188.0455 33 1 5 188.0455 33 Treatment 6 1 0.2244243 2.533554 0.0060782 147.8641 12.8921922 160.7624 0.415 0 160.7624 1 6 201.8052 33 1 6 201.8052 33 Treatment 7 1 0.2244243 2.533554 1.2868034 158.6839 15.4257457 175.3965 0.718 0 175.3965 1 7 186.9941 33 1 7 186.9941 33 Treatment 8 1 0.2244243 2.533554 -1.0393871 148.0755 17.9592993 164.9955 0.697 0 164.9955 1 8 190.1734 33 1 8 190.1734 33 Treatment 9 1 0.2244243 2.533554 0.3857991 150.9601 20.4928529 171.8387 0.928 0 171.8387 1 9 163.4426 33 1 9 163.4426 33 Treatment 10 1 0.2244243 2.533554 1.9275765 131.4119 23.0264064 156.3659 0.829 0 156.3659 We can look at the amount of missing data: prop.table(table(is.na(data_w_missing$y_missing)))
##
## FALSE  TRUE
## 0.735 0.265

## MAR Missing Data

Data that are missing at random (MAR) can be generated as well.

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
reg_weights = c(4, 0.5, 0.75, 0, 0.33),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30,
var_level = 1),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
randomeffect = list(int_id = list(variance = 8, var_level = 2),
time_id = list(variance = 3, var_level = 2)),
missing_data = list(new_outcome = 'y_missing', miss_cov = 'weight',
mar_prop = seq(from = 0, to = .9, length.out = 200),
type = 'mar'),
sample_size = list(level1 = 10, level2 = 20)
)

data_w_missing <- sim_arguments %>%
simulate_fixed(data = NULL, .) %>%
simulate_randomeffect(sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments) %>%
generate_missing(sim_arguments)

head(data_w_missing, n = 10)
X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id int_id time_id error fixed_outcome random_effects y miss_prop miss_prob missing y_missing
1 0 231.1471 33 1 0 231.1471 33 Treatment 1 1 0.2244243 2.533554 2.3826970 177.6903 0.2244243 180.2974 0.8683417 0.4186856 1 NA
1 1 158.6388 33 1 1 158.6388 33 Treatment 2 1 0.2244243 2.533554 -0.3191794 123.8091 2.7579779 126.2479 0.2035176 0.7042782 0 126.2479
1 2 171.6605 33 1 2 171.6605 33 Treatment 3 1 0.2244243 2.533554 0.7167563 134.0753 5.2915315 140.0836 0.3165829 0.8222003 0 140.0836
1 3 176.4105 33 1 3 176.4105 33 Treatment 4 1 0.2244243 2.533554 -0.3144774 138.1379 7.8250850 145.6485 0.3844221 0.7070236 0 145.6485
1 4 176.2812 33 1 4 176.2812 33 Treatment 5 1 0.2244243 2.533554 -0.3251463 138.5409 10.3586386 148.5744 0.3798995 0.1005621 1 NA
1 5 188.0455 33 1 5 188.0455 33 Treatment 6 1 0.2244243 2.533554 0.0060782 147.8641 12.8921922 160.7624 0.5562814 0.2103423 1 NA
1 6 201.8052 33 1 6 201.8052 33 Treatment 7 1 0.2244243 2.533554 1.2868034 158.6839 15.4257457 175.3965 0.7236181 0.3023945 1 NA
1 7 186.9941 33 1 7 186.9941 33 Treatment 8 1 0.2244243 2.533554 -1.0393871 148.0755 17.9592993 164.9955 0.5201005 0.6374396 0 164.9955
1 8 190.1734 33 1 8 190.1734 33 Treatment 9 1 0.2244243 2.533554 0.3857991 150.9601 20.4928529 171.8387 0.5743719 0.6387437 0 171.8387
1 9 163.4426 33 1 9 163.4426 33 Treatment 10 1 0.2244243 2.533554 1.9275765 131.4119 23.0264064 156.3659 0.2442211 0.8129966 0 156.3659

We can look at the amount of missing data:

prop.table(table(is.na(data_w_missing$y_missing))) ## ## FALSE TRUE ## 0.555 0.445 ## Dropout Missing Data Dropout missing data is possible when simulating data from a longitudinal design. When requesting dropout missing data, observations are removed after a specific point in the time cycle. For example, perhaps an individual stops being a part of the study after the third measurement. To request this type of missing data, the type = 'dropout' can be specified. In addition, one new argument is needed, clust_var. This argument represents the id associated with the clusters with longitudinal data. Below is an example: set.seed(321) sim_arguments <- list( formula = y ~ 1 + time + weight + age + treat + (1 + time| id), reg_weights = c(4, 0.5, 0.75, 0, 0.33), fixed = list(time = list(var_type = 'time'), weight = list(var_type = 'continuous', mean = 180, sd = 30), age = list(var_type = 'ordinal', levels = 30:60, var_level = 2), treat = list(var_type = 'factor', levels = c('Treatment', 'Control'), var_level = 2)), randomeffect = list(int_id = list(variance = 8, var_level = 2), time_id = list(variance = 3, var_level = 2)), missing_data = list(miss_prop = .25, new_outcome = 'y_missing', clust_var = 'id', type = 'dropout'), sample_size = list(level1 = 10, level2 = 20) ) data_w_missing <- sim_arguments %>% simulate_fixed(data = NULL, .) %>% simulate_randomeffect(sim_arguments) %>% simulate_error(sim_arguments) %>% generate_response(sim_arguments) %>% generate_missing(sim_arguments) head(data_w_missing, n = 10) X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id int_id time_id error fixed_outcome random_effects y missing y_missing 1 0 231.1471 33 1 0 231.1471 33 Treatment 1 1 0.2244243 2.533554 2.3826970 177.6903 0.2244243 180.2974 0 180.2974 1 1 158.6388 33 1 1 158.6388 33 Treatment 2 1 0.2244243 2.533554 -0.3191794 123.8091 2.7579779 126.2479 0 126.2479 1 2 171.6605 33 1 2 171.6605 33 Treatment 3 1 0.2244243 2.533554 0.7167563 134.0753 5.2915315 140.0836 0 140.0836 1 3 176.4105 33 1 3 176.4105 33 Treatment 4 1 0.2244243 2.533554 -0.3144774 138.1379 7.8250850 145.6485 0 145.6485 1 4 176.2812 33 1 4 176.2812 33 Treatment 5 1 0.2244243 2.533554 -0.3251463 138.5409 10.3586386 148.5744 0 148.5744 1 5 188.0455 33 1 5 188.0455 33 Treatment 6 1 0.2244243 2.533554 0.0060782 147.8641 12.8921922 160.7624 0 160.7624 1 6 201.8052 33 1 6 201.8052 33 Treatment 7 1 0.2244243 2.533554 1.2868034 158.6839 15.4257457 175.3965 0 175.3965 1 7 186.9941 33 1 7 186.9941 33 Treatment 8 1 0.2244243 2.533554 -1.0393871 148.0755 17.9592993 164.9955 0 164.9955 1 8 190.1734 33 1 8 190.1734 33 Treatment 9 1 0.2244243 2.533554 0.3857991 150.9601 20.4928529 171.8387 0 171.8387 1 9 163.4426 33 1 9 163.4426 33 Treatment 10 1 0.2244243 2.533554 1.9275765 131.4119 23.0264064 156.3659 1 NA We can look at the amount of missing data: prop.table(table(is.na(data_w_missing$y_missing)))
##
## FALSE  TRUE
##  0.74  0.26

We can also look now at the amount of missing by the time variable.

prop.table(table(is.na(data_w_missing$y_missing), data_w_missing$time))
##
##            0    1    2    3    4    5    6    7    8    9
##   FALSE 0.10 0.10 0.10 0.10 0.10 0.09 0.07 0.06 0.02 0.00
##   TRUE  0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.04 0.08 0.10

### Specify location of dropout

If additional control is desired, specifying the location of the dropout is possible. In this situation, a vector is passed to dropout_location that specifies for each individual (more generally each level 2 unit) where the dropout occurs. Below is an example of this.

set.seed(321)

sim_arguments <- list(
formula = y ~ 1 + time + weight + age + treat + (1 + time| id),
reg_weights = c(4, 0.5, 0.75, 0, 0.33),
fixed = list(time = list(var_type = 'time'),
weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60, var_level = 2),
treat = list(var_type = 'factor',
levels = c('Treatment', 'Control'),
var_level = 2)),
randomeffect = list(int_id = list(variance = 8, var_level = 2),
time_id = list(variance = 3, var_level = 2)),
missing_data = list(new_outcome = 'y_missing',
dropout_location = c(3, 9, 1, 6, 7, 8, 6, 9, 2, 4, 6, 5, 8, 9, 4, 5,
6, 7, 2, 9),
clust_var = 'id', type = 'dropout'),
sample_size = list(level1 = 10, level2 = 20)
)

data_w_missing <- sim_arguments %>%
simulate_fixed(data = NULL, .) %>%
simulate_randomeffect(sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments) %>%
generate_missing(sim_arguments)

head(data_w_missing, n = 10)
X.Intercept. time weight age treat time1 weight1 age1 treat1 level1_id id int_id time_id error fixed_outcome random_effects y missing y_missing
1 0 231.1471 33 1 0 231.1471 33 Treatment 1 1 0.2244243 2.533554 2.3826970 177.6903 0.2244243 180.2974 0 180.2974
1 1 158.6388 33 1 1 158.6388 33 Treatment 2 1 0.2244243 2.533554 -0.3191794 123.8091 2.7579779 126.2479 0 126.2479
1 2 171.6605 33 1 2 171.6605 33 Treatment 3 1 0.2244243 2.533554 0.7167563 134.0753 5.2915315 140.0836 0 140.0836
1 3 176.4105 33 1 3 176.4105 33 Treatment 4 1 0.2244243 2.533554 -0.3144774 138.1379 7.8250850 145.6485 1 NA
1 4 176.2812 33 1 4 176.2812 33 Treatment 5 1 0.2244243 2.533554 -0.3251463 138.5409 10.3586386 148.5744 1 NA
1 5 188.0455 33 1 5 188.0455 33 Treatment 6 1 0.2244243 2.533554 0.0060782 147.8641 12.8921922 160.7624 1 NA
1 6 201.8052 33 1 6 201.8052 33 Treatment 7 1 0.2244243 2.533554 1.2868034 158.6839 15.4257457 175.3965 1 NA
1 7 186.9941 33 1 7 186.9941 33 Treatment 8 1 0.2244243 2.533554 -1.0393871 148.0755 17.9592993 164.9955 1 NA
1 8 190.1734 33 1 8 190.1734 33 Treatment 9 1 0.2244243 2.533554 0.3857991 150.9601 20.4928529 171.8387 1 NA
1 9 163.4426 33 1 9 163.4426 33 Treatment 10 1 0.2244243 2.533554 1.9275765 131.4119 23.0264064 156.3659 1 NA

We can look at the amount of missing data:

prop.table(table(is.na(data_w_missing$y_missing))) ## ## FALSE TRUE ## 0.58 0.42 We can also look now at the amount of missing by the time variable. prop.table(table(is.na(data_w_missing$y_missing), data_w_missing\$time))
##
##             0     1     2     3     4     5     6     7     8     9
##   FALSE 0.100 0.095 0.085 0.080 0.070 0.060 0.040 0.030 0.020 0.000
##   TRUE  0.000 0.005 0.015 0.020 0.030 0.040 0.060 0.070 0.080 0.100

# Model Fit Arguments

The default behavior when fitting models to the generated data is to fit a model with the same formula as the generated data (i.e. formula simulation arguments). Secondly, the default model functions are lm/glm for single level simulation and lmer/glmer for multilevel simulation. Finally, the default regression weights specified in reg_weights are to compute statistics for power, type I error rates, and precision.

These defaults can be overriden by specifying simulation arguments through the named list called model_fit. Some examples were given in the Tidy Simulation vignette and the options that are able to be specified will depend on the type of model fitting function specified. A few examples are given for various model types, however, the user is directed to the documentation of the modeling functions specified through the model_function argument.