simglmsimglmIntroducing a new framework for simulation and power analysis with the simglm package, tidy simulation. The package now has helper functions in which the first argument is always the data and the second argument is always the simulation arguments. A second vignette is written that contains a more exhuastive list of simluation arguments that are allowed. This vignette will give the basics needed to specify simulation arguments to generate simulated data.
There are four primary functions to be used for basic simulation functionality. These include:
simulate_fixed: simulate fixed effectssimulate_error: simulate random errorsimulate_randomeffects: simulate random effects or cluster residualsgenerate_response: based on fixed, error, and random effects, generate outcome variable Yj given the following Yj=g(Xjβ+Zjbj+ej).We will dive into what each of these component represent in a second, but first let’s start with an example that simulates a linear regression model.
Let us assume for this first example that our outcome of interest is continuous and that the data is not clustered. In this example, our model would look like: Yj=Xjβ+ej. In this equation, Yj represents the continuous outcome, Xjβ represents the fixed portion of the model comprised of regression coefficients (β) and a design matrix (Xj), and ej represents random error.
The fixed portion of the model represents variables that are treated as fixed. This means that the values observed in the data are the only values we are interested in, will generalize to, or that we consider values of interest in our population. Let us consider the following formula: y ~ 1 + x1 + x2. In this formula there are a total of two varaibles that are assumed to be fixed, x1 and x2. These variables together with an intercept would make up the design matrix Xj. Let’s generate some example data using the simglm package and the simulate_fixed function.
library(simglm)
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + x1 + x2,
fixed = list(x1 = list(var_type = 'continuous', mean = 180, sd = 30),
x2 = list(var_type = 'continuous', mean = 40, sd = 5)),
sample_size = 10
)
simulate_fixed(data = NULL, sim_arguments)## X.Intercept. x1 x2 level1_id
## 1 1 231.1471 41.73851 1
## 2 1 158.6388 47.42296 2
## 3 1 171.6605 40.94163 3
## 4 1 176.4105 52.21630 4
## 5 1 176.2812 34.23280 5
## 6 1 188.0455 35.97664 6
## 7 1 201.8052 42.28035 7
## 8 1 186.9941 42.10166 8
## 9 1 190.1734 42.88792 9
## 10 1 163.4426 42.23178 10The first three columns of the resulting data frame is the design matrix described above, Xj. You may have noticed that the simulate_fixed function needs three elements defined in the simulation arguments (called sim_arguments) above. These elements are:
formula: this argument represents a R formula that is used to represent the model that is wished to be simulated. For the simulate_fixed argument, only the right hand side is used.fixed: these represent the specific details for generating the variables (other than the intercept) in the right hand side of the formula simulation argument. Each variable is specified as its own list by name in the formula argument and the var_type specifies the type of variable to generate. The vary_type argument is required for each fixed variable to simulate. Optional arguments, for example mean = and sd = above, will be discussed in more detail later.sample_size: this argument tells the function how many responses to generate.The columns x1 and x2 would represent variables that we would gather if these data were real. To reflect a real life scenario, consider the following fixed simuation.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'continuous', mean = 40, sd = 5)),
sample_size = 10
)
simulate_fixed(data = NULL, sim_arguments)## X.Intercept. weight age level1_id
## 1 1 231.1471 41.73851 1
## 2 1 158.6388 47.42296 2
## 3 1 171.6605 40.94163 3
## 4 1 176.4105 52.21630 4
## 5 1 176.2812 34.23280 5
## 6 1 188.0455 35.97664 6
## 7 1 201.8052 42.28035 7
## 8 1 186.9941 42.10166 8
## 9 1 190.1734 42.88792 9
## 10 1 163.4426 42.23178 10Now instead of the variables being called x1 and x2, they now reflect variables weight (measured in lbs) and age (measured continuously, not rounded to whole digits). If we wished to change the 'age' variable to be rounded to a whole integer, we could change the variable type to 'ordinal' as such.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60)),
sample_size = 10
)
simulate_fixed(data = NULL, sim_arguments)## X.Intercept. weight age level1_id
## 1 1 231.1471 44 1
## 2 1 158.6388 38 2
## 3 1 171.6605 31 3
## 4 1 176.4105 40 4
## 5 1 176.2812 60 5
## 6 1 188.0455 47 6
## 7 1 201.8052 33 7
## 8 1 186.9941 43 8
## 9 1 190.1734 52 9
## 10 1 163.4426 31 10When specifying a var_type as 'ordinal', an additional levels argument is needed to determine which values are possible for the simulation. The last type of variable that is useful to discuss now would be factor or categorical variables. These variables can be generated by setting the var_type = 'factor'.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
sample_size = 10
)
simulate_fixed(data = NULL, sim_arguments)## X.Intercept. weight age sex sex_orig level1_id
## 1 1 231.1471 44 0 female 1
## 2 1 158.6388 38 0 female 2
## 3 1 171.6605 31 0 female 3
## 4 1 176.4105 40 0 female 4
## 5 1 176.2812 60 1 male 5
## 6 1 188.0455 47 0 female 6
## 7 1 201.8052 33 0 female 7
## 8 1 186.9941 43 1 male 8
## 9 1 190.1734 52 0 female 9
## 10 1 163.4426 31 0 female 10The required arguments when a factor variable are identical to an ordinal variable, however for factor variables the levels argument can either be an integer or can be a character vector where the character labels are specified directly. As you can see from the output, when a character vector is used, two variables are returned, one that contains the variable represented numerically and another that is the variable represented as a character vector. More details on this behavior to follow.
The simulation of random error (ej from the equation above) is a bit simpler than generating the fixed effects. Suppose for example, we want to simply simulate random errors that are normally distributed with a mean of 0 and a variance of 1. This can be done with the simulate_error function.
## error level1_id
## 1 1.7049032 1
## 2 -0.7120386 2
## 3 -0.2779849 3
## 4 -0.1196490 4
## 5 -0.1239606 5
## 6 0.2681838 6
## 7 0.7268415 7
## 8 0.2331354 8
## 9 0.3391139 9
## 10 -0.5519147 10The simulate_error function only needs to know how many data values to simulate. By default, the rnorm function is used to generate random error and this function assumes a mean of 0 and standard deviation of 1 by default. I personally prefer the slightly more verbose code however.
set.seed(321)
sim_arguments <- list(
error = list(variance = 1),
sample_size = 10
)
simulate_error(data = NULL, sim_arguments)## error level1_id
## 1 1.7049032 1
## 2 -0.7120386 2
## 3 -0.2779849 3
## 4 -0.1196490 4
## 5 -0.1239606 5
## 6 0.2681838 6
## 7 0.7268415 7
## 8 0.2331354 8
## 9 0.3391139 9
## 10 -0.5519147 10This code makes it clearer when the variance of the errors is wished to be specified as some other value. For example:
set.seed(321)
sim_arguments <- list(
error = list(variance = 25),
sample_size = 10
)
simulate_error(data = NULL, sim_arguments)## error level1_id
## 1 8.5245161 1
## 2 -3.5601928 2
## 3 -1.3899246 3
## 4 -0.5982451 4
## 5 -0.6198031 5
## 6 1.3409189 6
## 7 3.6342074 7
## 8 1.1656771 8
## 9 1.6955694 9
## 10 -2.7595733 10Now that we have seen the basics of simulating fixed variables (covariates) and random error, we can now generate the response by combining the previous two steps and then using the generate_response function. What makes this a tidy simulation is that the pipe from magrittr, %>% can be used to combine steps into a simulation pipeline.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.3, -0.1, 0.5)
)
simulate_fixed(data = NULL, sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments)## X.Intercept. weight age sex sex_orig level1_id error fixed_outcome
## 1 1 231.1471 44 0 female 1 4.5862776 66.94413
## 2 1 158.6388 38 0 female 2 -0.5353077 45.79165
## 3 1 171.6605 31 0 female 3 4.9416770 50.39814
## 4 1 176.4105 40 0 female 4 -5.3611940 50.92316
## 5 1 176.2812 60 1 male 5 -3.7900764 49.38435
## 6 1 188.0455 47 0 female 6 0.4750036 53.71365
## 7 1 201.8052 33 0 female 7 -11.6546559 59.24157
## 8 1 186.9941 43 1 male 8 2.0875799 54.29822
## 9 1 190.1734 52 0 female 9 -5.6016371 53.85202
## 10 1 163.4426 31 0 female 10 -2.3734235 47.93277
## random_effects y
## 1 0 71.53041
## 2 0 45.25635
## 3 0 55.33981
## 4 0 45.56196
## 5 0 45.59428
## 6 0 54.18866
## 7 0 47.58692
## 8 0 56.38580
## 9 0 48.25039
## 10 0 45.55934The only additional argument that is needed for the generate_response function is the reg_weights argument. This argument represents the regression coeficients associated with β in the equation Yj=Xjβ+ej. The regression coefficients are multiplied by the design matrix to generate the column labeled “fixed_outcome” in the output. The output also contains the column, “random_effects” which are all 0 here indicating there are no random effects and the response variable, “y”.
Non-normal outcomes are possible with simglm. Two non-normal outcomes are currently supported with more support coming in the future. Binary and count outcomes are supported and can be specified with the outcome_type simulation argument. If outcome_type = 'logistic' or outcome_type = 'binary' then a binary outcome is generated (ie. 0/1 variable) using the rbinom function. If outcome_type = 'count' or outcome_type = 'poisson' then the outcome is transformed to be a count variable (ie. discrete variable; 0, 1, 2, etc.).
Here is an example of generating a binary outcome.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.3, -0.1, 0.5),
outcome_type = 'binary'
)
simulate_fixed(data = NULL, sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments)## X.Intercept. weight age sex sex_orig level1_id error fixed_outcome
## 1 1 231.1471 44 0 female 1 4.5862776 66.94413
## 2 1 158.6388 38 0 female 2 -0.5353077 45.79165
## 3 1 171.6605 31 0 female 3 4.9416770 50.39814
## 4 1 176.4105 40 0 female 4 -5.3611940 50.92316
## 5 1 176.2812 60 1 male 5 -3.7900764 49.38435
## 6 1 188.0455 47 0 female 6 0.4750036 53.71365
## 7 1 201.8052 33 0 female 7 -11.6546559 59.24157
## 8 1 186.9941 43 1 male 8 2.0875799 54.29822
## 9 1 190.1734 52 0 female 9 -5.6016371 53.85202
## 10 1 163.4426 31 0 female 10 -2.3734235 47.93277
## random_effects untransformed_outcome y
## 1 0 71.53041 1
## 2 0 45.25635 1
## 3 0 55.33981 1
## 4 0 45.56196 1
## 5 0 45.59428 1
## 6 0 54.18866 1
## 7 0 47.58692 1
## 8 0 56.38580 1
## 9 0 48.25039 1
## 10 0 45.55934 1And finally, an example of generating a count outcome. Note, the weight variable here has been grand mean centered in the generation (ie. mean = 0). This helps to ensure that the counts are not too large.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 0, sd = 30),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.01, 0.5),
outcome_type = 'count'
)
simulate_fixed(data = NULL, sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments)## X.Intercept. weight sex sex_orig level1_id error fixed_outcome
## 1 1 51.147097 1 male 1 -4.0233584 3.011471
## 2 1 -21.361157 1 male 2 2.2803457 2.286388
## 3 1 -8.339547 0 female 3 2.1016629 1.916605
## 4 1 -3.589471 1 male 4 2.8879225 2.464105
## 5 1 -3.718819 1 male 5 2.2317803 2.462812
## 6 1 8.045513 0 female 6 4.5862776 2.080455
## 7 1 21.805245 0 female 7 -0.5353077 2.218052
## 8 1 6.994062 0 female 8 4.9416770 2.069941
## 9 1 10.173416 1 male 9 -5.3611940 2.601734
## 10 1 -16.557440 0 female 10 -3.7900764 1.834426
## random_effects untransformed_outcome y
## 1 0 -1.011887 0
## 2 0 4.566734 90
## 3 0 4.018267 58
## 4 0 5.352028 194
## 5 0 4.694592 114
## 6 0 6.666733 743
## 7 0 1.682745 7
## 8 0 7.011618 1061
## 9 0 -2.759460 0
## 10 0 -1.955651 0Now that the basics of tidy simulation have been shown in the context of a linear regression model, let’s explore an example in which the power for this model is to be evaluated. In particular, suppose we are interested in estimating empirical power for the three fixed effects based on the following formula: y ~ 1 + weight + age + sex. More specifically, we are interested in estimating power for “weight”, “age”, and “sex” variables. A few additional functions are needed for this step including:
model_fit: this function will fit a model to the data.extract_coefficients: this function will extract the fixed coefficients based on the model fitted.To fit a model and extract coefficients, we could do the following building off the example from the previous section:
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.3, -0.1, 0.5)
)
simulate_fixed(data = NULL, sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments) %>%
model_fit(sim_arguments) %>%
extract_coefficients()## # A tibble: 4 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -1.24 19.1 -0.0650 0.950
## 2 weight 0.328 0.103 3.19 0.0189
## 3 age -0.198 0.271 -0.733 0.491
## 4 sex 2.87 5.93 0.484 0.646This output contains the model output for a single data simulation, more specifically we can see the parameter name, parameter estimate, the standard error for the parameter estimate, the test statistics, and the p-value. These were estimated using the lm function based on the same formula defined in sim_arguments. It is possible to specify your own formula, model fitting function, and regression weights to the model_fit function. For example, suppose we knew that weight was an important predictor, but are unable to collect it in real life. We could then specify an alternative model when evaluating power, but include the variable in the data generation step.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.3, -0.1, 0.5),
model_fit = list(formula = y ~ 1 + age + sex,
model_function = 'lm'),
reg_weights_model = c(2, -0.1, 0.5)
)
simulate_fixed(data = NULL, sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments) %>%
model_fit(sim_arguments) %>%
extract_coefficients()## # A tibble: 3 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 49.7 15.9 3.13 0.0167
## 2 age 0.0488 0.394 0.124 0.905
## 3 sex -1.25 8.79 -0.143 0.891Notice that we now get very different parameter estimates for this single data generation process which reflects the contribution of the variable “weight” that is not taken into account in the model fitting.
When evaluating empirical power, it is essential to replicate the analysis just like a Monte Carlo study to avoid extreme simulation conditions. The simglm package offers functions to aid in the replication given the simulation conditions and the desired simulation framework. To do this, two additional functions are used:
replicate_simulation: this function replicates the simulationcompute_statistics: this function compute desired statistics across the replications.set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 10,
reg_weights = c(2, 0.3, -0.1, 0.5),
model_fit = list(formula = y ~ 1 + age + sex,
model_function = 'lm'),
reg_weights_model = c(2, -0.1, 0.5),
replications = 10,
extract_coefficients = TRUE
)
replicate_simulation(sim_arguments) %>%
compute_statistics(sim_arguments)## New names:
## * term -> term...1
## * term -> term...3
## * crit_value -> crit_value...6
## * term -> term...7
## * crit_value -> crit_value...10
## * ...## # A tibble: 3 x 15
## term...1 avg_estimate term...3 power avg_test_stat crit_value...6 term...7
## <chr> <dbl> <chr> <dbl> <dbl> <dbl> <chr>
## 1 (Interc~ 47.2 (Interc~ 0.6 3.37 1.96 (Interc~
## 2 age 0.0852 age 0.3 0.102 1.96 age
## 3 sex -0.509 sex 0.1 -0.108 1.96 sex
## # ... with 8 more variables: type_1_error <dbl>, avg_adjtest_stat <dbl>,
## # crit_value...10 <dbl>, term...11 <chr>, param_estimate_sd <dbl>,
## # avg_standard_error <dbl>, precision_ratio <dbl>, replications <dbl>As can be seen from the output, the default behavior is to return statistics for power, average test statistic, type I error rate, adjusted average test statistic, standard deviation of parameter estimate, average standard error, precision ration (standard deviation of parameter estimate divided by average standard error), and the number of replications for each term.
To generate this output, only the number of replications is needed. Here only 10 replications were done, in practice many more replications would be done to ensure there are accurate results.
The default power parameter values used are: Normal distribution, two-tailed alternative hypotheses, and alpha of 0.05. Additional power parameters can be passed directly to override default values by including a power argument within the simulation arguments. For example, if a t-distribution with one degree of freedom an alpha of 0.02 is desired, these can be added as follows:
Note: A user would likely want to change plan(sequential) to something like plan(mutisession) or plan(multicore) to run these in parallel. plan(sequential) is used here for vignette processing.
set.seed(321)
library(future)
plan(sequential)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex,
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
error = list(variance = 25),
sample_size = 50,
reg_weights = c(2, 0.3, -0.1, 0.5),
model_fit = list(formula = y ~ 1 + age + sex,
model_function = 'lm'),
reg_weights_model = c(2, -0.1, 0.5),
replications = 1000,
power = list(
dist = 'qt',
alpha = .02,
opts = list(df = 1)
),
extract_coefficients = TRUE
)
replicate_simulation(sim_arguments) %>%
compute_statistics(sim_arguments)## New names:
## * term -> term...1
## * term -> term...3
## * crit_value -> crit_value...6
## * term -> term...7
## * crit_value -> crit_value...10
## * ...## # A tibble: 3 x 15
## term...1 avg_estimate term...3 power avg_test_stat crit_value...6 term...7
## <chr> <dbl> <chr> <dbl> <dbl> <dbl> <chr>
## 1 (Interc~ 55.9 (Interc~ 0 7.26 31.8 (Interc~
## 2 age -0.0994 age 0 -0.603 31.8 age
## 3 sex 0.561 sex 0 0.189 31.8 sex
## # ... with 8 more variables: type_1_error <dbl>, avg_adjtest_stat <dbl>,
## # crit_value...10 <dbl>, term...11 <chr>, param_estimate_sd <dbl>,
## # avg_standard_error <dbl>, precision_ratio <dbl>, replications <dbl>Nested designs are ones in which data belong to multiple levels. An example could be individuals nested within neighborhoods. In this example, a specific individual is tied directly to one neighborhood. These types of data often include correlations between individuals within a neighborhood that need to be taken into account when modeling the data. These types of data can be generated with simglm.
To do this, the formula syntax introduced above is modified slightly to include information on the nesting structure. In the example below, the nesting structure in the formula is specified in the portion within the parentheses. How the part within parentheses could be read is, add a random intercept effect for each neighborhood. This random intercept effect is similar to random error found in regression models, except the error is associated with neighborhoods and is the same values for all individuals within that neighborhood. The simulation arguments, randomeffect provides information about this term, where the variance can be specified.
Finally, the sample_size argument needs to be modified to include information about the two levels of nested. You could read the sample_size = list(level1 = 10, level2 = 20) argument below as: create 20 neighborhoods (ie. level2 = 20) and within each neighborhood have 10 individuals (ie. level1 = 10). Therefore the total sample size (ie. rows in the data) would be 200.
set.seed(321)
sim_arguments <- list(
formula = y ~ 1 + weight + age + sex + (1 | neighborhood),
reg_weights = c(4, -0.03, 0.2, 0.33),
fixed = list(weight = list(var_type = 'continuous', mean = 180, sd = 30),
age = list(var_type = 'ordinal', levels = 30:60),
sex = list(var_type = 'factor', levels = c('male', 'female'))),
randomeffect = list(int_neighborhood = list(variance = 8, var_level = 2)),
sample_size = list(level1 = 10, level2 = 20)
)
nested_data <- sim_arguments %>%
simulate_fixed(data = NULL, .) %>%
simulate_randomeffect(sim_arguments) %>%
simulate_error(sim_arguments) %>%
generate_response(sim_arguments)
head(nested_data, n = 10)## X.Intercept. weight age sex sex_orig level1_id neighborhood
## 1 1 231.1471 39 0 female 1 1
## 2 1 158.6388 30 0 female 2 1
## 3 1 171.6605 35 0 female 3 1
## 4 1 176.4105 42 0 female 4 1
## 5 1 176.2812 46 1 male 5 1
## 6 1 188.0455 34 0 female 6 1
## 7 1 201.8052 43 0 female 7 1
## 8 1 186.9941 38 0 female 8 1
## 9 1 190.1734 53 1 male 9 1
## 10 1 163.4426 41 1 male 10 1
## int_neighborhood error fixed_outcome random_effects y
## 1 -2.018169 0.03970688 4.865587 -2.018169 2.887125
## 2 -2.018169 1.10030811 5.240835 -2.018169 4.322974
## 3 -2.018169 1.48822364 5.850186 -2.018169 5.320241
## 4 -2.018169 -0.29997335 7.107684 -2.018169 4.789542
## 5 -2.018169 0.47789336 8.241565 -2.018169 6.701289
## 6 -2.018169 -0.38778144 5.158635 -2.018169 2.752684
## 7 -2.018169 -0.35268182 6.545843 -2.018169 4.174992
## 8 -2.018169 0.93313126 5.990178 -2.018169 4.905141
## 9 -2.018169 -1.39912533 9.224798 -2.018169 5.807503
## 10 -2.018169 -0.01603532 7.626723 -2.018169 5.592519## [1] 200The vignette simulation_arguments contains more example of specifying random effects and additional nesting designs including three levels of nested and cross-classified models.