Package Introduction

Common usage

A minimal call of mmrm(), consisting of only formula and data arguments will produce an object of class mmrm, mmrm_fit, and mmrm_tmb.

Here we fit a mmrm model with us (unstructured) covariance structure specified, as well as the defaults of reml = TRUE and control = mmrm_control().

library(mmrm)
fit <- mmrm(
  formula = FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data
)

The code specifies an MMRM with the given covariates and an unstructured covariance matrix for the timepoints (also called visits in the clinical trial context, here given by AVISIT) within the subjects (here USUBJID). While by default this uses restricted maximum likelihood (REML), it is also possible to use ML, see ?mmrm.

Printing the object will show you output which should be familiar to anyone who has used any popular modeling functions such as stats::lm(), stats::glm(), glmmTMB::glmmTMB(), and lme4::nlmer(). From this print out we see the function call, the data used, the covariance structure with number of variance parameters, as well as the likelihood method, and model deviance achieved. Additionally the user is provided a printout of the estimated coefficients and the model convergence information:

fit
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3386.45
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                   30.77747548                    1.53049977 
#>                     RACEWhite                     SEXFemale 
#>                    5.64356535                    0.32606192 
#>                      ARMCDTRT                    AVISITVIS2 
#>                    3.77423004                    4.83958845 
#>                    AVISITVIS3                    AVISITVIS4 
#>                   10.34211288                   15.05389826 
#>           ARMCDTRT:AVISITVIS2           ARMCDTRT:AVISITVIS3 
#>                   -0.04192625                   -0.69368537 
#>           ARMCDTRT:AVISITVIS4 
#>                    0.62422703 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

The summary() method then provides the coefficients table with Satterthwaite degrees of freedom as well as the covariance matrix estimate:

summary(fit)
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Method:      Satterthwaite
#> Vcov Method: Asymptotic
#> Inference:   REML
#> 
#> Model selection criteria:
#>      AIC      BIC   logLik deviance 
#>   3406.4   3439.3  -1693.2   3386.4 
#> 
#> Coefficients: 
#>                                Estimate Std. Error        df t value Pr(>|t|)
#> (Intercept)                    30.77748    0.88656 218.80000  34.715  < 2e-16
#> RACEBlack or African American   1.53050    0.62448 168.67000   2.451 0.015272
#> RACEWhite                       5.64357    0.66561 157.14000   8.479 1.56e-14
#> SEXFemale                       0.32606    0.53195 166.13000   0.613 0.540744
#> ARMCDTRT                        3.77423    1.07415 145.55000   3.514 0.000589
#> AVISITVIS2                      4.83959    0.80172 143.88000   6.037 1.27e-08
#> AVISITVIS3                     10.34211    0.82269 155.56000  12.571  < 2e-16
#> AVISITVIS4                     15.05390    1.31281 138.47000  11.467  < 2e-16
#> ARMCDTRT:AVISITVIS2            -0.04193    1.12932 138.56000  -0.037 0.970439
#> ARMCDTRT:AVISITVIS3            -0.69369    1.18765 158.17000  -0.584 0.559996
#> ARMCDTRT:AVISITVIS4             0.62423    1.85085 129.72000   0.337 0.736463
#>                                  
#> (Intercept)                   ***
#> RACEBlack or African American *  
#> RACEWhite                     ***
#> SEXFemale                        
#> ARMCDTRT                      ***
#> AVISITVIS2                    ***
#> AVISITVIS3                    ***
#> AVISITVIS4                    ***
#> ARMCDTRT:AVISITVIS2              
#> ARMCDTRT:AVISITVIS3              
#> ARMCDTRT:AVISITVIS4              
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance estimate:
#>         VIS1    VIS2    VIS3    VIS4
#> VIS1 40.5537 14.3960  4.9747 13.3867
#> VIS2 14.3960 26.5715  2.7855  7.4745
#> VIS3  4.9747  2.7855 14.8979  0.9082
#> VIS4 13.3867  7.4745  0.9082 95.5568

Common customizations

From the high-level mmrm() interface, common changes to the default function call can be specified.

Control Function

For fine control, mmrm_control() is provided. This function allows the user to choose the adjustment method for the degrees of freedom and the coefficients covariance matrix, specify optimization routines, number of cores to be used on Unix systems for trying several optimizers in parallel, provide a vector of starting parameter values, decide the action to be taken when the defined design matrix is singular, not drop unobserved visit levels. For example:

mmrm_control(
  method = "Kenward-Roger",
  optimizer = c("L-BFGS-B", "BFGS"),
  n_cores = 2,
  start = c(0, 1, 1, 0, 1, 0),
  accept_singular = FALSE,
  drop_visit_levels = FALSE
)

Note that this control list can either be passed via the control argument to mmrm, or selected controls can be directly specified in the mmrm call. We will see this below.

REML or ML

Users can specify if REML should be used (default) or if ML should be used in optimization.

fit_ml <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  reml = FALSE
)
fit_ml
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   ML
#> Deviance:    3397.934
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                    30.9663423                     1.5086851 
#>                     RACEWhite                      ARMCDTRT 
#>                     5.6133151                     3.7761037 
#>                    AVISITVIS2                    AVISITVIS3 
#>                     4.8270155                    10.3353319 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                    15.0487715                    -0.0156154 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                    -0.6663598                     0.6317222 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

Optimizer

Users can specify which optimizer should be used, changing from the default of four optimizers, which starts with L-BFGS-B and proceeds through the other choices if optimization fails to converge. Other choices are BFGS, CG, nlminb and other user-defined custom optimizers.

L-BFGS-B, BFGS and CG are all implemented with stats::optim() and the Hessian is not used, while nlminb is using stats::nlminb() which in turn uses both the gradient and the Hessian (by default but can be switch off) for the optimization.

fit_opt <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  optimizer = "BFGS"
)
fit_opt
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3387.373
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                    30.9676902                     1.5046744 
#>                     RACEWhite                      ARMCDTRT 
#>                     5.6131048                     3.7755423 
#>                    AVISITVIS2                    AVISITVIS3 
#>                     4.8285855                    10.3331770 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                    15.0525706                    -0.0173504 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                    -0.6675190                     0.6309586 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message:

Covariance Structure

Covariance structures supported by the mmrm are being continuously developed. For a complete list and description please visit the covariance vignette. Below we see the function call for homogeneous compound symmetry (cs).

fit_cs <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + cs(AVISIT | USUBJID),
  data = fev_data,
  reml = FALSE
)
fit_cs
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + cs(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  compound symmetry (2 variance parameters)
#> Inference:   ML
#> Deviance:    3536.989
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                    31.4207077                     0.5357237 
#>                     RACEWhite                      ARMCDTRT 
#>                     5.4546329                     3.4305212 
#>                    AVISITVIS2                    AVISITVIS3 
#>                     4.8326353                    10.2395076 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                    15.0672680                     0.2801641 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                    -0.5894964                     0.7939750 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

The time points have to be unique for each subject. That is, there cannot be time points with multiple observations for any subject. The rationale is that these observations would need to be correlated, but it is not possible within the currently implemented covariance structure framework to do that correctly. Moreover, for non-spatial covariance structures, the time variable must be coded as a factor.

Weighting

Users can perform weighted MMRM by specifying a numeric vector weights with positive values.

fit_wt <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  weights = fev_data$WEIGHT
)
fit_wt
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     fev_data$WEIGHT
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3476.526
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                   31.20065229                    1.18452837 
#>                     RACEWhite                      ARMCDTRT 
#>                    5.36525917                    3.39695951 
#>                    AVISITVIS2                    AVISITVIS3 
#>                    4.85890820                   10.03942420 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                   14.79354054                    0.03418184 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                    0.01308088                    0.86701567 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

Grouped Covariance Structure

Grouped covariance structures are supported by themmrm package. Covariance matrices for each group are identically structured (unstructured, compound symmetry, etc) but the estimates are allowed to vary across groups. We use the form cs(time | group / subject) to specify the group variable.

Here is an example of how we use ARMCD as group variable.

fit_cs <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + cs(AVISIT | ARMCD / USUBJID),
  data = fev_data,
  reml = FALSE
)
VarCorr(fit_cs)
#> $PBO
#>           VIS1      VIS2      VIS3      VIS4
#> VIS1 37.823638  3.601296  3.601296  3.601296
#> VIS2  3.601296 37.823638  3.601296  3.601296
#> VIS3  3.601296  3.601296 37.823638  3.601296
#> VIS4  3.601296  3.601296  3.601296 37.823638
#> 
#> $TRT
#>          VIS1     VIS2     VIS3     VIS4
#> VIS1 49.58110 10.98112 10.98112 10.98112
#> VIS2 10.98112 49.58110 10.98112 10.98112
#> VIS3 10.98112 10.98112 49.58110 10.98112
#> VIS4 10.98112 10.98112 10.98112 49.58110

We can see that the estimated covariance matrices are different in different ARMCD groups.

Adjustment Method

In additional to the residual and Between-Within degrees of freedom, both Satterthwaite and Kenward-Roger adjustment methods are available. The default is Satterthwaite adjustment of the degrees of freedom. To use e.g. the Kenward-Roger adjustment of the degrees of freedom as well as the coefficients covariance matrix, use the method argument:

A list of all allowed method is

1. “Kenward-Roger”
2. “Satterthwaite”
3. “Residual”
4. “Between-Within”

fit_kr <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  method = "Kenward-Roger"
)

Note that this requires reml = TRUE, i.e. Kenward-Roger adjustment is not possible when using maximum likelihood inference. While this adjustment choice is not visible in the print() result of the fitted model (because the initial model fit is not affected by the choice of the adjustment method), looking at the summary we see the method and the correspondingly adjusted standard errors and degrees of freedom:

summary(fit_kr)
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Covariance:  unstructured (10 variance parameters)
#> Method:      Kenward-Roger
#> Vcov Method: Kenward-Roger
#> Inference:   REML
#> 
#> Model selection criteria:
#>      AIC      BIC   logLik deviance 
#>   3407.4   3440.2  -1693.7   3387.4 
#> 
#> Coefficients: 
#>                                Estimate Std. Error        df t value Pr(>|t|)
#> (Intercept)                    30.96770    0.83335 187.91000  37.160  < 2e-16
#> RACEBlack or African American   1.50465    0.62901 169.95000   2.392  0.01784
#> RACEWhite                       5.61310    0.67139 158.87000   8.360 2.98e-14
#> ARMCDTRT                        3.77556    1.07910 146.27000   3.499  0.00062
#> AVISITVIS2                      4.82859    0.80408 143.66000   6.005 1.49e-08
#> AVISITVIS3                     10.33317    0.82303 155.66000  12.555  < 2e-16
#> AVISITVIS4                     15.05256    1.30180 138.39000  11.563  < 2e-16
#> ARMCDTRT:AVISITVIS2            -0.01737    1.13154 138.39000  -0.015  0.98777
#> ARMCDTRT:AVISITVIS3            -0.66753    1.18714 158.21000  -0.562  0.57470
#> ARMCDTRT:AVISITVIS4             0.63094    1.83319 129.64000   0.344  0.73127
#>                                  
#> (Intercept)                   ***
#> RACEBlack or African American *  
#> RACEWhite                     ***
#> ARMCDTRT                      ***
#> AVISITVIS2                    ***
#> AVISITVIS3                    ***
#> AVISITVIS4                    ***
#> ARMCDTRT:AVISITVIS2              
#> ARMCDTRT:AVISITVIS3              
#> ARMCDTRT:AVISITVIS4              
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Covariance estimate:
#>         VIS1    VIS2    VIS3    VIS4
#> VIS1 40.7335 14.2740  5.1411 13.5288
#> VIS2 14.2740 26.2243  2.6391  7.3219
#> VIS3  5.1411  2.6391 14.9497  1.0341
#> VIS4 13.5288  7.3219  1.0341 95.6006

For one-dimensional contrasts as in the coefficients table above, the degrees of freedom are the same for Kenward-Roger and Satterthwaite. However, Kenward-Roger uses adjusted standard errors, hence the p-values are different.

Note that if you would like to match SAS results for an unstructured covariance model, you can use the linear Kenward-Roger approximation:

fit_kr_lin <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  method = "Kenward-Roger",
  vcov = "Kenward-Roger-Linear"
)

This is due to the different parametrization of the unstructured covariance matrix, see the Kenward-Roger vignette for details.

Variance-covariance for Coefficients

There are multiple variance-covariance estimator available for the coefficients, including:

1. “Asymptotic”
2. “Empirical” (Cluster Robust Sandwich)
3. “Empirical-Jackknife”
4. “Empirical-Bias-Reduced”
5. “Kenward-Roger”
6. “Kenward-Roger-Linear”

Please note that, not all combinations of variance-covariance for coefficients and method of degrees of freedom are possible, e.g. “Kenward-Roger” and “Kenward-Roger-Linear” are available only when the degrees of freedom method is “Kenward-Roger”.

Details can be found in Coefficients Covariance Matrix Adjustment vignette and Weighted Least Square Empirical Covariance.

An example of using other variance-covariance is:

fit_emp <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  method = "Satterthwaite",
  vcov = "Empirical"
)

Keeping Unobserved Visits

Sometimes not all possible time points are observed in a given data set. When using a structured covariance matrix, e.g. with auto-regressive structure, then it can be relevant to keep the correct distance between the observed time points.

Consider the following example where we have deliberately removed the VIS3 observations from our initial example data set fev_data to obtain sparse_data. We first fit the model where we do not drop the visit level explicitly, using the drop_visit_levels = FALSE choice. Second we fit the model by default without this option.

sparse_data <- fev_data[fev_data$AVISIT != "VIS3", ]
sparse_result <- mmrm(
  FEV1 ~ RACE + ar1(AVISIT | USUBJID),
  data = sparse_data,
  drop_visit_levels = FALSE
)

dropped_result <- mmrm(
  FEV1 ~ RACE + ar1(AVISIT | USUBJID),
  data = sparse_data
)
#> In AVISIT there are dropped visits: VIS3

We see that we get a message about the dropped visit level by default. Now we can compare the estimated correlation matrices:

cov2cor(VarCorr(sparse_result))
#>            VIS1      VIS2      VIS3       VIS4
#> VIS1 1.00000000 0.4051834 0.1641736 0.06652042
#> VIS2 0.40518341 1.0000000 0.4051834 0.16417360
#> VIS3 0.16417360 0.4051834 1.0000000 0.40518341
#> VIS4 0.06652042 0.1641736 0.4051834 1.00000000
cov2cor(VarCorr(dropped_result))
#>            VIS1      VIS2       VIS4
#> VIS1 1.00000000 0.1468464 0.02156386
#> VIS2 0.14684640 1.0000000 0.14684640
#> VIS4 0.02156386 0.1468464 1.00000000

We see that when using the default, second result, we just drop the VIS3 from the covariance matrix. As a consequence, we model the correlation between VIS2 and VIS4 the same as the correlation between VIS1 and VIS2. Hence we get a smaller correlation estimate here compared to the first result, which includes VIS3 explicitly.

Extraction of model features

Similar to model objects created in other packages, components of mmrm and mmrm_tmb objects can be accessed with standard methods. Additionally, component() is provided to allow deeper and more precise access for those interested in digging through model output. Complete documentation of standard model output methods supported for mmrm_tmb objects can be found at the package website.

Summary

The summary method for mmrm objects provides easy access to frequently needed model components.

fit <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data
)
fit_summary <- summary(fit)

From this summary object, you can easily retrieve the coefficients table.

fit_summary$coefficients
#>                                  Estimate Std. Error       df     t value
#> (Intercept)                   30.96769899  0.8293349 187.9132 37.34040185
#> RACEBlack or African American  1.50464863  0.6206596 169.9454  2.42427360
#> RACEWhite                      5.61309565  0.6630909 158.8700  8.46504747
#> ARMCDTRT                       3.77555734  1.0762774 146.2690  3.50797778
#> AVISITVIS2                     4.82858803  0.8017144 143.6593  6.02282805
#> AVISITVIS3                    10.33317002  0.8224414 155.6572 12.56401918
#> AVISITVIS4                    15.05255715  1.3128602 138.3916 11.46546844
#> ARMCDTRT:AVISITVIS2           -0.01737409  1.1291645 138.3926 -0.01538668
#> ARMCDTRT:AVISITVIS3           -0.66753189  1.1865359 158.2106 -0.56258887
#> ARMCDTRT:AVISITVIS4            0.63094392  1.8507884 129.6377  0.34090549
#>                                   Pr(>|t|)
#> (Intercept)                   7.122411e-89
#> RACEBlack or African American 1.638725e-02
#> RACEWhite                     1.605553e-14
#> ARMCDTRT                      6.001485e-04
#> AVISITVIS2                    1.366921e-08
#> AVISITVIS3                    1.927523e-25
#> AVISITVIS4                    8.242709e-22
#> ARMCDTRT:AVISITVIS2           9.877459e-01
#> ARMCDTRT:AVISITVIS3           5.745112e-01
#> ARMCDTRT:AVISITVIS4           7.337266e-01

Other model parameters and metadata available in the summary object is as follows:

str(fit_summary)
#> List of 15
#>  $ cov_type        : chr "us"
#>  $ reml            : logi TRUE
#>  $ n_groups        : int 1
#>  $ n_theta         : int 10
#>  $ n_subjects      : int 197
#>  $ n_timepoints    : int 4
#>  $ n_obs           : int 537
#>  $ beta_vcov       : num [1:10, 1:10] 0.688 -0.207 -0.163 -0.569 -0.422 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:10] "(Intercept)" "RACEBlack or African American" "RACEWhite" "ARMCDTRT" ...
#>   .. ..$ : chr [1:10] "(Intercept)" "RACEBlack or African American" "RACEWhite" "ARMCDTRT" ...
#>  $ varcor          : num [1:4, 1:4] 40.73 14.27 5.14 13.53 14.27 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:4] "VIS1" "VIS2" "VIS3" "VIS4"
#>   .. ..$ : chr [1:4] "VIS1" "VIS2" "VIS3" "VIS4"
#>  $ method          : chr "Satterthwaite"
#>  $ vcov            : chr "Asymptotic"
#>  $ coefficients    : num [1:10, 1:5] 30.97 1.5 5.61 3.78 4.83 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : chr [1:10] "(Intercept)" "RACEBlack or African American" "RACEWhite" "ARMCDTRT" ...
#>   .. ..$ : chr [1:5] "Estimate" "Std. Error" "df" "t value" ...
#>  $ n_singular_coefs: int 0
#>  $ aic_list        :List of 4
#>   ..$ AIC     : num 3407
#>   ..$ BIC     : num 3440
#>   ..$ logLik  : num -1694
#>   ..$ deviance: num 3387
#>  $ call            : language mmrm(formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID), data = fev_data)
#>  - attr(*, "class")= chr "summary.mmrm"

Residuals

The residuals method for mmrm objects can be used to provide three different types of residuals:

1. Response or raw residuals - the difference between the observed and fitted or predicted value. MMRMs can allow for heteroscedasticity, so these residuals should be interpreted with caution.

fit <- mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data
)
residuals_resp <- residuals(fit, type = "response")

2. Pearson residuals - the raw residuals scaled by the estimated standard deviation of the response. This residual type is better suited to identifying outlying observations and the appropriateness of the covariance structure, compared to the raw residuals.

residuals_pearson <- residuals(fit, type = "pearson")

3. Normalized or scaled residuals - the raw residuals are ‘de-correlated’ based on the Cholesky decomposition of the variance-covariance matrix. These residuals should approximately follow the standard normal distribution, and therefore can be used to check for normality (@galecki2013linear).

residuals_norm <- residuals(fit, type = "normalized")

broom extensions

mmrm also contains S3 methods methods for tidy, glance and augment which were introduced by broom. Note that these methods will work also without loading the broom package. Please see ?mmrm_tidiers for the detailed documentation.

For example, we can apply the tidy method to return a summary table of coefficient estimates:

fit |>
  tidy()
#> # A tibble: 10 × 6
#>    term                          estimate std.error    df statistic  p.value
#>    <chr>                            <dbl>     <dbl> <dbl>     <dbl>    <dbl>
#>  1 (Intercept)                    31.0        0.829  188.   37.3    7.12e-89
#>  2 RACEBlack or African American   1.50       0.621  170.    2.42   1.64e- 2
#>  3 RACEWhite                       5.61       0.663  159.    8.47   1.61e-14
#>  4 ARMCDTRT                        3.78       1.08   146.    3.51   6.00e- 4
#>  5 AVISITVIS2                      4.83       0.802  144.    6.02   1.37e- 8
#>  6 AVISITVIS3                     10.3        0.822  156.   12.6    1.93e-25
#>  7 AVISITVIS4                     15.1        1.31   138.   11.5    8.24e-22
#>  8 ARMCDTRT:AVISITVIS2            -0.0174     1.13   138.   -0.0154 9.88e- 1
#>  9 ARMCDTRT:AVISITVIS3            -0.668      1.19   158.   -0.563  5.75e- 1
#> 10 ARMCDTRT:AVISITVIS4             0.631      1.85   130.    0.341  7.34e- 1

We can also specify some details to request confidence intervals of specific confidence level:

fit |>
  tidy(conf.int = TRUE, conf.level = 0.9)
#> # A tibble: 10 × 8
#>    term           estimate std.error    df statistic  p.value conf.low conf.high
#>    <chr>             <dbl>     <dbl> <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
#>  1 (Intercept)     31.0        0.829  188.   37.3    7.12e-89   29.6       32.3 
#>  2 ARMCDTRT         3.78       1.08   146.    3.51   6.00e- 4    2.01       5.55
#>  3 ARMCDTRT:AVIS…  -0.0174     1.13   138.   -0.0154 9.88e- 1   -1.87       1.84
#>  4 ARMCDTRT:AVIS…  -0.668      1.19   158.   -0.563  5.75e- 1   -2.62       1.28
#>  5 ARMCDTRT:AVIS…   0.631      1.85   130.    0.341  7.34e- 1   -2.41       3.68
#>  6 AVISITVIS2       4.83       0.802  144.    6.02   1.37e- 8    3.51       6.15
#>  7 AVISITVIS3      10.3        0.822  156.   12.6    1.93e-25    8.98      11.7 
#>  8 AVISITVIS4      15.1        1.31   138.   11.5    8.24e-22   12.9       17.2 
#>  9 RACEBlack or …   1.50       0.621  170.    2.42   1.64e- 2    0.484      2.53
#> 10 RACEWhite        5.61       0.663  159.    8.47   1.61e-14    4.52       6.70

Or we can apply the glance method to return a summary table of goodness of fit statistics:

fit |>
  glance()
#> # A tibble: 1 × 4
#>     AIC   BIC logLik deviance
#>   <dbl> <dbl>  <dbl>    <dbl>
#> 1 3407. 3440. -1694.    3387.

Finally, we can use the augment method to return a merged tibble of the data, fitted values and residuals:

fit |>
  augment()
#> # A tibble: 537 × 8
#>    .rownames  FEV1 RACE                     ARMCD AVISIT USUBJID .fitted  .resid
#>        <dbl> <dbl> <fct>                    <fct> <fct>  <fct>     <dbl>   <dbl>
#>  1         2  40.0 Black or African Americ… TRT   VIS2   PT1        40.0  -1.09 
#>  2         4  20.5 Black or African Americ… TRT   VIS4   PT1        20.5 -31.4  
#>  3         6  31.5 Asian                    PBO   VIS2   PT2        31.5  -4.34 
#>  4         7  36.9 Asian                    PBO   VIS3   PT2        36.9  -4.42 
#>  5         8  48.8 Asian                    PBO   VIS4   PT2        48.8   2.79 
#>  6        10  36.0 Black or African Americ… PBO   VIS2   PT3        36.0  -1.31 
#>  7        12  37.2 Black or African Americ… PBO   VIS4   PT3        37.2 -10.4  
#>  8        13  33.9 Asian                    TRT   VIS1   PT4        33.9  -0.851
#>  9        14  33.7 Asian                    TRT   VIS2   PT4        33.7  -5.81 
#> 10        16  54.5 Asian                    TRT   VIS4   PT4        54.5   4.02 
#> # ℹ 527 more rows

Also here we can specify details for the prediction intervals and type of residuals via the arguments:

fit |>
  augment(interval = "confidence", type.residuals = "normalized")
#> # A tibble: 537 × 11
#>    .rownames  FEV1 RACE       ARMCD AVISIT USUBJID .fitted .lower .upper .se.fit
#>        <dbl> <dbl> <fct>      <fct> <fct>  <fct>     <dbl>  <dbl>  <dbl>   <dbl>
#>  1         2  40.0 Black or … TRT   VIS2   PT1        40.0   40.0   40.0       0
#>  2         4  20.5 Black or … TRT   VIS4   PT1        20.5   20.5   20.5       0
#>  3         6  31.5 Asian      PBO   VIS2   PT2        31.5   31.5   31.5       0
#>  4         7  36.9 Asian      PBO   VIS3   PT2        36.9   36.9   36.9       0
#>  5         8  48.8 Asian      PBO   VIS4   PT2        48.8   48.8   48.8       0
#>  6        10  36.0 Black or … PBO   VIS2   PT3        36.0   36.0   36.0       0
#>  7        12  37.2 Black or … PBO   VIS4   PT3        37.2   37.2   37.2       0
#>  8        13  33.9 Asian      TRT   VIS1   PT4        33.9   33.9   33.9       0
#>  9        14  33.7 Asian      TRT   VIS2   PT4        33.7   33.7   33.7       0
#> 10        16  54.5 Asian      TRT   VIS4   PT4        54.5   54.5   54.5       0
#> # ℹ 527 more rows
#> # ℹ 1 more variable: .resid <dbl>

Other components

Specific model quantities not supported by methods can be retrieved with the component() function. The default will output all supported components.

For example, a user may want information about convergence:

component(fit, name = c("convergence", "evaluations", "conv_message"))
#> $convergence
#> [1] 0
#> 
#> $evaluations
#> function gradient 
#>       17       17 
#> 
#> $conv_message
#> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

or the original low-level call:

component(fit, name = "call")
#> mmrm(formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID), 
#>     data = fev_data)

the user could also ask for all provided components by not specifying the name argument.

component(fit)

Lower level functions

Low-level mmrm

The lower level function which is called by mmrm() is fit_mmrm(). This function is exported and can be used directly. It is similar to mmrm() but lacks some post-processing and support for Satterthwaite and Kenward-Roger calculations. It may be useful for other packages that want to fit the model without the adjustment calculations.

fit_mmrm(
  formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  weights = rep(1, nrow(fev_data)),
  reml = TRUE,
  control = mmrm_control()
)
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        fev_data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     rep(1, nrow(fev_data))
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3387.373
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                   30.96769899                    1.50464863 
#>                     RACEWhite                      ARMCDTRT 
#>                    5.61309565                    3.77555734 
#>                    AVISITVIS2                    AVISITVIS3 
#>                    4.82858803                   10.33317002 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                   15.05255715                   -0.01737409 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                   -0.66753189                    0.63094392 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

Hypothesis testing

This package supports estimation of one- and multi-dimensional contrasts (t-test and F-test calculation) with the df_1d() and df_md() functions. Both functions utilize the chosen adjustment method from the initial mmrm call for the calculation of degrees of freedom and (for Kenward-Roger methods) the variance estimates for the test-statistics.

One-dimensional contrasts

Compute the test of a one-dimensional (vector) contrast for a mmrm object with Satterthwaite degrees of freedom.

fit <- mmrm(
  formula = FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data
)

contrast <- numeric(length(component(fit, "beta_est")))
contrast[3] <- 1

df_1d(fit, contrast)
#> $est
#> [1] 5.643565
#> 
#> $se
#> [1] 0.6656093
#> 
#> $df
#> [1] 157.1382
#> 
#> $t_stat
#> [1] 8.478795
#> 
#> $p_val
#> [1] 1.564869e-14

This works similarly when choosing a Kenward-Roger adjustment:

fit_kr <- mmrm(
  formula = FEV1 ~ RACE + SEX + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = fev_data,
  method = "Kenward-Roger"
)

df_1d(fit_kr, contrast)
#> $est
#> [1] 5.643565
#> 
#> $se
#> [1] 0.6740941
#> 
#> $df
#> [1] 157.1382
#> 
#> $t_stat
#> [1] 8.372073
#> 
#> $p_val
#> [1] 2.931654e-14

We see that because this is a one-dimensional contrast, the degrees of freedoms are identical for Satterthwaite and Kenward-Roger. However, the standard errors are different and therefore the p-values are different.

Additional options for the degrees of freedom method are Residual and Between-Within.

Multi-dimensional contrasts

Compute the test of a multi-dimensional (matrix) contrast for the above defined mmrm object with Satterthwaite degrees of freedom:

contrast <- matrix(data = 0, nrow = 2, ncol = length(component(fit, "beta_est")))
contrast[1, 2] <- contrast[2, 3] <- 1

df_md(fit, contrast)
#> $num_df
#> [1] 2
#> 
#> $denom_df
#> [1] 165.5553
#> 
#> $f_stat
#> [1] 36.91143
#> 
#> $p_val
#> [1] 5.544575e-14

And for the Kenward-Roger adjustment:

df_md(fit_kr, contrast)
#> $num_df
#> [1] 2
#> 
#> $denom_df
#> [1] 165.5728
#> 
#> $f_stat
#> [1] 35.99422
#> 
#> $p_val
#> [1] 1.04762e-13

We see that for the multi-dimensional contrast we get slightly different denominator degrees of freedom for the two adjustment methods.

Also the simpler Residual and Between-Within method choices can be used of course together with multidimensional contrasts.

Support for emmeans

This package includes methods that allow mmrm objects to be used with the emmeans package. emmeans computes estimated marginal means (also called least-square means) for the coefficients of the MMRM. For example, in order to see the least-square means by visit and by treatment arm:

library(emmeans)
#> mmrm() registered as emmeans extension
lsmeans_by_visit <- emmeans(fit, ~ ARMCD | AVISIT)
lsmeans_by_visit
#> AVISIT = VIS1:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     33.3 0.755 148     31.8     34.8
#>  TRT     37.1 0.763 143     35.6     38.6
#> 
#> AVISIT = VIS2:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     38.2 0.612 147     37.0     39.4
#>  TRT     41.9 0.602 143     40.7     43.1
#> 
#> AVISIT = VIS3:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     43.7 0.462 130     42.8     44.6
#>  TRT     46.8 0.509 130     45.7     47.8
#> 
#> AVISIT = VIS4:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     48.4 1.189 134     46.0     50.7
#>  TRT     52.8 1.188 133     50.4     55.1
#> 
#> Results are averaged over the levels of: RACE, SEX 
#> Confidence level used: 0.95

Note that the degrees of freedom choice is inherited here from the initial mmrm fit. Furthermore, we can also obtain the differences between the treatment arms for each visit by applying pairs() on the object returned by emmeans() earlier:

pairs(lsmeans_by_visit, reverse = TRUE)
#> AVISIT = VIS1:
#>  contrast  estimate    SE  df t.ratio p.value
#>  TRT - PBO     3.77 1.074 146   3.514  0.0006
#> 
#> AVISIT = VIS2:
#>  contrast  estimate    SE  df t.ratio p.value
#>  TRT - PBO     3.73 0.859 145   4.346  <.0001
#> 
#> AVISIT = VIS3:
#>  contrast  estimate    SE  df t.ratio p.value
#>  TRT - PBO     3.08 0.690 131   4.467  <.0001
#> 
#> AVISIT = VIS4:
#>  contrast  estimate    SE  df t.ratio p.value
#>  TRT - PBO     4.40 1.681 133   2.617  0.0099
#> 
#> Results are averaged over the levels of: RACE, SEX

(This is similar like the pdiff option in SAS PROC MIXED.) Note that we use here the reverse argument to obtain treatment minus placebo results, instead of placebo minus treatment results.

Tidymodels

Tidymodels

mmrm is compatible to work in a tidymodels workflow. The following is an example of how such a workflow would be constructed.

library(tidymodels)

Direct fit

First we define the direct method to fit an mmrm model using the parsnip package functions linear_reg() and set_engine().

• linear_reg() defines a model that can predict numeric values from predictors using a linear function
• set_engine() is used to specify which package or system will be used to fit the model, along with any arguments specific to that software. We can set the method to adjust degrees of freedom directly in the call.

model <- linear_reg() |>
  set_engine("mmrm", method = "Satterthwaite") |>
  fit(FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID), fev_data)
model
#> parsnip model object
#> 
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     weights
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3387.373
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                   30.96769899                    1.50464863 
#>                     RACEWhite                      ARMCDTRT 
#>                    5.61309565                    3.77555734 
#>                    AVISITVIS2                    AVISITVIS3 
#>                    4.82858803                   10.33317002 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                   15.05255715                   -0.01737409 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                   -0.66753189                    0.63094392 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

We can also pass in the full mmrm_control object into the set_engine() call:

model_with_control <- linear_reg() |>
  set_engine("mmrm", control = mmrm_control(method = "Satterthwaite")) |>
  fit(FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID), fev_data)

Predictions

Lastly, we can also obtain predictions with the predict() method:

predict(model, new_data = fev_data)
#> # A tibble: 800 × 1
#>    .pred
#>    <dbl>
#>  1  32.5
#>  2  40.0
#>  3  45.7
#>  4  20.5
#>  5  28.0
#>  6  31.5
#>  7  36.9
#>  8  48.8
#>  9  30.7
#> 10  36.0
#> # ℹ 790 more rows

Note that we need to explicitly pass new_data because the method definition does not allow to default it to the data set we used for the model fitting automatically.

By using the type = "numeric" default of predict() as above we cannot further customize the calculations. We obtain predicted values without confidence intervals or standard errors.

On the other hand, when using type = "raw" we can customize the calculations via the opts list:

predict(
  model,
  new_data = fev_data,
  type = "raw",
  opts = list(se.fit = TRUE, interval = "prediction", nsim = 10L)
)
#>          fit        se      lwr      upr
#> 1   32.47877  6.597042 19.54881 45.40873
#> 2   39.97105  0.000000 39.97105 39.97105
#> 3   45.70508  4.144638 37.58174 53.82842
#> 4   20.48379  0.000000 20.48379 20.48379
#> 5   28.01243  5.597568 17.04140 38.98346
#> 6   31.45522  0.000000 31.45522 31.45522
#> 7   36.87889  0.000000 36.87889 36.87889
#> 8   48.80809  0.000000 48.80809 48.80809
#> 9   30.73774  5.748557 19.47077 42.00471
#> 10  35.98699  0.000000 35.98699 35.98699
#> 11  42.64153  3.851961 35.09182 50.19123
#> 12  37.16444  0.000000 37.16444 37.16444
#> 13  33.89229  0.000000 33.89229 33.89229
#> 14  33.74637  0.000000 33.74637 33.74637
#> 15  44.04155  3.783748 36.62554 51.45756
#> 16  54.45055  0.000000 54.45055 54.45055
#> 17  32.31386  0.000000 32.31386 32.31386
#> 18  37.31982  4.600556 28.30289 46.33674
#> 19  46.79361  0.000000 46.79361 46.79361
#> 20  41.71154  0.000000 41.71154 41.71154
#> 21  31.17198  6.148204 19.12172 43.22224
#> 22  36.63341  4.995059 26.84327 46.42354
#> 23  39.02423  0.000000 39.02423 39.02423
#> 24  47.26333 10.675479 26.33978 68.18689
#> 25  31.93050  0.000000 31.93050 31.93050
#> 26  32.90947  0.000000 32.90947 32.90947
#> 27  41.27523  3.750135 33.92510 48.62536
#> 28  48.28031  0.000000 48.28031 48.28031
#> 29  32.23021  0.000000 32.23021 32.23021
#> 30  35.91080  0.000000 35.91080 35.91080
#> 31  45.54898  0.000000 45.54898 45.54898
#> 32  53.02877  0.000000 53.02877 53.02877
#> 33  47.16898  0.000000 47.16898 47.16898
#> 34  46.64287  0.000000 46.64287 46.64287
#> 35  50.84665  3.780742 43.43653 58.25677
#> 36  58.09713  0.000000 58.09713 58.09713
#> 37  33.21881  6.132123 21.20007 45.23755
#> 38  37.68412  4.985898 27.91194 47.45630
#> 39  44.97613  0.000000 44.97613 44.97613
#> 40  47.67506 10.649621 26.80218 68.54793
#> 41  44.32755  0.000000 44.32755 44.32755
#> 42  38.97813  0.000000 38.97813 38.97813
#> 43  43.72862  0.000000 43.72862 43.72862
#> 44  46.43393  0.000000 46.43393 46.43393
#> 45  40.34576  0.000000 40.34576 40.34576
#> 46  42.76568  0.000000 42.76568 42.76568
#> 47  40.11155  0.000000 40.11155 40.11155
#> 48  49.71974 10.500331 29.13948 70.30001
#> 49  41.46341  5.994247 29.71490 53.21192
#> 50  45.73510  4.956644 36.02026 55.44995
#> 51  53.31791  0.000000 53.31791 53.31791
#> 52  56.07641  0.000000 56.07641 56.07641
#> 53  32.16382  6.124539 20.15995 44.16770
#> 54  37.14256  4.982080 27.37786 46.90726
#> 55  41.90837  0.000000 41.90837 41.90837
#> 56  47.46284 10.589307 26.70819 68.21750
#> 57  27.78883  6.082642 15.86707 39.71059
#> 58  34.13887  5.009749 24.31994 43.95780
#> 59  34.65663  0.000000 34.65663 34.65663
#> 60  39.07791  0.000000 39.07791 39.07791
#> 61  31.18775  5.649359 20.11521 42.26029
#> 62  35.89612  0.000000 35.89612 35.89612
#> 63  41.31608  3.825641 33.81796 48.81420
#> 64  47.67264  0.000000 47.67264 47.67264
#> 65  22.65440  0.000000 22.65440 22.65440
#> 66  36.35488  4.697484 27.14798 45.56178
#> 67  45.20175  3.876015 37.60490 52.79860
#> 68  40.85376  0.000000 40.85376 40.85376
#> 69  32.60048  0.000000 32.60048 32.60048
#> 70  33.64329  0.000000 33.64329 33.64329
#> 71  44.00451  3.770784 36.61391 51.39511
#> 72  40.92278  0.000000 40.92278 40.92278
#> 73  32.14831  0.000000 32.14831 32.14831
#> 74  46.43604  0.000000 46.43604 46.43604
#> 75  41.34973  0.000000 41.34973 41.34973
#> 76  66.30382  0.000000 66.30382 66.30382
#> 77  42.79902  5.733930 31.56073 54.03732
#> 78  47.95358  0.000000 47.95358 47.95358
#> 79  53.97364  0.000000 53.97364 53.97364
#> 80  56.89204 10.707631 35.90547 77.87861
#> 81  46.35384  5.783292 35.01880 57.68889
#> 82  56.64544  0.000000 56.64544 56.64544
#> 83  49.70872  0.000000 49.70872 49.70872
#> 84  60.40497  0.000000 60.40497 60.40497
#> 85  45.98525  0.000000 45.98525 45.98525
#> 86  51.90911  0.000000 51.90911 51.90911
#> 87  41.50787  0.000000 41.50787 41.50787
#> 88  53.42727  0.000000 53.42727 53.42727
#> 89  23.86859  0.000000 23.86859 23.86859
#> 90  35.98563  0.000000 35.98563 35.98563
#> 91  43.60626  0.000000 43.60626 43.60626
#> 92  44.77520 10.473678 24.24717 65.30323
#> 93  29.59773  0.000000 29.59773 29.59773
#> 94  35.50688  0.000000 35.50688 35.50688
#> 95  55.42944  0.000000 55.42944 55.42944
#> 96  52.10530  0.000000 52.10530 52.10530
#> 97  31.69644  0.000000 31.69644 31.69644
#> 98  32.16159  0.000000 32.16159 32.16159
#> 99  51.04735  0.000000 51.04735 51.04735
#> 100 55.85987  0.000000 55.85987 55.85987
#> 101 49.11706  0.000000 49.11706 49.11706
#> 102 49.25544  0.000000 49.25544 49.25544
#> 103 51.72211  0.000000 51.72211 51.72211
#> 104 69.99128  0.000000 69.99128 69.99128
#> 105 22.07169  0.000000 22.07169 22.07169
#> 106 36.35845  4.752453 27.04381 45.67309
#> 107 46.08393  0.000000 46.08393 46.08393
#> 108 52.42288  0.000000 52.42288 52.42288
#> 109 37.69466  0.000000 37.69466 37.69466
#> 110 44.59400  0.000000 44.59400 44.59400
#> 111 52.08897  0.000000 52.08897 52.08897
#> 112 58.22961  0.000000 58.22961 58.22961
#> 113 37.22824  0.000000 37.22824 37.22824
#> 114 34.39863  0.000000 34.39863 34.39863
#> 115 45.88949  3.819649 38.40312 53.37587
#> 116 36.34012  0.000000 36.34012 36.34012
#> 117 45.44182  0.000000 45.44182 45.44182
#> 118 41.54847  0.000000 41.54847 41.54847
#> 119 43.92172  0.000000 43.92172 43.92172
#> 120 61.83243  0.000000 61.83243 61.83243
#> 121 27.25656  0.000000 27.25656 27.25656
#> 122 34.77803  4.694712 25.57657 43.97950
#> 123 45.65133  0.000000 45.65133 45.65133
#> 124 44.56078 10.465319 24.04913 65.07243
#> 125 33.19334  0.000000 33.19334 33.19334
#> 126 39.72671  4.654594 30.60387 48.84955
#> 127 45.59637  3.804733 38.13923 53.05351
#> 128 41.66826  0.000000 41.66826 41.66826
#> 129 27.12753  0.000000 27.12753 27.12753
#> 130 31.74858  0.000000 31.74858 31.74858
#> 131 44.57711  3.832919 37.06473 52.08949
#> 132 41.60000  0.000000 41.60000 41.60000
#> 133 39.45250  0.000000 39.45250 39.45250
#> 134 32.61823  0.000000 32.61823 32.61823
#> 135 34.62445  0.000000 34.62445 34.62445
#> 136 45.90515  0.000000 45.90515 45.90515
#> 137 36.17780  0.000000 36.17780 36.17780
#> 138 39.79796  0.000000 39.79796 39.79796
#> 139 45.87019  3.756764 38.50707 53.23332
#> 140 50.08272  0.000000 50.08272 50.08272
#> 141 36.27753  5.787517 24.93420 47.62085
#> 142 44.64316  0.000000 44.64316 44.64316
#> 143 44.88252  3.940445 37.15939 52.60564
#> 144 39.73529  0.000000 39.73529 39.73529
#> 145 34.06164  0.000000 34.06164 34.06164
#> 146 40.18592  0.000000 40.18592 40.18592
#> 147 41.17584  0.000000 41.17584 41.17584
#> 148 57.76669  0.000000 57.76669 57.76669
#> 149 38.18460  0.000000 38.18460 38.18460
#> 150 38.61735  4.664087 29.47591 47.75880
#> 151 47.19893  0.000000 47.19893 47.19893
#> 152 48.18237 11.002531 26.61780 69.74693
#> 153 37.32785  0.000000 37.32785 37.32785
#> 154 37.89476  4.633594 28.81308 46.97644
#> 155 43.16048  0.000000 43.16048 43.16048
#> 156 41.40349  0.000000 41.40349 41.40349
#> 157 30.15733  0.000000 30.15733 30.15733
#> 158 35.84353  0.000000 35.84353 35.84353
#> 159 40.95250  0.000000 40.95250 40.95250
#> 160 46.76086 10.165464 26.83691 66.68480
#> 161 36.09804  5.672920 24.97932 47.21676
#> 162 41.37928  0.000000 41.37928 41.37928
#> 163 50.17316  0.000000 50.17316 50.17316
#> 164 45.35226  0.000000 45.35226 45.35226
#> 165 39.06491  0.000000 39.06491 39.06491
#> 166 39.39597  4.678873 30.22555 48.56640
#> 167 43.69372  3.825391 36.19610 51.19135
#> 168 42.11960  0.000000 42.11960 42.11960
#> 169 29.81042  0.000000 29.81042 29.81042
#> 170 42.57055  0.000000 42.57055 42.57055
#> 171 47.81652  0.000000 47.81652 47.81652
#> 172 68.06024  0.000000 68.06024 68.06024
#> 173 35.62071  0.000000 35.62071 35.62071
#> 174 40.83933  4.673032 31.68036 49.99831
#> 175 45.83438  3.820220 38.34689 53.32188
#> 176 51.72310 10.386162 31.36659 72.07960
#> 177 33.89134  0.000000 33.89134 33.89134
#> 178 36.42808  0.000000 36.42808 36.42808
#> 179 37.57519  0.000000 37.57519 37.57519
#> 180 58.46873  0.000000 58.46873 58.46873
#> 181 19.54516  0.000000 19.54516 19.54516
#> 182 31.13541  0.000000 31.13541 31.13541
#> 183 40.89955  0.000000 40.89955 40.89955
#> 184 42.08852 10.663482 21.18848 62.98856
#> 185 22.18809  0.000000 22.18809 22.18809
#> 186 41.05857  0.000000 41.05857 41.05857
#> 187 37.32452  0.000000 37.32452 37.32452
#> 188 47.28633 11.876279 24.00925 70.56341
#> 189 35.28933  5.751681 24.01625 46.56242
#> 190 43.12432  0.000000 43.12432 43.12432
#> 191 41.99349  0.000000 41.99349 41.99349
#> 192 49.12250 10.656310 28.23651 70.00848
#> 193 44.03080  0.000000 44.03080 44.03080
#> 194 38.66417  0.000000 38.66417 38.66417
#> 195 53.45993  0.000000 53.45993 53.45993
#> 196 53.14026 11.322147 30.94926 75.33127
#> 197 29.81948  0.000000 29.81948 29.81948
#> 198 30.43859  0.000000 30.43859 30.43859
#> 199 40.18095  0.000000 40.18095 40.18095
#> 200 48.07272 10.730881 27.04058 69.10486
#> 201 26.78578  0.000000 26.78578 26.78578
#> 202 34.55115  0.000000 34.55115 34.55115
#> 203 44.66449  3.818151 37.18105 52.14793
#> 204 40.06421  0.000000 40.06421 40.06421
#> 205 36.66313  5.692612 25.50582 47.82045
#> 206 43.09329  0.000000 43.09329 43.09329
#> 207 46.09670  3.862445 38.52645 53.66695
#> 208 45.71567  0.000000 45.71567 45.71567
#> 209 40.74992  0.000000 40.74992 40.74992
#> 210 44.74635  0.000000 44.74635 44.74635
#> 211 47.51420  3.806254 40.05408 54.97432
#> 212 53.24620 10.356163 32.94849 73.54390
#> 213 40.35635  6.341861 27.92653 52.78617
#> 214 45.16757  5.066752 35.23691 55.09822
#> 215 50.02199  3.953162 42.27393 57.77005
#> 216 56.03985 10.747892 34.97437 77.10533
#> 217 40.14674  0.000000 40.14674 40.14674
#> 218 48.75859  0.000000 48.75859 48.75859
#> 219 46.43462  0.000000 46.43462 46.43462
#> 220 53.05319 10.740726 32.00175 74.10462
#> 221 29.33990  0.000000 29.33990 29.33990
#> 222 36.24424  4.623367 27.18261 45.30588
#> 223 42.39946  3.778756 34.99324 49.80569
#> 224 47.93165  0.000000 47.93165 47.93165
#> 225 36.59443  5.532229 25.75146 47.43740
#> 226 41.11632  0.000000 41.11632 41.11632
#> 227 47.05889  0.000000 47.05889 47.05889
#> 228 52.24599  0.000000 52.24599 52.24599
#> 229 45.12054  5.799549 33.75364 56.48745
#> 230 54.14236  0.000000 54.14236 54.14236
#> 231 50.44618  0.000000 50.44618 50.44618
#> 232 58.53593 10.933464 37.10674 79.96513
#> 233 37.53657  0.000000 37.53657 37.53657
#> 234 44.28282  4.604412 35.25834 53.30730
#> 235 49.45840  0.000000 49.45840 49.45840
#> 236 59.12866  0.000000 59.12866 59.12866
#> 237 40.31268  0.000000 40.31268 40.31268
#> 238 39.66049  0.000000 39.66049 39.66049
#> 239 50.89726  0.000000 50.89726 50.89726
#> 240 56.13116  0.000000 56.13116 56.13116
#> 241 32.82981  0.000000 32.82981 32.82981
#> 242 46.53837  0.000000 46.53837 46.53837
#> 243 44.46016  3.775122 37.06106 51.85927
#> 244 51.81265  0.000000 51.81265 51.81265
#> 245 27.76886  5.876695 16.25075 39.28698
#> 246 29.91939  0.000000 29.91939 29.91939
#> 247 40.70944  3.900847 33.06392 48.35496
#> 248 44.37942 10.776593 23.25768 65.50115
#> 249 43.47695  5.922196 31.86966 55.08424
#> 250 51.05656  0.000000 51.05656 51.05656
#> 251 50.50059  0.000000 50.50059 50.50059
#> 252 64.11388  0.000000 64.11388 64.11388
#> 253 32.21843  0.000000 32.21843 32.21843
#> 254 29.64732  0.000000 29.64732 29.64732
#> 255 42.48707  3.773569 35.09101 49.88312
#> 256 45.09919  0.000000 45.09919 45.09919
#> 257 39.75659  0.000000 39.75659 39.75659
#> 258 37.28894  0.000000 37.28894 37.28894
#> 259 44.80145  0.000000 44.80145 44.80145
#> 260 65.95920  0.000000 65.95920 65.95920
#> 261 33.43439  0.000000 33.43439 33.43439
#> 262 33.57042  0.000000 33.57042 33.57042
#> 263 39.91543  0.000000 39.91543 39.91543
#> 264 49.57098  0.000000 49.57098 49.57098
#> 265 38.91634  0.000000 38.91634 38.91634
#> 266 36.69011  0.000000 36.69011 36.69011
#> 267 45.66665  0.000000 45.66665 45.66665
#> 268 52.07431  0.000000 52.07431 52.07431
#> 269 42.21411  0.000000 42.21411 42.21411
#> 270 45.02901  0.000000 45.02901 45.02901
#> 271 50.22530  3.791640 42.79383 57.65678
#> 272 56.56084 10.278348 36.41565 76.70603
#> 273 30.98338  0.000000 30.98338 30.98338
#> 274 44.72932  0.000000 44.72932 44.72932
#> 275 40.68711  0.000000 40.68711 40.68711
#> 276 34.71530  0.000000 34.71530 34.71530
#> 277 27.30752  0.000000 27.30752 27.30752
#> 278 37.31585  0.000000 37.31585 37.31585
#> 279 42.23592  3.759776 34.86689 49.60495
#> 280 44.83000  0.000000 44.83000 44.83000
#> 281 32.93042  0.000000 32.93042 32.93042
#> 282 44.91911  0.000000 44.91911 44.91911
#> 283 45.68636  0.000000 45.68636 45.68636
#> 284 65.98800  0.000000 65.98800 65.98800
#> 285 46.60130  0.000000 46.60130 46.60130
#> 286 40.89786  0.000000 40.89786 40.89786
#> 287 46.66708  0.000000 46.66708 46.66708
#> 288 53.97575 11.149688 32.12277 75.82874
#> 289 34.51394  6.010715 22.73316 46.29473
#> 290 40.31956  4.975761 30.56725 50.07187
#> 291 43.83270  0.000000 43.83270 43.83270
#> 292 44.11604  0.000000 44.11604 44.11604
#> 293 38.29612  0.000000 38.29612 38.29612
#> 294 42.38074  4.635575 33.29518 51.46630
#> 295 51.38570  0.000000 51.38570 51.38570
#> 296 56.20979  0.000000 56.20979 56.20979
#> 297 35.93809  5.716738 24.73349 47.14269
#> 298 43.45819  0.000000 43.45819 43.45819
#> 299 38.38741  0.000000 38.38741 38.38741
#> 300 56.42818  0.000000 56.42818 56.42818
#> 301 39.05050  0.000000 39.05050 39.05050
#> 302 44.66707  4.620403 35.61124 53.72289
#> 303 49.86836  3.779447 42.46078 57.27594
#> 304 54.09200  0.000000 54.09200 54.09200
#> 305 31.40521  0.000000 31.40521 31.40521
#> 306 46.13330  0.000000 46.13330 46.13330
#> 307 45.29845  0.000000 45.29845 45.29845
#> 308 28.06936  0.000000 28.06936 28.06936
#> 309 39.05960  5.609423 28.06533 50.05387
#> 310 42.50283  0.000000 42.50283 42.50283
#> 311 46.45368  0.000000 46.45368 46.45368
#> 312 64.97366  0.000000 64.97366 64.97366
#> 313 30.67876  6.146096 18.63263 42.72489
#> 314 35.63991  5.014149 25.81236 45.46746
#> 315 41.27878  3.894391 33.64592 48.91165
#> 316 43.97847  0.000000 43.97847 43.97847
#> 317 35.33466  0.000000 35.33466 35.33466
#> 318 39.34378  0.000000 39.34378 39.34378
#> 319 41.27633  0.000000 41.27633 41.27633
#> 320 50.74572 10.179152 30.79495 70.69649
#> 321 34.18429  5.600519 23.20748 45.16111
#> 322 39.83058  0.000000 39.83058 39.83058
#> 323 43.49673  0.000000 43.49673 43.49673
#> 324 44.06114  0.000000 44.06114 44.06114
#> 325 41.43742  0.000000 41.43742 41.43742
#> 326 40.68384  4.621386 31.62609 49.74159
#> 327 46.16954  0.000000 46.16954 46.16954
#> 328 54.24024  0.000000 54.24024 54.24024
#> 329 36.61831  0.000000 36.61831 36.61831
#> 330 42.09272  0.000000 42.09272 42.09272
#> 331 50.69556  0.000000 50.69556 50.69556
#> 332 51.72563  0.000000 51.72563 51.72563
#> 333 32.08271  6.196018 19.93874 44.22668
#> 334 36.39974  5.035745 26.52986 46.26962
#> 335 41.38610  3.907053 33.72842 49.04378
#> 336 53.89947  0.000000 53.89947 53.89947
#> 337 31.94884  6.056050 20.07920 43.81848
#> 338 36.34138  4.976628 26.58737 46.09539
#> 339 39.94420  0.000000 39.94420 39.94420
#> 340 56.42482  0.000000 56.42482 56.42482
#> 341 41.86385  0.000000 41.86385 41.86385
#> 342 34.56420  0.000000 34.56420 34.56420
#> 343 38.68927  0.000000 38.68927 38.68927
#> 344 62.88743  0.000000 62.88743 62.88743
#> 345 28.85343  0.000000 28.85343 28.85343
#> 346 38.89808  4.719128 29.64876 48.14741
#> 347 49.29495  0.000000 49.29495 49.29495
#> 348 48.90884 10.547147 28.23681 69.58087
#> 349 28.74029  0.000000 28.74029 28.74029
#> 350 36.38836  4.693034 27.19018 45.58653
#> 351 43.59994  0.000000 43.59994 43.59994
#> 352 57.38616  0.000000 57.38616 57.38616
#> 353 35.36824  0.000000 35.36824 35.36824
#> 354 43.06110  0.000000 43.06110 43.06110
#> 355 31.27551  0.000000 31.27551 31.27551
#> 356 54.13245  0.000000 54.13245 54.13245
#> 357 25.97050  0.000000 25.97050 25.97050
#> 358 34.04514  4.685170 24.86238 43.22791
#> 359 40.67015  3.822493 33.17820 48.16210
#> 360 44.36054 10.502085 23.77684 64.94425
#> 361 39.69391  6.032977 27.86950 51.51833
#> 362 44.79720  4.966395 35.06325 54.53116
#> 363 51.17493  0.000000 51.17493 51.17493
#> 364 48.44043  0.000000 48.44043 48.44043
#> 365 43.33128  0.000000 43.33128 43.33128
#> 366 46.45918  4.614377 37.41517 55.50320
#> 367 55.93546  0.000000 55.93546 55.93546
#> 368 54.15312  0.000000 54.15312 54.15312
#> 369 33.79699  5.577173 22.86593 44.72805
#> 370 40.60252  0.000000 40.60252 40.60252
#> 371 44.44715  0.000000 44.44715 44.44715
#> 372 40.54161  0.000000 40.54161 40.54161
#> 373 33.95563  0.000000 33.95563 33.95563
#> 374 36.84083  4.598866 27.82721 45.85444
#> 375 43.67802  0.000000 43.67802 43.67802
#> 376 42.76023  0.000000 42.76023 42.76023
#> 377 34.18486  5.773366 22.86927 45.50045
#> 378 42.82678  0.000000 42.82678 42.82678
#> 379 39.59218  0.000000 39.59218 39.59218
#> 380 47.93427 10.750071 26.86451 69.00402
#> 381 33.49216  0.000000 33.49216 33.49216
#> 382 35.39266  0.000000 35.39266 35.39266
#> 383 42.88943  3.750905 35.53780 50.24107
#> 384 42.36266  0.000000 42.36266 42.36266
#> 385 48.54368  0.000000 48.54368 48.54368
#> 386 43.94366  0.000000 43.94366 43.94366
#> 387 50.98218  3.856793 43.42301 58.54136
#> 388 47.91204  0.000000 47.91204 47.91204
#> 389 20.72928  0.000000 20.72928 20.72928
#> 390 28.00599  0.000000 28.00599 28.00599
#> 391 40.19255  0.000000 40.19255 40.19255
#> 392 37.79360  0.000000 37.79360 37.79360
#> 393 32.17343  5.839839 20.72756 43.61931
#> 394 36.75177  0.000000 36.75177 36.75177
#> 395 42.75025  3.876767 35.15193 50.34858
#> 396 47.37158 10.625479 26.54602 68.19713
#> 397 34.59822  0.000000 34.59822 34.59822
#> 398 39.32034  0.000000 39.32034 39.32034
#> 399 40.65702  0.000000 40.65702 40.65702
#> 400 48.52002 10.155415 28.61577 68.42426
#> 401 40.41786  5.747438 29.15309 51.68263
#> 402 43.03255  0.000000 43.03255 43.03255
#> 403 54.65715  0.000000 54.65715 54.65715
#> 404 55.54195 10.725181 34.52098 76.56291
#> 405 35.55742  0.000000 35.55742 35.55742
#> 406 43.70215  0.000000 43.70215 43.70215
#> 407 42.52157  0.000000 42.52157 42.52157
#> 408 54.89337  0.000000 54.89337 54.89337
#> 409 32.03460  0.000000 32.03460 32.03460
#> 410 29.45107  0.000000 29.45107 29.45107
#> 411 45.35138  0.000000 45.35138 45.35138
#> 412 45.33026 10.680741 24.39639 66.26412
#> 413 38.73784  0.000000 38.73784 38.73784
#> 414 39.30063  4.628047 30.22982 48.37143
#> 415 41.42283  0.000000 41.42283 41.42283
#> 416 47.32385  0.000000 47.32385 47.32385
#> 417 40.41284  5.765932 29.11182 51.71386
#> 418 47.55310  0.000000 47.55310 47.55310
#> 419 49.06509  0.000000 49.06509 49.06509
#> 420 53.79511 10.835026 32.55885 75.03137
#> 421 29.22591  0.000000 29.22591 29.22591
#> 422 40.08175  0.000000 40.08175 40.08175
#> 423 45.68142  0.000000 45.68142 45.68142
#> 424 41.47403  0.000000 41.47403 41.47403
#> 425 37.55612  6.313792 25.18132 49.93092
#> 426 41.98123  5.069835 32.04453 51.91792
#> 427 42.51970  0.000000 42.51970 42.51970
#> 428 69.36099  0.000000 69.36099 69.36099
#> 429 42.39760  0.000000 42.39760 42.39760
#> 430 43.72376  0.000000 43.72376 43.72376
#> 431 49.47601  0.000000 49.47601 49.47601
#> 432 51.94188  0.000000 51.94188 51.94188
#> 433 31.77722  5.712935 20.58007 42.97437
#> 434 40.59100  0.000000 40.59100 40.59100
#> 435 39.97833  0.000000 39.97833 39.97833
#> 436 31.69049  0.000000 31.69049 31.69049
#> 437 33.99809  5.718537 22.78996 45.20622
#> 438 37.20517  0.000000 37.20517 37.20517
#> 439 46.28740  0.000000 46.28740 46.28740
#> 440 49.81365 10.541258 29.15316 70.47413
#> 441 35.15913  6.240422 22.92813 47.39014
#> 442 40.63997  5.064102 30.71451 50.56543
#> 443 46.80529  3.944402 39.07441 54.53618
#> 444 41.58720  0.000000 41.58720 41.58720
#> 445 32.17365  0.000000 32.17365 32.17365
#> 446 37.07479  4.643827 27.97306 46.17653
#> 447 40.69375  0.000000 40.69375 40.69375
#> 448 47.52336 10.294250 27.34700 67.69972
#> 449 32.28771  0.000000 32.28771 32.28771
#> 450 41.76205  0.000000 41.76205 41.76205
#> 451 40.06768  0.000000 40.06768 40.06768
#> 452 47.21582 10.357533 26.91543 67.51622
#> 453 29.14213  0.000000 29.14213 29.14213
#> 454 39.50989  0.000000 39.50989 39.50989
#> 455 43.32349  0.000000 43.32349 43.32349
#> 456 47.16756  0.000000 47.16756 47.16756
#> 457 40.93020  0.000000 40.93020 40.93020
#> 458 42.19406  0.000000 42.19406 42.19406
#> 459 41.21057  0.000000 41.21057 41.21057
#> 460 49.76205 10.573444 29.03849 70.48562
#> 461 38.54330  0.000000 38.54330 38.54330
#> 462 41.44104  4.674910 32.27838 50.60369
#> 463 43.96324  0.000000 43.96324 43.96324
#> 464 42.67652  0.000000 42.67652 42.67652
#> 465 22.79584  0.000000 22.79584 22.79584
#> 466 33.91004  4.713715 24.67133 43.14875
#> 467 41.58421  3.863378 34.01212 49.15629
#> 468 44.31106 10.826159 23.09218 65.52994
#> 469 31.43559  0.000000 31.43559 31.43559
#> 470 38.85064  0.000000 38.85064 38.85064
#> 471 48.24288  0.000000 48.24288 48.24288
#> 472 46.15940 10.680516 25.22597 67.09283
#> 473 44.71302  0.000000 44.71302 44.71302
#> 474 51.85370  0.000000 51.85370 51.85370
#> 475 50.77548  3.827838 43.27306 58.27791
#> 476 58.11432 10.539851 37.45659 78.77205
#> 477 30.56757  0.000000 30.56757 30.56757
#> 478 35.65607  4.672016 26.49909 44.81305
#> 479 41.25037  3.806722 33.78933 48.71141
#> 480 45.88736 10.385438 25.53228 66.24245
#> 481 37.37624  6.196290 25.23173 49.52074
#> 482 41.66978  5.041648 31.78833 51.55123
#> 483 45.99979  3.925392 38.30617 53.69342
#> 484 59.90473  0.000000 59.90473 59.90473
#> 485 33.87728  6.250441 21.62664 46.12792
#> 486 37.28988  5.057180 27.37799 47.20177
#> 487 49.76150  0.000000 49.76150 49.76150
#> 488 46.60552 11.240815 24.57393 68.63711
#> 489 47.21985  0.000000 47.21985 47.21985
#> 490 40.34525  0.000000 40.34525 40.34525
#> 491 48.29793  0.000000 48.29793 48.29793
#> 492 54.57153 11.120461 32.77583 76.36723
#> 493 36.13680  5.576096 25.20786 47.06575
#> 494 44.39634  0.000000 44.39634 44.39634
#> 495 41.71421  0.000000 41.71421 41.71421
#> 496 47.37535  0.000000 47.37535 47.37535
#> 497 42.03797  0.000000 42.03797 42.03797
#> 498 37.56100  0.000000 37.56100 37.56100
#> 499 45.11793  0.000000 45.11793 45.11793
#> 500 52.86788 10.658388 31.97782 73.75794
#> 501 34.62530  0.000000 34.62530 34.62530
#> 502 45.28206  0.000000 45.28206 45.28206
#> 503 44.51505  3.782415 37.10165 51.92844
#> 504 63.57761  0.000000 63.57761 63.57761
#> 505 35.80878  0.000000 35.80878 35.80878
#> 506 40.93038  4.619125 31.87707 49.98370
#> 507 45.85156  3.781237 38.44047 53.26265
#> 508 52.67314  0.000000 52.67314 52.67314
#> 509 35.88734  0.000000 35.88734 35.88734
#> 510 38.73222  0.000000 38.73222 38.73222
#> 511 46.70361  0.000000 46.70361 46.70361
#> 512 53.65398  0.000000 53.65398 53.65398
#> 513 36.71543  0.000000 36.71543 36.71543
#> 514 43.89170  4.681157 34.71680 53.06659
#> 515 49.56246  3.823862 42.06782 57.05709
#> 516 54.83060 10.432472 34.38333 75.27787
#> 517 37.85241  5.735177 26.61167 49.09315
#> 518 41.54317  0.000000 41.54317 41.54317
#> 519 51.67909  0.000000 51.67909 51.67909
#> 520 51.76691 10.726745 30.74287 72.79094
#> 521 27.40130  0.000000 27.40130 27.40130
#> 522 30.33517  0.000000 30.33517 30.33517
#> 523 37.73092  0.000000 37.73092 37.73092
#> 524 29.11668  0.000000 29.11668 29.11668
#> 525 30.03596  5.581874 19.09569 40.97623
#> 526 32.08830  0.000000 32.08830 32.08830
#> 527 41.66067  0.000000 41.66067 41.66067
#> 528 53.90815  0.000000 53.90815 53.90815
#> 529 34.02622  5.583689 23.08239 44.97005
#> 530 35.06937  0.000000 35.06937 35.06937
#> 531 47.17615  0.000000 47.17615 47.17615
#> 532 56.49347  0.000000 56.49347 56.49347
#> 533 34.02880  5.592064 23.06856 44.98904
#> 534 38.88006  0.000000 38.88006 38.88006
#> 535 47.54070  0.000000 47.54070 47.54070
#> 536 43.53705  0.000000 43.53705 43.53705
#> 537 31.82054  0.000000 31.82054 31.82054
#> 538 39.62816  0.000000 39.62816 39.62816
#> 539 44.95543  0.000000 44.95543 44.95543
#> 540 21.11543  0.000000 21.11543 21.11543
#> 541 34.74671  0.000000 34.74671 34.74671
#> 542 43.27308  4.643477 34.17203 52.37413
#> 543 49.29538  3.801055 41.84545 56.74531
#> 544 56.69249  0.000000 56.69249 56.69249
#> 545 22.73126  0.000000 22.73126 22.73126
#> 546 32.50075  0.000000 32.50075 32.50075
#> 547 42.37206  0.000000 42.37206 42.37206
#> 548 42.89847  0.000000 42.89847 42.89847
#> 549 55.62582  0.000000 55.62582 55.62582
#> 550 45.38998  0.000000 45.38998 45.38998
#> 551 52.66743  0.000000 52.66743 52.66743
#> 552 56.87348 13.266696 30.87124 82.87573
#> 553 30.66032  6.219698 18.46993 42.85070
#> 554 37.44228  5.077309 27.49094 47.39362
#> 555 34.18931  0.000000 34.18931 34.18931
#> 556 45.59740  0.000000 45.59740 45.59740
#> 557 28.89198  0.000000 28.89198 28.89198
#> 558 38.46147  0.000000 38.46147 38.46147
#> 559 42.42099  3.758768 35.05394 49.78804
#> 560 49.90357  0.000000 49.90357 49.90357
#> 561 39.74586  5.650912 28.67028 50.82144
#> 562 44.14167  0.000000 44.14167 44.14167
#> 563 49.91712  3.835237 42.40020 57.43405
#> 564 55.24278  0.000000 55.24278 55.24278
#> 565 36.24790  6.341405 23.81898 48.67683
#> 566 41.05912  5.066433 31.12909 50.98914
#> 567 45.91354  3.957584 38.15682 53.67027
#> 568 51.93141 10.750508 30.86080 73.00201
#> 569 27.38001  0.000000 27.38001 27.38001
#> 570 33.63251  0.000000 33.63251 33.63251
#> 571 44.70168  3.816977 37.22054 52.18282
#> 572 39.34410  0.000000 39.34410 39.34410
#> 573 26.98575  0.000000 26.98575 26.98575
#> 574 24.04175  0.000000 24.04175 24.04175
#> 575 42.16648  0.000000 42.16648 42.16648
#> 576 44.75380  0.000000 44.75380 44.75380
#> 577 31.55469  0.000000 31.55469 31.55469
#> 578 44.42696  0.000000 44.42696 44.42696
#> 579 44.10343  0.000000 44.10343 44.10343
#> 580 48.06505 10.636330 27.21823 68.91188
#> 581 34.87547  5.567314 23.96373 45.78720
#> 582 37.87445  0.000000 37.87445 37.87445
#> 583 48.31828  0.000000 48.31828 48.31828
#> 584 50.21520  0.000000 50.21520 50.21520
#> 585 41.94615  0.000000 41.94615 41.94615
#> 586 39.62690  0.000000 39.62690 39.62690
#> 587 46.69763  0.000000 46.69763 46.69763
#> 588 49.44653 11.200123 27.49469 71.39836
#> 589 38.01775  5.716119 26.81436 49.22113
#> 590 43.75255  0.000000 43.75255 43.75255
#> 591 47.38873  0.000000 47.38873 47.38873
#> 592 52.70780 10.559801 32.01096 73.40463
#> 593 32.43412  0.000000 32.43412 32.43412
#> 594 43.07163  0.000000 43.07163 43.07163
#> 595 42.99551  0.000000 42.99551 42.99551
#> 596 53.82759  0.000000 53.82759 53.82759
#> 597 39.45747  6.093558 27.51431 51.40062
#> 598 42.93167  5.023402 33.08598 52.77735
#> 599 50.64802  0.000000 50.64802 50.64802
#> 600 63.44051  0.000000 63.44051 63.44051
#> 601 34.48949  0.000000 34.48949 34.48949
#> 602 40.08056  0.000000 40.08056 40.08056
#> 603 41.86656  3.760166 34.49677 49.23635
#> 604 47.46553  0.000000 47.46553 47.46553
#> 605 32.03992  5.900764 20.47464 43.60521
#> 606 37.11697  0.000000 37.11697 37.11697
#> 607 44.12071  3.891287 36.49393 51.74749
#> 608 36.25120  0.000000 36.25120 36.25120
#> 609 29.20171  0.000000 29.20171 29.20171
#> 610 31.53773  0.000000 31.53773 31.53773
#> 611 42.35683  0.000000 42.35683 42.35683
#> 612 64.78352  0.000000 64.78352 64.78352
#> 613 32.72757  0.000000 32.72757 32.72757
#> 614 37.50022  0.000000 37.50022 37.50022
#> 615 42.76167  3.774605 35.36358 50.15976
#> 616 57.03861  0.000000 57.03861 57.03861
#> 617 36.32475  0.000000 36.32475 36.32475
#> 618 40.15241  4.606191 31.12444 49.18038
#> 619 41.46725  0.000000 41.46725 41.46725
#> 620 59.01411  0.000000 59.01411 59.01411
#> 621 30.14970  0.000000 30.14970 30.14970
#> 622 34.91740  0.000000 34.91740 34.91740
#> 623 52.13900  0.000000 52.13900 52.13900
#> 624 58.73839  0.000000 58.73839 58.73839
#> 625 35.83185  0.000000 35.83185 35.83185
#> 626 41.04423  4.630729 31.96817 50.12029
#> 627 45.82688  3.788474 38.40160 53.25215
#> 628 56.41409  0.000000 56.41409 56.41409
#> 629 37.80184  5.603360 26.81945 48.78422
#> 630 43.55593  0.000000 43.55593 43.55593
#> 631 44.26320  0.000000 44.26320 44.26320
#> 632 59.25579  0.000000 59.25579 59.25579
#> 633 28.47314  0.000000 28.47314 28.47314
#> 634 47.47581  0.000000 47.47581 47.47581
#> 635 44.01685  3.838137 36.49424 51.53946
#> 636 49.57489 11.420569 27.19098 71.95879
#> 637 39.38085  5.735344 28.13978 50.62192
#> 638 46.47483  0.000000 46.47483 46.47483
#> 639 51.22677  0.000000 51.22677 51.22677
#> 640 45.82777  0.000000 45.82777 45.82777
#> 641 33.43408  5.845841 21.97644 44.89172
#> 642 39.06783  0.000000 39.06783 39.06783
#> 643 42.98333  3.876119 35.38628 50.58039
#> 644 48.01822 10.646154 27.15215 68.88430
#> 645 29.99542  0.000000 29.99542 29.99542
#> 646 35.69583  4.661185 26.56008 44.83159
#> 647 41.11547  3.792202 33.68289 48.54805
#> 648 54.17796  0.000000 54.17796 54.17796
#> 649 39.32289  5.814683 27.92632 50.71946
#> 650 44.55743  0.000000 44.55743 44.55743
#> 651 47.26282  3.853349 39.71039 54.81524
#> 652 62.59579  0.000000 62.59579 62.59579
#> 653 31.80300  5.533889 20.95678 42.64923
#> 654 35.48396  0.000000 35.48396 35.48396
#> 655 44.07768  0.000000 44.07768 44.07768
#> 656 46.57837  0.000000 46.57837 46.57837
#> 657 47.67979  0.000000 47.67979 47.67979
#> 658 47.73388  4.693847 38.53411 56.93365
#> 659 50.94631  3.840944 43.41820 58.47442
#> 660 58.47218 10.736580 37.42887 79.51549
#> 661 22.15439  0.000000 22.15439 22.15439
#> 662 35.14301  4.747843 25.83741 44.44862
#> 663 42.82000  3.902246 35.17174 50.46827
#> 664 46.24563 11.140610 24.41044 68.08083
#> 665 34.27765  0.000000 34.27765 34.27765
#> 666 36.90059  0.000000 36.90059 36.90059
#> 667 43.05627  3.763618 35.67971 50.43282
#> 668 40.54285  0.000000 40.54285 40.54285
#> 669 29.09494  0.000000 29.09494 29.09494
#> 670 37.21768  0.000000 37.21768 37.21768
#> 671 43.08491  0.000000 43.08491 43.08491
#> 672 46.50100 10.178571 26.55137 66.45063
#> 673 27.12174  0.000000 27.12174 27.12174
#> 674 34.11916  0.000000 34.11916 34.11916
#> 675 45.56320  3.869846 37.97845 53.14796
#> 676 48.00823 10.671138 27.09318 68.92327
#> 677 35.93048  5.761078 24.63898 47.22199
#> 678 40.80230  0.000000 40.80230 40.80230
#> 679 45.89269  0.000000 45.89269 45.89269
#> 680 43.69153  0.000000 43.69153 43.69153
#> 681 28.56569  5.758633 17.27898 39.85240
#> 682 29.22869  0.000000 29.22869 29.22869
#> 683 40.67646  3.935694 32.96264 48.39028
#> 684 55.68362  0.000000 55.68362 55.68362
#> 685 31.90698  0.000000 31.90698 31.90698
#> 686 37.31061  0.000000 37.31061 37.31061
#> 687 40.75546  0.000000 40.75546 40.75546
#> 688 49.50911 10.264240 29.39157 69.62665
#> 689 42.19474  0.000000 42.19474 42.19474
#> 690 44.87228  0.000000 44.87228 44.87228
#> 691 47.55198  0.000000 47.55198 47.55198
#> 692 56.68097 10.179204 36.73010 76.63184
#> 693 50.62894  0.000000 50.62894 50.62894
#> 694 45.47551  0.000000 45.47551 45.47551
#> 695 48.62168  0.000000 48.62168 48.62168
#> 696 56.58212 11.403253 34.23216 78.93208
#> 697 29.66493  0.000000 29.66493 29.66493
#> 698 34.57406  0.000000 34.57406 34.57406
#> 699 42.45295  3.759694 35.08408 49.82181
#> 700 38.11676  0.000000 38.11676 38.11676
#> 701 33.77204  0.000000 33.77204 33.77204
#> 702 34.26148  0.000000 34.26148 34.26148
#> 703 45.29511  3.843793 37.76141 52.82880
#> 704 58.81037  0.000000 58.81037 58.81037
#> 705 31.46668  6.138068 19.43629 43.49708
#> 706 36.78469  4.989530 27.00539 46.56399
#> 707 39.88119  0.000000 39.88119 39.88119
#> 708 47.32261 10.635509 26.47740 68.16783
#> 709 31.62708  0.000000 31.62708 31.62708
#> 710 37.03239  4.617467 27.98232 46.08246
#> 711 42.69162  3.769091 35.30433 50.07890
#> 712 48.22049  0.000000 48.22049 48.22049
#> 713 42.58829  0.000000 42.58829 42.58829
#> 714 45.80410  4.599826 36.78861 54.81959
#> 715 49.33262  0.000000 49.33262 49.33262
#> 716 53.74331  0.000000 53.74331 53.74331
#> 717 29.71857  0.000000 29.71857 29.71857
#> 718 30.45651  0.000000 30.45651 30.45651
#> 719 38.29800  0.000000 38.29800 38.29800
#> 720 45.15328 10.384176 24.80067 65.50589
#> 721 36.81040  0.000000 36.81040 36.81040
#> 722 37.61606  4.651945 28.49842 46.73371
#> 723 42.35045  0.000000 42.35045 42.35045
#> 724 39.39860  0.000000 39.39860 39.39860
#> 725 36.09876  6.126345 24.09134 48.10617
#> 726 40.94066  4.989280 31.16185 50.71947
#> 727 49.73629  0.000000 49.73629 49.73629
#> 728 41.58082  0.000000 41.58082 41.58082
#> 729 43.58901  0.000000 43.58901 43.58901
#> 730 40.16762  0.000000 40.16762 40.16762
#> 731 46.70338  3.814471 39.22716 54.17961
#> 732 53.94830 10.750356 32.87799 75.01862
#> 733 39.60913  5.778258 28.28395 50.93430
#> 734 41.08206  0.000000 41.08206 41.08206
#> 735 49.65683  3.929652 41.95486 57.35881
#> 736 69.37409  0.000000 69.37409 69.37409
#> 737 34.12096  5.593960 23.15700 45.08492
#> 738 41.27625  0.000000 41.27625 41.27625
#> 739 44.76138  0.000000 44.76138 44.76138
#> 740 39.69815  0.000000 39.69815 39.69815
#> 741 38.44296  0.000000 38.44296 38.44296
#> 742 48.20586  0.000000 48.20586 48.20586
#> 743 47.54082  3.959125 39.78108 55.30056
#> 744 35.50735  0.000000 35.50735 35.50735
#> 745 32.08153  0.000000 32.08153 32.08153
#> 746 37.16398  4.670384 28.01020 46.31777
#> 747 42.75619  3.806533 35.29552 50.21686
#> 748 47.39510 10.384921 27.04103 67.74917
#> 749 44.69256  0.000000 44.69256 44.69256
#> 750 41.45664  4.743254 32.16004 50.75325
#> 751 42.18689  0.000000 42.18689 42.18689
#> 752 51.68534 11.039633 30.04806 73.32262
#> 753 37.01741  0.000000 37.01741 37.01741
#> 754 38.26920  0.000000 38.26920 38.26920
#> 755 49.28806  0.000000 49.28806 49.28806
#> 756 50.67485 10.475900 30.14247 71.20724
#> 757 40.45953  0.000000 40.45953 40.45953
#> 758 45.10337  0.000000 45.10337 45.10337
#> 759 45.58250  0.000000 45.58250 45.58250
#> 760 62.96989  0.000000 62.96989 62.96989
#> 761 30.78252  0.000000 30.78252 30.78252
#> 762 41.58139  4.669080 32.43016 50.73262
#> 763 48.87398  3.830319 41.36669 56.38126
#> 764 44.69667  0.000000 44.69667 44.69667
#> 765 32.72491  0.000000 32.72491 32.72491
#> 766 45.78702  0.000000 45.78702 45.78702
#> 767 48.74886  0.000000 48.74886 48.74886
#> 768 84.08449  0.000000 84.08449 84.08449
#> 769 28.60809  5.786327 17.26710 39.94908
#> 770 30.19495  0.000000 30.19495 30.19495
#> 771 36.78573  0.000000 36.78573 36.78573
#> 772 61.03588  0.000000 61.03588 61.03588
#> 773 20.36749  0.000000 20.36749 20.36749
#> 774 35.22480  0.000000 35.22480 35.22480
#> 775 37.42847  0.000000 37.42847 37.42847
#> 776 30.20501  0.000000 30.20501 30.20501
#> 777 41.72819  5.735646 30.48653 52.96985
#> 778 49.12862  0.000000 49.12862 49.12862
#> 779 47.31234  0.000000 47.31234 47.31234
#> 780 57.08286 10.616485 36.27493 77.89078
#> 781 19.28388  0.000000 19.28388 19.28388
#> 782 30.00682  0.000000 30.00682 30.00682
#> 783 39.69711  3.902999 32.04738 47.34685
#> 784 49.21768  0.000000 49.21768 49.21768
#> 785 31.42637  6.179983 19.31383 43.53891
#> 786 36.73485  5.027381 26.88136 46.58833
#> 787 42.72556  3.905426 35.07107 50.38006
#> 788 40.13353  0.000000 40.13353 40.13353
#> 789 42.34534  0.000000 42.34534 42.34534
#> 790 52.32575  0.000000 52.32575 52.32575
#> 791 46.92223  3.809508 39.45573 54.38873
#> 792 69.26254  0.000000 69.26254 69.26254
#> 793 40.35635  6.341861 27.92653 52.78617
#> 794 45.16757  5.066752 35.23691 55.09822
#> 795 50.02199  3.953162 42.27393 57.77005
#> 796 56.03985 10.747892 34.97437 77.10533
#> 797 35.70341  0.000000 35.70341 35.70341
#> 798 41.64454  0.000000 41.64454 41.64454
#> 799 43.29513  3.750232 35.94481 50.64545
#> 800 54.25081  0.000000 54.25081 54.25081

The result is now a matrix, because that is what the predict() method returns for mmrm objects. Note that this cannot be changed to return a tibble at the moment.

Similarly, we can also use the augment() method to add predicted values to a new data set:

augment(model, new_data = fev_data) |>
  select(USUBJID, AVISIT, .resid, .pred)
#> # A tibble: 800 × 4
#>    USUBJID AVISIT .resid .pred
#>    <fct>   <fct>   <dbl> <dbl>
#>  1 PT1     VIS1       NA  32.5
#>  2 PT1     VIS2        0  40.0
#>  3 PT1     VIS3       NA  45.7
#>  4 PT1     VIS4        0  20.5
#>  5 PT2     VIS1       NA  28.0
#>  6 PT2     VIS2        0  31.5
#>  7 PT2     VIS3        0  36.9
#>  8 PT2     VIS4        0  48.8
#>  9 PT3     VIS1       NA  30.7
#> 10 PT3     VIS2        0  36.0
#> # ℹ 790 more rows

Note that here we cannot customize the predict options as this is currently not supported by the augment() method in parsnip.

Using mmrm in workflows

We can leverage the workflows package in order to fit the same model.

• First we define the specification for linear regression with the mmrm engine.
• Second we define the workflow, by defining the outcome and predictors that will be used in the formula. We then add the model using the formula.
• Lastly, we fit the model

mmrm_spec <- linear_reg() |>
  set_engine("mmrm", method = "Satterthwaite")

mmrm_wflow <- workflow() |>
  add_variables(outcomes = FEV1, predictors = c(RACE, ARMCD, AVISIT, USUBJID)) |>
  add_model(mmrm_spec, formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID))

mmrm_wflow |>
  fit(data = fev_data)
#> ══ Workflow [trained] ══════════════════════════════════════════════════════════
#> Preprocessor: Variables
#> Model: linear_reg()
#> 
#> ── Preprocessor ────────────────────────────────────────────────────────────────
#> Outcomes: FEV1
#> Predictors: c(RACE, ARMCD, AVISIT, USUBJID)
#> 
#> ── Model ───────────────────────────────────────────────────────────────────────
#> mmrm fit
#> 
#> Formula:     FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data:        data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     weights
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3387.373
#> 
#> Coefficients: 
#>                   (Intercept) RACEBlack or African American 
#>                   30.96769899                    1.50464863 
#>                     RACEWhite                      ARMCDTRT 
#>                    5.61309565                    3.77555734 
#>                    AVISITVIS2                    AVISITVIS3 
#>                    4.82858803                   10.33317002 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                   15.05255715                   -0.01737409 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                   -0.66753189                    0.63094392 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

We can separate out the data preparation step from the modeling step using the recipes package. Here we are converting the ARMCD variable into a dummy variable and creating an interaction term with the new dummy variable and each visit.

mmrm_recipe <- recipe(FEV1 ~ ., data = fev_data) |>
  step_dummy(ARMCD) |>
  step_interact(terms = ~ starts_with("ARMCD"):AVISIT)

Using prep() and juice() we can see what the transformed data that will be used in the model fit looks like.

mmrm_recipe |>
  prep() |>
  juice()
#> # A tibble: 800 × 13
#>    USUBJID AVISIT RACE       SEX   FEV1_BL WEIGHT VISITN VISITN2  FEV1 ARMCD_TRT
#>    <fct>   <fct>  <fct>      <fct>   <dbl>  <dbl>  <int>   <dbl> <dbl>     <dbl>
#>  1 PT1     VIS1   Black or … Fema…    25.3  0.677      1  -0.626  NA           1
#>  2 PT1     VIS2   Black or … Fema…    25.3  0.801      2   0.184  40.0         1
#>  3 PT1     VIS3   Black or … Fema…    25.3  0.709      3  -0.836  NA           1
#>  4 PT1     VIS4   Black or … Fema…    25.3  0.809      4   1.60   20.5         1
#>  5 PT2     VIS1   Asian      Male     45.0  0.465      1   0.330  NA           0
#>  6 PT2     VIS2   Asian      Male     45.0  0.233      2  -0.820  31.5         0
#>  7 PT2     VIS3   Asian      Male     45.0  0.360      3   0.487  36.9         0
#>  8 PT2     VIS4   Asian      Male     45.0  0.507      4   0.738  48.8         0
#>  9 PT3     VIS1   Black or … Fema…    43.5  0.682      1   0.576  NA           0
#> 10 PT3     VIS2   Black or … Fema…    43.5  0.892      2  -0.305  36.0         0
#> # ℹ 790 more rows
#> # ℹ 3 more variables: ARMCD_TRT_x_AVISITVIS2 <dbl>,
#> #   ARMCD_TRT_x_AVISITVIS3 <dbl>, ARMCD_TRT_x_AVISITVIS4 <dbl>

We can pass the covariance structure as well in the set_engine() definition. This allows for more flexibility on presetting different covariance structures in the pipeline while keeping the data preparation step independent.

mmrm_spec_with_cov <- linear_reg() |>
  set_engine(
    "mmrm",
    method = "Satterthwaite",
    covariance = as.cov_struct(~ us(AVISIT | USUBJID))
  )

We combine these steps into a workflow:

(mmrm_wflow_nocov <- workflow() |>
  add_model(mmrm_spec_with_cov, formula = FEV1 ~ SEX) |>
  add_recipe(mmrm_recipe))
#> ══ Workflow ════════════════════════════════════════════════════════════════════
#> Preprocessor: Recipe
#> Model: linear_reg()
#> 
#> ── Preprocessor ────────────────────────────────────────────────────────────────
#> 2 Recipe Steps
#> 
#> • step_dummy()
#> • step_interact()
#> 
#> ── Model ───────────────────────────────────────────────────────────────────────
#> Linear Regression Model Specification (regression)
#> 
#> Engine-Specific Arguments:
#>   method = Satterthwaite
#>   covariance = as.cov_struct(~us(AVISIT | USUBJID))
#> 
#> Computational engine: mmrm

Last step is to fit the data with the workflow object

(fit_tidy <- fit(mmrm_wflow_nocov, data = fev_data))
#> ══ Workflow [trained] ══════════════════════════════════════════════════════════
#> Preprocessor: Recipe
#> Model: linear_reg()
#> 
#> ── Preprocessor ────────────────────────────────────────────────────────────────
#> 2 Recipe Steps
#> 
#> • step_dummy()
#> • step_interact()
#> 
#> ── Model ───────────────────────────────────────────────────────────────────────
#> mmrm fit
#> 
#> Formula:     FEV1 ~ SEX
#> Data:        data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     weights
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3699.803
#> 
#> Coefficients: 
#> (Intercept)   SEXFemale 
#> 42.80540973  0.04513432 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

To retrieve the fit object from within the workflow object run the following

fit_tidy |>
  hardhat::extract_fit_engine()
#> mmrm fit
#> 
#> Formula:     FEV1 ~ SEX
#> Data:        data (used 537 observations from 197 subjects with maximum 4 
#> timepoints)
#> Weights:     weights
#> Covariance:  unstructured (10 variance parameters)
#> Inference:   REML
#> Deviance:    3699.803
#> 
#> Coefficients: 
#> (Intercept)   SEXFemale 
#> 42.80540973  0.04513432 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

Acknowledgments

The mmrm package is based on previous work internal in Roche, namely the tern and tern.mmrm packages which were based on lme4. The work done in the rbmi package has been important since it used glmmTMB for fitting MMRMs.

We would like to thank Ben Bolker from the glmmTMB team for multiple discussions when we tried to get the Satterthwaite degrees of freedom implemented with glmmTMB (see https://github.com/glmmTMB/glmmTMB/blob/satterthwaite_df/glmmTMB/vignettes/satterthwaite_unstructured_example2.Rmd). Also Ben helped us significantly with an example showing how to use TMB for a random effect vector (https://github.com/bbolker/tmb-case-studies/tree/master/vectorMixed).

References