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:

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.

Starting Values

The starting values will affect the optimization result. A better starting value usually can make the optimization more efficient. In mmrm we provide two starting value functions, one is std_start and the other is emp_start. std_start will try to use the identity matrix as the covariance, however there are convergence problems for ar1 and ar1h if the identity matrix is provided, thus for these two covariance structures we use \(\rho=0.5\) instead. emp_start will try to use the empirical covariance matrix of the residuals of the ordinary least squares model as the starting value for unstructured covariance structure. If some timepoints are missing from data, identity matrix will be used for that submatrix. The correlation between existing and non-existing timepoints are set to 0.

As the starting values will affect the result, please be cautious on choosing the starting values.

Example of Default Starting Value Fails

Here we provide an example where the std_start fails. In the following code chunk, we will create a dummy dataset for mmrm analysis.

In the generated data, the variance is not in the same scale across visits.

Using emp_start as the starting value, mmrm will converge fast.

However, if we use std_start, there will be convergence problems. We can also force a specific optimization algorithm and add control details, here e.g. choosing nlminb with increased maximum number of function evaluations and iterations.

REML or ML

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

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.

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).

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.

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.

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”

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:

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:

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.

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

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.

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

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

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.
  1. 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.
  1. 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).

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:

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

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

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

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

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:

or the original low-level call:

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

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.

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.

This works similarly when choosing a Kenward-Roger adjustment:

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:

And for the Kenward-Roger adjustment:

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:

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:

(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.

To further obtain the confidence interval of the least square mean differences, we can apply confint() on the result returned by pairs() .

This is similar to the LSMEANS in SAS, with CL and DIFF options.

Support for car

This package includes methods that allow mmrm objects to be used with the car::Anova function. Anova conducts type II/III hypothesis testing for the effect in mmrm models. For example, in order to see if the used covariates are related to the response:

Note that the degrees of freedom choice is inherited here from the initial mmrm fit. In addition, please note that if you see results that are slightly different from SAS, it could be because the reference level is set differently for categorical covariates. We can also use type III hypothesis testing:

Tidymodels

Tidymodels

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

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.

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

Predictions

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

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  5.912741 20.89001 44.06753
#> 2   39.97105  0.000000 39.97105 39.97105
#> 3   45.70508  4.035906 37.79485 53.61531
#> 4   20.48379  0.000000 20.48379 20.48379
#> 5   28.01243  5.596962 17.04258 38.98227
#> 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  5.707393 19.55145 41.92402
#> 10  35.98699  0.000000 35.98699 35.98699
#> 11  42.64153  3.880319 35.03624 50.24681
#> 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  3.809048 36.57595 51.50715
#> 16  54.45055  0.000000 54.45055 54.45055
#> 17  32.31386  0.000000 32.31386 32.31386
#> 18  37.31982  4.727282 28.05451 46.58512
#> 19  46.79361  0.000000 46.79361 46.79361
#> 20  41.71154  0.000000 41.71154 41.71154
#> 21  31.17198  6.125535 19.16615 43.17781
#> 22  36.63341  5.156900 26.52607 46.74075
#> 23  39.02423  0.000000 39.02423 39.02423
#> 24  47.26333  9.838596 27.98004 66.54663
#> 25  31.93050  0.000000 31.93050 31.93050
#> 26  32.90947  0.000000 32.90947 32.90947
#> 27  41.27523  3.770645 33.88490 48.66556
#> 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  3.791595 43.41526 58.27804
#> 36  58.09713  0.000000 58.09713 58.09713
#> 37  33.21881  6.101077 21.26092 45.17670
#> 38  37.68412  5.136786 27.61621 47.75204
#> 39  44.97613  0.000000 44.97613 44.97613
#> 40  47.67506  9.829325 28.40993 66.94018
#> 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  9.688168 30.73129 68.70820
#> 49  41.46341  5.999508 29.70459 53.22223
#> 50  45.73510  5.097137 35.74490 55.72531
#> 51  53.31791  0.000000 53.31791 53.31791
#> 52  56.07641  0.000000 56.07641 56.07641
#> 53  32.16382  6.091635 20.22444 44.10321
#> 54  37.14256  5.127849 27.09216 47.19296
#> 55  41.90837  0.000000 41.90837 41.90837
#> 56  47.46284  9.811555 28.23255 66.69314
#> 57  27.78883  6.120827 15.79223 39.78543
#> 58  34.13887  5.181902 23.98253 44.29521
#> 59  34.65663  0.000000 34.65663 34.65663
#> 60  39.07791  0.000000 39.07791 39.07791
#> 61  31.18775  5.686639 20.04214 42.33336
#> 62  35.89612  0.000000 35.89612 35.89612
#> 63  41.31608  3.862475 33.74577 48.88639
#> 64  47.67264  0.000000 47.67264 47.67264
#> 65  22.65440  0.000000 22.65440 22.65440
#> 66  36.35488  4.868379 26.81303 45.89672
#> 67  45.20175  3.892415 37.57275 52.83074
#> 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  3.809385 36.53825 51.47076
#> 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  5.658634 31.70830 53.88974
#> 78  47.95358  0.000000 47.95358 47.95358
#> 79  53.97364  0.000000 53.97364 53.97364
#> 80  56.89204  9.742736 37.79663 75.98746
#> 81  46.35384  5.685601 35.21027 57.49741
#> 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  9.654905 25.85193 63.69846
#> 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  4.913836 26.72751 45.98939
#> 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
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#> 483 45.99979  3.963307 38.23185 53.76773
#> 484 59.90473  0.000000 59.90473 59.90473
#> 485 33.87728  6.266356 21.59544 46.15911
#> 486 37.28988  5.289179 26.92328 47.65648
#> 487 49.76150  0.000000 49.76150 49.76150
#> 488 46.60552 10.011086 26.98415 66.22689
#> 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  9.690632 35.57824 73.56482
#> 493 36.13680  5.594379 25.17202 47.10158
#> 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  9.663191 33.92837 71.80739
#> 501 34.62530  0.000000 34.62530 34.62530
#> 502 45.28206  0.000000 45.28206 45.28206
#> 503 44.51505  3.813649 37.04043 51.98966
#> 504 63.57761  0.000000 63.57761 63.57761
#> 505 35.80878  0.000000 35.80878 35.80878
#> 506 40.93038  4.731397 31.65702 50.20375
#> 507 45.85156  3.801097 38.40155 53.30157
#> 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  4.768765 34.54509 53.23830
#> 515 49.56246  3.822896 42.06972 57.05519
#> 516 54.83060  9.666709 35.88420 73.77700
#> 517 37.85241  5.651515 26.77564 48.92917
#> 518 41.54317  0.000000 41.54317 41.54317
#> 519 51.67909  0.000000 51.67909 51.67909
#> 520 51.76691  9.753753 32.64990 70.88391
#> 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  5.578964 19.10139 40.97053
#> 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  5.600234 23.04996 45.00247
#> 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  5.579114 23.09394 44.96366
#> 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  4.761738 33.94024 52.60591
#> 543 49.29538  3.814731 41.81865 56.77212
#> 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  9.971143 37.33040 76.41657
#> 553 30.66032  6.271714 18.36798 42.95265
#> 554 37.44228  5.313096 27.02880 47.85576
#> 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  3.772239 35.02753 49.81444
#> 560 49.90357  0.000000 49.90357 49.90357
#> 561 39.74586  5.687601 28.59837 50.89335
#> 562 44.14167  0.000000 44.14167 44.14167
#> 563 49.91712  3.872352 42.32745 57.50679
#> 564 55.24278  0.000000 55.24278 55.24278
#> 565 36.24790  6.337597 23.82644 48.66937
#> 566 41.05912  5.232697 30.80322 51.31502
#> 567 45.91354  3.968932 38.13458 53.69251
#> 568 51.93141  9.851064 32.62368 71.23914
#> 569 27.38001  0.000000 27.38001 27.38001
#> 570 33.63251  0.000000 33.63251 33.63251
#> 571 44.70168  3.870596 37.11545 52.28791
#> 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  9.653418 29.14470 66.98541
#> 581 34.87547  5.563398 23.97141 45.77953
#> 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  9.697597 30.43959 68.45347
#> 589 38.01775  5.630528 26.98211 49.05338
#> 590 43.75255  0.000000 43.75255 43.75255
#> 591 47.38873  0.000000 47.38873 47.38873
#> 592 52.70780  9.715725 33.66532 71.75027
#> 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  6.082469 27.53605 51.37889
#> 598 42.93167  5.154112 32.82979 53.03354
#> 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  3.776035 34.46566 49.26745
#> 604 47.46553  0.000000 47.46553 47.46553
#> 605 32.03992  5.735557 20.79844 43.28141
#> 606 37.11697  0.000000 37.11697 37.11697
#> 607 44.12071  3.907079 36.46297 51.77844
#> 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  3.778132 35.35667 50.16667
#> 616 57.03861  0.000000 57.03861 57.03861
#> 617 36.32475  0.000000 36.32475 36.32475
#> 618 40.15241  4.717823 30.90564 49.39917
#> 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  4.735212 31.76338 50.32507
#> 627 45.82688  3.805665 38.36791 53.28584
#> 628 56.41409  0.000000 56.41409 56.41409
#> 629 37.80184  5.546148 26.93159 48.67209
#> 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  3.876057 36.41992 51.61378
#> 636 49.57489  9.772639 30.42087 68.72891
#> 637 39.38085  5.710671 28.18814 50.57356
#> 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  5.760149 22.14439 44.72376
#> 642 39.06783  0.000000 39.06783 39.06783
#> 643 42.98333  3.875913 35.38668 50.57998
#> 644 48.01822  9.731482 28.94487 67.09158
#> 645 29.99542  0.000000 29.99542 29.99542
#> 646 35.69583  4.740593 26.40444 44.98722
#> 647 41.11547  3.806696 33.65449 48.57646
#> 648 54.17796  0.000000 54.17796 54.17796
#> 649 39.32289  5.719220 28.11342 50.53235
#> 650 44.55743  0.000000 44.55743 44.55743
#> 651 47.26282  3.895636 39.62751 54.89812
#> 652 62.59579  0.000000 62.59579 62.59579
#> 653 31.80300  5.537108 20.95047 42.65554
#> 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  4.786444 38.35262 57.11514
#> 659 50.94631  3.846320 43.40766 58.48496
#> 660 58.47218  9.717386 39.42645 77.51790
#> 661 22.15439  0.000000 22.15439 22.15439
#> 662 35.14301  4.881795 25.57487 44.71116
#> 663 42.82000  3.907268 35.16190 50.47811
#> 664 46.24563  9.775317 27.08637 65.40490
#> 665 34.27765  0.000000 34.27765 34.27765
#> 666 36.90059  0.000000 36.90059 36.90059
#> 667 43.05627  3.769928 35.66735 50.44519
#> 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  9.577673 27.72911 65.27290
#> 673 27.12174  0.000000 27.12174 27.12174
#> 674 34.11916  0.000000 34.11916 34.11916
#> 675 45.56320  3.880031 37.95848 53.16793
#> 676 48.00823  9.713551 28.97002 67.04644
#> 677 35.93048  5.605558 24.94379 46.91717
#> 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  5.770737 17.25525 39.87613
#> 682 29.22869  0.000000 29.22869 29.22869
#> 683 40.67646  3.939045 32.95608 48.39685
#> 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  9.593235 30.70671 68.31150
#> 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  9.596940 37.87131 75.49063
#> 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  9.749187 37.47406 75.69018
#> 697 29.66493  0.000000 29.66493 29.66493
#> 698 34.57406  0.000000 34.57406 34.57406
#> 699 42.45295  3.779255 35.04575 49.86015
#> 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  3.853063 37.74324 52.84697
#> 704 58.81037  0.000000 58.81037 58.81037
#> 705 31.46668  6.111158 19.48904 43.44433
#> 706 36.78469  5.144500 26.70166 46.86773
#> 707 39.88119  0.000000 39.88119 39.88119
#> 708 47.32261  9.826144 28.06373 66.58150
#> 709 31.62708  0.000000 31.62708 31.62708
#> 710 37.03239  4.729226 27.76328 46.30150
#> 711 42.69162  3.788915 35.26548 50.11775
#> 712 48.22049  0.000000 48.22049 48.22049
#> 713 42.58829  0.000000 42.58829 42.58829
#> 714 45.80410  4.698963 36.59430 55.01390
#> 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  9.615571 26.30711 63.99946
#> 721 36.81040  0.000000 36.81040 36.81040
#> 722 37.61606  4.733690 28.33820 46.89393
#> 723 42.35045  0.000000 42.35045 42.35045
#> 724 39.39860  0.000000 39.39860 39.39860
#> 725 36.09876  6.016152 24.30732 47.89020
#> 726 40.94066  5.098526 30.94773 50.93358
#> 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  3.815795 39.22456 54.18221
#> 732 53.94830  9.667795 34.99977 72.89683
#> 733 39.60913  5.741144 28.35669 50.86156
#> 734 41.08206  0.000000 41.08206 41.08206
#> 735 49.65683  3.914188 41.98517 57.32850
#> 736 69.37409  0.000000 69.37409 69.37409
#> 737 34.12096  5.582665 23.17914 45.06279
#> 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  3.905595 39.88600 55.19564
#> 744 35.50735  0.000000 35.50735 35.50735
#> 745 32.08153  0.000000 32.08153 32.08153
#> 746 37.16398  4.752728 27.84881 46.47916
#> 747 42.75619  3.807270 35.29408 50.21830
#> 748 47.39510  9.660100 28.46165 66.32855
#> 749 44.69256  0.000000 44.69256 44.69256
#> 750 41.45664  4.864206 31.92298 50.99031
#> 751 42.18689  0.000000 42.18689 42.18689
#> 752 51.68534  9.765987 32.54436 70.82632
#> 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  9.622862 31.81439 69.53532
#> 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  4.811277 32.15146 51.01132
#> 763 48.87398  3.849305 41.32948 56.41848
#> 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  5.692304 17.45138 39.76480
#> 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  5.646700 30.66087 52.79552
#> 778 49.12862  0.000000 49.12862 49.12862
#> 779 47.31234  0.000000 47.31234 47.31234
#> 780 57.08286  9.739411 37.99396 76.17175
#> 781 19.28388  0.000000 19.28388 19.28388
#> 782 30.00682  0.000000 30.00682 30.00682
#> 783 39.69711  3.895230 32.06260 47.33162
#> 784 49.21768  0.000000 49.21768 49.21768
#> 785 31.42637  6.221006 19.23342 43.61932
#> 786 36.73485  5.172897 26.59615 46.87354
#> 787 42.72556  3.946954 34.98967 50.46145
#> 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  3.895248 39.28769 54.55678
#> 792 69.26254  0.000000 69.26254 69.26254
#> 793 40.35635  6.336839 27.93638 52.77633
#> 794 45.16757  5.233093 34.91089 55.42424
#> 795 50.02199  3.964035 42.25262 57.79136
#> 796 56.03985  9.851431 36.73140 75.34830
#> 797 35.70341  0.000000 35.70341 35.70341
#> 798 41.64454  0.000000 41.64454 41.64454
#> 799 43.29513  3.774074 35.89808 50.69218
#> 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:

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.

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

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.

We combine these steps into a workflow:

Last step is to fit the data with the workflow object

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

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