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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.

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.

gen_data <- function(
    n = 100,
    mu = -100 / 52,
    delta = 50 / 52,
    mua = 2000,
    sigmaa = 300,
    sigmab = 60,
    corab = 0.2,
    sigma = 10,
    times = c(0, 2, 6, 12, 24, 36, 52, 70, 88, 104)) {
  nt <- length(times)
  out <- data.frame(
    pts = rep(seq_len(n * 2), each = nt),
    trt = rep(c("Treatment", "Placebo"), rep(n * nt, 2)),
    time = rep(times, n * 2)
  )

  covab <- corab * sigmaa * sigmab # cov between a and b
  cov <- matrix(c(sigmaa^2, covab, covab, sigmab^2), ncol = 2) # Cov matrix for the slope and intercept
  si <- rbind(
    MASS::mvrnorm(n, mu = c(mua, mu + delta), Sigma = cov),
    MASS::mvrnorm(n, mu = c(mua, mu + delta), Sigma = cov)
  )
  idx <- rep(seq_len(n * 2), each = nt)
  out$fev1 <- si[idx, 1] + si[idx, 2] * times + rnorm(n * nt * 2, sd = sigma)
  out$trt <- factor(out$trt)
  out$time <- factor(out$time)
  out$pts <- factor(out$pts)
  return(out)
}
set.seed(123)
out <- gen_data()

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

vapply(split(out$fev1, out$time), sd, FUN.VALUE = 1)
#>         0         2         6        12        24        36        52        70 
#>  278.6079  319.0589  482.4172  799.9107 1491.1440 2194.5776 3140.0768 4204.9355 
#>        88       104 
#> 5272.6041 6221.2195

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

mmrm(fev1 ~ trt * time + us(time | pts), data = out, start = emp_start)
#> mmrm fit
#> 
#> Formula:     fev1 ~ trt * time + us(time | pts)
#> Data:        out (used 2000 observations from 200 subjects with maximum 10 
#> timepoints)
#> Covariance:  unstructured (55 variance parameters)
#> Inference:   REML
#> Deviance:    19154.63
#> 
#> Coefficients: 
#>          (Intercept)         trtTreatment                time2 
#>         1962.6980059           11.3831958            0.2130684 
#>                time6               time12               time24 
#>           -1.6901336           -1.1984277           -7.3954489 
#>               time36               time52               time70 
#>          -11.4078895          -15.8040920          -22.5556524 
#>               time88              time104   trtTreatment:time2 
#>          -28.2068895          -33.6608067            6.0436315 
#>   trtTreatment:time6  trtTreatment:time12  trtTreatment:time24 
#>           27.3686373           49.4246567          107.3638488 
#>  trtTreatment:time36  trtTreatment:time52  trtTreatment:time70 
#>          161.4310444          233.6438342          316.7387101 
#>  trtTreatment:time88 trtTreatment:time104 
#>          397.9895967          471.8871913 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmch

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.

mmrm(
  fev1 ~ trt * time + us(time | pts),
  data = out,
  start = std_start,
  optimizer = "nlminb",
  optimizer_control = list(eval.max = 1000, iter.max = 1000)
)

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 
#>                   30.96768936                    1.50467465 
#>                     RACEWhite                      ARMCDTRT 
#>                    5.61310613                    3.77554452 
#>                    AVISITVIS2                    AVISITVIS3 
#>                    4.82858600                   10.33317622 
#>                    AVISITVIS4           ARMCDTRT:AVISITVIS2 
#>                   15.05257117                   -0.01735504 
#>           ARMCDTRT:AVISITVIS3           ARMCDTRT:AVISITVIS4 
#>                   -0.66752133                    0.63095590 
#> 
#> Model Inference Optimization:
#> Converged with code 0 and message: No message provided.

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")
  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.
residuals_pearson <- residuals(fit, type = "pearson")
  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).
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    1.99       5.56
#>  3 ARMCDTRT:AVIS…  -0.0174     1.13   138.   -0.0154 9.88e- 1   -1.89       1.85
#>  4 ARMCDTRT:AVIS…  -0.668      1.19   158.   -0.563  5.75e- 1   -2.63       1.30
#>  5 ARMCDTRT:AVIS…   0.631      1.85   130.    0.341  7.34e- 1   -2.44       3.70
#>  6 AVISITVIS2       4.83       0.802  144.    6.02   1.37e- 8    3.50       6.16
#>  7 AVISITVIS3      10.3        0.822  156.   12.6    1.93e-25    8.97      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    0.478      2.53
#> 10 RACEWhite        5.61       0.663  159.    8.47   1.61e-14    4.52       6.71

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.

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
#> Welcome to emmeans.
#> Caution: You lose important information if you filter this package's results.
#> See '? untidy'
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.

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.

confint(pairs(lsmeans_by_visit, reverse = TRUE))
#> AVISIT = VIS1:
#>  contrast  estimate    SE  df lower.CL upper.CL
#>  TRT - PBO     3.77 1.074 146     1.65     5.90
#> 
#> AVISIT = VIS2:
#>  contrast  estimate    SE  df lower.CL upper.CL
#>  TRT - PBO     3.73 0.859 145     2.03     5.43
#> 
#> AVISIT = VIS3:
#>  contrast  estimate    SE  df lower.CL upper.CL
#>  TRT - PBO     3.08 0.690 131     1.72     4.44
#> 
#> AVISIT = VIS4:
#>  contrast  estimate    SE  df lower.CL upper.CL
#>  TRT - PBO     4.40 1.681 133     1.07     7.72
#> 
#> Results are averaged over the levels of: RACE, SEX 
#> Confidence level used: 0.95

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:

library(car)
#> Loading required package: carData
#> mmrm() registered as car::Anova extension
Anova(fit, type = "II")
#> Analysis of Fixed Effect Table (Type II F tests)
#>              Num Df Denom Df F Statistic   Pr(>=F)    
#> RACE              2   165.56      36.911 5.545e-14 ***
#> SEX               1   166.13       0.376    0.5407    
#> ARMCD             1   169.12      31.421 8.296e-08 ***
#> AVISIT            3   148.65     142.822 < 2.2e-16 ***
#> ARMCD:AVISIT      3   147.91       0.258    0.8555    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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:

Anova(fit, type = "III")
#> Analysis of Fixed Effect Table (Type III F tests)
#>              Num Df Denom Df F Statistic   Pr(>=F)    
#> RACE              2   165.56      36.911 5.545e-14 ***
#> SEX               1   166.13       0.376    0.5407    
#> ARMCD             1   168.52      31.663 7.502e-08 ***
#> AVISIT            3   148.12     142.112 < 2.2e-16 ***
#> ARMCD:AVISIT      3   147.91       0.258    0.8555    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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.
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  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
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#> 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:

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