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*epsmchThe 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 95.5568Common customizations
From the high-level mmrm()
interface, common changes to the default function call can be
specified.
Control Function
For fine control, mmrm_control()
is provided. This function allows the user to choose the adjustment
method for the degrees of freedom and the coefficients covariance
matrix, specify optimization routines, number of cores to be used on
Unix systems for trying several optimizers in parallel, provide a vector
of starting parameter values, decide the action to be taken when the
defined design matrix is singular, not drop unobserved visit levels. For
example:
mmrm_control(
method = "Kenward-Roger",
optimizer = c("L-BFGS-B", "BFGS"),
n_cores = 2,
start = c(0, 1, 1, 0, 1, 0),
accept_singular = FALSE,
drop_visit_levels = FALSE
)Note that this control list can either be passed via the
control argument to mmrm, or selected controls
can be directly specified in the mmrm call. We will see
this below.
REML or ML
Users can specify if REML should be used (default) or if ML should be used in optimization.
fit_ml <- mmrm(
formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
data = fev_data,
reml = FALSE
)
fit_ml
#> mmrm fit
#>
#> Formula: FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data: fev_data (used 537 observations from 197 subjects with maximum 4
#> timepoints)
#> Covariance: unstructured (10 variance parameters)
#> Inference: ML
#> Deviance: 3397.934
#>
#> Coefficients:
#> (Intercept) RACEBlack or African American
#> 30.9663423 1.5086851
#> RACEWhite ARMCDTRT
#> 5.6133151 3.7761037
#> AVISITVIS2 AVISITVIS3
#> 4.8270155 10.3353319
#> AVISITVIS4 ARMCDTRT:AVISITVIS2
#> 15.0487715 -0.0156154
#> ARMCDTRT:AVISITVIS3 ARMCDTRT:AVISITVIS4
#> -0.6663598 0.6317222
#>
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmchOptimizer
Users can specify which optimizer should be used, changing from the
default of four optimizers, which starts with L-BFGS-B and
proceeds through the other choices if optimization fails to converge.
Other choices are BFGS, CG,
nlminb and other user-defined custom optimizers.
L-BFGS-B, BFGS and CG are all
implemented with stats::optim() and the Hessian is not
used, while nlminb is using stats::nlminb()
which in turn uses both the gradient and the Hessian (by default but can
be switch off) for the optimization.
fit_opt <- mmrm(
formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
data = fev_data,
optimizer = "BFGS"
)
fit_opt
#> mmrm fit
#>
#> Formula: FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data: fev_data (used 537 observations from 197 subjects with maximum 4
#> timepoints)
#> Covariance: unstructured (10 variance parameters)
#> Inference: REML
#> Deviance: 3387.373
#>
#> Coefficients:
#> (Intercept) RACEBlack or African American
#> 30.9676902 1.5046744
#> RACEWhite ARMCDTRT
#> 5.6131048 3.7755423
#> AVISITVIS2 AVISITVIS3
#> 4.8285855 10.3331770
#> AVISITVIS4 ARMCDTRT:AVISITVIS2
#> 15.0525706 -0.0173504
#> ARMCDTRT:AVISITVIS3 ARMCDTRT:AVISITVIS4
#> -0.6675190 0.6309586
#>
#> Model Inference Optimization:
#> Converged with code 0 and message:Covariance Structure
Covariance structures supported by the mmrm are being
continuously developed. For a complete list and description please visit
the covariance vignette. Below we see the
function call for homogeneous compound symmetry (cs).
fit_cs <- mmrm(
formula = FEV1 ~ RACE + ARMCD * AVISIT + cs(AVISIT | USUBJID),
data = fev_data,
reml = FALSE
)
fit_cs
#> mmrm fit
#>
#> Formula: FEV1 ~ RACE + ARMCD * AVISIT + cs(AVISIT | USUBJID)
#> Data: fev_data (used 537 observations from 197 subjects with maximum 4
#> timepoints)
#> Covariance: compound symmetry (2 variance parameters)
#> Inference: ML
#> Deviance: 3536.989
#>
#> Coefficients:
#> (Intercept) RACEBlack or African American
#> 31.4207077 0.5357237
#> RACEWhite ARMCDTRT
#> 5.4546329 3.4305212
#> AVISITVIS2 AVISITVIS3
#> 4.8326353 10.2395076
#> AVISITVIS4 ARMCDTRT:AVISITVIS2
#> 15.0672680 0.2801641
#> ARMCDTRT:AVISITVIS3 ARMCDTRT:AVISITVIS4
#> -0.5894964 0.7939750
#>
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmchThe 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*epsmchGrouped 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 49.58110We 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
- “Kenward-Roger”
- “Satterthwaite”
- “Residual”
- “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 95.6006For 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:
- “Asymptotic”
- “Empirical” (Cluster Robust Sandwich)
- “Empirical-Jackknife”
- “Empirical-Bias-Reduced”
- “Kenward-Roger”
- “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: VIS3We 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 1.00000000We 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-01Other 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:
- 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")- 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")- 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- 1We can also specify some details to request confidence intervals of specific confidence level:
fit |>
tidy(conf.int = TRUE, conf.level = 0.9)
#> # A tibble: 10 × 8
#> term estimate std.error df statistic p.value conf.low conf.high
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 31.0 0.829 188. 37.3 7.12e-89 29.6 32.3
#> 2 ARMCDTRT 3.78 1.08 146. 3.51 6.00e- 4 2.01 5.55
#> 3 ARMCDTRT:AVIS… -0.0174 1.13 138. -0.0154 9.88e- 1 -1.87 1.84
#> 4 ARMCDTRT:AVIS… -0.668 1.19 158. -0.563 5.75e- 1 -2.62 1.28
#> 5 ARMCDTRT:AVIS… 0.631 1.85 130. 0.341 7.34e- 1 -2.41 3.68
#> 6 AVISITVIS2 4.83 0.802 144. 6.02 1.37e- 8 3.51 6.15
#> 7 AVISITVIS3 10.3 0.822 156. 12.6 1.93e-25 8.98 11.7
#> 8 AVISITVIS4 15.1 1.31 138. 11.5 8.24e-22 12.9 17.2
#> 9 RACEBlack or … 1.50 0.621 170. 2.42 1.64e- 2 0.484 2.53
#> 10 RACEWhite 5.61 0.663 159. 8.47 1.61e-14 4.52 6.70Or 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 rowsAlso here we can specify details for the prediction intervals and type of residuals via the arguments:
fit |>
augment(interval = "confidence", type.residuals = "normalized")
#> # A tibble: 537 × 11
#> .rownames FEV1 RACE ARMCD AVISIT USUBJID .fitted .lower .upper .se.fit
#> <dbl> <dbl> <fct> <fct> <fct> <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 2 40.0 Black or … TRT VIS2 PT1 40.0 40.0 40.0 0
#> 2 4 20.5 Black or … TRT VIS4 PT1 20.5 20.5 20.5 0
#> 3 6 31.5 Asian PBO VIS2 PT2 31.5 31.5 31.5 0
#> 4 7 36.9 Asian PBO VIS3 PT2 36.9 36.9 36.9 0
#> 5 8 48.8 Asian PBO VIS4 PT2 48.8 48.8 48.8 0
#> 6 10 36.0 Black or … PBO VIS2 PT3 36.0 36.0 36.0 0
#> 7 12 37.2 Black or … PBO VIS4 PT3 37.2 37.2 37.2 0
#> 8 13 33.9 Asian TRT VIS1 PT4 33.9 33.9 33.9 0
#> 9 14 33.7 Asian TRT VIS2 PT4 33.7 33.7 33.7 0
#> 10 16 54.5 Asian TRT VIS4 PT4 54.5 54.5 54.5 0
#> # ℹ 527 more rows
#> # ℹ 1 more variable: .resid <dbl>Other components
Specific model quantities not supported by methods can be retrieved
with the component()
function. The default will output all supported components.
For example, a user may want information about convergence:
component(fit, name = c("convergence", "evaluations", "conv_message"))
#> $convergence
#> [1] 0
#>
#> $evaluations
#> function gradient
#> 17 17
#>
#> $conv_message
#> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"or the original low-level call:
component(fit, name = "call")
#> mmrm(formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
#> data = fev_data)the user could also ask for all provided components by not specifying
the name argument.
component(fit)Lower level functions
Low-level mmrm
The lower level function which is called by mmrm()
is fit_mmrm().
This function is exported and can be used directly. It is similar to mmrm()
but lacks some post-processing and support for Satterthwaite and
Kenward-Roger calculations. It may be useful for other packages that
want to fit the model without the adjustment calculations.
fit_mmrm(
formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
data = fev_data,
weights = rep(1, nrow(fev_data)),
reml = TRUE,
control = mmrm_control()
)
#> mmrm fit
#>
#> Formula: FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID)
#> Data: fev_data (used 537 observations from 197 subjects with maximum 4
#> timepoints)
#> Weights: rep(1, nrow(fev_data))
#> Covariance: unstructured (10 variance parameters)
#> Inference: REML
#> Deviance: 3387.373
#>
#> Coefficients:
#> (Intercept) RACEBlack or African American
#> 30.96769899 1.50464863
#> RACEWhite ARMCDTRT
#> 5.61309565 3.77555734
#> AVISITVIS2 AVISITVIS3
#> 4.82858803 10.33317002
#> AVISITVIS4 ARMCDTRT:AVISITVIS2
#> 15.05255715 -0.01737409
#> ARMCDTRT:AVISITVIS3 ARMCDTRT:AVISITVIS4
#> -0.66753189 0.63094392
#>
#> Model Inference Optimization:
#> Converged with code 0 and message: convergence: rel_reduction_of_f <= factr*epsmchHypothesis 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-14This 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-14We 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-14And 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-13We 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.
fit <- mmrm(
formula = FEV1 ~ RACE + ARMCD * AVISIT + us(AVISIT | USUBJID),
data = fev_data
)
if (require(emmeans)) {
emmeans(fit, ~ ARMCD | AVISIT)
}
#> Loading required package: emmeans
#> mmrm() registered as emmeans extension
#> AVISIT = VIS1:
#> ARMCD emmean SE df lower.CL upper.CL
#> PBO 33.3 0.757 149 31.8 34.8
#> TRT 37.1 0.764 144 35.6 38.6
#>
#> AVISIT = VIS2:
#> ARMCD emmean SE df lower.CL upper.CL
#> PBO 38.2 0.608 150 37.0 39.4
#> TRT 41.9 0.598 146 40.7 43.1
#>
#> AVISIT = VIS3:
#> ARMCD emmean SE df lower.CL upper.CL
#> PBO 43.7 0.462 131 42.8 44.6
#> TRT 46.8 0.507 130 45.8 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
#> Confidence level used: 0.95Note that the degrees of freedom choice is inherited here from the
initial mmrm fit.
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*epsmchWe 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 rowsNote that we need to explicitly pass new_data because
the method definition does not allow to default it to the data set we
used for the model fitting automatically.
By using the type = "numeric" default of
predict() as above we cannot further customize the
calculations. We obtain predicted values without confidence intervals or
standard errors.
On the other hand, when using type = "raw" we can
customize the calculations via the opts list:
predict(
model,
new_data = fev_data,
type = "raw",
opts = list(se.fit = TRUE, interval = "prediction", nsim = 10L)
)
#> fit se lwr upr
#> 1 32.47877 6.308284 20.11476 44.84278
#> 2 39.97105 0.000000 39.97105 39.97105
#> 3 45.70508 4.119563 37.63088 53.77927
#> 4 20.48379 0.000000 20.48379 20.48379
#> 5 28.01243 5.746923 16.74867 39.27619
#> 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 5.830991 19.30921 42.16627
#> 10 35.98699 0.000000 35.98699 35.98699
#> 11 42.64153 3.935990 34.92713 50.35593
#> 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 3.880636 36.43564 51.64746
#> 16 54.45055 0.000000 54.45055 54.45055
#> 17 32.31386 0.000000 32.31386 32.31386
#> 18 37.31982 4.818414 27.87590 46.76373
#> 19 46.79361 0.000000 46.79361 46.79361
#> 20 41.71154 0.000000 41.71154 41.71154
#> 21 31.17198 6.415757 18.59733 43.74663
#> 22 36.63341 5.191749 26.45777 46.80905
#> 23 39.02423 0.000000 39.02423 39.02423
#> 24 47.26333 10.146921 27.37574 67.15093
#> 25 31.93050 0.000000 31.93050 31.93050
#> 26 32.90947 0.000000 32.90947 32.90947
#> 27 41.27523 3.848714 33.73189 48.81857
#> 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 3.865962 43.26951 58.42380
#> 36 58.09713 0.000000 58.09713 58.09713
#> 37 33.21881 6.392655 20.68944 45.74819
#> 38 37.68412 5.165132 27.56065 47.80759
#> 39 44.97613 0.000000 44.97613 44.97613
#> 40 47.67506 10.131090 27.81849 67.53163
#> 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 9.868482 30.37787 69.06162
#> 49 41.46341 6.123754 29.46107 53.46575
#> 50 45.73510 5.133167 35.67428 55.79592
#> 51 53.31791 0.000000 53.31791 53.31791
#> 52 56.07641 0.000000 56.07641 56.07641
#> 53 32.16382 6.386361 19.64679 44.68086
#> 54 37.14256 5.153306 27.04226 47.24285
#> 55 41.90837 0.000000 41.90837 41.90837
#> 56 47.46284 10.111043 27.64557 67.28012
#> 57 27.78883 6.183820 15.66876 39.90889
#> 58 34.13887 5.189785 23.96708 44.31066
#> 59 34.65663 0.000000 34.65663 34.65663
#> 60 39.07791 0.000000 39.07791 39.07791
#> 61 31.18775 5.786900 19.84563 42.52987
#> 62 35.89612 0.000000 35.89612 35.89612
#> 63 41.31608 3.905510 33.66142 48.97074
#> 64 47.67264 0.000000 47.67264 47.67264
#> 65 22.65440 0.000000 22.65440 22.65440
#> 66 36.35488 4.918011 26.71575 45.99400
#> 67 45.20175 3.979893 37.40130 53.00219
#> 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 3.917483 36.32638 51.68263
#> 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 5.946641 31.14382 54.45423
#> 78 47.95358 0.000000 47.95358 47.95358
#> 79 53.97364 0.000000 53.97364 53.97364
#> 80 56.89204 10.029431 37.23472 76.54937
#> 81 46.35384 5.894326 34.80118 57.90651
#> 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 9.810224 25.54751 64.00288
#> 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 4.908061 26.73883 45.97807
#> 107 46.08393 0.000000 46.08393 46.08393
#> 108 52.42288 0.000000 52.42288 52.42288
#> 109 37.69466 0.000000 37.69466 37.69466
#> 110 44.59400 0.000000 44.59400 44.59400
#> 111 52.08897 0.000000 52.08897 52.08897
#> 112 58.22961 0.000000 58.22961 58.22961
#> 113 37.22824 0.000000 37.22824 37.22824
#> 114 34.39863 0.000000 34.39863 34.39863
#> 115 45.88949 3.994092 38.06121 53.71777
#> 116 36.34012 0.000000 36.34012 36.34012
#> 117 45.44182 0.000000 45.44182 45.44182
#> 118 41.54847 0.000000 41.54847 41.54847
#> 119 43.92172 0.000000 43.92172 43.92172
#> 120 61.83243 0.000000 61.83243 61.83243
#> 121 27.25656 0.000000 27.25656 27.25656
#> 122 34.77803 4.824956 25.32129 44.23477
#> 123 45.65133 0.000000 45.65133 45.65133
#> 124 44.56078 9.762471 25.42669 63.69487
#> 125 33.19334 0.000000 33.19334 33.19334
#> 126 39.72671 4.811618 30.29611 49.15731
#> 127 45.59637 3.911735 37.92951 53.26323
#> 128 41.66826 0.000000 41.66826 41.66826
#> 129 27.12753 0.000000 27.12753 27.12753
#> 130 31.74858 0.000000 31.74858 31.74858
#> 131 44.57711 4.014584 36.70867 52.44555
#> 132 41.60000 0.000000 41.60000 41.60000
#> 133 39.45250 0.000000 39.45250 39.45250
#> 134 32.61823 0.000000 32.61823 32.61823
#> 135 34.62445 0.000000 34.62445 34.62445
#> 136 45.90515 0.000000 45.90515 45.90515
#> 137 36.17780 0.000000 36.17780 36.17780
#> 138 39.79796 0.000000 39.79796 39.79796
#> 139 45.87019 3.860275 38.30419 53.43619
#> 140 50.08272 0.000000 50.08272 50.08272
#> 141 36.27753 5.984281 24.54855 48.00650
#> 142 44.64316 0.000000 44.64316 44.64316
#> 143 44.88252 3.948870 37.14287 52.62216
#> 144 39.73529 0.000000 39.73529 39.73529
#> 145 34.06164 0.000000 34.06164 34.06164
#> 146 40.18592 0.000000 40.18592 40.18592
#> 147 41.17584 0.000000 41.17584 41.17584
#> 148 57.76669 0.000000 57.76669 57.76669
#> 149 38.18460 0.000000 38.18460 38.18460
#> 150 38.61735 4.899978 29.01357 48.22113
#> 151 47.19893 0.000000 47.19893 47.19893
#> 152 48.18237 9.824028 28.92763 67.43711
#> 153 37.32785 0.000000 37.32785 37.32785
#> 154 37.89476 4.832820 28.42260 47.36691
#> 155 43.16048 0.000000 43.16048 43.16048
#> 156 41.40349 0.000000 41.40349 41.40349
#> 157 30.15733 0.000000 30.15733 30.15733
#> 158 35.84353 0.000000 35.84353 35.84353
#> 159 40.95250 0.000000 40.95250 40.95250
#> 160 46.76086 9.591960 27.96096 65.56075
#> 161 36.09804 5.774069 24.78108 47.41501
#> 162 41.37928 0.000000 41.37928 41.37928
#> 163 50.17316 0.000000 50.17316 50.17316
#> 164 45.35226 0.000000 45.35226 45.35226
#> 165 39.06491 0.000000 39.06491 39.06491
#> 166 39.39597 4.843286 29.90331 48.88864
#> 167 43.69372 3.882174 36.08480 51.30265
#> 168 42.11960 0.000000 42.11960 42.11960
#> 169 29.81042 0.000000 29.81042 29.81042
#> 170 42.57055 0.000000 42.57055 42.57055
#> 171 47.81652 0.000000 47.81652 47.81652
#> 172 68.06024 0.000000 68.06024 68.06024
#> 173 35.62071 0.000000 35.62071 35.62071
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#> 565 36.24790 6.582562 23.34632 49.14949
#> 566 41.05912 5.233084 30.80246 51.31577
#> 567 45.91354 4.012394 38.04940 53.77769
#> 568 51.93141 10.222967 31.89476 71.96805
#> 569 27.38001 0.000000 27.38001 27.38001
#> 570 33.63251 0.000000 33.63251 33.63251
#> 571 44.70168 3.991518 36.87845 52.52491
#> 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 9.656265 29.13912 66.99099
#> 581 34.87547 5.715429 23.67343 46.07750
#> 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 9.935487 29.97333 68.91972
#> 589 38.01775 5.935291 26.38479 49.65070
#> 590 43.75255 0.000000 43.75255 43.75255
#> 591 47.38873 0.000000 47.38873 47.38873
#> 592 52.70780 9.997480 33.11309 72.30250
#> 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 6.183711 27.33762 51.57732
#> 598 42.93167 5.136470 32.86437 52.99896
#> 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 3.855089 34.31072 49.42239
#> 604 47.46553 0.000000 47.46553 47.46553
#> 605 32.03992 5.890081 20.49557 43.58427
#> 606 37.11697 0.000000 37.11697 37.11697
#> 607 44.12071 3.969819 36.34001 51.90141
#> 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 3.852128 35.21164 50.31170
#> 616 57.03861 0.000000 57.03861 57.03861
#> 617 36.32475 0.000000 36.32475 36.32475
#> 618 40.15241 4.803152 30.73840 49.56641
#> 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 4.802091 31.63230 50.45615
#> 627 45.82688 3.875352 38.23133 53.42243
#> 628 56.41409 0.000000 56.41409 56.41409
#> 629 37.80184 5.711561 26.60738 48.99629
#> 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 3.947202 36.28048 51.75322
#> 636 49.57489 9.897077 30.17697 68.97280
#> 637 39.38085 5.871740 27.87245 50.88925
#> 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 6.041565 21.59283 45.27533
#> 642 39.06783 0.000000 39.06783 39.06783
#> 643 42.98333 3.940666 35.25977 50.70689
#> 644 48.01822 10.077238 28.26720 67.76925
#> 645 29.99542 0.000000 29.99542 29.99542
#> 646 35.69583 4.806546 26.27517 45.11649
#> 647 41.11547 3.872923 33.52469 48.70626
#> 648 54.17796 0.000000 54.17796 54.17796
#> 649 39.32289 5.838425 27.87979 50.76599
#> 650 44.55743 0.000000 44.55743 44.55743
#> 651 47.26282 3.952821 39.51543 55.01020
#> 652 62.59579 0.000000 62.59579 62.59579
#> 653 31.80300 5.695003 20.64100 42.96500
#> 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 4.839727 38.24819 57.21957
#> 659 50.94631 3.910646 43.28159 58.61104
#> 660 58.47218 9.847027 39.17236 77.77199
#> 661 22.15439 0.000000 22.15439 22.15439
#> 662 35.14301 4.917278 25.50533 44.78070
#> 663 42.82000 3.966951 35.04492 50.59508
#> 664 46.24563 10.015960 26.61471 65.87656
#> 665 34.27765 0.000000 34.27765 34.27765
#> 666 36.90059 0.000000 36.90059 36.90059
#> 667 43.05627 3.854842 35.50092 50.61162
#> 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 9.615864 27.65426 65.34775
#> 673 27.12174 0.000000 27.12174 27.12174
#> 674 34.11916 0.000000 34.11916 34.11916
#> 675 45.56320 3.942434 37.83618 53.29023
#> 676 48.00823 9.844927 28.71252 67.30393
#> 677 35.93048 5.767766 24.62587 47.23509
#> 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 5.977435 16.85014 40.28125
#> 682 29.22869 0.000000 29.22869 29.22869
#> 683 40.67646 3.963285 32.90856 48.44436
#> 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 9.611354 30.67120 68.34702
#> 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 9.628190 37.81006 75.55188
#> 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 10.188397 36.61323 76.55101
#> 697 29.66493 0.000000 29.66493 29.66493
#> 698 34.57406 0.000000 34.57406 34.57406
#> 699 42.45295 3.876374 34.85540 50.05050
#> 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 3.909403 37.63282 52.95740
#> 704 58.81037 0.000000 58.81037 58.81037
#> 705 31.46668 6.403466 18.91612 44.01725
#> 706 36.78469 5.175349 26.64120 46.92819
#> 707 39.88119 0.000000 39.88119 39.88119
#> 708 47.32261 10.130739 27.46673 67.17850
#> 709 31.62708 0.000000 31.62708 31.62708
#> 710 37.03239 4.796137 27.63213 46.43265
#> 711 42.69162 3.861075 35.12405 50.25918
#> 712 48.22049 0.000000 48.22049 48.22049
#> 713 42.58829 0.000000 42.58829 42.58829
#> 714 45.80410 4.774344 36.44656 55.16164
#> 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 9.609132 26.31973 63.98683
#> 721 36.81040 0.000000 36.81040 36.81040
#> 722 37.61606 4.824581 28.16006 47.07207
#> 723 42.35045 0.000000 42.35045 42.35045
#> 724 39.39860 0.000000 39.39860 39.39860
#> 725 36.09876 6.222617 23.90265 48.29486
#> 726 40.94066 5.173903 30.79999 51.08132
#> 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 3.896282 39.06681 54.33996
#> 732 53.94830 9.853809 34.63519 73.26142
#> 733 39.60913 5.926982 27.99246 51.22580
#> 734 41.08206 0.000000 41.08206 41.08206
#> 735 49.65683 3.946977 41.92090 57.39277
#> 736 69.37409 0.000000 69.37409 69.37409
#> 737 34.12096 5.773555 22.80500 45.43692
#> 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 3.929751 39.83865 55.24299
#> 744 35.50735 0.000000 35.50735 35.50735
#> 745 32.08153 0.000000 32.08153 32.08153
#> 746 37.16398 4.810780 27.73503 46.59294
#> 747 42.75619 3.882535 35.14656 50.36582
#> 748 47.39510 9.673661 28.43508 66.35513
#> 749 44.69256 0.000000 44.69256 44.69256
#> 750 41.45664 4.862447 31.92642 50.98687
#> 751 42.18689 0.000000 42.18689 42.18689
#> 752 51.68534 10.053956 31.97995 71.39073
#> 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 9.706431 31.65060 69.69911
#> 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 4.863942 32.04824 51.11454
#> 763 48.87398 3.943725 41.14442 56.60353
#> 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 5.950337 16.94564 40.27054
#> 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 5.972956 30.02141 53.43497
#> 778 49.12862 0.000000 49.12862 49.12862
#> 779 47.31234 0.000000 47.31234 47.31234
#> 780 57.08286 10.061664 37.36236 76.80336
#> 781 19.28388 0.000000 19.28388 19.28388
#> 782 30.00682 0.000000 30.00682 30.00682
#> 783 39.69711 3.927072 32.00019 47.39403
#> 784 49.21768 0.000000 49.21768 49.21768
#> 785 31.42637 6.265464 19.14629 43.70645
#> 786 36.73485 5.162849 26.61585 46.85384
#> 787 42.72556 3.968060 34.94831 50.50282
#> 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 4.021447 39.04034 54.80412
#> 792 69.26254 0.000000 69.26254 69.26254
#> 793 40.35635 6.582729 27.45444 53.25826
#> 794 45.16757 5.233652 34.90980 55.42534
#> 795 50.02199 4.006159 42.17006 57.87392
#> 796 56.03985 10.222214 36.00468 76.07502
#> 797 35.70341 0.000000 35.70341 35.70341
#> 798 41.64454 0.000000 41.64454 41.64454
#> 799 43.29513 3.859917 35.72983 50.86043
#> 800 54.25081 0.000000 54.25081 54.25081The 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 rowsNote 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*epsmchWe 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: mmrmLast 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*epsmchTo 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*epsmchAcknowledgments
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).