Introduction to tern.mmrm

Statistical model

Under the linear mixed model (LMM) framework, an outcome vector ${\bf Y}$ is modeled as $$\bf Y = X \boldsymbol \beta + Z b + {\boldsymbol \varepsilon}$$ where $\bf X$ is the matrix for fixed effects, $\bf Z$ is the matrix for random effects, $\mathbf b \sim N(0, \mathbf D(\boldsymbol \theta))$, and ${\boldsymbol \varepsilon} \sim N(0, \mathbf R(\boldsymbol \theta))$. Letting $\mathbf V = \mathbf Z \mathbf D \mathbf Z^T + \mathbf R$, the marginal and conditional models are then given by $\mathbf Y \sim N(\bf X \boldsymbol \beta, V)$ and $\mathbf Y | \mathbf b \sim N(\bf X \boldsymbol \beta + Z b, R)$, respectively.

An MMRM is a special case of a LMM such that $\bf Y$ is a collection of measurements made on a set of individuals over time, i.e. $\mathbf Y = ({\bf Y}_1^T, {\bf Y}_2^T, ...)^T$ where $\mathbf Y_i = \mathbf X_i \boldsymbol \beta + \mathbf Z_i \mathbf b_i + {\boldsymbol \varepsilon}_i$. In our context of clinical trials, individuals are patients identified by their unique subject id, measurements are of treatment response, and fixed effects include treatment arm, categorical visit, treatment by visit interaction, and potentially other covariates (e.g. age, gender).

Estimation

The parameters $\boldsymbol \beta$, $\bf b$, and $\boldsymbol \theta$ can be estimated by maximizing the penalized and restricted maximum likelihoods. The average treatment response at each visit is often of particular interest. However, simply comparing predicted marginal responses $E(Y_{ij})$ averaged across treatment x visit groups will not account for potential imbalance in covariates. (Imbalance may occur even in randomized trials due to patients dying or dropping out over time.) For a more fair comparison, least-squares (LS) mean:

1. establishes a reference grid where each cell represents a unique combination of the factor (covariate) levels,
2. calculates the predicted marginal response for each cell, and then
3. takes a weighted average of the predicted marginal responses.

The following simple example illustrates the concept of LS means. Suppose we have a longitudinal clinical trial where three factors are considered: treatment (A and B), visit (1, 2, …), and gender (M and F). The marginal predicted response (sample size) for each reference cell is as follows.

The average predicted responses for arms A and B at Visit 1 are $(100 \times 20 + 50 \times 30) / 50 = 70$ and $(90 \times 35 + 40 \times 15) / 50 = 75$, respectively. The overall mean is higher in arm B than in arm A even though the mean is lower within each gender category. This seeming contradiction is caused by the imbalance in the data. LS mean calculates the weighted average across cells of the same treatment and visit. In this example, this is equivalent to taking the weighted average over each column. One may assign equal weights to each cell, i.e. $0.5 \times 100 + 0.5 \times 50 = 75$ and $0.5 \times 90 + 0.5 \times 40 = 65$. Alternatively, one may assign weights proportional to the observed frequencies of the factor combinations, i.e. $0.55 \times 100 + 0.45 \times 50 = 77.5$ and $0.55 \times 90 + 0.45 \times 40 = 67.5$. In both cases, the LS mean of response is lower in arm B than in arm A.

LS means are calculated in tern.mmrm via the R package emmeans. Users have the option to weigh marginal predicted responses with either equal weights or proportional weights. Note that for proportional weights, the weights are calculated at each visit by taking into account the observed frequencies of factor combinations at that time. Therefore, even though covariate imbalance may vary over time, LS mean provides an adjusted analysis of treatment response at all visits.

Inference

Performing inference on estimated parameters (e.g. calculating p-values) is less straightforward for MMRM. This is because the exact null distributions for parameter effects are unknown. SAS addresses this issue by utilizing Satterthwaite’s method to approximate the adjusted degrees of freedom for $F$ and $t$ tests. lme4 and lmerTest have also implemented Satterthwaite’s method. Unfortunately we found that these are not robust in their convergence behavior. Compared to lme4, the R package nlme can consider more flexible covariance structures. However, we have chosen not to use this package because it does not provide exact Satterthwaite adjusted degrees of freedom and the available approximate degrees of freedom can differ substantially. Therefore we built the new package mmrm. With mmrm, tern.mmrm is able to replicate outputs from SAS.

Covariance structure

Users of tern.mmrm have currently the following options for the covariance structure $\mathbf V_i$:

• Unstructured: $$V_{ij} = \theta_{ij}$$
• Homogeneous AR(1): $$(\mathbf V_i)_{jk} = \sigma^2 \rho^{|j-k|}$$
• Heterogeneous AR(1): $$(\mathbf V_i)_{jk} = \sigma_j \sigma_k \rho^{|j-k|}$$
• Heterogeneous Toeplitz: $$(\mathbf V_i)_{jk}=\sigma_j \sigma_k \theta_{|j-k|}$$
• Heterogeneous Ante-Dependence: $$(\mathbf V_i)_{jk} = \sigma_j \sigma_k \prod_{i=j}^{k}\rho_i$$

Setup

Our example dataset consists of several variables: subject ID (USUBJID), visit number (AVISIT), treatment (ARMCD = TRT or PBO), 3-category race, sex, FEV1 at baseline (%), and FEV1 at study visits (%). FEV1 (forced expired volume in one second) is a measure of how quickly the lungs can be emptied. Low levels of FEV1 may indicate chronic obstructive pulmonary disease (COPD). The scientific question at hand is whether treatment leads to an increase in FEV1 over time after adjusting for baseline covariates.

library(tern.mmrm)
#> Loading required package: tern
#> Loading required package: rtables
#> Loading required package: formatters
#> 
#> Attaching package: 'formatters'
#> The following object is masked from 'package:base':
#> 
#>     %||%
#> Loading required package: magrittr
#> 
#> Attaching package: 'rtables'
#> The following object is masked from 'package:utils':
#> 
#>     str
#> Registered S3 method overwritten by 'tern':
#>   method   from 
#>   tidy.glm broom
data(mmrm_test_data)
head(mmrm_test_data)
#> # A tibble: 6 × 7
#>   USUBJID AVISIT ARMCD RACE                      SEX    FEV1_BL  FEV1
#>   <fct>   <fct>  <fct> <fct>                     <fct>    <dbl> <dbl>
#> 1 PT1     VIS1   TRT   Black or African American Female    25.3  NA  
#> 2 PT1     VIS2   TRT   Black or African American Female    25.3  40.0
#> 3 PT1     VIS3   TRT   Black or African American Female    25.3  NA  
#> 4 PT1     VIS4   TRT   Black or African American Female    25.3  20.5
#> 5 PT2     VIS1   PBO   Asian                     Male      45.0  NA  
#> 6 PT2     VIS2   PBO   Asian                     Male      45.0  31.5

Model fitting via tern.mmrm

Fitting MMRM is easy in tern.mmrm. By default, the model fitting function fit_mmrm assumes unstructured correlation and proportional weights when calculating LS means.

mmrm_results <- fit_mmrm(
  vars = list(
    response = "FEV1",
    covariates = c("RACE", "SEX", "FEV1_BL"),
    id = "USUBJID",
    arm = "ARMCD",
    visit = "AVISIT"
  ),
  data = mmrm_test_data,
  cor_struct = "unstructured",
  weights_emmeans = "proportional"
)

The resulting object consists among other things the estimated random and fixed effects (fit), the estimated LS means for a particular treatment at a particular visit (lsmeans), and the difference in LS means at a particular visit (lsmeans$contrasts). Based on the output, there is evidence that supports treatment leading to an increase in FEV1 over placebo at all study visits.

mmrm_results$lsmeans$contrasts
#>   ARMCD AVISIT estimate        se       df lower_cl upper_cl   t_stat
#> 1   TRT   VIS1 3.983290 1.0454036 142.3210 1.916765 6.049815 3.810289
#> 2   TRT   VIS2 3.930758 0.8135131 142.2576 2.322622 5.538895 4.831831
#> 3   TRT   VIS3 2.983718 0.6656674 129.6093 1.666737 4.300699 4.482295
#> 4   TRT   VIS4 4.404001 1.6604869 132.8789 1.119595 7.688407 2.652235
#>        p_value relative_reduc
#> 1 2.058960e-04    -0.12101832
#> 2 3.460313e-06    -0.10424997
#> 3 1.606280e-05    -0.06893864
#> 4 8.969722e-03    -0.09154581

Other correlation structures and weighting schemes can be considered.

Model fitting manually in R

One can reproduce these results manually in R. For example, to fit the unstructured model:

library(mmrm)
#> 
#> Attaching package: 'mmrm'
#> The following object is masked from 'package:tern.mmrm':
#> 
#>     fit_mmrm
library(emmeans)
#> Welcome to emmeans.
#> Caution: You lose important information if you filter this package's results.
#> See '? untidy'

fit <- mmrm(
  formula = FEV1 ~ RACE + SEX + FEV1_BL + ARMCD * AVISIT + us(AVISIT | USUBJID),
  data = mmrm_test_data
)
emmeans::emmeans(fit, pairwise ~ ARMCD | AVISIT, weights = "proportional")
#> $emmeans
#> AVISIT = VIS1:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     32.9 0.732 141     31.5     34.4
#>  TRT     36.9 0.743 141     35.4     38.4
#> 
#> AVISIT = VIS2:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     37.7 0.576 142     36.6     38.8
#>  TRT     41.6 0.570 140     40.5     42.8
#> 
#> AVISIT = VIS3:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     43.3 0.441 127     42.4     44.2
#>  TRT     46.3 0.494 130     45.3     47.2
#> 
#> AVISIT = VIS4:
#>  ARMCD emmean    SE  df lower.CL upper.CL
#>  PBO     48.1 1.170 133     45.8     50.4
#>  TRT     52.5 1.170 133     50.2     54.8
#> 
#> Results are averaged over the levels of: RACE, SEX 
#> Confidence level used: 0.95 
#> 
#> $contrasts
#> AVISIT = VIS1:
#>  contrast  estimate    SE  df t.ratio p.value
#>  PBO - TRT    -3.98 1.050 142  -3.810  0.0002
#> 
#> AVISIT = VIS2:
#>  contrast  estimate    SE  df t.ratio p.value
#>  PBO - TRT    -3.93 0.814 142  -4.832  <.0001
#> 
#> AVISIT = VIS3:
#>  contrast  estimate    SE  df t.ratio p.value
#>  PBO - TRT    -2.98 0.666 130  -4.482  <.0001
#> 
#> AVISIT = VIS4:
#>  contrast  estimate    SE  df t.ratio p.value
#>  PBO - TRT    -4.40 1.660 133  -2.652  0.0090
#> 
#> Results are averaged over the levels of: RACE, SEX

Compared to manual implementation, tern.mmrm users can avoid specifying the formula manually. Note also that tern.mmrm has a default optimizer which is in fact the same as the one shown here. If the default fails to converge, it automatically attempts to use multiple other optimizers, delivering results should there be a consensus.

Model fitting via SAS

proc import datafile="location_of_data.csv"
     out=dat
     dbms=csv
     replace;
     getnames=yes;
run;

proc mixed data= dat;
       class ARMCD(ref="PBO") AVISIT(ref="VIS1") USUBJID SEX(ref="Male") RACE(ref="Asian");
       Model FEV1 = ARMCD  RACE  SEX  FEV1_BL  AVISIT  ARMCD*AVISIT    / solution ddfm=satterthwaite;
       repeated AVISIT / type=UN subject=USUBJID;
       lsmeans ARMCD*AVISIT/pdiff obsmargins cl;
run;

Model fitting

A key part in the analysis of data is model selection, where often the desire is to select a parsimonious model that adequately fits the data. We provide four of the same information criteria as found in SAS PROC MIXED.

mmrm_results$diagnostics
#> $`REML criterion`
#> [1] 3361.379
#> 
#> $AIC
#> [1] 3381.379
#> 
#> $AICc
#> [1] 3381.807
#> 
#> $BIC
#> [1] 3414.211

Each criterion is specifically calculated as follows:

1. REML criterion: $-2 l$ where $l$ is the REML log-likelihood. This criterion has no penalty for complexity, so generally decreases as model becomes more complex.
2. AIC: $-2l + 2d$ where $d$ is the number of covariance parameters.
3. AICc: $-2l + 2dn^*/(n^*-d-1)$ where $n^*$ is the number of observations minus the number of fixed effects, or $d + 2$ if it is smaller than that. AICc is a finite-sample corrected version of AIC.
4. BIC: $-2l + d \log n$ where $n$ is the number of subjects as opposed to the number of observations.

An objective approach to model selection is to minimize the AIC, AICc, or BIC. For more details on these information criteria (including references), see SAS Documentation.

Residual analysis

Another key part in the analysis of data is verifying model assumptions. g_mmrm_diagnostic(fit, type = ...) provides several handy residual plots, each of which is diagnostic for a different aspect of MMRMs.

• Marginal fitted values ($\mathbf X_i \widehat {\boldsymbol \beta}$) vs. Marginal residuals ($\mathbf Y_i - \mathbf X_i \widehat{\boldsymbol \beta}$)
  • Argument: type = "fit-residual"
  • Diagnostic for: homoskedasticity of marginal residuals
• QQ plot of standardized marginal residuals ($\widehat{\mathbf V}_i^{-1/2} (\mathbf Y_i - \mathbf X_i \widehat{\boldsymbol \beta})$)
  • Argument: type = q-q-residual
  • Diagnostic for: normality of marginal residuals