# Marginal Effects for Mixed Effects Models

    library(knitr)
    library(data.table)
    #> data.table 1.18.2.1 using 12 threads (see ?getDTthreads).  Latest news: r-datatable.com
    library(brms)
    #> Loading required package: Rcpp
    #> Loading 'brms' package (version 2.23.0). Useful instructions
    #> can be found by typing help('brms'). A more detailed introduction
    #> to the package is available through vignette('brms_overview').
    #>
    #> Attaching package: 'brms'
    #>
    #> The following object is masked from 'package:stats':
    #>
    #>     ar
    library(brmsmargins)

This vignette provides a brief overview of how to calculate marginal effects for Bayesian regression models involving only mixed effects (i.e., fixed and random) and fit using the `brms` package.

## Integrating out Random Effects

A random intercept logistic regression model where a binary (0/1) outcome, \(Y\) is observed at the \(i^{th}\) assessment for the \(j^{th}\) person and there are \(p\) variables included in the regression model can be written as:

\[ \hat{\pi}_{ij} = g \left(P \left( Y_{ij} = 1 \Big| X_{ij} = x_{ij}, u_j \right) \right) = \beta_0 + \sum_{k = 1}^p x_{ij,k} \beta_k + u_j \]

where \(g(\cdot)\) indicates the link function, here the logit

\[ \mu = g(\pi) = ln\left(\frac{\pi}{1 - \pi}\right) \]

and \(g^{-1}(\cdot)\) is the inverse link function:

\[ \pi = g^{-1}(\mu) = \frac{1}{1 + exp(-\mu)} \]

A conditional predicted probability, conditional on the random effect can be calculated as:

\[ \hat{\pi}_{ij}(u_j = 0) = P\left(Y_{ij} = 1 \Big| X_{ij} = x_{ij}, u_j = 0 \right) = g^{-1} \left( \beta_0 + \sum_{k = 1}^p x_{ij,k} \beta_k + 0 \right) \]

However, to correctly calculate a prediction that is marginal to the random effects, the random effects must be integrated out. Not set at a specific value or set at their mean (0).

\[ \hat{\pi}_{ij} = P\left(Y_{ij} = 1 \Big| X_{ij} = x_{ij} \right) = \int_{-\infty}^{\infty} g^{-1} \left( \beta_0 + \sum_{k = 1}^p x_{ij,k} \beta_k + u \right)f(u)du \]

Integrating out the random effects analytically can quickly become complex. For example, it rapidly becomes more complex when there are multiple random effects, such as if there is more than one grouping or clustering variable. It also can become more complex when different distributions are used / assumed.

Monte Carlo integration is a convenient, numerical approach that uses random samples to approximate the integral. Continuing the simple example of a logistic regression model where the only random effect is a random intercept, \(u_j\) and where we assume that \(u_j \sim \mathcal{N}(0, \sigma^{2}_u)\), we could draw \(Q\) random samples, say 100, from \(\mathcal{N}(0, \sigma^{2}_u)\), call these \(RE_a\), then Monte Carlo integration would be:

\[ \hat{\pi}_{ij} = P\left(Y_{ij} = 1 \Big| X_{ij} = x_{ij} \right) = \frac{\displaystyle \sum_{a = 1}^{Q} g^{-1} \left( \beta_0 + \sum_{k = 1}^p x_{ij,k} \beta_k + RE_a \right)}{Q} \]

This approach works for most generalized linear mixed models, although the outcome would not be a probability, necessarily, but whatever the result of the inverse link function is.

In a Bayesian framework, this approach would be repeated for each posterior draw as both the regression coefficients and \(RE_a\) differs. Because this is repeated across each posterior draw, a very large number of random draws, \(Q\), for the Monte Carlo integration is probably not needed. Although a modest number, say \(Q = 100\), would have a relatively large amount of simulation error, it is random error and when repeated across typically thousands of posterior draws, the impact is likely diminished.

Once we have these marginal predictions, we can calculate marginal effects using numerical derivatives as:

\[ \frac{P\left(Y_{ij} = 1 \Big| X_{ij} = x_{ij} + h \right) - P\left(Y_{ij} = 1 \Big| X_{ij} = x_{ij} \right)}{h} \]

which for a continuous variable provides an approximation of the derivative, often quite good as long as \(h\) is sufficiently small.

## Using `brmsmargins()`

A simpler introduction and very brief overview and motivation is available in the vignette for fixed effects only. When there are fixed and random effects, calculating average marginal effects (AMEs) is more complicated. Generally, predictions are conditional on the random effects. To deal with this, we need to integrate out the random effects for every prediction. Please note that this is quite computationally demanding, at least as currently implemented. For every predicted value and each posterior draw, random samples from the model estimated random effects distribution are drawn, added, back transformed, and averaged.

Thus, if you wanted AMEs across a dataset of 1,000 people, with 2,000 posterior draws, and you wanted to use 100 points for the numerical integration, a total of 200 million (1,000 x 2,000 x 100) values are calculated. The Monte Carlo integration is implemented in C++ code to try to help speed up the process, but it is not “quick” and also may be memory intensive.

Because of the complexity involved, only limited types of mixed effects models are supported.

## Mixed Effects Logistic Regression

We will simulate some multilevel binary data for our mixed effects logistic regression model with individual differences in both the intercept and slope.

    d <- withr::with_seed(
      seed = 12345, code = {
        nGroups <- 100
        nObs <- 20
        theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
        theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
        theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
        theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
        theta.location[, 1] <- theta.location[, 1] - 2.5
        theta.location[, 2] <- theta.location[, 2] + 1
        d <- data.table(
          x = rep(rep(0:1, each = nObs / 2), times = nGroups))
        d[, ID := rep(seq_len(nGroups), each = nObs)]

        for (i in seq_len(nGroups)) {
          d[ID == i, y := rbinom(
            n = nObs,
            size = 1,
            prob = plogis(theta.location[i, 1] + theta.location[i, 2] * x))
            ]
        }
        copy(d)
      })

    mlogit <- brms::brm(
      y ~ 1 + x + (1 + x | ID), family = "bernoulli",
      data = d, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      prior = prior(normal(-2.5, 1), class = "Intercept") +
        prior(normal(1, .5), class = "b") +
        prior(student_t(3, 1.25, 1), class = "sd", coef = "Intercept", group = "ID") +
        prior(student_t(3, .5, .5), class = "sd", coef = "x", group = "ID"),
      silent = 2, refresh = 0,
      chains = 4L, cores = 4L, backend = "cmdstanr")

    summary(mlogit)
    #> Warning: There were 24 divergent transitions after warmup. Increasing
    #> adapt_delta above 0.8 may help. See
    #> http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
    #>  Family: bernoulli
    #>   Links: mu = logit
    #> Formula: y ~ 1 + x + (1 + x | ID)
    #>    Data: d (Number of observations: 2000)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~ID (Number of levels: 100)
    #>                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sd(Intercept)        1.19      0.18     0.87     1.56 1.00     1130     1850
    #> sd(x)                0.54      0.25     0.06     1.00 1.02      279      506
    #> cor(Intercept,x)    -0.20      0.39    -0.78     0.77 1.00     1625     1284
    #>
    #> Regression Coefficients:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept    -2.49      0.19    -2.88    -2.14 1.00     1270     2120
    #> x             0.96      0.17     0.63     1.30 1.00     2403     2217
    #>
    #> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
    #> and Tail_ESS are effective sample size measures, and Rhat is the potential
    #> scale reduction factor on split chains (at convergence, Rhat = 1).

### AMEs

Now we can use `brmsmargins()`. By default, it will only use the fixed effects. To integrate out random effects, we specify `effects = "integrateoutRE"`. The number of values used for numerical integration are set via the argument, `k`, here `k = 100L`, the default. More details are in: `?brmsmargins:::.predict`

    h <- .001
    ame1 <- brmsmargins(
      mlogit,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame1$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.111 | 0.11 | 0.06 | 0.163 | NA | NA | 0.99 | HDI | NA | NA | AME x

We can follow a similar process getting discrete predictions at x held at 0 or 1. In this instance, the summary of predictions is more interesting as well, since they are at meaningfully different values of `x`. They also agree quite closely with the average probability at different `x` values calculated in the data.

    ame2 <- brmsmargins(
      mlogit,
      at = data.frame(x = c(0, 1)),
      contrasts = cbind("AME x" = c(-1, 1)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

Here is a summary of the predictions.

    knitr::kable(ame2$Summary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.118 | 0.117 | 0.075 | 0.175 | NA | NA | 0.99 | HDI | NA | NA
0.228 | 0.228 | 0.161 | 0.305 | NA | NA | 0.99 | HDI | NA | NA

    knitr::kable(ame2$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.11 | 0.109 | 0.06 | 0.166 | NA | NA | 0.99 | HDI | NA | NA | AME x

    knitr::kable(d[, .(M = mean(y)), by = .(ID, x)][, .(M = mean(M)), by = x])

x | M
--- | ---
0 | 0.117
1 | 0.228

Note that when integrating out random effects, the random seed is quite important. If the `seed` argument is not specified, `brmsmargins()` will randomly select one. This would not matter when generating predictions only from fixed effects, but when using random samples to integrate out random effects, if different random seeds are used for different predictions, you would expect some (small) differences even for the same input data for prediction. This may not be an issue for predictions on their own. However, when numerically approximating a derivative by a very small difference in predictions, such as with `h = .001` tiny differences are magnified. To see the impact, consider this example where we explicitly set multiple random seeds, one for each row of the data used for predictions. In both cases, we use exactly `x = 0`, so the difference is due to Monte Carlo variation only, but with `k = 10L` the small error, when divided by `h = .001` becomes very large, impossibly so.

    h <- .001
    ame.error <- brmsmargins(
      mlogit,
      add = data.frame(x = c(0, 0)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 10L, seed = c(1234, 54321))

    knitr::kable(ame.error$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
-0.717 | -0.371 | -167.14 | 174.157 | NA | NA | 0.99 | HDI | NA | NA | AME x

This disappears when we use the same seed for each row of the data used for predictions. Here we get all zeros for the difference, as we would expect. Note that you do not need to specify a seed for each row of the data. You can specify one seed (or rely on `brmsmargins()` default), which will then be used for all rows of the data.

    h <- .001
    ame.noerror <- brmsmargins(
      mlogit,
      add = data.frame(x = c(0, 0)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 10L, seed = c(1234, 1234))

    knitr::kable(ame.noerror$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0 | 0 | 0 | 0 | NA | NA | 0.99 | HDI | NA | NA | AME x

### Marginal Coefficients

The fixed effects coefficients are conditional on the random effects. To aid interpretation, we also can calculate marginal coefficients or population averaged coefficients. The function to do this is `marginalcoef()` which uses the method described by Hedeker and colleagues (2018). Here is an example and comparison to results using a single level logistic regression that ignores the clustering in the data.

    ## calculate marginal coefficients
    mc.logit <- marginalcoef(mlogit, CI = 0.95, seed = 1234)

    ## calculate single level logistic regression
    glm.logit <- glm(y ~ 1 + x, family = "binomial", data = d)
    glm.logit <- as.data.table(cbind(Est = coef(glm.logit), confint(glm.logit)))
    #> Waiting for profiling to be done...

Now we can view and compare the results.

    knitr::kable(cbind(
      mc.logit$Summary[, .(
        MargCoef = sprintf("%0.3f", round(M, 3)),
        MargCoefCI = sprintf("[%0.3f, %0.3f]", round(LL, 3), round(UL, 3)))],
      glm.logit[, .(
        GLMCoef = sprintf("%0.3f", round(Est, 3)),
        GLMCI = sprintf("[%0.3f, %0.3f]", round(`2.5 %`, 3), round(`97.5 %`, 3)))]))

MargCoef | MargCoefCI | GLMCoef | GLMCI
--- | --- | --- | ---
-2.022 | [-2.389, -1.666] | -2.021 | [-2.219, -1.833]
0.798 | [0.536, 1.075] | 0.802 | [0.561, 1.047]

## Mixed Effects Poisson Regression

We will simulate some multilevel poisson data for our mixed effects poisson regression model with individual differences in both the intercept and slope.

    dpoisson <- withr::with_seed(
      seed = 12345, code = {
        nGroups <- 100
        nObs <- 20
        theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
        theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
        theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
        theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
        theta.location[, 1] <- theta.location[, 1] - 2.5
        theta.location[, 2] <- theta.location[, 2] + 1
        d <- data.table(
          x = rep(rep(0:1, each = nObs / 2), times = nGroups))
        d[, ID := rep(seq_len(nGroups), each = nObs)]

        for (i in seq_len(nGroups)) {
          d[ID == i, y := rpois(
            n = nObs,
            lambda = exp(theta.location[i, 1] + theta.location[i, 2] * x))
            ]
        }
        copy(d)
      })

    mpois <- brms::brm(
      y ~ 1 + x + (1 + x | ID), family = "poisson",
      data = dpoisson, seed = 1234,
      prior = prior(normal(-2.5, 1), class = "Intercept") +
        prior(normal(1, .5), class = "b") +
        prior(student_t(3, 1.25, 1), class = "sd", coef = "Intercept", group = "ID") +
        prior(student_t(3, .5, .5), class = "sd", coef = "x", group = "ID"),
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      chains = 4L, cores = 4L, backend = "cmdstanr",
      silent = 2, refresh = 0, adapt_delta = 0.99)

    summary(mpois)
    #>  Family: poisson
    #>   Links: mu = log
    #> Formula: y ~ 1 + x + (1 + x | ID)
    #>    Data: dpoisson (Number of observations: 2000)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~ID (Number of levels: 100)
    #>                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sd(Intercept)        1.19      0.15     0.92     1.51 1.00     1370     2517
    #> sd(x)                0.38      0.19     0.03     0.75 1.01      355      891
    #> cor(Intercept,x)    -0.07      0.40    -0.73     0.83 1.00     1614     1003
    #>
    #> Regression Coefficients:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept    -2.47      0.18    -2.84    -2.13 1.00     1860     2334
    #> x             0.97      0.15     0.68     1.28 1.00     4302     3080
    #>
    #> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
    #> and Tail_ESS are effective sample size measures, and Rhat is the potential
    #> scale reduction factor on split chains (at convergence, Rhat = 1).

### AMEs

We use `brmsmargins()` in the same way as for the mixed effects logistic regression. Here is an example with a numeric derivative treating `x` as continuous.

    h <- .001
    ame1.pois <- brmsmargins(
      mpois,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame1.pois$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.333 | 0.312 | 0.141 | 0.729 | NA | NA | 0.99 | HDI | NA | NA | AME x

Here is an example treating `x` as discrete.

    ame2.pois <- brmsmargins(
      mpois,
      at = data.frame(x = c(0, 1)),
      contrasts = cbind("AME x" = c(-1, 1)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame2.pois$ContrastSummary)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.2942814 | 0.2782104 | 0.1138293 | 0.6221603 | NA | NA | 0.99 | HDI | NA | NA | AME x

### Marginal Coefficients

Just as for mixed effects logistic regression, we can calculate marginal or population averaged coefficients for mixed effects poisson regression using the same process as described by Hedeker and colleagues (2018). Here is an example and comparison to results using a single level poisson regression that ignores the clustering in the data.

    ## calculate marginal coefficients
    mc.pois <- marginalcoef(mpois, CI = 0.95, seed = 1234)

    ## calculate single level poisson regression
    glm.pois <- glm(y ~ 1 + x, family = "poisson", data = dpoisson)
    glm.pois <- as.data.table(cbind(Est = coef(glm.pois), confint(glm.pois)))
    #> Waiting for profiling to be done...

Now we can view and compare the results.

    knitr::kable(cbind(
      mc.pois$Summary[, .(
        MargCoef = sprintf("%0.3f", round(M, 3)),
        MargCoefCI = sprintf("[%0.3f, %0.3f]", round(LL, 3), round(UL, 3)))],
      glm.pois[, .(
        GLMCoef = sprintf("%0.3f", round(Est, 3)),
        GLMCI = sprintf("[%0.3f, %0.3f]", round(`2.5 %`, 3), round(`97.5 %`, 3)))]))

MargCoef | MargCoefCI | GLMCoef | GLMCI
--- | --- | --- | ---
-1.778 | [-2.261, -1.265] | -1.858 | [-2.019, -1.705]
0.989 | [0.705, 1.282] | 0.983 | [0.802, 1.170]

## Mixed Effects Negative Binomial Regression

Negative binomial models work the same way as for poisson models. We use the same dataset, just for demonstration.

    dnb <- withr::with_seed(
      seed = 12345, code = {
        nGroups <- 100
        nObs <- 20
        theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
        theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
        theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
        theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
        theta.location[, 1] <- theta.location[, 1] - 2.5
        theta.location[, 2] <- theta.location[, 2] + 1
        d <- data.table(
          x = rep(rep(0:1, each = nObs / 2), times = nGroups))
        d[, ID := rep(seq_len(nGroups), each = nObs)]

        for (i in seq_len(nGroups)) {
          d[ID == i, y := rnbinom(
            n = nObs,
            size = 5,
            mu = exp(theta.location[i, 1] + theta.location[i, 2] * x))
            ]
        }
        copy(d)
      })

    mnb <- brms::brm(
      y ~ 1 + x + (1 + x | ID), family = "negbinomial",
      data = dnb, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      chains = 4L, cores = 4L, backend = "cmdstanr",
      silent = 2, refresh = 0, adapt_delta = 0.99)

    summary(mnb)
    #>  Family: negbinomial
    #>   Links: mu = log
    #> Formula: y ~ 1 + x + (1 + x | ID)
    #>    Data: dnb (Number of observations: 2000)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~ID (Number of levels: 100)
    #>                  Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sd(Intercept)        1.19      0.17     0.89     1.56 1.01     1038     2109
    #> sd(x)                0.52      0.21     0.06     0.92 1.02      350      436
    #> cor(Intercept,x)    -0.30      0.31    -0.78     0.45 1.01     1496     1350
    #>
    #> Regression Coefficients:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept    -2.51      0.19    -2.92    -2.15 1.00     1619     1933
    #> x             1.01      0.17     0.69     1.38 1.00     2231     2341
    #>
    #> Further Distributional Parameters:
    #>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> shape  1969.21  48049.95     2.72  1417.87 1.00     1580      767
    #>
    #> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
    #> and Tail_ESS are effective sample size measures, and Rhat is the potential
    #> scale reduction factor on split chains (at convergence, Rhat = 1).

### AMEs

We use `brmsmargins()` in the same way as for the mixed effects poisson regression. Here is an example with a numeric derivative treating `x` as continuous.

    h <- .001
    ame1.nb <- brmsmargins(
      mnb,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame1.nb$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.297 | 0.28 | 0.125 | 0.65 | NA | NA | 0.99 | HDI | NA | NA | AME x

Here is an example treating `x` as discrete.

    ame2.nb <- brmsmargins(
      mnb,
      at = data.frame(x = c(0, 1)),
      contrasts = cbind("AME x" = c(-1, 1)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame2.nb$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.258 | 0.245 | 0.11 | 0.552 | NA | NA | 0.99 | HDI | NA | NA | AME x

### Marginal Coefficients

Negative binomial models cannot be fit by the `glm()` function in `R` so we just show the population averaged values from `brms`.

    ## calculate marginal coefficients
    mc.nb <- marginalcoef(mnb, CI = 0.95, seed = 1234)

View the results.

    knitr::kable(
      mc.nb$Summary[, .(
        MargCoef = sprintf("%0.3f", round(M, 3)),
        MargCoefCI = sprintf("[%0.3f, %0.3f]", round(LL, 3), round(UL, 3)))])

MargCoef | MargCoefCI
--- | ---
-1.805 | [-2.296, -1.264]
0.929 | [0.590, 1.301]

## Mixed Effects Binomial Regression

Binomial data works largely the same. Here we have an example dataset of a trial with participants randomized to treatment (1) or control (0) coded in `trt`, continuous age in years ranging from 8 to 16. The outcome is binomial, such as in cognitive tasks where there are multiple trials as the number of correct trials at each timepoint are recorded. In this example, the number of trials completed each time varies some across participants. We fit a binomial model in `brms` where the outcome is the number of correct trials and the total number of trials is stored in `total`.

    dbinom <- withr::with_seed(
      seed = 12345, code = {
      library(data.table)
      n <- 200L
      d <- CJ(ID = 1:n, time = 0:2)[, `:=`(
          trt = rep(rbinom(n, 1, .5), each = 3),
          age = rep(runif(n, 8, 16), each = 3),
          u   = rep(rnorm(n, 0, .6), each = 3)
        )][, total := sample(10:15, .N, TRUE)][
        , p := plogis(-1 + 0.8 * trt + 0.25 * (age - 12) + 0.3 * time + u)][
        , correct := rbinom(.N, total, p)][
        , .(ID, trt, age, time, total, correct)]
      copy(d)
    })

    mbinom <- brms::brm(
      correct | trials(total) ~ 1 + trt + age + (1 | ID), family = "binomial",
      data = dbinom, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      chains = 4L, cores = 4L, backend = "cmdstanr",
      silent = 2, refresh = 0, adapt_delta = 0.99)

    summary(mbinom)
    #>  Family: binomial
    #>   Links: mu = logit
    #> Formula: correct | trials(total) ~ 1 + trt + age + (1 | ID)
    #>    Data: dbinom (Number of observations: 600)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~ID (Number of levels: 200)
    #>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> sd(Intercept)     0.59      0.04     0.50     0.67 1.00     1721     2572
    #>
    #> Regression Coefficients:
    #>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept    -3.39      0.28    -3.93    -2.85 1.00     1798     2301
    #> trt           0.69      0.10     0.49     0.89 1.00     1828     2495
    #> age           0.24      0.02     0.19     0.28 1.00     1707     2320
    #>
    #> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
    #> and Tail_ESS are effective sample size measures, and Rhat is the potential
    #> scale reduction factor on split chains (at convergence, Rhat = 1).

### AMEs

We use `brmsmargins()` in the same way as for the other models. Here is an example with a numeric derivative treating `age` as continuous. What we can see is that one year of age is associated with getting 0.628 more trials correct. However, given that the exact number of trials varies, we may rather have the probability of getting a trial correct, rather than have units on the number of successes scale.

    h <- .001
    ame1.binom <- brmsmargins(
      mbinom,
      add = data.frame(age = c(0, h)),
      contrasts = cbind("AME age" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame1.binom$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.628 | 0.63 | 0.482 | 0.76 | NA | NA | 0.99 | HDI | NA | NA | AME age

To do this for continuous variables, we need to still use the `add` argument to `brmsmargins`, but we want to modify the data so that all of the `total`s are 1 as for 1 trial, it will be the probability. We can do this by passing the data to `newdata` (which normally just uses the model frame as its default). Now we can see that a one year increase in age is associated with an increase in the average marginal probability of getting a trial correct. While the direction will be the same, the scale is now the probability scale, rather than an average number of trials scale.

    h <- .001
    tmpd <- model.frame(mbinom)
    tmpd$total <- 1
    ame2.binom <- brmsmargins(
      mbinom,
      add = data.frame(age = c(0, h)),
      newdata = tmpd,
      contrasts = cbind("AME age" = c(-1 / h, 1 / h)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame2.binom$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.05 | 0.05 | 0.038 | 0.06 | NA | NA | 0.99 | HDI | NA | NA | AME age

Here is an example treating `trt` (treatment) as discrete. Note that when using `at` rather than modify the `newdata`, we can also simply add `total = 1` to the `at` call. Either would work in this case. Again we can see that the average marginal probability of getting a trial correct is higher in the treatment than in the control group.

    ame3.binom <- brmsmargins(
      mbinom,
      at = data.frame(trt = c(0, 1), total = 1),
      contrasts = cbind("AME trt" = c(-1, 1)),
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame3.binom$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.15 | 0.15 | 0.093 | 0.203 | NA | NA | 0.99 | HDI | NA | NA | AME trt

## Centered Categorical Predictors

In mixed effects models, it is common to center continuous predictors. Consider a longitudinal study where sleep was collected nightly for a week in multiple people. Hours of sleep duration will have variation that exists between people (different people have different typical or average sleep durations) and within people (the same person will sleep different amounts on different nights).

Entering observed hours of sleep duration as a predictor in the model will conflate associations between sleep duration and the outcome that exist at the between person and within person level.

Here is a small example for two people with observed sleep (“Sleep”), the between person component, the average sleep per person (“BSleep”), and the within person sleep, the person centered sleep duration (“WSleep”).

ID | Sleep | BSleep | WSleep
--- | --- | --- | ---
1 | 6 | 7 | -1
1 | 7 | 7 | 0
1 | 8 | 7 | +1
2 | 4 | 5 | -1
2 | 5 | 5 | 0
2 | 6 | 5 | +1

These values are obtained by averaging the observed sleep values by ID. Then by taking the difference between the observed sleep duration values and the between person mean sleep values, thus:

\[ WSleep = Sleep - BSleep \]

Both `BSleep` and `WSleep` could be included as predictor variables in a model, or if the interest is solely in the within person association, `WSleep` only included.

This type of centering is relatively common in mixed effects or multilevel models, at least for continuous predictors. The presence, or absence, of such centering for continuous variables has relatively little bearing on calculating the average marginal effects (AMEs) as generally both variables remain continuous and a derivative makes sense.

In contrast to continuous predictors where it is common, it is relatively uncommon to center categorical predictors. However, research (Yaremych, Preacher, & Hedeker, 2021) highlights how the same rationale that applies to continuous predictors, in regards to the benefits of centering, equally apply to categorical predictors. One area where particular attention has been paid to the importance of centering, whether continuous or categorical predictors, are in individual participant data meta analyses (IPDMA) with one stage analyses. In these IPDMA where there are treatment x covariate interactions to evaluate effect modifiers, centering the predictor / covariate is critical for avoiding ‘ecological bias’ (Hua et al., 2017).

Yaremych et al (2021) showed that an equivalent algebraic approach to creating centered continuous variables can be applied to categorical variables, after coding (e.g., using dummy coding). In the same study used for the continuous example, suppose that whether or not people had a good night’s sleep was recorded dummy coded as yes = 1 and no = 0. This table shows how it might be centered.

ID | GoodSleep | BSleep | WSleep
--- | --- | --- | ---
1 | 0 | 0.50 | -0.50
1 | 0 | 0.50 | -0.50
1 | 1 | 0.50 | +0.50
1 | 1 | 0.50 | +0.50
2 | 0 | 0.75 | -0.75
2 | 1 | 0.75 | +0.25
2 | 1 | 0.75 | +0.25
2 | 1 | 0.75 | +0.25
3 | 0 | 0 | 0
3 | 0 | 0 | 0
3 | 0 | 0 | 0
3 | 0 | 0 | 0

We can include these new variables as predictors in a model just as any other predictors. However, in contrast to the continuous case where for the AMEs we tend to calculate a derivative, the categorical case is more challenging. Typically, for categorical variables, AMEs are calculated as the AME moving from one category to another category. This can easily be accomplished in `brmsmargins` using the `at` argument. However, when a categorical variable is centered by ID, the values for one category are not constant by ID.

To address, this, `brmsmargins` implements an additional argument, `wat` for within level `at`. This argument is used to give the values to be used at each ID level. The `wat` argument is used in conjunction with the `at` argument. The `at` argument continues to be used for the general setup, with `wat` used for the specific values.

This is more easily shown than said. Here we simulate some multilevel data with a binary predictor, `x` and a binary outcome, `y`.

    d <- withr::with_seed(
      seed = 12345, code = {
        nGroups <- 100
        nObs <- 20
        theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
        theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
        theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
        theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
        theta.location[, 1] <- theta.location[, 1] - 2.5
        theta.location[, 2] <- theta.location[, 2] + 1

        d <- data.table(ID = seq_len(nGroups))
        d <- d[, .(x = rbinom(n = nObs, size = 1, prob = runif(1, 0, 1))), by = ID]

        for (i in seq_len(nGroups)) {
          d[ID == i, y := rbinom(
            n = nObs,
            size = 1,
            prob = plogis(theta.location[i, 1] + theta.location[i, 2] * x))
            ]
        }
        copy(d)
      })

Now we can create a between and within person version of `x`, which we will call `xb` and `xw`. We also fit the `brms` model, using `xw`. In this instance we are not interested in any between person effects, so that variable is omitted. Any variation not explained by it will go into the random intercept, anyways.

    ## within person centering binary predictor
    d[, xb := mean(x), by = ID]
    d[, xw := x - xb]

    mlogitcent <- brms::brm(
      y ~ 1 + xw + (1 + xw | ID), family = "bernoulli",
      data = d, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      silent = 2, refresh = 0,
      chains = 4L, cores = 4L, backend = "cmdstanr")

Now comes the new part. Based on how we coded our predictor, `x`, we will create the two values by ID. In this case, `x` was coded as 0 and 1. So we start with 0 as the low value and 1 as the high value. Then we need to take the difference between 0 and 1 with the mean by ID. We (arbitrarily) label these `a` and `b`. We could have chosen any label, `min` and `max` or `low` and `high` or whatever we wanted. It just needs to be distinct and something that `brmsmargins()` can match between the `at` and the `wat` arguments. The resulting data get turned into a single long `data.table`, called `xwid`. We use this to create a list, saved as `usewat`. The list must have two elements at a minimum:

1. A named element, named “ID” that contains a character string with the name of the ID variable.
2. An element named after the centered variable, with a `data.frame` or `data.table` containing the values to use for each ID and for each label we chose, here `a` and `b`.

The table shows a few examples of what the dataset looks like.

    xwid <- melt(
      d[, .(a = 0 - na.omit(xb)[1],
            b = 1 - na.omit(xb)[1]),
        by = ID], id.vars = "ID")

    usewat <- list(ID = "ID", xw = xwid)

    knitr::kable(xwid[ID %in% c(1, 2, 4, 100)][order(ID)], digits = 2)

ID | variable | value
--- | --- | ---
1 | a | 0.00
1 | b | 1.00
2 | a | -0.30
2 | b | 0.70
4 | a | -1.00
4 | b | 0.00
100 | a | -0.15
100 | b | 0.85

This new dataset shows what values to use for moving from one category of `x` to another category of `x` looks like, for each ID. Note that sometimes these are the same. In the case of ID 1, all `x` values were identical and thus all `xw` values will be 0. This is appropriate because for that ID, there was no variation on `x` and so we have no idea what the “true” change for ID 1 would be if moving from one category to another.

Now we can use `brmsmargins()` to calculate the AME. We must still specify the `at` argument. Here we use our arbitrary labels of `a` and `b` for the variable, `xw`. We also pass our list to the argument `wat`. Internally, `brmsmargins()` will substitute `a` for all the values for `a` by ID from our list, and similarly for `b`. The reason for this separation is in the `at` argument, one might have specific values of other variables as well, had there been other predictors in the model.

The resulting summary gives the average marginal predictions for `xw = a` and `xw = b` which in our case refers to when `xw` is at 0 on `x` for that ID and when it is at 1 for `x` for that ID.

    ame.cent <- brmsmargins(
      object = mlogitcent,
      at = expand.grid(xw = c("a", "b")),
      wat = usewat,
      contrasts = cbind("AME xw" = c(-1, 1)),
      effects = "integrateoutRE", k = 20L, seed = 1234)

    knitr::kable(ame.cent$Summary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.122 | 0.119 | 0.055 | 0.213 | NA | NA | 0.99 | HDI | NA | NA
0.231 | 0.229 | 0.132 | 0.364 | NA | NA | 0.99 | HDI | NA | NA

Just as in other examples, we can get a summary of the contrast of these two values, our AME.

    knitr::kable(ame.cent$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.109 | 0.108 | 0.044 | 0.183 | NA | NA | 0.99 | HDI | NA | NA | AME xw

## Interactions and Marginal Effects

As a final example, we will look at a model with an interaction. Specifically we have a binary outcome, `y`, measured repeatedly. A binary predictor, `x` measured repeatedly within units and group membership, `Group`, a binary variable that is constant within units so only varies between units.

To map these to a concrete potential research question, suppose that 100 people were randomized 1:1 to either a waitlist control group (Group = 0) or an emotion regulation intervention designed to help manage stress (Group = 1). After completing the intervention, participants completed 20 days of daily diaries. Sleep was recorded each day and categorized as a good night’ sleep (`y` = 1) or a poor night’s sleep (`y` = 0). Each day participants also reported whether they experienced no stressor (`x` = 0) or any stressor (`x` = 1).

The goal is to examine whether the intervention group moderates participants’ stress reactivity, defined as the association between experiencing a stressor or not on a given day and whether they have good or poor sleep that night. Based on clinical judgement and the intervention cost, it is estimated that any probability difference of plus or minus 2% is practically equivalent, so the region of practical equivalence (ROPE) is set at -0.02 to +0.02. A probability difference of 5% or more is considered the minimally important difference (MID), the smallest difference that is clinically meaningful. Thus the MID is set at less than or equal to -0.05 or greater than or equal to +0.05.

To show this example, we first simulate a dataset, using a similar process as in previous examples.

    d <- withr::with_seed(
      seed = 12345, code = {
        nGroups <- 100
        nObs <- 20
        theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
        theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
        theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
        theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
        theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
        theta.location[, 1] <- theta.location[, 1] - 2.5
        theta.location[, 2] <- theta.location[, 2] + rep(c(-1, 1), each = nGroups / 2)

        d <- data.table(ID = seq_len(nGroups), group = rep(0:1, each = nGroups / 2))
        d <- d[, .(x = rbinom(n = nObs, size = 1, prob = runif(1, 0, 1))), by = .(ID, group)]

        for (i in seq_len(nGroups)) {
          d[ID == i, y := rbinom(
            n = nObs,
            size = 1,
            prob = plogis(theta.location[i, 1] + theta.location[i, 2] * x))
            ]
        }
        copy(d)
      })

Following the earlier example, we center our binary predictor, `x` (whether participants experienced any stressor that day). This separates the effects of experiencing a stressor within a person from the fact that some people may experience stressors most days or rarely experience any stressors. We call the between person variable, the average proportion of days experiencing a stressor, `xb`. We call the within person variable, `xw`. As in the example on centering within person categorical variables, we also create a dataset with the within person hypothetical values for each person, `usewat`.

    ## within person centering binary predictor
    d[, xb := mean(x), by = ID]
    d[, xw := x - xb]

    xwid <- melt(
      d[, .(a = 0 - na.omit(xb)[1],
            b = 1 - na.omit(xb)[1]),
        by = ID], id.vars = "ID")

    usewat <- list(ID = "ID", xw = xwid)

Group is measured between people so we do not need to center it. We include a random slope by `xw` (within person experience or not of a stressor) and an interaction between `xw` and `Group` to evaluate whether the association between experiencing a stressor on a given day, at the within person level, is associated with a good or poor night sleep and whether that differs between people randomized to the waitlist control or the intervention. The model can be fit as in the following code.

    mlogitcentint <- brms::brm(
      y ~ 1 + xw * group + xb + (1 + xw | ID), family = "bernoulli",
      data = d, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      silent = 2, refresh = 0,
      chains = 4L, cores = 4L, backend = "cmdstanr")

We will generate average marginal predicted probabilities for each group and with within person stressor exposure held at none and any. The grid of possibilities is created and stored in `use.at` and shown in the following table.

    use.at <- expand.grid(group = 0:1, xw = c("a", "b"))
    knitr::kable(use.at, digits = 0)

group | xw
--- | ---
0 | a
1 | a
0 | b
1 | b

From this grid, there are three contrasts of interest.

1. What is the average marginal effect (AME) of within person stressor exposure on good or poor sleep that night if everyone were randomized to the waitlist control group?
2. What is the AME of within person stressor exposure on good or poor sleep that night if everyone were randomized to the intervention group?
3. What is the difference in the AME of within person stressor exposure between the intervention and waitlist control group?

A quick way to create this contrast matrix is to start with our contrast codes for the AME: -1 and +1. We use the kronecker product with a 2 x 2 identity matrix to expand this into the contrast codes needed for our 1st and 2nd questions. Then we take the difference in contrast weights between these two which are the weights for the 3rd question. The code to do this and the resulting contrast matrix with labels is shown in the following code and table.

    contr.mat <- cbind(xw = c(-1, 1)) %x% diag(2)
    contr.mat <- cbind(
      contr.mat,
      contr.mat[, 2] - contr.mat[, 1])

    colnames(contr.mat) <- c(
      paste0("AME xw: Grp ", 0:1),
      "delta AME xw")

    knitr::kable(contr.mat, digits = 0)

AME xw: Grp 0 | AME xw: Grp 1 | delta AME xw
--- | --- | ---
-1 | 0 | 1
0 | -1 | -1
1 | 0 | -1
0 | 1 | 1

Now we can use `brmsmargins()` with the model, our grid of values to generate average marginal predictions, and our contrast matrix to generate the AMEs and answer our three questions. We specify here that we want 95% credible intervals, and we specify the desired ROPE and MID regions as well as the seed so that we can reproduce the results exactly. Once run, we can see the average marginal predictions for our grid of values in `use.at` in the following table.

    ame.centint <- brmsmargins(
      object = mlogitcentint,
      at = use.at,
      wat = usewat,
      contrasts = contr.mat, CI = 0.95,
      ROPE = c(-.02, .02), MID = c(-.05, .05),
      effects = "integrateoutRE", k = 20L, seed = 1234)

    knitr::kable(ame.centint$Summary[, .(M, LL, UL, CI, CIType)], digits = 3)

M | LL | UL | CI | CIType
--- | --- | --- | --- | ---
0.116 | 0.054 | 0.187 | 0.95 | HDI
0.143 | 0.070 | 0.213 | 0.95 | HDI
0.045 | 0.017 | 0.079 | 0.95 | HDI
0.267 | 0.170 | 0.379 | 0.95 | HDI

Finally we can look at the contrast summary to answer our three questions of interest: the two simple AMEs of within person stressor exposure and the difference between intervention conditions in those AMEs. These are shown in the following table.

From the results, we can see that if everyone were randomized to the waitlist control group, we would expect a lower probability of a good night sleep on stressor compared to non stressor days, at the within person level. This effect is unlikely to be practically equivalent to 0 (< 5% of the posterior falls in the ROPE) but it also is not clearly exceeding the MID with < 90% of the posterior distribution exceeding the MID.

In comparison, if everyone were randomized to the intervention group, we would expect a better night sleep on stressor compared to non stressor days at the within person level. This is unlikely to be an effect practically equivalent to zero (< 5% of the posterior falls in the ROPE) and is likely to be at least a MID, as most of the posterior exceeds the MID.

Finally, we can see that the difference in the AMEs between the control and intervention groups is quite large, with none of the posterior distribution falling in the ROPE and nearly all of it exceeding the MID.

We can also see that our contrast coding did indeed yield the difference between the two simple AMEs and can confirm with ‘by hand’ calculations that the mean contrast estimate for the group difference is indeed the mean for the AME for the intervention (Grp 1) minus the mean for the AME for the waitlist control (Grp 0). Collectively, this suggests that within people, days they are exposed to a stressor or not is associated with their sleep that night, and that the emotion regulation intervention moderates the association between within person stressor exposure and sleep.

    knitr::kable(ame.centint$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
-0.072 | -0.069 | -0.134 | -0.012 | 3.175 | 73.675 | 0.95 | HDI | [-0.02, 0.02] | [-Inf, -0.05] | [0.05, Inf] | AME xw: Grp 0
0.123 | 0.122 | 0.039 | 0.211 | 0.775 | 96.100 | 0.95 | HDI | [-0.02, 0.02] | [-Inf, -0.05] | [0.05, Inf] | AME xw: Grp 1
0.195 | 0.192 | 0.096 | 0.297 | 0.025 | 99.850 | 0.95 | HDI | [-0.02, 0.02] | [-Inf, -0.05] | [0.05, Inf] | delta AME xw

## References

Potentially useful references.

- Hedeker, D., du Toit, S. H., Demirtas, H., & Gibbons, R. D. (2018). A note on marginalization of regression parameters from mixed models of binary outcomes. Biometrics, 74(1), 354-361.
- Hua, H., Burke, D. L., Crowther, M. J., Ensor, J., Tudur Smith, C., & Riley, R. D. (2017). One-stage individual participant data meta-analysis models: estimation of treatment-covariate interactions must avoid ecological bias by separating out within-trial and across-trial information. Statistics in Medicine, 36(5), 772–789. https://doi.org/10.1002/sim.7171
- Mize, T. D., Doan, L., & Long, J. S. (2019). A general framework for comparing predictions and marginal effects across models. Sociological Methodology, 49(1), 152-189.
- Norton, E. C., Dowd, B. E., & Maciejewski, M. L. (2019). Marginal effects—quantifying the effect of changes in risk factors in logistic regression models. JAMA, 321(13), 1304-1305.
- Pavlou, M., Ambler, G., Seaman, S., & Omar, R. Z. (2015). A note on obtaining correct marginal predictions from a random intercepts model for binary outcomes. BMC Medical Research Methodology, 15(1), 1-6.
- Yaremych, H. E., Preacher, K. J., & Hedeker, D. (2021). Centering categorical predictors in multilevel models: Best practices and interpretation. Psychological Methods. https://doi.org/10.1037/met0000434