# Marginal Effects for Location Scale Models

    library(knitr)
    library(data.table)
    #> data.table 1.17.8 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 location scale regression models, involving fixed effects only or mixed effects (i.e., fixed and random) and fit using the `brms` package.

A simpler introduction and very brief overview and motivation for marginal effects is available in the vignette for fixed effects only.

This vignette will focus on Gaussian location scale models fit with `brms`. Gaussian location scale models in `brms` have two distributional parameters (dpar):

- the mean or location (often labeled mu) of the distribution, which is the default parameter and has been examined in the other vignettes.
- the variability or scale (often labeled sigma) of the distribution, which is not modeled as an outcome by default, but can be.

Location scale models allow things like assumptions of homogeneity of variance to be relaxed. In repeated measures data, random effects for the scale allow calculating and predicting intraindividual variability (IIV).

## AMEs for Fixed Effects Location Scale Models

To start with, we will look at a fixed effects only location scale model. We will simulate a dataset.

    d <- withr::with_seed(
      seed = 12345, code = {
        nObs <- 1000L
        d <- data.table(
          grp = rep(0:1, each = nObs / 2L),
          x = rnorm(nObs, mean = 0, sd = 0.25))
        d[, y := rnorm(nObs,
                       mean = x + grp,
                       sd = exp(1 + x + grp))]
        copy(d)
      })

    ls.fe <- brm(bf(
      y ~ 1 + x + grp,
      sigma ~ 1 + x + grp),
      family = "gaussian",
      data = d, seed = 1234,
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      silent = 2, refresh = 0,
      chains = 4L, cores = 4L, backend = "cmdstanr")
    #> Loading required namespace: rstan

    summary(ls.fe)
    #>  Family: gaussian
    #>   Links: mu = identity; sigma = log
    #> Formula: y ~ 1 + x + grp
    #>          sigma ~ 1 + x + grp
    #>    Data: d (Number of observations: 1000)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Regression Coefficients:
    #>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept          -0.09      0.13    -0.33     0.15 1.00     4631     3198
    #> sigma_Intercept     1.01      0.03     0.95     1.07 1.00     5701     2972
    #> x                   1.63      0.43     0.77     2.48 1.00     4872     3001
    #> grp                 1.02      0.33     0.37     1.67 1.00     3008     3006
    #> sigma_x             0.85      0.09     0.67     1.02 1.00     5074     2576
    #> sigma_grp           1.01      0.04     0.92     1.10 1.00     5409     2984
    #>
    #> 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).

Now we can use `brmsmargins()`. By default, it will be for the location parameter, the mean. As this is a Gaussian linear model with no transformations and not interactions, the AMEs are the same as the regression coefficients.

Here is an example continuous AME.

    h <- .001
    ame1 <- brmsmargins(
      ls.fe,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      CI = 0.95, CIType = "ETI",
      effects = "fixedonly")

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

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1.632 | 1.63 | 0.775 | 2.482 | NA | NA | 0.95 | ETI | NA | NA | AME x

Here is an AME for discrete / categorical predictors.

    ame2 <- brmsmargins(
      ls.fe,
      at = data.frame(grp = c(0, 1)),
      contrasts = cbind("AME grp" = c(-1, 1)),
      CI = 0.95, CIType = "ETI",
      effects = "fixedonly")

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

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1.023 | 1.024 | 0.371 | 1.673 | NA | NA | 0.95 | ETI | NA | NA | AME grp

In `brms` the scale parameter for Gaussian models, `sigma` uses a log link function. Therefore when back transformed to the original scale, the AMEs will not be the same as the regression coefficients which are on the link scale (log transformed).

We specify that we want AMEs for `sigma` by setting: `dpar = "sigma"`. Here is a continuous example.

    h <- .001
    ame3 <- brmsmargins(
      ls.fe,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      CI = 0.95, CIType = "ETI", dpar = "sigma",
      effects = "fixedonly")

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

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
4.469 | 4.462 | 3.521 | 5.436 | NA | NA | 0.95 | ETI | NA | NA | AME x

Here is a discrete / categorical example.

    ame4 <- brmsmargins(
      ls.fe,
      at = data.frame(grp = c(0, 1)),
      contrasts = cbind("AME grp" = c(-1, 1)),
      CI = 0.95, CIType = "ETI", dpar = "sigma",
      effects = "fixedonly")

    knitr::kable(ame4$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
4.905 | 4.899 | 4.405 | 5.452 | NA | NA | 0.95 | ETI | NA | NA | AME grp

These results are comparable to the mean difference in standard deviation by `grp`. Note that in general, these may not closely align. However, in this instance as `x` and `grp` were simulated to be uncorrelated, the simple unadjusted results match the regression results closely.

    d[, .(SD = sd(y)), by = grp][, diff(SD)]

[1] 4.976021

## AMEs for Mixed Effects Location Scale Models

We will simulate some multilevel location scale data for model and fit the mixed effects location scale model.

    dmixed <- 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
        dmixed <- data.table(
          x = rep(rep(0:1, each = nObs / 2), times = nGroups))
        dmixed[, ID := rep(seq_len(nGroups), each = nObs)]

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

    ls.me <- brm(bf(
      y ~ 1 + x + (1 + x | ID),
      sigma ~ 1 + x + (1 + x | ID)),
      family = "gaussian",
      data = dmixed, 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") +
        prior(normal(-1.5, 1), class = "Intercept", dpar = "sigma") +
        prior(normal(1, .5), class = "b", dpar = "sigma") +
        prior(student_t(3, 1.25, 1), class = "sd", coef = "Intercept", group = "ID", dpar = "sigma") +
        prior(student_t(3, .5, .5), class = "sd", coef = "x", group = "ID", dpar = "sigma"),
      save_pars = save_pars(group = TRUE, latent = FALSE, all = TRUE),
      silent = 2, refresh = 0,
      chains = 4L, cores = 4L, backend = "cmdstanr")
    #> Warning: 19 of 4000 (0.0%) transitions hit the maximum treedepth limit of 10.
    #> See https://mc-stan.org/misc/warnings for details.

Note that this model has not achieved good convergence, but as it already took about 6 minutes to run, for the sake of demonstration we continue. In practice, one would want to make adjustments to ensure good convergence and an adequate effective sample size.

    summary(ls.me)
    #>  Family: gaussian
    #>   Links: mu = identity; sigma = log
    #> Formula: y ~ 1 + x + (1 + x | ID)
    #>          sigma ~ 1 + x + (1 + x | ID)
    #>    Data: dmixed (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
    #> sd(Intercept)                    1.20      0.10     1.03     1.41 1.03      194
    #> sd(x)                            0.43      0.04     0.36     0.52 1.00      552
    #> sd(sigma_Intercept)              1.27      0.10     1.10     1.47 1.01      428
    #> sd(sigma_x)                      0.50      0.05     0.40     0.61 1.00     1419
    #> cor(Intercept,x)                -0.40      0.12    -0.61    -0.14 1.00      577
    #> cor(sigma_Intercept,sigma_x)    -0.35      0.11    -0.55    -0.12 1.00     1399
    #>                              Tail_ESS
    #> sd(Intercept)                     320
    #> sd(x)                            1083
    #> sd(sigma_Intercept)              1015
    #> sd(sigma_x)                      2107
    #> cor(Intercept,x)                  960
    #> cor(sigma_Intercept,sigma_x)     2360
    #>
    #> Regression Coefficients:
    #>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    #> Intercept          -2.56      0.11    -2.78    -2.33 1.03      116      265
    #> sigma_Intercept    -1.50      0.14    -1.78    -1.23 1.04      120      276
    #> x                   0.94      0.06     0.83     1.05 1.01      430     1302
    #> sigma_x             0.97      0.06     0.85     1.09 1.00      860     1944
    #>
    #> 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).

We use `brmsmargins()` similar as for other mixed effects models. For more details see the vignette on marginal effects for mixed effects models.

Here is an example treating `x` as continuous using only the fixed effects for the AME for the scale parameter, `sigma`.

    h <- .001
    ame1a.lsme <- brmsmargins(
      ls.me,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      dpar = "sigma",
      effects = "fixedonly")

    knitr::kable(ame1a.lsme$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.401 | 0.397 | 0.272 | 0.554 | NA | NA | 0.99 | HDI | NA | NA | AME x

Here is the example again, this time integrating out the random effects, which results in a considerable difference in the estimate of the AME.

    h <- .001
    ame1b.lsme <- brmsmargins(
      ls.me,
      add = data.frame(x = c(0, h)),
      contrasts = cbind("AME x" = c(-1 / h, 1 / h)),
      dpar = "sigma",
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame1b.lsme$ContrastSummary, digits = 3)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.791 | 0.75 | 0.357 | 1.596 | NA | NA | 0.99 | HDI | NA | NA | AME x

Here is an example treating `x` as discrete, using only the fixed effects.

    ame2a.lsme <- brmsmargins(
      ls.me,
      at = data.frame(x = c(0, 1)),
      contrasts = cbind("AME x" = c(-1, 1)),
      dpar = "sigma",
      effects = "fixedonly")

    knitr::kable(ame2a.lsme$ContrastSummary)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.3713619 | 0.368748 | 0.2533034 | 0.5077644 | NA | NA | 0.99 | HDI | NA | NA | AME x

Here is the example again, this time integrating out the random effects, likely the more appropriate estimate for most use cases.

    ame2b.lsme <- brmsmargins(
      ls.me,
      at = data.frame(x = c(0, 1)),
      contrasts = cbind("AME x" = c(-1, 1)),
      dpar = "sigma",
      effects = "integrateoutRE", k = 100L, seed = 1234)

    knitr::kable(ame2b.lsme$ContrastSummary)

M | Mdn | LL | UL | PercentROPE | PercentMID | CI | CIType | ROPE | MID | Label
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0.7016545 | 0.6668309 | 0.3358997 | 1.416038 | NA | NA | 0.99 | HDI | NA | NA | AME x

This also is relatively close calculating all the individual standard deviations and taking their differences, then averaging.

    dmixed[, .(SD = sd(y)), by = .(ID, x)
           ][, .(SDdiff = diff(SD)), by = ID][, mean(SDdiff)]
    #> [1] 0.6281889