Marginal Effects for Location Scale Models

library(knitr)
library(data.table)
#> data.table 1.14.2 using 24 threads (see ?getDTthreads).  Latest news: r-datatable.com
library(brms)
#> Loading required package: Rcpp
#> Loading 'brms' package (version 2.17.0). 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,
  silent = 2, refresh = 0,
  chains = 4L, cores = 4L, backend = "cmdstanr")
#> Compiling Stan program...

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
#> 
#> Population-Level Effects: 
#>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept          -0.09      0.12    -0.33     0.15 1.00     4829     3553
#> sigma_Intercept     1.01      0.03     0.94     1.07 1.00     4696     3023
#> x                   1.62      0.45     0.75     2.49 1.00     4470     2830
#> grp                 1.02      0.35     0.34     1.69 1.00     2526     2684
#> sigma_x             0.85      0.09     0.67     1.02 1.00     4878     3309
#> sigma_grp           1.01      0.05     0.92     1.09 1.00     4425     2871
#> 
#> 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
1.623	1.631	0.746	2.492	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
1.016	1.02	0.343	1.69	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
4.463	4.456	3.488	5.442	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
4.907	4.905	4.409	5.436	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,
  silent = 2, refresh = 0,
  chains = 4L, cores = 4L, backend = "cmdstanr")
#> Compiling Stan program...
#> Warning: 102 of 4000 (3.0%) 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
#> 
#> Group-Level Effects: 
#> ~ID (Number of levels: 100) 
#>                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> sd(Intercept)                    1.19      0.09     1.03     1.37 1.02      211
#> sd(x)                            0.42      0.04     0.35     0.51 1.00      862
#> sd(sigma_Intercept)              1.27      0.09     1.11     1.47 1.01      536
#> sd(sigma_x)                      0.50      0.05     0.41     0.61 1.00     1569
#> cor(Intercept,x)                -0.40      0.12    -0.61    -0.16 1.00      577
#> cor(sigma_Intercept,sigma_x)    -0.36      0.11    -0.55    -0.13 1.00     1516
#>                              Tail_ESS
#> sd(Intercept)                     519
#> sd(x)                            1634
#> sd(sigma_Intercept)               819
#> sd(sigma_x)                      2418
#> cor(Intercept,x)                 1159
#> cor(sigma_Intercept,sigma_x)     2200
#> 
#> Population-Level Effects: 
#>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept          -2.55      0.12    -2.77    -2.33 1.03      105      300
#> sigma_Intercept    -1.48      0.13    -1.72    -1.22 1.02      217      388
#> x                   0.94      0.06     0.83     1.06 1.00      572     1550
#> sigma_x             0.97      0.06     0.85     1.09 1.00     1031     1892
#> 
#> 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
0.408	0.404	0.283	0.555	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
0.804	0.766	0.391	1.575	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
0.3782473	0.3750965	0.2673526	0.509232	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
0.7127988	0.6795497	0.3513727	1.382269	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

- AMEs for Fixed Effects Location Scale Models
- AMEs for Mixed Effects Location Scale Models