---
title: "Marginal Effects for Fixed Effects Models"
output:
  html_document:
    toc: true
    toc_float:
      collapsed: false
      smooth_scroll: true
    toc_depth: 3
vignette: >
  %\VignetteIndexEntry{Marginal Effects for Fixed Effects Models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---




```r
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 regression models involving 
only fixed effects and fit using the `brms` package.

## What are marginal effects?

Marginal effects can be used to describe how an outcome is 
predicted to change with a change in a predictor (or predictors).
It is a derivative. For convenience, typically calculated numerically
rather than analytically.

To motivate marginal effects, we can look at some regression 
models fit in a frequentist framework for simplicity and speed.
Here we use the `mtcars` dataset built into `R`. First, we 
can look at a linear regression model of the association between
`mpg` and `hp`. Here we can see the estimated regression coefficient 
for `mpg`.


```r
m.linear <- lm(hp ~ am + mpg, data = mtcars)

coef(m.linear)["mpg"]
#>       mpg 
#> -11.19988
```

In linear models with no interactions, no (non linear) transformations,
and a linear link function, the regression coefficient is the 
predicted change in the outcome for a one unit change in the predictor,
regardless of any other values. For example, here we can look at the 
predicted difference in the outcome for a one unit difference in `mpg`
from 0 to 1, holding `am = 0`.


```r
yhat <- predict(
  m.linear,
  newdata = data.frame(am = 0, mpg = c(0, 1)),
  type = "response")

diff(yhat)
#>         2 
#> -11.19988
```

We can look at the same estimate but moving `mpg` from 
10 to 11 instead 0 to 1, holding `am = 1`.


```r
yhat <- predict(
  m.linear,
  newdata = data.frame(am = 1, mpg = c(10, 11)),
  type = "response")

diff(yhat)
#>         2 
#> -11.19988
```

All of these quantities are identical. In this case, the regression 
coefficient can be interpreted as a marginal effect: the expected change
in the outcome for a one unit shift in `mpg`, regardless of the 
value of `am` and regardless of the values where `mpg` is evaluated.

This convenient property does not hold for many types of models.
Next consider a logistic regression model. The regression 
coefficient, shown below, is on the log odds scale, not the 
probability scale. This is not convenient for interpretation,
as the log odds scale is not the same scale as our outcome.


```r
m.logistic <- glm(vs ~ am + mpg, data = mtcars, family = binomial())

coef(m.logistic)["mpg"]
#>       mpg 
#> 0.6809205
```

We can find predicted differences on the probability scale.
Here moving `mpg` from 10 to 11 holding `am = 0`.


```r
yhat <- predict(
  m.logistic,
  newdata = data.frame(am = 0, mpg = c(10, 11)),
  type = "response")

diff(yhat)
#>           2 
#> 0.002661989
```

We can look at the same estimate but moving `mpg` from 
20 to 21 instead 10 to 11 again holding `am = 0`.


```r
yhat <- predict(
  m.logistic,
  newdata = data.frame(am = 0, mpg = c(20, 21)),
  type = "response")

diff(yhat)
#>         2 
#> 0.1175344
```

We can look at the same estimate moving `mpg` from 
20 to 21 as before, but this time holding `am = 1`.


```r
yhat <- predict(
  m.logistic,
  newdata = data.frame(am = 1, mpg = c(20, 21)),
  type = "response")

diff(yhat)
#>          2 
#> 0.08606869
```

All the estimates in this case differ. The association between `mpg` and 
**probability** of `vs` is not linear.
Marginal effects provide a way to get results on the response scale, 
which can aid interpretation. 

A common type of marginal effect is an average marginal effect (AME).
To calculate an AME numerically, we can get predicted probabilities 
from a model for every observation in the dataset. For continuous variables,
we might use a very small difference to approximate the derivative.
For categorical variables, we might calculate a discrete difference.

### Average Marginal Effect (AME)

Here is an example of a continuous AME.
`h` is a value near to zero used for the numerical 
derivative. We take all the values observed in the dataset
for the first set of predicted probabilities. Then we take the
observed values + `h` and calculate new predicted probabilities.
The difference, divided by `h` is the "instantaneous" (i.e., derivative)
on the probability scale for a one unit shift in the predictor, `mpg`,
for each person. When we average all of these, we get the AME.


```r
h <- .001

nd.1 <- nd.0 <- model.frame(m.logistic)
nd.1$mpg <- nd.1$mpg + h

yhat.0 <- predict(
  m.logistic,
  newdata = nd.0,
  type = "response")

yhat.1 <- predict(
  m.logistic,
  newdata = nd.1,
  type = "response")

mean((yhat.1 - yhat.0) / h)
#> [1] 0.06922997
```

Here is an example of a discrete AME. The variable,
`am` only takes two values: 0 or 1. So we calculate 
predicted probabilities if everyone had `am = 0` and then
again if everyone had `am = 1`.


```r
nd.1 <- nd.0 <- model.frame(m.logistic)
nd.0$am <- 0
nd.1$am <- 1

yhat.0 <- predict(
  m.logistic,
  newdata = nd.0,
  type = "response")

yhat.1 <- predict(
  m.logistic,
  newdata = nd.1,
  type = "response")

mean((yhat.1 - yhat.0))
#> [1] -0.2618203
```

In both these examples, we are averaging across the different values 
observed in the dataset. In a frequentist framework, additional details
are needed to calculate uncertainty intervals. In a Bayesian framework,
uncertainty intervals can be calculated readily by summarizing the 
posterior.

## AMEs for Logistic Regression

The main function for users to use is `brmsmargins()`. Here is an 
example calculating AMEs for `mpg` and `am`. First we will fit the same
logistic regression model using `brms`.


```r
bayes.logistic <- brm(
  vs ~ am + mpg, data = mtcars,
  family = "bernoulli", seed = 1234,
  silent = 2, refresh = 0,
  chains = 4L, cores = 4L, backend = "cmdstanr")
#> Compiling Stan program...
```


```r
summary(bayes.logistic)
#>  Family: bernoulli 
#>   Links: mu = logit 
#> Formula: vs ~ am + mpg 
#>    Data: mtcars (Number of observations: 32) 
#>   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   -16.03      5.49   -28.79    -7.15 1.00     1797     1897
#> am           -3.77      1.82    -7.87    -0.71 1.00     1689     1766
#> mpg           0.86      0.30     0.38     1.57 1.00     1707     1842
#> 
#> 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()`. We give it the model object,
a `data.frame` of the values to be added, first 0, then (0 + h),
and a contrast matrix. The default is a 99 percent credible interval,
which we override here to 0.95. We use highest density intervals, 
which are the default. We also could have selected "ETI" for 
equal tail intervals. `brmsmargins()` will return a list
with the posterior of each prediction, a summary of the posterior 
for the predictions, the posterior for the contrasts, and a 
summary of the posterior for the contrasts. Here we just have the 
one contrast, but multiple could have been specified.


```r
h <- .001
ame1 <- brmsmargins(
  bayes.logistic,
  add = data.frame(mpg = c(0, 0 + h)),
  contrasts = cbind("AME MPG" = c(-1 / h, 1 / h)),
  CI = 0.95, CIType = "HDI")

kable(ame1$ContrastSummary, digits = 3)
```



|     M|  Mdn|    LL|    UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label   |
|-----:|----:|-----:|-----:|-----------:|----------:|----:|:------|:----|:---|:-------|
| 0.071| 0.07| 0.054| 0.091|          NA|         NA| 0.95|HDI    |NA   |NA  |AME MPG |

Now we can look at how we could calculate a discrete AME.
This time we use the `at` argument instead of the `add` 
argument as we want to hold `am` at specific values, 
not add 0 and 1 to the observed `am` values.
Because 0 and 1 are meaningful values of `am`,
we also look at the summary of the posterior for the predictions.
These predictions average across all values of `mpg`.


```r
ame2 <- brmsmargins(
  bayes.logistic,
  at = data.frame(am = c(0, 1)),
  contrasts = cbind("AME am" = c(-1, 1)),
  CI = 0.95, CIType = "HDI")

kable(ame2$Summary, digits = 3)
```



|     M|   Mdn|    LL|    UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |
|-----:|-----:|-----:|-----:|-----------:|----------:|----:|:------|:----|:---|
| 0.543| 0.544| 0.419| 0.653|          NA|         NA| 0.95|HDI    |NA   |NA  |
| 0.284| 0.277| 0.180| 0.409|          NA|         NA| 0.95|HDI    |NA   |NA  |


```r
kable(ame2$ContrastSummary)
```



|         M|        Mdn|         LL|         UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label  |
|---------:|----------:|----------:|----------:|-----------:|----------:|----:|:------|:----|:---|:------|
| -0.258856| -0.2648354| -0.4252779| -0.0862889|          NA|         NA| 0.95|HDI    |NA   |NA  |AME am |

Note that by default, `brmsmargins()` uses the model frame 
from the model object as the dataset. This, however, can be overridden.
You can give it any (valid) dataset and it will add or override the chosen
values and average across the predictions from the different rows of 
the dataset.


## AMEs for Poisson Regression

Here is a short example for Poisson regression used for 
count outcomes. We use a dataset drawn from:
https://stats.oarc.ucla.edu/r/dae/poisson-regression/


```r

d <- fread("https://stats.oarc.ucla.edu/stat/data/poisson_sim.csv")
d[, prog := factor(prog, levels = 1:3, labels = c("General", "Academic", "Vocational"))]

bayes.poisson <- brm(
  num_awards ~ prog + math, data = d,
  family = "poisson", seed = 1234,
  silent = 2, refresh = 0,
  chains = 4L, cores = 4L, backend = "cmdstanr")
#> Compiling Stan program...
```


```r
summary(bayes.poisson)
#>  Family: poisson 
#>   Links: mu = log 
#> Formula: num_awards ~ prog + math 
#>    Data: d (Number of observations: 200) 
#>   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         -5.31      0.66    -6.66    -4.03 1.00     1933     2312
#> progAcademic       1.15      0.38     0.46     1.97 1.00     2088     1947
#> progVocational     0.39      0.47    -0.48     1.36 1.00     2174     2024
#> math               0.07      0.01     0.05     0.09 1.00     2592     2120
#> 
#> 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).
```

AME for a continuous variable, using default CI interval and type.


```r
h <- .001
ame1.p <- brmsmargins(
  bayes.poisson,
  add = data.frame(math = c(0, 0 + h)),
  contrasts = cbind("AME math" = c(-1 / h, 1 / h)))

kable(ame1.p$ContrastSummary, digits = 3)
```



|     M|   Mdn|    LL|    UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label    |
|-----:|-----:|-----:|-----:|-----------:|----------:|----:|:------|:----|:---|:--------|
| 0.044| 0.044| 0.026| 0.065|          NA|         NA| 0.99|HDI    |NA   |NA  |AME math |

AME for a categorical variable. Here we calculate pairwise contrasts
for all three program types. These are the predicted number of awards.


```r
ame2.p <- brmsmargins(
  bayes.poisson,
  at = data.frame(
    prog = factor(1:3,
                  labels = c("General", "Academic", "Vocational"))),
  contrasts = cbind(
    "AME General v Academic" = c(1, -1, 0),
    "AME General v Vocational" = c(1, 0, -1),
    "AME Academic v Vocational" = c(0, 1, -1)))
    
kable(ame2.p$Summary, digits = 3)
```



|     M|   Mdn|    LL|    UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |
|-----:|-----:|-----:|-----:|-----------:|----------:|----:|:------|:----|:---|
| 0.263| 0.253| 0.076| 0.526|          NA|         NA| 0.99|HDI    |NA   |NA  |
| 0.781| 0.779| 0.600| 0.988|          NA|         NA| 0.99|HDI    |NA   |NA  |
| 0.380| 0.368| 0.126| 0.710|          NA|         NA| 0.99|HDI    |NA   |NA  |


```r
kable(ame2.p$ContrastSummary, digits = 3)
```



|      M|    Mdn|     LL|     UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label                     |
|------:|------:|------:|------:|-----------:|----------:|----:|:------|:----|:---|:-------------------------|
| -0.518| -0.524| -0.820| -0.176|          NA|         NA| 0.99|HDI    |NA   |NA  |AME General v Academic    |
| -0.117| -0.111| -0.503|  0.280|          NA|         NA| 0.99|HDI    |NA   |NA  |AME General v Vocational  |
|  0.400|  0.406|  0.019|  0.726|          NA|         NA| 0.99|HDI    |NA   |NA  |AME Academic v Vocational |


## AMEs for Negative Binomial Regression

Here is a short example for Negative Binomial regression used for 
count outcomes. We use the same setup as for the Poisson regression example.


```r
d <- read.csv("https://stats.oarc.ucla.edu/stat/data/poisson_sim.csv")
d$prog <- factor(d$prog, levels = 1:3, labels = c("General", "Academic", "Vocational"))

bayes.nb <- brm(
  num_awards ~ prog + math, data = d,
  family = "negbinomial", seed = 1234,
  silent = 2, refresh = 0,
  chains = 4L, cores = 4L, backend = "cmdstanr")
#> Compiling Stan program...
```


```r
summary(bayes.nb)
#>  Family: negbinomial 
#>   Links: mu = log; shape = identity 
#> Formula: num_awards ~ prog + math 
#>    Data: d (Number of observations: 200) 
#>   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         -5.39      0.70    -6.82    -4.06 1.00     2813     2387
#> progAcademic       1.14      0.38     0.42     1.89 1.00     2187     2343
#> progVocational     0.40      0.47    -0.49     1.32 1.00     2155     2093
#> math               0.07      0.01     0.05     0.09 1.00     3139     2500
#> 
#> Family Specific Parameters: 
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> shape    19.82     35.82     1.91   112.92 1.00     2453     2214
#> 
#> 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).
```

AME for a continuous variable, using default CI interval and type.


```r
h <- .001
ame1.nb <- brmsmargins(
  bayes.nb,
  add = data.frame(math = c(0, 0 + h)),
  contrasts = cbind("AME math" = c(-1 / h, 1 / h)))

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



|     M|   Mdn|    LL|   UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label    |
|-----:|-----:|-----:|----:|-----------:|----------:|----:|:------|:----|:---|:--------|
| 0.045| 0.045| 0.022| 0.07|          NA|         NA| 0.99|HDI    |NA   |NA  |AME math |

AME for a categorical variable. Here we calculate pairwise contrasts
for all three program types. These are the predicted number of awards.


```r
ame2.nb <- brmsmargins(
  bayes.nb,
  at = data.frame(
    prog = factor(1:3,
                  labels = c("General", "Academic", "Vocational"))),
  contrasts = cbind(
    "AME General v Academic" = c(1, -1, 0),
    "AME General v Vocational" = c(1, 0, -1),
    "AME Academic v Vocational" = c(0, 1, -1)))
    
kable(ame2.nb$Summary, digits = 3)
```



|     M|   Mdn|    LL|    UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |
|-----:|-----:|-----:|-----:|-----------:|----------:|----:|:------|:----|:---|
| 0.264| 0.251| 0.072| 0.562|          NA|         NA| 0.99|HDI    |NA   |NA  |
| 0.781| 0.778| 0.565| 1.010|          NA|         NA| 0.99|HDI    |NA   |NA  |
| 0.388| 0.374| 0.146| 0.756|          NA|         NA| 0.99|HDI    |NA   |NA  |


```r
kable(ame2.nb$ContrastSummary, digits = 3)
```



|      M|    Mdn|     LL|     UL| PercentROPE| PercentMID|   CI|CIType |ROPE |MID |Label                     |
|------:|------:|------:|------:|-----------:|----------:|----:|:------|:----|:---|:-------------------------|
| -0.517| -0.523| -0.840| -0.127|          NA|         NA| 0.99|HDI    |NA   |NA  |AME General v Academic    |
| -0.124| -0.120| -0.532|  0.270|          NA|         NA| 0.99|HDI    |NA   |NA  |AME General v Vocational  |
|  0.392|  0.405| -0.024|  0.768|          NA|         NA| 0.99|HDI    |NA   |NA  |AME Academic v Vocational |

## References

These references may be useful.

- 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.
- 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.
