Although regression models are frequently used in empirical research
to study relationships among variables, often the quantity of
substantive interest is not one of the coefficients of the model, but
rather a quantity derived from the coefficients, such as predicted
values or average marginal effects. The usual method for estimating the
uncertainty of the derived quantities is an approximation known as the
“delta method”. The delta method involves two approximations: 1) that
the variance of the derived quantity can be represented as a first-order
Taylor series, and 2) that the resulting estimate is normally
distributed^{1}.In many cases, especially with nonlinear
models, these approximation can fail badly. `clarify`

implements an alternative to the delta method—simulation-based
inference—which involves simulating the sampling distributions of the
derived quantities.

The methodology `clarify`

relies on is described in King, Tomz, and Wittenberg (2000) and Rainey (2023). Similar functionality exists in
the `CLARIFY`

package in Stata [Tomz,
Wittenberg, and King (2003)]^{2} and used to be available in the
`Zelig`

R package (Imai, King, and Lau
2008), though there are differences in these implementations.
`clarify`

provides additional flexibility by allowing the
user to request any derived quantity, in addition to providing shortcuts
for common quantities, including predictions at representative values,
average marginal effects, and average dose-response functions (described
below). `clarify`

relies on and can be seen as a companion to
the `marginaleffects`

package, which offers similar
functionality but uses the delta method for calculating uncertainty.

`clarify`

There are four steps to using `clarify`

:

Fit the model to the data using modeling functions in supported packages

Use

`sim()`

to take draws from the sampling distribution of the estimated model coefficientsUse

`sim_apply()`

or its wrappers`sim_setx()`

,`sim_ame()`

, and`sim_adrf()`

to compute derived quantities using each simulated set of coefficientsUse

`summary()`

and`plot()`

to summarize and visualize the distribution of the derived quantities and perform inference on them

In the sections below, we’ll describe how to implement these steps in
detail. First, we’ll load `clarify`

using
`library()`

.

`library(clarify)`

For a running example, we’ll use the `lalonde`

dataset in
the `MatchIt`

package, which contains data on 614
participants enrolled in a job training program or sampled from a
survey. The treatment variable is `treat`

and the outcome is
`re78`

, and all other variables are confounders. Although the
original intent was to estimate the effect of `treat`

on
`re78`

, we’ll use it more generally to demonstrate all of
`clarify`

’s capabilities. In addition, we’ll use a
transformation of the outcome variable to demonstrate applications to
nonlinear models.

```
data("lalonde", package = "MatchIt")
$re78_0 <- ifelse(lalonde$re78 == 0, 1, 0)
lalonde
head(lalonde)
#> treat age educ race married nodegree re74 re75 re78 re78_0
#> NSW1 1 37 11 black 1 1 0 0 9930.0460 0
#> NSW2 1 22 9 hispan 0 1 0 0 3595.8940 0
#> NSW3 1 30 12 black 0 0 0 0 24909.4500 0
#> NSW4 1 27 11 black 0 1 0 0 7506.1460 0
#> NSW5 1 33 8 black 0 1 0 0 289.7899 0
#> NSW6 1 22 9 black 0 1 0 0 4056.4940 0
```

The first step is to fit the model. `clarify`

can operate
on a large set of models (those supported by
`marginaleffects`

), including generalized linear models,
multinomial models, multivariate models, and instrumental variable
models, many of which are available in other R packages. Even if
`clarify`

does not offer direct support for a given model,
there are ways to use its functionality regardless (explained in more
detail below).

Because we are computing derived quantities, it is not critical to parameterize the model in such a way that the coefficients are interpretable. Below, we’ll fit a probit regression model for the outcome given the treatment and confounders. Coefficients in probit regression do not have a straightforward interpretation, but that’s okay; our quantities of interest can be expressed as derived quantities–functions of the model parameters, such as predictions, counterfactual predictions, and averages and contrasts of them.

```
<- glm(re78_0 ~ treat + age + educ + race + married +
fit + re74 + re75, data = lalonde,
nodegree family = binomial("probit"))
```

After fitting the model, we will use `sim()`

to draw
coefficients from their sampling distribution. The sampling distribution
is assumed to be multivariate normal or multivariate t with appropriate
degrees of freedom, with a mean vector equal to the coefficient vector
and a covariance matrix equal to the asymptotic covariance matrix
extracted from the model. The arguments to `sim()`

are listed
below:

`sim(fit = , n = , vcov = , coefs = , dist = )`

`fit`

– the fitted model object, the output of the call to the fitting function (e.g.,`glm()`

)`n`

– the number of simulated values to draw; by default, 1000. More values will yield more replicable and precise results at the cost of speed.`vcov`

– either the covariance matrix of the estimated coefficients, a function used to extract it from the model (e.g.,`sandwich::vcovHC()`

for the robust covariance matrix), or a string or formula giving a code for extracting the covariance matrix (see`marginaleffects::get_vcov()`

for details). If left unspecified, the default covariance matrix will be extracted from the model.`coefs`

– either a vector of coefficients to be sampled or a function to extract them from the fitted model. If left unspecified, the default coefficients will be extracted from the model. Typically this does not need to be specified.`dist`

– the name of the distribution from which to draw the sampled coefficients. Can be`"normal"`

for a normal distribution or`t(#)`

for a t-distribution, where`#`

represents the degrees of freedom. If left unspecified,`sim()`

will decide on which distribution makes sense given the characteristics of the model (the decision is made by`insight::get_df()`

with`type = "wald"`

). Typically this does not need to be specified.

If your model is not supported by `clarify`

, you can omit
the `fit`

argument and just specify the `vcov`

and
`coefs`

argument, which will draw the coefficients from the
distribution named in `dist`

(`"normal"`

by
default).

`sim()`

uses a random number generator to draw the sampled
coefficients from the sampling distribution, so a seed should be set
using `set.seed()`

to ensure results are replicable across
sessions.

The output of the call to `sim()`

is a
`clarify_sim`

object, which contains the sampled
coefficients, the original model fit object if supplied, and the
coefficients and covariance matrix used to sample.

```
set.seed(1234)
# Drawing simulated coefficients using an HC2 robust
# covariance matrix
<- sim(fit, vcov = "HC2")
s
s#> A `clarify_sim` object
#> - 10 coefficients, 1000 simulated values
#> - sampled distribution: multivariate normal
#> - original fitting function call:
#>
#> glm(formula = re78_0 ~ treat + age + educ + race + married +
#> nodegree + re74 + re75, family = binomial("probit"), data = lalonde)
```

After sampling the coefficients, we will compute derived quantities
on each set of sampled coefficient and store the result, which
represents a “posterior” distribution of the derived quantity, as well
as on the original coefficients, which are used as the final estimates.
The core functionality is provided by `sim_apply()`

, which
accepts a `clarify_sim`

object from `sim()`

and a
function to compute and return one or more derived quantities, then
applies that function to each set of simulated coefficients. The
arguments to `sim_apply()`

are below:

`sim_apply(sim = , FUN = , verbose = , cl = , ...)`

`sim`

– a`clarify_sim`

object; the output of a call to`sim()`

.`FUN`

– a function that takes in either a model fit object or a vector of coefficients and returns one or more derived quantities. The first argument should be named`fit`

to take in a model fit object or`coefs`

to take in coefficients.`verbose`

– whether to display a progress bar.`cl`

– an argument that controls parallel processing, which can be the number of cores to use or a cluster object resulting from`parallel::makeCluster()`

.`...`

- further arguments to`FUN`

.

The `FUN`

argument can be specified in one of two ways:
either as a function that takes in a model fit object or a function that
takes in a vector of coefficients. The latter will always work but the
former only works for supported models. When the function takes in a
model fit object, `sim_apply()`

will first insert each set of
sampled coefficients into the model fit object and then supply the
modified model to `FUN`

.

For example, let’s say our derived quantity of interest is the
predicted probability of the outcome for participant PSID1. We would
specified our `FUN`

function as follows:

```
<- function(fit) {
sim_fun1 predict(fit, newdata = lalonde["PSID1",], type = "response")
}
```

The `fit`

object supplied to this function will be one in
which the coefficients have been set to their values in a draw from
their sampling distribution as generated by `sim()`

. We then
supply the function to `sim_apply()`

to simulate the sampling
distribution of the predicted value of interest:

```
<- sim_apply(s, FUN = sim_fun1, verbose = FALSE)
est1
est1#> A `clarify_est` object (from `sim_apply()`)
#> - 1000 simulated values
#> - 1 quantity estimated:
#> PSID1 0.02377322
```

The resulting `clarify_est`

object contains the simulated
estimates in matrix form as well as the estimate computed on the
original coefficients. We’ll examine the sampling distribution shortly,
but first we’ll demonstrate computing a derived quantity from the
coefficients directly.

The `race`

variable is a factor, and the
`black`

category is used as the reference level, so it’s not
immediately clear whether there is a difference between the coefficients
`racehispan`

and `racewhite`

, which represent the
non-reference categories `hispan`

and `white`

. To
compare these two directly, we can use `sim_apply()`

to
compute a derived quantity that corresponds to the difference between
them.

```
<- function(coefs) {
sim_fun2 <- unname(coefs["racehispan"])
hispan <- unname(coefs["racewhite"])
white
c("w - h" = white - hispan)
}
<- sim_apply(s, FUN = sim_fun2, verbose = FALSE)
est2
est2#> A `clarify_est` object (from `sim_apply()`)
#> - 1000 simulated values
#> - 1 quantity estimated:
#> w - h 0.1089068
```

The function supplied to `FUN`

can be arbitrarily
complicated and return as many derived quantities as you want, though
the slower each run of `FUN`

is, the longer it will take to
simulate the derived quantities. Using parallel processing by supplying
an argument to `cl`

can sometimes dramatically speed up
evaluation.

There are several functions in `clarify`

that serve as
convenience wrappers for `sim_apply()`

to automate some
common derived quantities of interest. These include

`sim_setx()`

– computing predicted values and first differences at representative or user-specified values of the predictors`sim_ame()`

– computing average marginal means, contrasts of average marginal means, and average marginal effects`sim_adrf()`

– computing average dose-response functions and average marginal effects functions

These are described in their own sections below. In addition, there
are functions that have methods for `clarify_est`

objects,
including `cbind()`

for combining two
`clarify_est`

objects together and `transform()`

for computing quantities that are derived from the already-computed
derived quantities. These are also described in their own sections
below.

To examine the uncertainty around and perform inference on our
estimated quantities, we can use `plot()`

and
`summary()`

on the `clarify_est`

object.

`plot()`

displays a density plot of the resulting
estimates across the simulations, with markers identifying the point
estimate (computed using the original model coefficients as recommended
by Rainey (2023) ) and, optionally,
uncertainty bounds (which function like confidence or credible interval
bounds). The arguments to `plot()`

are below:

`plot(x = , parm = , ci = , level = , method = , reference =)`

`x`

– the`clarify_est`

object (the output of a call to`sim_apply()`

).`parm`

– the names or indices of the quantities to be plotted if more than one was estimated in`sim_apply()`

; if unspecified, all will be plotted.`ci`

– whether to display lines at the uncertainty bounds. The default is`TRUE`

to display them.`level`

– if`ci`

is`TRUE`

, the desired two-sided confidence level. The default is .95 so that that the bounds are at the .025 and .975 quantiles when`method`

(see below) is`"quantile"`

.`method`

– if`ci`

is`TRUE`

, the method used to compute the bounds. Allowable methods include a Normal approximation (`"wald"`

) or using the quantiles of the resulting distribution (`"quantile"`

). The Normal approximation involves multiplying the standard deviation of the estimates (i.e., which functions like the standard error of the sampling distribution) by the critical Z-statistic computed using`(1-level)/2`

to create a symmetric margin of error around the point estimate. The default is`"quantile"`

to instead use quantile-based bounds, which are more appropriate when the distribution is non-Normal. However, quantile-based bounds may require more simulations to stabilize.`reference`

– whether to display a normal density over the plot for each estimate, The default is`FALSE`

to omit them.

Below, we plot the first estimate we computed above, the predicted probability for participant PSID1:

`plot(est1, reference = TRUE, ci = FALSE)`

From the plot, we can see that the distribution of simulated values
is non-Normal, asymmetrical, and not centered around the estimate, with
no values falling below 0 because the outcome is a predicted
probability. Given its non-Normality, the quantile-based bounds are
clearly more appropriate, as the bounds computed from the Normal
approximation would be outside the bounds of the estimate. The plot
itself is a `ggplot`

object that can be modified using
`ggplot2`

syntax.

We can use `summary()`

to display the value of the point
estimate, the uncertainty bounds, and other statistics that describe the
distribution of estimates. The arguments to `summary()`

are
below:

`summary(object = , parm = , level = , method = , null = )`

`object`

– the`clarify_est`

object (the output of a call to`sim_apply()`

).`parm`

– the names or indices of the quantities to be displayed if more than one was estimated in`sim_apply()`

; if unspecified, all will be displayed.`level`

– the desired two-sided confidence level. The default is .95 so that that the bounds are at the .025 and .975 quantiles when method (see below) is`"quantile"`

.`method`

– the method used to compute the uncertainty bounds. Allowable methods include a Normal approximation (`"wald"`

) or using the quantiles of the resulting distribution (`"quantile"`

). See`plot()`

above.`null`

– an optional argument specifying the null value in a hypothesis test for the estimates. If specified, a p-value will be computed using either a standard Z-test (if`method`

is`"quantile"`

) or an inversion of the uncertainty interval. The default is not to display any p-values.

We’ll use `summary()`

with the default arguments on our
first `clarify_est`

object to view the point estimate and
quantile-based uncertainty bounds.

```
summary(est1)
#> Estimate 2.5 % 97.5 %
#> PSID1 0.02377 0.00339 0.10539
```

Our second estimated quantity, the difference between two regression coefficients, is closer to Normally distributed, as the plot below demonstrates (and would be expected theoretically), so we’ll use the Normal approximation to test the hypothesis that difference differs from 0.

`plot(est2, reference = TRUE, ci = FALSE)`

```
summary(est2, method = "wald", null = 0)
#> Estimate 2.5 % 97.5 % Std. Error Z value P-value
#> w - h 0.109 -0.307 0.525 0.212 0.51 0.61
```

The uncertainty intervals and p-values in the `summary()`

output are computed using the Normal approximation because we set
`method = "wald"`

, and the p-value for the test that our
estimate is equal to 0 is returned because we set `null = 0`

.
The computed bounds are very close to the quantile-based bound
(displayed in the plot, and can be requested in `summary()`

by setting `method = "quantile"`

) because the simulated
sampling distribution is close to Normal. Note that the Normal
approximation should be used only when the simulated sampling
distribution is both close to Normally distributed and centered around
the estimate (i.e., when the mean of the simulated values [red vertical
line] coincides with the estimate computed on the original coefficients
[black vertical line]).

`sim_apply()`

wrappers: `sim_setx()`

,
`sim_ame()`

, `sim_adrf()`

`sim_apply()`

can be used to compute the simulated
sampling distribution for any arbitrary derived quantity of interest,
but there are some quantities that are common in applied research and
may otherwise be somewhat challenging to program on their own, so
`clarify`

provides shortcut functions to make computing these
quantities simple. These functions include `sim_setx()`

,
`sim_ame()`

, and `sim_adrf()`

. Each of these can
only be used when regression models compatible with `clarify`

are supplied to the original call to `sim()`

.

Like `sim_apply()`

, each of these functions is named
`sim_*()`

, which signifies that they are to be used on an
object produced by `sim()`

(i.e., a `clarify_sim`

object). (Multiple calls to these functions can be applied to the same
`clarify_sim`

object and combined; see the
`cbind()`

section below.) These functions are described
below.

`sim_setx()`

: predictions at representative values`sim_setx()`

provides an interface to compute predictions
at representative and user-supplied values of the predictors. For
example, we might want to know what the effect of treatment is for a
“typical” individual, which corresponds to the contrast between two
model-based predictions (i.e., one under treatment and one under control
for a unit with “typical” covariate values). This functionality mirrors
the `setx()`

and `setx1()`

functionality of
`Zelig`

and provides similar functionality to functions in
`modelbased`

, `emmeans`

, `effects`

, and
`ggeffects`

.

For each covariate, the user can specify whether they want
predictions at specific values or at “typical” values, which are defined
in `clarify`

as the mode for unordered categorical and binary
variables, the median for ordered categorical variables, and the mean
for continuous variables. Predictions for multiple covariate
combinations can be requested by specifying values that will be used to
create a grid of covariate values. In addition, the “first difference”,
defined here as the difference between predictions for two covariate
combinations, can be computed.

The arguments to `sim_setx()`

are as follows:

`sim_setx(sim = , x = , x1 = , outcome = , type = , verbose = , cl = )`

`sim`

– a`clarify_sim`

object; the output of a call to`sim()`

.`x`

– a named list containing the requested values of the predictors, e.g.,`list(v1 = 1:4, v2 = "A")`

. Any predictors not named will be set at their “typical” value as defined above.`x1`

– an optional named list similar to`x`

except with the value of one predictor changed. When specified, the first difference is computed between the covariate combination defined in`x`

(and only one combination is allowed when`x1`

is specified) and the covariate combination defined in`x1`

.`outcome`

– a string containing the name of the outcome of interest when a multivariate (multiple outcome) model is supplied to`sim()`

or the outcome category of interest when a multinomial model is supplied to`sim()`

. For univariate (single outcome) and binary outcomes, this is ignored.`type`

– a string containing the type of predicted value to return. In most cases, this can be left unspecified to request predictions on the scale of the outcome.`verbose`

– whether to display a progress bar.`cl`

– an argument that controls parallel processing, which can be the number of cores to use or a cluster object resulting from`parallel::makeCluster()`

.

Here, we’ll use `sim_setx()`

to examine predicted values
of the outcome for control and treated units, at `re75`

set
to 0 and 20000, and `race`

set to “black”.

```
<- sim_setx(s,
est3 x = list(treat = 0:1,
re75 = c(0, 20000),
race = "black"),
verbose = FALSE)
```

When we use `summary()`

on the resulting output, we can
see the estimates and their uncertainty intervals (calculated using
quantiles by default). To see the complete grid of the predictor values
used in the predictions, which helps to identify the “typical” values of
the other predictors, we can access the `"setx"`

attribute of
the object:

```
attr(est3, "setx")
#> treat age educ race married nodegree re74
#> treat = 0, re75 = 0 0 27.36319 10.26873 black 0 1 4557.547
#> treat = 1, re75 = 0 1 27.36319 10.26873 black 0 1 4557.547
#> treat = 0, re75 = 20000 0 27.36319 10.26873 black 0 1 4557.547
#> treat = 1, re75 = 20000 1 27.36319 10.26873 black 0 1 4557.547
#> re75
#> treat = 0, re75 = 0 0
#> treat = 1, re75 = 0 0
#> treat = 0, re75 = 20000 20000
#> treat = 1, re75 = 20000 20000
```

We can plot the distributions of the simulated values using
`plot()`

, which also separates the predictions by the
predictor values (it’s often clearer without the uncertainty bounds).
The `var`

argument controls which variable is used for
facetting the plots.

`plot(est3, var = "re75", ci = FALSE)`

We can see again how a delta method or Normal approximation may not have yielded valid uncertainty intervals given the non-Normality of the distributions.

If a continuous variable with many levels is included in the grid of
the predictors, something like a dose-response function for a typical
unit can be generated. Below, we set `re75`

to vary from 0 to
20000 in steps of 2000.

```
<- sim_setx(s,
est4 x = list(treat = 0:1,
re75 = seq(0, 20000, by = 2000),
race = "black"),
verbose = FALSE)
```

When we plot the output, we can see how the predictions varies across
the levels of `re75`

:

`plot(est4)`

We’ll return to display average dose-response functions using
`sim_adrf()`

later.

Finally, we can use `sim_setx()`

to compute first
differences, the contrast between two covariate combinations. We supply
one covariate profile to `x`

and another to `x1`

,
and `sim_setx()`

simulates the two predicted values and their
difference. Below, we simulate first difference for a treated and
control unit who have `re75`

of 0 and typical values of all
other covariates:

```
<- sim_setx(s,
est5 x = list(treat = 0, re75 = 0),
x1 = list(treat = 1, re75 = 0),
verbose = FALSE)
```

When we use `summary()`

, we see the estimates for the
predicted values and their first difference (“FD”):

```
summary(est5)
#> Estimate 2.5 % 97.5 %
#> treat = 0 0.2316 0.1658 0.3117
#> treat = 1 0.1667 0.0930 0.2728
#> FD -0.0649 -0.1346 0.0209
```

It is possible to compute first differences without using
`x1`

using `transform()`

, which we describe
later.

`sim_ame()`

: average marginal means and effectsUsing predicted values and effects at representative values is one
way to summarize regression models, but another way is to compute
average marginal means (AMMs) and average marginal effects (AMEs). The
definitions for these terms may vary and the names for these concepts
differ across sources, but here we define AMMs as the average of the
predicted values for all units after setting one predictor to a chosen
value, and we define AMEs for binary predictors as the contrast of two
AMMs and for continuous predictors as the average of instantaneous rate
of change corresponding to a small change in the predictor from its
observed values across all units^{3}.

The arguments to `sim_ame()`

are as follows:

```
sim_ame(sim = , var = , subset = , contrast = , outcome = ,
type = , eps = , verbose = , cl = )
```

`sim`

– a`clarify_sim`

object; the output of a call to`sim()`

.`var`

– the name of focal variable over which to compute the AMMs or AMEs, or a list containing the values for which AMMs should be computed.`subset`

– a logical vector, evaluated in the original dataset used to fit the model, defining a subset of units for which the AMMs or AMEs are to be computed.`contrast`

– the name of an effect measure used to contrast AMMs. For continuous outcomes,`"diff"`

requests the difference in means, but others are available for binary outcomes, including`"rr"`

for the risk ratio,`"or"`

for the odds ratio, and`"nnt"`

for the number needed to treat. If not specified, only AMMs will be computed if the variable named in`var`

is binary. Ignored when the variable named in`var`

is continuous because the AME is the only quantity computed. When`var`

names a multi-category categorical variable,`contrast`

cannot be used; see the section describing`transfom()`

for computing contrasts with them.`outcome`

– a string containing the name of the outcome of interest when a multivariate (multiple outcome) model is supplied to`sim()`

or the outcome category of interest when a multinomial model is supplied to`sim()`

. For univariate (single outcome) and binary outcomes, this is ignored.`type`

– a string containing the type of predicted value to return. In most cases, this can be left unspecified to request predictions on the scale of the outcome.`eps`

– the value by which the observed values of the variable named in`var`

are changed when it is continuous to compute the AME. This usually does not need to be specified.`verbose`

– whether to display a progress bar.`cl`

– an argument that controls parallel processing, which can be the number of cores to use or a cluster object resulting from`parallel::makeCluster()`

.

Here, we’ll use `sim_ame()`

to compute the AME of
`treat`

just among those who were treated (in causal
inference, this is known as the average treatment effect in the treated,
or ATT). We’ll request our estimate to be on the risk ratio scale.

```
<- sim_ame(s,
est6 var = "treat", subset = treat == 1,
contrast = "rr", verbose = FALSE)
```

We can use `summary()`

to display the estimates and their
uncertainty intervals. Here, we’ll also use `null`

to include
a test for the null hypothesis that the risk ratio is equal to 1.

```
summary(est6, null = c(`RR` = 1))
#> Estimate 2.5 % 97.5 % P-value
#> E[Y(0)] 0.319 0.250 0.400 .
#> E[Y(1)] 0.244 0.189 0.306 .
#> RR 0.765 0.559 1.082 0.13
```

Here we see the estimates for the AMMs, `E[Y(0)]`

for the
expected value of the outcome setting `treat`

to 0 and
`E[Y(1)]`

for the expected value of the outcome setting
`treat`

to 1, and the risk ratio `RR`

. The p-value
on the test for the risk ratio aligns with the uncertainty interval
containing 1.

If we instead wanted the risk difference or odds ratio, we would not
have to re-compute the AMMs. Instead, we can use
`transform()`

to compute a new derived quantity from the
computed AMMs. The section on `transform()`

demonstrates
this.

We can compute the AME for a continuous predictor. Here, we’ll
consider `age`

(just for demonstration; this analysis does
not have a valid interpretation).

`<- sim_ame(s, var = "age", verbose = FALSE) est7 `

We can use `summary()`

to display the AME estimate and its
uncertainty interval.

```
summary(est7)
#> Estimate 2.5 % 97.5 %
#> dY/d(age) 0.00618 0.00257 0.00965
```

The AME is named `dY/d(age)`

, which signifies that a
derivative has been computed (more precisely, the average of the
unit-specific derivatives). This estimate can be interpreted like a
slope in a linear regression model, but as a singe summary of the effect
of a predictor it is too coarse to capture nonlinear relationships. The
section below explains how to compute average dose-response functions
for continuous predictors, which provide a more complete picture of
their effects on an outcome.

`sim_adrf()`

: average dose-response functionsA dose-response function for an individual is the relationship between the set value of a continuous focal predictor and the expected outcome. The average does-response function (ADRF) is the average of the dose-response functions across all units. Essentially, it is a function that relates the value of the predictor to the corresponding AMM of the outcome, the average value of the outcome were all units to be set to that level of the predictor. ADRFs are the continuous analogue to AMMs and can be used to provide additional detail about the effect of a continuous predictor beyond a single AME.

A related quantity is the average marginal effect function (AMEF),
which describes the relationship between a continuous focal predictor
and the AME at that level of the predictor. That is, rather than
describing how the outcome changes as a function of the predictor, it
describes how the *effect* of the predictor on the outcome
changes as a function of the predictor. It is essentially the derivative
of the ADRF and can be used to identify at what points along the ADRF
the predictor has an effect.

The ADRF and AMEF can be computed using `sim_adrf()`

. The
arguments are below:

```
sim_adrf(sim = , var = , subset = , contrast = , at = ,
n = , outcome = , type = , eps = , verbose = ,
cl = )
```

`sim`

– a`clarify_sim`

object; the output of a call to`sim()`

.`var`

– the name of focal variable over which to compute the ADRF or AMEF.`subset`

– a logical vector, evaluated in the original dataset used to fit the model, defining a subset of units for which the ARDF or AMEF is to be computed.`contrast`

– either`"adrf"`

or`"amef"`

to request the ADRF or AMEF, respectively. The default is to request the ADRF.`at`

– the values of the focal predictor at which to compute the ADRF or AMEF. This should be a vector of values that the focal predictor can take on. If unspecified, a vector of`n`

(see below) equally-spaced values from the minimum to the maximum value of the predictor will be used. This should typically be used only if quantities are desired over a subset of the values of the focal predictor.`n`

– if`at`

is unspecified, the number of points along the range of the focal predictor at which to compute the ADRF or AMEF. More yields smoother functions, but will take longer and require more memory. The default is 21.`outcome`

– a string containing the name of the outcome of interest when a multivariate (multiple outcome) model is supplied to`sim()`

or the outcome category of interest when a multinomial model is supplied to`sim()`

. For univariate (single outcome) and binary outcomes, this is ignored.`type`

– a string containing the type of predicted value to return. In most cases, this can be left unspecified to request predictions on the scale of the outcome.`eps`

– the value by which the observed values of the variable named in`var`

are changed when it is continuous to compute the AMEF. This usually does not need to be specified.`verbose`

– whether to display a progress bar.`cl`

– an argument that controls parallel processing, which can be the number of cores to use or a cluster object resulting from`parallel::makeCluster()`

.

Here, we’ll consider `age`

(just for demonstration; this
analysis does not have a valid interpretation) and compute the ADRF and
AMEF of `age`

on the outcome. We’ll only examine ages between
18 and 50, even though the range of `age`

goes slightly
beyond these values. First, we’ll compute the ADRF of `age`

,
which examines how the outcome would vary on average if one set all
unit’s value of `age`

to each value between 18 and 50 (here
we only use even ages).

```
<- sim_adrf(s, var = "age", contrast = "adrf",
est8 at = seq(18, 50, by = 2),
verbose = FALSE)
```

We can plot the ADRF using `plot()`

.

`plot(est8)`

From the plot, we can see that as `age`

increases, the
expected outcome also increases.

We can also examine the AMMs at the requested ages using
`summary()`

, which will display all the estimated AMMs by
default, so we will request just the first 4 (`age`

s 18 to
24):

```
summary(est8, parm = 1:4)
#> Estimate 2.5 % 97.5 %
#> E[Y(18)] 0.178 0.143 0.227
#> E[Y(20)] 0.189 0.155 0.233
#> E[Y(22)] 0.200 0.169 0.240
#> E[Y(24)] 0.211 0.182 0.249
```

Next we’ll compute the AMEF, the effect of `age`

at each
level of `age`

.

```
<- sim_adrf(s, var = "age", contrast = "amef",
est9 at = seq(18, 50, by = 2),
verbose = FALSE)
```

We can plot the AMEF using `plot()`

:

`plot(est9)`

From the plot, we can see the AME of `age`

increases
slightly but is mostly constant across values of `age`

, and
the uncertainty intervals for the AMEs consistently exclude 0.

Often, our quantities of interest are not just the outputs of the
functions above, but comparisons between them. For example, to test for
moderation of a treatment effect, we may want to compare AMEs in
multiple treatment groups. Or, it might be that we are interested in an
effect described using a different effect measure than the one
originally produced; for example, we may decide we want the risk
difference AME after computing the risk ratio AME. The functions
`transform()`

and `cbind()`

allow users to
transform quantities in a single `clarify_est`

object and
combine two `clarify_est`

objects. These are essential for
computing quantities that themselves are derived from the derived
quantities computed by the `sim_*()`

functions.

`transform()`

`transform()`

is a generic function in R that is typically
used to create a new variable in a data frame that is a function of
other columns. For example, to compute the binary outcome we used in our
model, we could have run the following:

`<- transform(lalonde, re78_0 = ifelse(re78 == 0, 1, 0)) lalonde `

(Users familiar with the `tidyverse`

will note the
similarities between `transform()`

and
`dplyr::mutate()`

; only `transform()`

can be used
with `clarify_est`

objects.)

Similarly, to compute a derived or transformed quantity from a
`clarify_est`

object, we can use `transform()`

.
Here, we’ll compute the risk difference AME of `treat`

;
previously, we used `sim_ame()`

to compute the AMMs and the
risk ratio.

`<- transform(est6, RD = `E[Y(1)]` - `E[Y(0)]`) est6 `

Note that we used tics (`) around the names of the AMMs; this is necessary when they contain special characters like parentheses or brackets.

When we run `summary()`

on the output, the new quantity,
which we named “RD”, will be displayed along with the other estimates.
We’ll also set a null value for this quantity.

```
summary(est6, null = c(`RR` = 1, `RD` = 0))
#> Estimate 2.5 % 97.5 % P-value
#> E[Y(0)] 0.3195 0.2501 0.4000 .
#> E[Y(1)] 0.2443 0.1892 0.3062 .
#> RR 0.7646 0.5592 1.0822 0.13
#> RD -0.0752 -0.1692 0.0216 0.13
```

One nice thing about using simulation-based inference with p-values computed from inverting the confidence intervals is that the p-values for the risk difference and risk ratio (and any other effect measure for comparing a pair of values) will always exactly align, thereby ensuring inference does not depend on the effect measure used.

The same value would be computed if we were to have called
`sim_ame()`

on the same `clarify_sim`

object and
requested the risk difference using `contrast = "diff"`

;
using `transform()`

saves time because the AMMs are already
computed and stored in the `clarify_est`

object.

`cbind()`

`cbind()`

is another generic R function that is typically
used to combine two or more datasets columnwise (i.e., to widen a
dataset). In `clarify`

, `cbind()`

can be used to
combine two `clarify_est`

objects so that the estimates can
be examined jointly and so that it is possible to compare them directly.
For example, let’s say we computed AMEs in two subgroups and wanted to
compare them. To do so, we would call `sim_ame()`

twice, one
for each subset:

```
# AME of treat with race = "black"
<- sim_ame(s, var = "treat", subset = race == "black",
est10b contrast = "diff", verbose = FALSE)
summary(est10b)
#> Estimate 2.5 % 97.5 %
#> E[Y(0)] 0.3370 0.2562 0.4279
#> E[Y(1)] 0.2596 0.2004 0.3258
#> Diff -0.0774 -0.1743 0.0215
# AME of treat with race = "hispan"
<- sim_ame(s, var = "treat", subset = race == "hispan",
est10h contrast = "diff", verbose = FALSE)
summary(est10h)
#> Estimate 2.5 % 97.5 %
#> E[Y(0)] 0.1745 0.1052 0.2875
#> E[Y(1)] 0.1225 0.0602 0.2376
#> Diff -0.0520 -0.1161 0.0169
```

Here, we computed the risk difference for the subgroups
`race == "black"`

and `race == "hispan"`

. If we
wanted to compare the risk differences, we could combine them and
compute a new quantity equal to their difference. We’ll do that
below.

First, we need to rename to quantities in each object so they don’t
overlap; we can do so using `names()`

, which has a special
method for `clarify_est`

objects.

```
names(est10b) <- paste(names(est10b), "b", sep = "_")
names(est10h) <- paste(names(est10h), "h", sep = "_")
```

Next, we use `cbind()`

to bind the objects together.

```
<- cbind(est10b, est10h)
est10 summary(est10)
#> Estimate 2.5 % 97.5 %
#> E[Y(0)]_b 0.3370 0.2562 0.4279
#> E[Y(1)]_b 0.2596 0.2004 0.3258
#> Diff_b -0.0774 -0.1743 0.0215
#> E[Y(0)]_h 0.1745 0.1052 0.2875
#> E[Y(1)]_h 0.1225 0.0602 0.2376
#> Diff_h -0.0520 -0.1161 0.0169
```

Finally, we can use `transform()`

to compute the
difference between the risk differences:

```
<- transform(est10, `Dh - Db` = Diff_h - Diff_b)
est10 summary(est10, parm = "Dh - Db")
#> Estimate 2.5 % 97.5 %
#> Dh - Db 0.02540 -0.00406 0.07855
```

Importantly, `cbind()`

can only be used to join together
`clarify_est`

objects computed using the same simulated
coefficients (i.e., resulting from the same call to `sim()`

).
This preserves the covariance among the estimated quantities, which is
critical for valid inference. That is, `sim()`

should only be
called once per model, and all derived quantities should be computed
using its output.

`clarify`

with multiply imputed dataSimulation-based inference in multiply imputed data is relatively straightforward. Simulated coefficients are drawn from the model estimated in each imputed dataset separately, and then the simulated coefficients are pooled into a single set of simulated coefficients. In Bayesian terms, this would be considered “mixing draws” and is the recommended approach for Bayesian analysis with multiply imputed data (Zhou and Reiter 2010).

Using `clarify`

with multiply imputed data is simple.
Rather than using `sim()`

, we use the function
`misim()`

. `misim()`

functions just like
`sim()`

except that it takes in a list of model fits (i.e,
containing a model fit to each imputed dataset) or an object containing
such a list (e.g., a `mira`

object from
`mice::with()`

or a `mimira`

object from
`MatchThem::with()`

). `misim()`

simulates
coefficient distributions within each imputed dataset and then appends
them together to a form a single combined set of coefficient draws.

`sim_apply()`

and its wrappers accept the output of
`misim()`

and compute the desired quantity using each set of
coefficients. When these function rely on using a dataset (e.g.,
`sim_ame()`

, which averages predicted outcomes across all
units in the dataset used to fit the model), they automatically know to
associate a given coefficient draw with the imputed dataset that was
used to fit the model that produced that draw. In user-written functions
supplied to the `FUN`

argument of `sim_apply()`

,
it is important to extract the dataset from the model fit. This is
demonstrated below.

The final estimates of the quantity of interest is computed as the mean of the estimates computed in each imputed dataset (i.e., using the original coefficients, not the simulated ones), which is the same quantity that would be computed using standard pooling rules. This is not always valid for noncollapsible estimates, like ratios, and so care should be taken to ensure the mean of the resulting estimates has a valid interpretation.

The arguments to `misim()`

are as follows:

`misim(fitlist = , n = , vcov = , coefs = , dist = )`

`fitslist`

– a list of model fits or an accepted object containing them (e.g.,`mira`

object from`mice::with()`

)`n`

– the number of simulations to run for each imputed dataset. The default is 1000, but fewer can be used because the total number of simulated quantities will be`m*n`

, where`m`

is the number of imputed datasets.`vcov`

,`coefs`

,`dist`

– the same as with`sim()`

, except that a list of such arguments can be supplied to be applied to each imputed dataset.

Below we illustrate using `misim()`

and
`sim_apply()`

with multiply imputed data. We’ll use the
`africa`

dataset from the `Amelia`

package.

```
library(Amelia)
data("africa", package = "Amelia")
# Multiple imputation
<- amelia(x = africa, m = 10, cs = "country",
a.out ts = "year", logs = "gdp_pc", p2s = 0)
# Fit model to each dataset
<- with(a.out, lm(gdp_pc ~ infl * trade))
model.list
# Simulate coefficients
<- misim(model.list, n = 200)
si
si#> A `clarify_misim` object
#> - 4 coefficients, 10 imputations with 200 simulated values each
#> - sampled distributions: multivariate t(116)
```

The function we’ll be applying to each imputed dataset will be one
that computes the average marginal effect of `infl`

. (We’ll
run the same analysis afterward using `sim_ame()`

.)

```
<- function(fit) {
sim_fun #Extract the original dataset using get_predictors()
<- insight::get_predictors(fit)
X
<- predict(fit, newdata = X)
p0
#Perturb infl slightly
<- predict(fit, newdata = transform(X, infl = infl + 1e-5))
p1
return(c(AME = mean((p1 - p0) / 1e-5)))
}
<- sim_apply(si, FUN = sim_fun, verbose = FALSE)
est_mi
summary(est_mi)
#> Estimate 2.5 % 97.5 %
#> AME -5.75 -8.91 -2.28
```

Note that `sim_apply()`

“knows” which imputation produced
each set of simulated coefficients, so using
`insight::get_predictors()`

on the `fit`

supplied
to `sim_fun()`

will use the right dataset. Care should be
taken when analyses restrict each imputed dataset in a different way
(e.g. when matching with a caliper in each one), as the resulting
imputations may not refer to a specific target population and mixing the
draws may be invalid.

Below, we can use `sim_ame()`

:

`<- sim_ame(si, var = "infl", verbose = FALSE) est_mi2 `

We get the same results, as expected.

Imai, Kosuke, Gary King, and Olivia Lau. 2008. “Toward a Common
Framework for Statistical Analysis and Development.” *Journal
of Computational and Graphical Statistics* 17 (4): 892–913. https://doi.org/10.1198/106186008X384898.

King, Gary, Michael Tomz, and Jason Wittenberg. 2000. “Making the
Most of Statistical Analyses: Improving Interpretation and
Presentation.” *American Journal of Political Science* 44
(2): 347–61. https://doi.org/10.2307/2669316.

Rainey, Carlisle. 2023. “A Careful Consideration of
CLARIFY: Simulation-Induced Bias in Point Estimates of
Quantities of Interest.” *Political Science Research and
Methods*, April, 1–10. https://doi.org/10.1017/psrm.2023.8.

Tomz, Michael, Jason Wittenberg, and Gary King. 2003. “Clarify:
Software for Interpreting and Presenting Statistical Results.”
*Journal of Statistical Software* 8 (January): 1–30. https://doi.org/10.18637/jss.v008.i01.

Zhou, Xiang, and Jerome P. Reiter. 2010. “A Note on Bayesian
Inference After Multiple Imputation.” *The American
Statistician* 64 (2): 159–63. https://doi.org/10.1198/tast.2010.09109.

In addition, most software that uses the delta method uses numerical differentiation, which also involves an approximation.↩︎

Despite the similar name, the R package

`clarify`

and the Stata package`CLARIFY`

differ in several ways, one of which is that the estimates reported by`clarify`

in R are those computed using the original model coefficients, whereas those reported by`CLARIFY`

in Stata are those computed as the average of the simulated distribution. The R implementation avoids the “simulation-induced bias” described by Rainey (2023).↩︎In

`marginaleffects`

, AMMs are computed using`predictions()`

(not with`marginalmeans()`

, which does something closer to`sim_setx()`

), AMEs for binary variables are computed using`comparisons()`

, and AMEs for continuous variables are computed using`marginaleffects()`

. AMMs are sometimes known as average “adjusted” or “counterfactual” predictions.↩︎