# Comparisons Between Predictions Made With Different Regressor Values

Source:`R/comparisons.R`

`comparisons.Rd`

Predict the outcome variable at different regressor values (e.g., college
graduates vs. others), and compare those predictions by computing a difference,
ratio, or some other function. `comparisons()`

can return many quantities of
interest, such as contrasts, differences, risk ratios, changes in log odds,
slopes, elasticities, etc.

`comparisons()`

: unit-level (conditional) estimates.`avg_comparisons()`

: average (marginal) estimates.

`variables`

identifies the focal regressors whose "effect" we are interested in. `comparison`

determines how predictions with different regressor values are compared (difference, ratio, odds, etc.). The `newdata`

argument and the `datagrid()`

function control where statistics are evaluated in the predictor space: "at observed values", "at the mean", "at representative values", etc.

See the comparisons vignette and package website for worked examples and case studies:

## Usage

```
comparisons(
model,
newdata = NULL,
variables = NULL,
comparison = "difference",
type = NULL,
vcov = TRUE,
by = FALSE,
conf_level = 0.95,
transform = NULL,
cross = FALSE,
wts = NULL,
hypothesis = NULL,
equivalence = NULL,
p_adjust = NULL,
df = Inf,
eps = NULL,
...
)
avg_comparisons(
model,
newdata = NULL,
variables = NULL,
type = NULL,
vcov = TRUE,
by = TRUE,
conf_level = 0.95,
comparison = "difference",
transform = NULL,
cross = FALSE,
wts = NULL,
hypothesis = NULL,
equivalence = NULL,
p_adjust = NULL,
df = Inf,
eps = NULL,
...
)
```

## Arguments

- model
Model object

- newdata
Grid of predictor values at which we evaluate the comparisons.

`NULL`

(default): Unit-level contrasts for each observed value in the original dataset (empirical distribution). See`insight::get_data()`

data frame: Unit-level contrasts for each row of the

`newdata`

data frame.string:

"mean": Contrasts at the Mean. Contrasts when each predictor is held at its mean or mode.

"median": Contrasts at the Median. Contrasts when each predictor is held at its median or mode.

"marginalmeans": Contrasts at Marginal Means.

"tukey": Contrasts at Tukey's 5 numbers.

"grid": Contrasts on a grid of representative numbers (Tukey's 5 numbers and unique values of categorical predictors).

`datagrid()`

call to specify a custom grid of regressors. For example:`newdata = datagrid(cyl = c(4, 6))`

:`cyl`

variable equal to 4 and 6 and other regressors fixed at their means or modes.`newdata = datagrid(mpg = fivenum)`

:`mpg`

variable held at Tukey's five numbers (using the`fivenum`

function), and other regressors fixed at their means or modes.See the Examples section and the datagrid documentation.

- variables
Focal variables

`NULL`

: compute comparisons for all the variables in the model object (can be slow).Character vector: subset of variables (usually faster).

Named list: names identify the subset of variables of interest, and values define the type of contrast to compute. Acceptable values depend on the variable type:

Factor or character variables:

"reference": Each factor level is compared to the factor reference (base) level

"all": All combinations of observed levels

"sequential": Each factor level is compared to the previous factor level

"pairwise": Each factor level is compared to all other levels

"minmax": The highest and lowest levels of a factor.

Vector of length 2 with the two values to compare.

Logical variables:

NULL: contrast between TRUE and FALSE

Numeric variables:

Numeric of length 1: Contrast for a gap of

`x`

, computed at the observed value plus and minus`x / 2`

. For example, estimating a`+1`

contrast compares adjusted predictions when the regressor is equal to its observed value minus 0.5 and its observed value plus 0.5.Numeric vector of length 2: Contrast between the 2nd element and the 1st element of the

`x`

vector.Data frame with the same number of rows as

`newdata`

, with two columns of "low" and "high" values to compare.Function which accepts a numeric vector and returns a data frame with two columns of "low" and "high" values to compare. See examples below.

"iqr": Contrast across the interquartile range of the regressor.

"sd": Contrast across one standard deviation around the regressor mean.

"2sd": Contrast across two standard deviations around the regressor mean.

"minmax": Contrast between the maximum and the minimum values of the regressor.

Examples:

`variables = list(gear = "pairwise", hp = 10)`

`variables = list(gear = "sequential", hp = c(100, 120))`

See the Examples section below for more.

- comparison
How should pairs of predictions be compared? Difference, ratio, odds ratio, or user-defined functions.

string: shortcuts to common contrast functions.

Supported shortcuts strings: difference, differenceavg, differenceavgwts, dydx, eyex, eydx, dyex, dydxavg, eyexavg, eydxavg, dyexavg, dydxavgwts, eyexavgwts, eydxavgwts, dyexavgwts, ratio, ratioavg, ratioavgwts, lnratio, lnratioavg, lnratioavgwts, lnor, lnoravg, lnoravgwts, expdydx, expdydxavg, expdydxavgwts

See the Comparisons section below for definitions of each transformation.

function: accept two equal-length numeric vectors of adjusted predictions (

`hi`

and`lo`

) and returns a vector of contrasts of the same length, or a unique numeric value.See the Transformations section below for examples of valid functions.

- type
string indicates the type (scale) of the predictions used to compute contrasts or slopes. This can differ based on the model type, but will typically be a string such as: "response", "link", "probs", or "zero". When an unsupported string is entered, the model-specific list of acceptable values is returned in an error message. When

`type`

is`NULL`

, the default value is used. This default is the first model-related row in the`marginaleffects:::type_dictionary`

dataframe.- vcov
Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values:

FALSE: Do not compute standard errors. This can speed up computation considerably.

TRUE: Unit-level standard errors using the default

`vcov(model)`

variance-covariance matrix.String which indicates the kind of uncertainty estimates to return.

Heteroskedasticity-consistent:

`"HC"`

,`"HC0"`

,`"HC1"`

,`"HC2"`

,`"HC3"`

,`"HC4"`

,`"HC4m"`

,`"HC5"`

. See`?sandwich::vcovHC`

Heteroskedasticity and autocorrelation consistent:

`"HAC"`

Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger"

Other:

`"NeweyWest"`

,`"KernHAC"`

,`"OPG"`

. See the`sandwich`

package documentation.

One-sided formula which indicates the name of cluster variables (e.g.,

`~unit_id`

). This formula is passed to the`cluster`

argument of the`sandwich::vcovCL`

function.Square covariance matrix

Function which returns a covariance matrix (e.g.,

`stats::vcov(model)`

)

- by
Aggregate unit-level estimates (aka, marginalize, average over). Valid inputs:

`FALSE`

: return the original unit-level estimates.`TRUE`

: aggregate estimates for each term.Character vector of column names in

`newdata`

or in the data frame produced by calling the function without the`by`

argument.Data frame with a

`by`

column of group labels, and merging columns shared by`newdata`

or the data frame produced by calling the same function without the`by`

argument.See examples below.

- conf_level
numeric value between 0 and 1. Confidence level to use to build a confidence interval.

- transform
string or function. Transformation applied to unit-level estimates and confidence intervals just before the function returns results. Functions must accept a vector and return a vector of the same length. Support string shortcuts: "exp", "ln"

- cross
`FALSE`

: Contrasts represent the change in adjusted predictions when one predictor changes and all other variables are held constant.`TRUE`

: Contrasts represent the changes in adjusted predictions when all the predictors specified in the`variables`

argument are manipulated simultaneously (a "cross-contrast").

- wts
string or numeric: weights to use when computing average contrasts or slopes. These weights only affect the averaging in

`avg_*()`

or with the`by`

argument, and not the unit-level estimates themselves.string: column name of the weights variable in

`newdata`

. When supplying a column name to`wts`

, it is recommended to supply the original data (including the weights variable) explicitly to`newdata`

.numeric: vector of length equal to the number of rows in the original data or in

`newdata`

(if supplied).

- hypothesis
specify a hypothesis test or custom contrast using a numeric value, vector, or matrix, a string, or a string formula.

Numeric:

Single value: the null hypothesis used in the computation of Z and p (before applying

`transform`

).Vector: Weights to compute a linear combination of (custom contrast between) estimates. Length equal to the number of rows generated by the same function call, but without the

`hypothesis`

argument.Matrix: Each column is a vector of weights, as describe above, used to compute a distinct linear combination of (contrast between) estimates. The column names of the matrix are used as labels in the output.

String formula to specify linear or non-linear hypothesis tests. If the

`term`

column uniquely identifies rows, terms can be used in the formula. Otherwise, use`b1`

,`b2`

, etc. to identify the position of each parameter. Examples:`hp = drat`

`hp + drat = 12`

`b1 + b2 + b3 = 0`

String:

"pairwise": pairwise differences between estimates in each row.

"reference": differences between the estimates in each row and the estimate in the first row.

"sequential": difference between an estimate and the estimate in the next row.

"revpairwise", "revreference", "revsequential": inverse of the corresponding hypotheses, as described above.

See the Examples section below and the vignette: https://vincentarelbundock.github.io/marginaleffects/articles/hypothesis.html

- equivalence
Numeric vector of length 2: bounds used for the two-one-sided test (TOST) of equivalence, and for the non-inferiority and non-superiority tests. See Details section below.

- p_adjust
Adjust p-values for multiple comparisons: "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", or "fdr". See stats::p.adjust

- df
Degrees of freedom used to compute p values and confidence intervals. A single numeric value between 1 and

`Inf`

. When`df`

is`Inf`

, the normal distribution is used. When`df`

is finite, the`t`

distribution is used. See insight::get_df for a convenient function to extract degrees of freedom. Ex:`slopes(model, df = insight::get_df(model))`

- eps
NULL or numeric value which determines the step size to use when calculating numerical derivatives: (f(x+eps)-f(x))/eps. When

`eps`

is`NULL`

, the step size is 0.0001 multiplied by the difference between the maximum and minimum values of the variable with respect to which we are taking the derivative. Changing`eps`

may be necessary to avoid numerical problems in certain models.- ...
Additional arguments are passed to the

`predict()`

method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the`marginaleffects`

website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the`?marginaleffects`

documentation for a non-exhaustive list of available arguments.

## Value

A `data.frame`

with one row per observation (per term/group) and several columns:

`rowid`

: row number of the`newdata`

data frame`type`

: prediction type, as defined by the`type`

argument`group`

: (optional) value of the grouped outcome (e.g., categorical outcome models)`term`

: the variable whose marginal effect is computed`dydx`

: slope of the outcome with respect to the term, for a given combination of predictor values`std.error`

: standard errors computed by via the delta method.`p.value`

: p value associated to the`estimate`

column. The null is determined by the`hypothesis`

argument (0 by default), and p values are computed before applying the`transform`

argument.

See `?print.marginaleffects`

for printing options.

## Standard errors using the delta method

Standard errors for all quantities estimated by `marginaleffects`

can be obtained via the delta method. This requires differentiating a function with respect to the coefficients in the model using a finite difference approach. In some models, the delta method standard errors can be sensitive to various aspects of the numeric differentiation strategy, including the step size. By default, the step size is set to `1e-8`

, or to `1e-4`

times the smallest absolute model coefficient, whichever is largest.

`marginaleffects`

can delegate numeric differentiation to the `numDeriv`

package, which allows more flexibility. To do this, users can pass arguments to the `numDeriv::jacobian`

function through a global option. For example:

`options(marginaleffects_numDeriv = list(method = "simple", method.args = list(eps = 1e-6)))`

`options(marginaleffects_numDeriv = list(method = "Richardson", method.args = list(eps = 1e-5)))`

`options(marginaleffects_numDeriv = NULL)`

See the "Standard Errors and Confidence Intervals" vignette on the `marginaleffects`

website for more details on the computation of standard errors:

https://vincentarelbundock.github.io/marginaleffects/articles/uncertainty.html

Note that the `inferences()`

function can be used to compute uncertainty estimates using a bootstrap or simulation-based inference. See the vignette:

https://vincentarelbundock.github.io/marginaleffects/articles/bootstrap.html

## Model-Specific Arguments

Some model types allow model-specific arguments to modify the nature of
marginal effects, predictions, marginal means, and contrasts. Please report
other package-specific `predict()`

arguments on Github so we can add them to
the table below.

https://github.com/vincentarelbundock/marginaleffects/issues

Package | Class | Argument | Documentation |

`brms` | `brmsfit` | `ndraws` | brms::posterior_predict |

`re_formula` | brms::posterior_predict | ||

`lme4` | `merMod` | `re.form` | lme4::predict.merMod |

`allow.new.levels` | lme4::predict.merMod | ||

`glmmTMB` | `glmmTMB` | `re.form` | glmmTMB::predict.glmmTMB |

`allow.new.levels` | glmmTMB::predict.glmmTMB | ||

`zitype` | glmmTMB::predict.glmmTMB | ||

`mgcv` | `bam` | `exclude` | mgcv::predict.bam |

`robustlmm` | `rlmerMod` | `re.form` | robustlmm::predict.rlmerMod |

`allow.new.levels` | robustlmm::predict.rlmerMod | ||

`MCMCglmm` | `MCMCglmm` | `ndraws` |

## comparison argument functions

The following transformations can be applied by supplying one of the shortcut strings to the
`comparison`

argument.
`hi`

is a vector of adjusted predictions for the "high" side of the
contrast. `lo`

is a vector of adjusted predictions for the "low" side of the
contrast. `y`

is a vector of adjusted predictions for the original data. `x`

is the predictor in the original data. `eps`

is the step size to use to
compute derivatives and elasticities.

Shortcut | Function |

difference | \(hi, lo) hi - lo |

differenceavg | \(hi, lo) mean(hi) - mean(lo) |

dydx | \(hi, lo, eps) (hi - lo)/eps |

eyex | \(hi, lo, eps, y, x) (hi - lo)/eps * (x/y) |

eydx | \(hi, lo, eps, y, x) ((hi - lo)/eps)/y |

dyex | \(hi, lo, eps, x) ((hi - lo)/eps) * x |

dydxavg | \(hi, lo, eps) mean((hi - lo)/eps) |

eyexavg | \(hi, lo, eps, y, x) mean((hi - lo)/eps * (x/y)) |

eydxavg | \(hi, lo, eps, y, x) mean(((hi - lo)/eps)/y) |

dyexavg | \(hi, lo, eps, x) mean(((hi - lo)/eps) * x) |

ratio | \(hi, lo) hi/lo |

ratioavg | \(hi, lo) mean(hi)/mean(lo) |

lnratio | \(hi, lo) log(hi/lo) |

lnratioavg | \(hi, lo) log(mean(hi)/mean(lo)) |

lnor | \(hi, lo) log((hi/(1 - hi))/(lo/(1 - lo))) |

lnoravg | \(hi, lo) log((mean(hi)/(1 - mean(hi)))/(mean(lo)/(1 - mean(lo)))) |

expdydx | \(hi, lo, eps) ((exp(hi) - exp(lo))/exp(eps))/eps |

expdydxavg | \(hi, lo, eps) mean(((exp(hi) - exp(lo))/exp(eps))/eps) |

## Bayesian posterior summaries

By default, credible intervals in bayesian models are built as equal-tailed intervals. This can be changed to a highest density interval by setting a global option:

`options("marginaleffects_posterior_interval" = "eti")`

`options("marginaleffects_posterior_interval" = "hdi")`

By default, the center of the posterior distribution in bayesian models is identified by the median. Users can use a different summary function by setting a global option:

`options("marginaleffects_posterior_center" = "mean")`

`options("marginaleffects_posterior_center" = "median")`

When estimates are averaged using the `by`

argument, the `tidy()`

function, or
the `summary()`

function, the posterior distribution is marginalized twice over.
First, we take the average *across* units but *within* each iteration of the
MCMC chain, according to what the user requested in `by`

argument or
`tidy()/summary()`

functions. Then, we identify the center of the resulting
posterior using the function supplied to the
`"marginaleffects_posterior_center"`

option (the median by default).

## Equivalence, Inferiority, Superiority

\(\theta\) is an estimate, \(\sigma_\theta\) its estimated standard error, and \([a, b]\) are the bounds of the interval supplied to the `equivalence`

argument.

Non-inferiority:

\(H_0\): \(\theta \leq a\)

\(H_1\): \(\theta > a\)

\(t=(\theta - a)/\sigma_\theta\)

p: Upper-tail probability

Non-superiority:

\(H_0\): \(\theta \geq b\)

\(H_1\): \(\theta < b\)

\(t=(\theta - b)/\sigma_\theta\)

p: Lower-tail probability

Equivalence: Two One-Sided Tests (TOST)

p: Maximum of the non-inferiority and non-superiority p values.

Thanks to Russell V. Lenth for the excellent `emmeans`

package and documentation which inspired this feature.

## Examples

```
if (FALSE) {
library(marginaleffects)
# Linear model
tmp <- mtcars
tmp$am <- as.logical(tmp$am)
mod <- lm(mpg ~ am + factor(cyl), tmp)
avg_comparisons(mod, variables = list(cyl = "reference"))
avg_comparisons(mod, variables = list(cyl = "sequential"))
avg_comparisons(mod, variables = list(cyl = "pairwise"))
# GLM with different scale types
mod <- glm(am ~ factor(gear), data = mtcars)
avg_comparisons(mod, type = "response")
avg_comparisons(mod, type = "link")
# Contrasts at the mean
comparisons(mod, newdata = "mean")
# Contrasts between marginal means
comparisons(mod, newdata = "marginalmeans")
# Contrasts at user-specified values
comparisons(mod, newdata = datagrid(am = 0, gear = tmp$gear))
comparisons(mod, newdata = datagrid(am = unique, gear = max))
m <- lm(mpg ~ hp + drat + factor(cyl) + factor(am), data = mtcars)
comparisons(m, variables = "hp", newdata = datagrid(FUN_factor = unique, FUN_numeric = median))
# Numeric contrasts
mod <- lm(mpg ~ hp, data = mtcars)
avg_comparisons(mod, variables = list(hp = 1))
avg_comparisons(mod, variables = list(hp = 5))
avg_comparisons(mod, variables = list(hp = c(90, 100)))
avg_comparisons(mod, variables = list(hp = "iqr"))
avg_comparisons(mod, variables = list(hp = "sd"))
avg_comparisons(mod, variables = list(hp = "minmax"))
# using a function to specify a custom difference in one regressor
dat <- mtcars
dat$new_hp <- 49 * (dat$hp - min(dat$hp)) / (max(dat$hp) - min(dat$hp)) + 1
modlog <- lm(mpg ~ log(new_hp) + factor(cyl), data = dat)
fdiff <- \(x) data.frame(x, x + 10)
avg_comparisons(modlog, variables = list(new_hp = fdiff))
# Adjusted Risk Ratio: see the contrasts vignette
mod <- glm(vs ~ mpg, data = mtcars, family = binomial)
avg_comparisons(mod, comparison = "lnratioavg", transform = exp)
# Adjusted Risk Ratio: Manual specification of the `comparison`
avg_comparisons(
mod,
comparison = function(hi, lo) log(mean(hi) / mean(lo)),
transform = exp)
# cross contrasts
mod <- lm(mpg ~ factor(cyl) * factor(gear) + hp, data = mtcars)
avg_comparisons(mod, variables = c("cyl", "gear"), cross = TRUE)
# variable-specific contrasts
avg_comparisons(mod, variables = list(gear = "sequential", hp = 10))
# hypothesis test: is the `hp` marginal effect at the mean equal to the `drat` marginal effect
mod <- lm(mpg ~ wt + drat, data = mtcars)
comparisons(
mod,
newdata = "mean",
hypothesis = "wt = drat")
# same hypothesis test using row indices
comparisons(
mod,
newdata = "mean",
hypothesis = "b1 - b2 = 0")
# same hypothesis test using numeric vector of weights
comparisons(
mod,
newdata = "mean",
hypothesis = c(1, -1))
# two custom contrasts using a matrix of weights
lc <- matrix(c(
1, -1,
2, 3),
ncol = 2)
comparisons(
mod,
newdata = "mean",
hypothesis = lc)
# `by` argument
mod <- lm(mpg ~ hp * am * vs, data = mtcars)
comparisons(mod, by = TRUE)
mod <- lm(mpg ~ hp * am * vs, data = mtcars)
avg_comparisons(mod, variables = "hp", by = c("vs", "am"))
library(nnet)
mod <- multinom(factor(gear) ~ mpg + am * vs, data = mtcars, trace = FALSE)
by <- data.frame(
group = c("3", "4", "5"),
by = c("3,4", "3,4", "5"))
comparisons(mod, type = "probs", by = by)
}
```