Package {pecanr}

Title:	Partial Eta-Squared for Crossed, Nested, and Mixed Linear Mixed Models
Version:	0.4.0
Description:	Computes partial eta-squared effect sizes for fixed effects in linear mixed models fitted with the 'lme4' package. Supports crossed, nested, and mixed (crossed-and-nested) random effects structures with any number of grouping factors. Mixed designs handle cases where grouping factors are simultaneously crossed with some variables and nested within others (e.g., photos nested within models, but both crossed with participants). Factor predictors are supported directly. Random slope variances are translated to the outcome scale using a variance decomposition approach, correctly accounting for predictor scaling and interaction terms. Both general and operative effect sizes are provided, with optional parametric bootstrap confidence intervals. For correlated predictors, per-predictor effect sizes use unique (semipartial) variance by default. Methods are based on Correll, Mellinger, McClelland, and Judd (2020) <doi:10.1016/j.tics.2019.12.009>, Correll, Mellinger, and Pedersen (2022) <doi:10.3758/s13428-021-01687-2>, and Rights and Sterba (2019) <doi:10.1037/met0000184>.
License:	MIT + file LICENSE
URL:	https://github.com/bcohen0901/pecanr
BugReports:	https://github.com/bcohen0901/pecanr/issues
Encoding:	UTF-8
Imports:	lme4, stats
Suggests:	spelling, testthat (≥ 3.0.0)
Config/testthat/edition:	3
Language:	en-US
Config/roxygen2/version:	8.0.0
NeedsCompilation:	no
Packaged:	2026-10-20 19:00:59 UTC; brandoncohen
Author:	Brandon Cohen [aut, cre], Joshua Correll [aut, ths]
Maintainer:	Brandon Cohen <brandon.cohen-1@colorado.edu>
Repository:	CRAN
Date/Publication:	2026-10-20 19:30:02 UTC

pecanr: Partial Eta-Squared for Crossed, Nested, and Mixed Linear Mixed Models

Description

Computes partial eta-squared effect sizes for fixed effects in linear mixed models fitted with the 'lme4' package. Supports crossed, nested, and mixed (crossed-and-nested) random effects structures with any number of grouping factors. Mixed designs handle cases where grouping factors are simultaneously crossed with some variables and nested within others (e.g., photos nested within models, but both crossed with participants). Factor predictors are supported directly. Random slope variances are translated to the outcome scale using a variance decomposition approach, correctly accounting for predictor scaling and interaction terms. Both general and operative effect sizes are provided, with optional parametric bootstrap confidence intervals. For correlated predictors, per-predictor effect sizes use unique (semipartial) variance by default. Methods are based on Correll, Mellinger, McClelland, and Judd (2020) doi:10.1016/j.tics.2019.12.009, Correll, Mellinger, and Pedersen (2022) doi:10.3758/s13428-021-01687-2, and Rights and Sterba (2019) doi:10.1037/met0000184.

Author(s)

Maintainer: Brandon Cohen brandon.cohen-1@colorado.edu (ORCID)

Authors:

• Brandon Cohen brandon.cohen-1@colorado.edu (ORCID)

• Joshua Correll joshua.correll@colorado.edu (ORCID) [thesis advisor]

See Also

Useful links:

• https://github.com/bcohen0901/pecanr

• Report bugs at https://github.com/bcohen0901/pecanr/issues

Batch Calculate Partial Eta-Squared for Multiple Effects

Description

Calculates partial eta-squared for all fixed effects (excluding the intercept) in a model, returning one row per model coefficient. For a multi-level factor or multi-df interaction this yields one value per contrast (per design-matrix column), not a single factor-level value.

Usage

batch_eta2p(
  model,
  data,
  design = c("crossed", "nested", "mixed"),
  subj_var = NULL,
  item_var = NULL,
  cross_vars = NULL,
  nest_vars = NULL,
  operative = FALSE,
  partial_predictors = TRUE,
  verbose = FALSE
)

Arguments

Value

A data frame with one row per effect containing:

Rows are sorted by eta2p in decreasing order.

Partial Eta-Squared for Linear Mixed Models

Description

Calculates partial eta-squared effect sizes for fixed effects in linear mixed models with crossed, nested, or mixed (crossed-and-nested) random effects.

Usage

eta2p(
  model,
  effect,
  data,
  design = c("crossed", "nested", "mixed"),
  subj_var = NULL,
  item_var = NULL,
  cross_vars = NULL,
  nest_vars = NULL,
  effect_level = NULL,
  var_x = NULL,
  slope = NULL,
  operative = FALSE,
  partial_predictors = TRUE,
  ci = FALSE,
  ci_level = 0.95,
  n_boot = 1000,
  seed = NULL,
  verbose = FALSE
)

## S3 method for class 'eta2p_lmm'
print(x, ...)

## S3 method for class 'eta2p_lmm'
summary(object, ...)

Arguments

Details

Variance decomposition

The function implements a variance decomposition approach for computing partial eta-squared in mixed models. The variance a fixed effect explains is

\mathrm{Var}_{\text{explained}} = b^2 \times \sigma^2_X,

where b is the (partial) fixed-effect coefficient and \sigma^2_X is the variance of the predictor column. Random slope variances are translated to the outcome scale by the same principle,

\sigma^2_{\text{slope}}(Y) = \sigma^2_b \times \sigma^2_X.

For an interaction, \sigma^2_X is the variance of the product of the mean-centered constituent predictors (see "Centering" below). The var_x argument bypasses this computation when the variance is known a priori.

Centering and intended scope

These quantities are exact when predictors are mean-centered, which is standard and recommended practice for the experimental and contrast-coded designs the method primarily targets (sum/contrast coding centers factors automatically). Two points:

• Interactions: components are centered automatically. For an interaction the function mean-centers each constituent predictor and then takes the variance of their product (it does not center the product itself). This makes the interaction effect size invariant to the location of the constituents: re-centering a predictor by a constant leaves the model fit and the interaction coefficient unchanged, and now also leaves the effect size unchanged. (The variance of the raw product depends on the constituents' means and so is not well defined as an effect size; this is why the centering is applied. The constituent predictors should still be centered before fitting for the coefficients themselves to be interpretable, but the effect-size value no longer depends on it.)

• Main effects / random slopes use \sigma^2_X, which equals E[X^2] only when X is centered. The function does not center main-effect predictors (doing so would not change \sigma^2_X = \mathrm{Var}(X), which is already location-invariant), but the random-slope translation \sigma^2_b \sigma^2_X is exact only for centered predictors; center continuous predictors before fitting.

Under correlated predictors, each b is a partial coefficient (it controls for the other predictors). By default (partial_predictors = TRUE) it is multiplied by the predictor's unique variance – the variance remaining after the part linearly predictable from the other predictors is removed – so the variance attributed to the predictor is

b^2 \, \mathrm{Var}(X \mid X_{others}) = b^2 \, \mathrm{Var}(X) \, \mathrm{tol}(X),

where \mathrm{tol}(X) = 1 - R^2_{X \sim X_{others}} is the predictor's tolerance. (The two forms are algebraically identical.) This yields a semipartial variance-explained measure: it reflects only the unique contribution of the predictor and declines as its redundancy with the others increases, matching the unique-variance target estimated by an explained- error (model-comparison) approach.

Setting partial_predictors = FALSE instead uses the predictor's total variance \sigma^2_X. The resulting per-predictor values are partial (conditional) effect sizes that do not isolate unique variance when predictors are correlated. For centered, orthogonal designs the two options coincide exactly, because the covariance among predictors is zero, so the choice matters only under collinearity.

The tolerance correction is also what makes the interaction handling location-invariant: centering an interaction's components is equivalent to the \mathrm{Var} \times \mathrm{tol} of the product, which is constant across re-centering. The choice is independent of the operative option, which concerns the error denominator rather than the numerator. See Rights & Sterba (2019) on the partial-versus-total distinction.

General vs. operative effect sizes

For general effect sizes (default), all variance components appear in the denominator. For operative effect sizes (operative = TRUE), only components that contribute to the standard error of the effect are included. In mixed designs, each grouping factor (both crossed and nested) is independently classified as "within" or "between" the focal effect; between-subjects intercept variances are excluded from the operative denominator.

The operative option concerns only the denominator; it is independent of the partial_predictors setting, which controls the numerator. The two compose: an operative effect size uses the same numerator a general one would (unique variance by default), paired with the reduced operative denominator.

Mixed designs

Use design = "mixed" when the model contains both crossed and nested random effects simultaneously. A canonical example is participants viewing multiple photos of each model (stimulus): photos are nested within models, but both levels are crossed with participants.

The corresponding lme4 model and eta2p call would be:

fit <- lmer(y ~ x + (1 | participant) + (1 | model) + (1 | photo:model),
            data = dat)
eta2p(fit, "x", dat,
      design     = "mixed",
      cross_vars = "participant",
      nest_vars  = c("photo", "model"))

Residual variance is attributed to the crossed side and counted exactly once; the nested calculator's residual is subtracted to prevent double-counting.

Crossed designs

Supports any number of grouping factors via cross_vars. The two-argument form (subj_var + item_var) is retained for backward compatibility and is equivalent to cross_vars = c(subj_var, item_var).

Confidence intervals

With ci = TRUE, a confidence interval is obtained by parametric bootstrap via lme4::bootMer: responses are simulated from the fitted model, the model is refit, and eta2p is recomputed on each replicate; ci_lower and ci_upper are the percentile limits. Because eta2p is a ratio of REML variance components with no closed-form sampling distribution, this bootstrap is the appropriate interval rather than an analytic (e.g. noncentral) formula. Set seed for reproducibility, and start with a small n_boot to gauge runtime on large models.

Value

An object of class "eta2p_lmm". Printing it shows the estimate on one line; summary() shows the full breakdown. It contains:

Functions

• print(eta2p_lmm): Concise print method: reports the estimate (and CI, if present) on one line. Use summary() for the full breakdown.

• summary(eta2p_lmm): Comprehensive summary: the estimate plus the numerator ingredients (coefficient, var(X), tolerance) and the error variance components that make up the denominator.

References

Correll, J., Mellinger, C., McClelland, G. H., & Judd, C. M. (2020). Avoid Cohen's 'Small', 'Medium', and 'Large' for Power Analysis. Trends in Cognitive Sciences, 24(3), 200–207. doi:10.1016/j.tics.2019.12.009

Correll, J., Mellinger, C., & Pedersen, E. J. (2022). Flexible approaches for estimating partial eta squared in mixed-effects models with crossed random factors. Behavior Research Methods, 54, 1626–1642. doi:10.3758/s13428-021-01687-2

Rights, J. D., & Sterba, S. K. (2019). Quantifying explained variance in multilevel models: An integrative framework for defining R-squared measures. Psychological Methods, 24(3), 309–338. doi:10.1037/met0000184

See Also

batch_eta2p to compute per-coefficient values for all fixed effects at once.

Examples

library(lme4)

#  Two crossed factors (backward-compatible call)
set.seed(42)
crossed_data <- data.frame(
  y         = rnorm(120),
  condition = rep(c(-0.5, 0.5), 60),
  subject   = factor(rep(1:20, each = 6)),
  item      = factor(rep(1:6, 20))
)
model_c <- lmer(y ~ condition + (1 | subject) + (1 | item),
                data = crossed_data)
res <- eta2p(model_c, "condition", crossed_data,
             design   = "crossed",
             subj_var = "subject",
             item_var = "item")
res                 # concise: just the estimate
summary(res)        # full breakdown: coefficient, var(X), error components

#  Sensitivity analysis: the effect size a hypothetical slope would imply,
#  holding the error structure fixed (useful alongside simr)
eta2p(model_c, "condition", crossed_data,
      design   = "crossed",
      subj_var = "subject",
      item_var = "item",
      slope    = 0.03)

#  Three crossed factors using cross_vars
set.seed(42)
three_way_data <- data.frame(
  y         = rnorm(180),
  condition = rep(c(-0.5, 0.5), 90),
  subject   = factor(rep(1:20, each = 9)),
  item      = factor(rep(rep(1:6, each = 3), 10)),
  rater     = factor(rep(1:3, 60))
)
model_3 <- lmer(y ~ condition +
                  (1 | subject) + (1 | item) + (1 | rater),
                data = three_way_data)
eta2p(model_3, "condition", three_way_data,
      design     = "crossed",
      cross_vars = c("subject", "item", "rater"))

#  Mixed design: photos nested within models, crossed with participants
# Participants each view multiple photos of multiple models.
# Photos are nested within models (each photo belongs to one model),
# but both levels are crossed with participants.
set.seed(42)
n_subj  <- 30; n_model <- 10; n_photo <- 3
mixed_data <- expand.grid(
  participant = factor(seq_len(n_subj)),
  photo_id    = factor(seq_len(n_model * n_photo))
)
mixed_data$model_id <- factor(
  rep(seq_len(n_model), each = n_photo)[as.integer(mixed_data$photo_id)]
)
mixed_data$x <- rnorm(nrow(mixed_data))
mixed_data$y <- rnorm(nrow(mixed_data))

model_m <- lmer(y ~ x +
                  (1 | participant) +
                  (1 | model_id) +
                  (1 | photo_id:model_id),
                data = mixed_data)
eta2p(model_m, "x", mixed_data,
      design     = "mixed",
      cross_vars = "participant",
      nest_vars  = c("photo_id", "model_id"))

#  Supply predictor variance directly (var_x)
eta2p(model_c, "condition", crossed_data,
      design     = "crossed",
      cross_vars = c("subject", "item"),
      var_x      = 1)   # +/-1 binary predictor: var = 1 by design

#  Factor predictor (no manual recoding needed)
crossed_data$grp <- factor(rep(c("A", "B"), 60))
model_f <- lmer(y ~ grp + (1 | subject) + (1 | item), data = crossed_data)
# accepts the variable name "grp"; resolves to the coefficient automatically
eta2p(model_f, "grp", crossed_data,
      design = "crossed", cross_vars = c("subject", "item"))

#  Operative effect size
eta2p(model_c, "condition", crossed_data,
      design = "crossed", cross_vars = c("subject", "item"),
      operative = TRUE)

#  Total-variance (raw) numerator instead of the default unique variance
eta2p(model_c, "condition", crossed_data,
      design = "crossed", cross_vars = c("subject", "item"),
      partial_predictors = FALSE)

#  Confidence interval by parametric bootstrap
eta2p(model_c, "condition", crossed_data,
      design = "crossed", cross_vars = c("subject", "item"),
      ci = TRUE, n_boot = 200, seed = 1)

#  All fixed effects at once (per-coefficient)
batch_eta2p(model_c, crossed_data,
            design = "crossed", cross_vars = c("subject", "item"))