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

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, and a single factor-level (omnibus) effect size can be obtained for a multi-level factor or multi-df interaction. 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) , Correll, Mellinger, and Pedersen (2022) , and Rights and Sterba (2019) .

pecanr

pecanr computes partial eta-squared (eta2p) effect sizes for fixed effects in linear mixed models fitted with lme4. It correctly handles crossed, nested, and mixed (crossed-and-nested) random effects structures – including random slopes – using a variance decomposition approach that translates slope variances to the outcome scale.

Why pecanr?

pecanr accounts for:

• Crossed designs – any number of grouping factors (subjects, items, raters, etc.)
• Nested designs – hierarchical structures with automatic level detection
• Mixed designs – grouping factors that are simultaneously nested within some variables and crossed with others (e.g., photos nested within models, but both crossed with participants)
• Random slopes – translated to the outcome scale via sigma^2_slope x sigma^2_X
• All random-effects syntaxes – single-bar (x | g), double-bar (x || g), explicit (1 | g) + (0 + x | g), and interaction-grouping (1 | a:b) terms are all matched correctly (as of 0.3.1)
• Factor predictors – supplied directly, by variable name or coefficient name, with no manual recoding to numeric
• Omnibus effect sizes – a single factor-level value for a multi-level factor or multi-df interaction, matching the omnibus F or chi-square test
• Operative effect sizes – excluding variance components that don’t contribute to the standard error of the tested effect
• Unique variance – per-predictor effect sizes use unique (semipartial) variance by default for correlated predictors; partial_predictors = FALSE for the raw total-variance numerator
• Confidence intervals – optional, via parametric bootstrap

Installation

Install the released version from CRAN:

install.packages("pecanr")

Or the development version from GitHub:

# install.packages("pak")
pak::pak("bcohen0901/pecanr")

Upgrading from 0.3.0 or earlier? Version 0.3.1 fixes two bugs that silently dropped random-effect variance components from the error term for models using the double-bar syntax (x || g), explicit (1 | g) + (0 + x | g) terms, or interaction-grouping terms like (1 | photo:model). Effect sizes for such models were previously too large and will be smaller (correct) after upgrading. Models using only single-bar terms are unaffected. See NEWS.md.

Usage

Crossed design (subjects x items)

library(lme4)
library(pecanr)

model <- lmer(y ~ condition + (1 | subject) + (1 | item), data = my_data)

eta2p(model, effect = "condition", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item"))

Three crossed factors

model3 <- lmer(y ~ condition + (1 | subject) + (1 | item) + (1 | rater),
               data = my_data)

eta2p(model3, effect = "condition", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item", "rater"))

Nested design

model_nested <- lmer(y ~ treatment + (1 | school/class), data = my_data)

eta2p(model_nested, effect = "treatment", data = my_data,
      design    = "nested",
      nest_vars = c("class", "school"))

Mixed design (nested-and-crossed)

Use design = "mixed" when some grouping factors are nested within others but all levels are crossed with additional factors. A common example is participants viewing multiple photos of each model: photos are nested within models, but both levels are crossed with participants.

model_mixed <- lmer(y ~ x + (1 | participant) + (1 | model) + (1 | photo:model),
                    data = my_data)

eta2p(model_mixed, effect = "x", data = my_data,
      design     = "mixed",
      cross_vars = "participant",
      nest_vars  = c("photo", "model"))

Factor predictors

Factor predictors work directly – pass either the variable name or a specific coefficient name. A factor with more than two levels maps to several coefficients, so request one contrast at a time, use eta2p_omnibus() for the whole factor, or use batch_eta2p().

model_f <- lmer(rating ~ group + (1 | subject) + (1 | item), data = my_data)

# variable name is resolved to its coefficient automatically
eta2p(model_f, effect = "group", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item"))

Omnibus effect size for a multi-level factor or interaction

eta2p_omnibus() returns a single factor-level partial eta-squared for a multi-df effect, corresponding to its omnibus F or chi-square test (rather than to individual contrasts).

model_e <- lmer(rating ~ group * emotion + (1 | subject) + (1 | item),
                data = my_data)

eta2p_omnibus(model_e, effect = "emotion", data = my_data,
              design     = "crossed",
              cross_vars = c("subject", "item"))

# interactions: give the term in colon form
eta2p_omnibus(model_e, effect = "group:emotion", data = my_data,
              design     = "crossed",
              cross_vars = c("subject", "item"))

Batch over all effects

batch_eta2p(model, data = my_data,
            design     = "crossed",
            cross_vars = c("subject", "item"))

Operative effect sizes

eta2p(model, effect = "condition", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item"),
      operative  = TRUE)

Unique variance and correlated predictors

By default, the variance attributed to a single predictor is its unique (semipartial) variance — its total variance times its tolerance, Var(X) * tol(X) — so each effect size reflects only the predictor's unique contribution and declines as its redundancy with the other predictors increases. For centered, orthogonal designs this is identical to using the total variance, so it matters only when predictors are correlated.

To use the total-variance (raw) numerator instead, set partial_predictors = FALSE:

eta2p(model, effect = "x1", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item"),
      partial_predictors = FALSE)

Confidence intervals

A parametric bootstrap confidence interval is available via ci = TRUE. Because the effect size is a ratio of REML variance components, the interval is obtained by simulating from the fitted model, refitting, and recomputing the effect size on each replicate. This refits the model n_boot times, so it is opt-in and can be slow on large models.

eta2p(model, effect = "condition", data = my_data,
      design     = "crossed",
      cross_vars = c("subject", "item"),
      ci         = TRUE,
      n_boot     = 1000,
      seed       = 1)

A note on interactions and centering

For an interaction, the variance entering the numerator is computed from the mean-centered constituent predictors automatically (the product itself is not centered). As a result the interaction effect size is invariant to the location of its constituent predictors – you do not need to center them yourself for the effect size to be well defined. Centering continuous predictors before fitting is still good practice for a separate reason: it makes the lower-order (main-effect) coefficients interpretable in models that contain interactions.

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.

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.

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.