Permutation tests for mixed-effects models

Why lmer doesn’t give p-values

lme4::lmer() deliberately omits p-values from its output. The reason is principled: in mixed-effects models, the degrees of freedom for fixed effects are not well-defined. Approximations exist (Satterthwaite, Kenward-Roger), but they rely on assumptions about the variance-covariance structure that may not hold in small or unbalanced datasets.

grouped_perm_glmm() sidesteps this entirely by computing p.perm — a fully nonparametric p-value based on permuting the outcome variable and refitting the model. No degree-of-freedom approximation is needed.

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When to use grouped_perm_glmm()

Use grouped_perm_glmm() when your data have:

• Repeated measures — the same subject observed at multiple time points or conditions
• Hierarchical structure — participants nested within sites, scans nested within participants, and so on
• Small samples — fewer than ~50 subjects, where asymptotic approximations are unreliable

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A worked example

We simulate a repeated-measures dataset with 20 subjects, each observed 4 times. The outcome depends on a continuous fixed effect and a binary group variable, with random intercepts per subject.

set.seed(42)
n_subj  <- 20
n_obs   <- 4

df <- data.frame(
  subject = rep(seq_len(n_subj), each = n_obs),
  time    = rep(seq_len(n_obs), times = n_subj),
  group   = rep(c("control", "case"), each = n_subj * n_obs / 2),
  y       = NA_real_
)

# Random intercepts + fixed effects
random_int  <- rep(rnorm(n_subj, sd = 1.2), each = n_obs)
df$y <- 0.8 * (df$group == "case") +
        0.3 * df$time +
        random_int +
        rnorm(nrow(df), sd = 0.8)

head(df, 8)
#>   subject time   group           y
#> 1       1    1 control  1.69983926
#> 2       1    2 control  0.82010339
#> 3       1    3 control  2.40761625
#> 4       1    4 control  3.81688990
#> 5       2    1 control  1.13851696
#> 6       2    2 control -0.42201311
#> 7       2    3 control  0.01654669
#> 8       2    4 control -0.88816827

Fit the permutation GLMM with 999 permutations:

res <- grouped_perm_glmm(
  tbl         = df,
  formla      = y ~ group + time + (1 | subject),
  var_to_perm = "y",
  permNum     = 999,
  seed        = 42
)
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res
#> # A tibble: 3 × 5
#>   term         effect estimate statistic  p.perm
#>   <chr>        <chr>     <dbl>     <dbl>   <dbl>
#> 1 (Intercept)  fixed     0.918     1.67  0.879  
#> 2 groupcontrol fixed    -0.153    -0.211 0.828  
#> 3 time         fixed     0.213     2.65  0.00501

The result contains only fixed-effect rows (random-effect parameters are filtered out). Columns are the same as grouped_perm_glm() except p.value is absent and effect is always "fixed".

---

Comparing to lme4 output directly

It is instructive to compare the permutation p-value against lmerTest’s Satterthwaite approximation:

library(lmerTest)
#> Loading required package: lme4
#> Loading required package: Matrix
#> 
#> Attaching package: 'lmerTest'
#> The following object is masked from 'package:lme4':
#> 
#>     lmer
#> The following object is masked from 'package:stats':
#> 
#>     step
fit <- lmer(y ~ group + time + (1 | subject), data = df)
summary(fit)$coefficients
#>                Estimate Std. Error       df    t value   Pr(>|t|)
#> (Intercept)   0.9182706  0.5501158 23.82026  1.6692313 0.10816365
#> groupcontrol -0.1528794  0.7240395 18.00000 -0.2111479 0.83514393
#> time          0.2130416  0.0805092 59.00000  2.6461768 0.01042014

The t-statistics will match exactly — ptestR fits the same model. Only the p-values differ: p.perm uses the empirical null distribution, while lmerTest uses an approximate t-distribution with estimated degrees of freedom.

---

Random effects structure

ptestR permutes only the outcome variable. The random-effects structure (grouping factors, random slopes) is held fixed. This means:

• Permutations are exchangeable at the observation level
• The between-subject variance is preserved across permutations
• The test is valid under exchangeability of residuals, which is a weaker assumption than normality

For more complex designs (e.g. cross-classified random effects, or permutation within clusters), manual permutation strategies may be needed — ptestR does not currently support restricted permutation schemes.

---

Random slopes

The formula interface is passed directly to lme4::lmer(), so random slopes work as expected:

res_slopes <- grouped_perm_glmm(
  tbl         = df,
  formla      = y ~ group + time + (1 + time | subject),
  var_to_perm = "y",
  permNum     = 499,
  seed        = 42
)
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00486158 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00274302 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00328921 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00201901 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00398618 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00255742 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.020955 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00862873 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00230504 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0201226 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
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#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00532124 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0037847 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00233226 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00200157 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00274921 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.00392532 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 0.0026986 (tol = 0.002, component 1)
#>   See ?lme4::convergence and ?lme4::troubleshooting.
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
#> boundary (singular) fit: see help('isSingular')
res_slopes
#> # A tibble: 3 × 5
#>   term         effect estimate statistic p.perm
#>   <chr>        <chr>     <dbl>     <dbl>  <dbl>
#> 1 (Intercept)  fixed    0.873      2.04  0.727 
#> 2 groupcontrol fixed   -0.0627    -0.104 0.928 
#> 3 time         fixed    0.213      2.22  0.0180

Note that more complex random-effects structures increase computation time proportionally with permNum.

---

Convergence warnings

With small datasets and complex random-effects structures, some permuted fits may produce convergence warnings from lme4. These are expected and do not affect the validity of p.perm — the permuted t-statistics from borderline fits still contribute to the null distribution. To suppress these warnings during bulk permutation runs:

suppressWarnings(
  grouped_perm_glmm(df, y ~ group + (1 | subject), "y", permNum = 999)
)

---

Computational cost

Each permutation refits the full model. For a dataset of 80 observations with a random intercept, 1 000 permutations typically takes a few seconds. For 10 000 permutations or larger datasets, consider parallelising across features rather than within a single call — see vignette("grouped-analysis").

---

Further reading

• vignette("getting-started") — grouped_perm_glm() and core concepts
• vignette("grouped-analysis") — applying permutation tests across many features using dplyr::group_by() and purrr::map()
• ?grouped_perm_glmm