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Finding interaction patterns

Hamid Bulut

2026-11-02

Source: vignettes/finding_interactions.Rmd
finding_interactions.Rmd

Overview

A common follow-up question is which strata depart most clearly from the additive expectation.

For this exploratory diagnostic, maihda() can compute the adjusted-model stratum random effects, intervals, and flags. When you use interactions = "BH", the flags use the Benjamini-Hochberg adjustment (Benjamini and Hochberg 1995). The diagnostic is stored in the analysis object and can be reused by the effect-decomposition and predicted-strata plots.

Run a standard analysis and choose the multiplicity rule

The default is adjust = "BH" (Benjamini-Hochberg): scanning all strata to see which ones interact is a screening question, so the flags are a conservative false-discovery-rate screen. Partial pooling shrinks a truly-null stratum’s Wald tail, so the screen under-flags rather than over-flags – it errs strict, never liberal (see ?maihda_interactions). For the uncorrected, per-stratum individual view, set interactions = "none".

The call below names the (default) Benjamini-Hochberg rule explicitly:

library(MAIHDA)
model_bh <- maihda(
  BMI ~ Age + Gender + Race + Education + (1 | Gender:Race:Education),
  data = maihda_health_data,
  interactions = "BH" # Benjamini-Hochberg adjustment (also the default)
)

The printed output reports how many strata were flagged and which adjustment rule was used. The full table is stored in model_bh$interactions.

model_bh$interactions
#> ── Intersectional interactions ─────────────────────────────────────────────────
#> 1 of 50 strata flagged (95% interval; BH-adjusted conservative p-values).
#> Model: adjusted (two-model); interaction on the link (latent) scale.
#> 
#>  stratum                       label   n interaction     se  lower upper
#>        8 male × White × Some College 328       1.359 0.3448 0.6836 2.035
#>    p_value p_adjusted flagged direction
#>  8.056e-05   0.004028    TRUE     above
#> 
#> Interaction BLUPs are shrunken (partially pooled) estimates; treat flags as
#>   exploratory. See ?maihda_interactions.
#>

Each row is one stratum. The main columns are:

  • interaction: the adjusted-model stratum random effect, on the model scale.
  • lower and upper: the interval for that random effect.
  • direction: whether the stratum is above or below the additive expectation.
  • flagged: whether the stratum passes the selected screening rule.

For frequentist fits, the table also includes the conditional standard error, p-value, and adjusted p-value when a correction is requested.

Highlight flagged strata

The plotting methods can reuse the stored diagnostic. Because model_bh was fitted with interactions = "BH", highlight_interactions = TRUE uses the Benjamini-Hochberg flags. In the effect-decomposition plot, the labels also follow that same flagged set.

plot(model_bh, type = "effect_decomp", highlight_interactions = TRUE)

The same flags can be reused in the predicted-strata view.

plot(model_bh, type = "predicted", highlight_interactions = TRUE)

If the analysis was fitted without a stored interaction diagnostic, pass the adjustment rule directly:

plot(model, type = "effect_decomp", highlight_interactions = "BH")

Is an interaction negligible? (equivalence / ROPE)

A flag asks whether an interaction differs from zero. Often the more useful question is whether it is large enough to matter. Pass a region of practical equivalence (rope, on the model scale) and each stratum is classified from its interval (Schuirmann 1987; Kruschke 2018) as relevant (interval entirely outside the region), negligible (entirely inside it), or inconclusive (straddling a bound):

maihda_interactions(model_bh, rope = 0.5)
#> ── Intersectional interactions ─────────────────────────────────────────────────
#> 1 of 50 strata flagged (95% interval; BH-adjusted conservative p-values).
#> Model: adjusted (two-model); interaction on the link (latent) scale.
#> Equivalence vs ROPE [-0.5, 0.5]: 1 relevant | 0 negligible | 49 inconclusive.
#> 
#>  stratum                       label   n interaction     se  lower upper
#>        8 male × White × Some College 328       1.359 0.3448 0.6836 2.035
#>    p_value p_adjusted flagged direction decision
#>  8.056e-05   0.004028    TRUE     above relevant
#> 
#> Interaction BLUPs are shrunken (partially pooled) estimates; treat flags as
#>   exploratory. See ?maihda_interactions.
#>

There is no default rope: the smallest interaction worth caring about is a substantive choice, made on the model’s (link) scale.

See also

References

Benjamini, Yoav, and Yosef Hochberg. 1995. “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society: Series B (Methodological) 57 (1): 289–300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x.
Kruschke, John K. 2018. “Rejecting or Accepting Parameter Values in Bayesian Estimation.” Advances in Methods and Practices in Psychological Science 1 (2): 270–80. https://doi.org/10.1177/2515245918771304.
Schuirmann, Donald J. 1987. “A Comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability.” Journal of Pharmacokinetics and Biopharmaceutics 15 (6): 657–80. https://doi.org/10.1007/BF01068419.