# Interpreting MAIHDA Plots and Diagnostics

#### Hamid Bulut

#### 2026-07-09

## Overview

`plot()` on a fitted MAIHDA model gives you several views of the same model, each answering a different question. This vignette explains, for every plot type, what it shows, how to read it, and what not to conclude from it. Calling `plot()` with no specified `type` draws them all.

Let’s start with fitting the model.

    library(MAIHDA)
    data("maihda_health_data")

    health_complete <- maihda_health_data[complete.cases(
      maihda_health_data[, c("BMI", "Age", "Gender", "Race", "Education")]
    ), ]

    model <- maihda(
      BMI ~ Age + Gender + Race + Education + (1 | Gender:Race:Education),
      data = health_complete
    )

We fit with `maihda()` rather than a bare `fit_maihda()` because `plot()` on the analysis routes each view to the model it is valid on: the VPC, predicted, and shrinkage views below read the null model (the total between-stratum inequality), while the effect-decomposition views read the adjusted model (whose fixed effects carry the additive part, so the stratum random effect is the pure interaction). `plot()` on a single `fit_maihda()` fit shows every view on that one model – with this formula that would be the adjusted model, whose VPC is the much smaller residual interaction share, not the headline between-stratum share.

## `vpc` – variance partition

The VPC plot shows how the total unexplained variance splits into a between-stratum component (the numerator of the VPC/ICC), any other random-effect components, and the within-stratum residual.

    plot(model, type = "vpc")

How to read it. The between-stratum slice is the share of unexplained variation that lies between intersectional groups. For non-Gaussian models this split is on the latent scale.

## `predicted` – stratum predictions with intervals

    plot(model, type = "predicted")

How to read it. Each point is a stratum’s model-based prediction with a 95% interval, ordered so you can see which intersections sit above or below the average. The estimates are shrunken toward the overall mean. For a binomial model the axis is the predicted probability instead of the outcome mean.

## `obs_vs_shrunken` – shrinkage made visible

    plot(model, type = "obs_vs_shrunken")

How to read it. This contrasts the raw observed stratum means with the model’s shrunken estimates. Points far from the diagonal are strata whose raw mean was pulled substantially toward the centre, typically the smallest, noisiest cells. It is a sanity check on how much the multilevel model is regularising.

## `effect_decomp` – additive vs. intersection-specific

    plot(model, type = "effect_decomp")

How to read it. This separates, for each stratum, the part of its deviation from the grand mean that is explained by the additive main effects from the intersection-specific part (the stratum random effect, what is left over after the additive effects). Large intersection-specific bars are the candidate “more/less than the sum of the parts” intersections, but treat them as hypotheses to probe, not as confirmed interactions, since they also absorb sample composition and estimation noise.

## `prediction_deviation` – the deviation dashboard

    plot(model, type = "prediction_deviation")

How to read it. This two-panel dashboard highlights the most notable cases or strata. What counts as “notable” depends on the model: the largest deviation from the mean prediction (Gaussian/Poisson), the largest absolute deviance residual (binomial), or the most surprising observation (ordinal).

## Group-comparison plots

When you fit across a higher-level grouping variable with `compare_maihda_groups()` (or `maihda(group = ...)`), extra plot types become available – `group_vpc`, `group_components`, `group_between_variance`, and `group_pcv` (also reachable as `type = "vpc"`, `"components"`, `"between_variance"`, and `"pcv"` on the comparison object). Those are covered in the group comparison vignette.

## Customizing the appearance

Every `plot()` call with a single `type` returns a plain ggplot object, so you are never locked into the package’s styling – restyle it with the usual ggplot2 grammar by adding a theme, overriding the labels, or dropping in another layer. Themes, `labs()`, and added layers compose cleanly:

    library(ggplot2)

    plot(model, type = "vpc") +
      theme_classic(base_size = 13) +
      labs(title = "Variance partition, restyled")

The views that map a fill or colour (`vpc`, `context_vpc`, `effect_decomp`) also accept a replacement palette. ggplot2 prints a harmless “Scale for fill is already present … which will replace the existing scale” message as it swaps the built-in palette out (suppressed here):

    plot(model, type = "vpc") +
      scale_fill_brewer(palette = "Set2")

A few plot types return something other than a single ggplot, so they restyle slightly differently:

- `prediction_deviation` is a two-panel patchwork. `+ theme_*()` styles only the active panel; use `&` to apply a theme to both panels at once:

      plot(model, type = "prediction_deviation") & theme_minimal()

- `type = "all"` returns a named list of ggplot objects (it prints them as a side effect), so pick one out to restyle it:

      plots <- plot(model)          # list: vpc, predicted, effect_decomp, ...
      plots$predicted + theme_bw()

## See also

- Introduction to MAIHDA – the end-to-end workflow.
- MAIHDA for binary outcomes – how these plots adapt to logistic models.