# MAIHDA for Binary Outcomes (Discriminatory Accuracy)

#### Hamid Bulut

#### 2026-07-09

## Why binary outcomes?

Many of the original MAIHDA applications target a binary health outcome – obese vs. not, hypertensive vs. not, screened vs. not. Merlo’s framing of discriminatory accuracy is, at heart, a question about a binary classifier: how well do the intersectional strata alone separate people who do and do not have the outcome?

For a binary outcome the model is a multilevel logistic regression:

\[ \text{logit}\,\Pr(y_{ij} = 1) = \beta_0 + u_j, \qquad u_j \sim N(0, \sigma_u^2), \]

where \(j\) indexes the intersectional stratum. MAIHDA reads off two complementary quantities from this model:

- the VPC/ICC, the share of the (latent) variation that lies between strata;
- the discriminatory accuracy (e.g. the AUC / C-statistic), how well stratum membership predicts the individual outcome.

A high between-stratum VPC can still go with only moderate discriminatory accuracy at the individual level – that contrast is the whole point of the “DA” in MAIHDA, and this vignette shows how to get both numbers.

    library(MAIHDA)
    data("maihda_health_data")

    # A two-level outcome: obese (Yes) vs. not (No)
    table(maihda_health_data$Obese)
    #>
    #>   No  Yes
    #> 1923 1077

## Fitting a logistic MAIHDA model

`fit_maihda()` detects a two-level outcome automatically. If you leave `family` at its default, the function switches to `family = "binomial"`, recodes the response to 0/1 for `glmer()`, and warns you so the change is never silent:

    model_null <- fit_maihda(
      Obese ~ 1 + (1 | Gender:Race:Education),
      data = maihda_health_data
    )
    #> Warning: The outcome variable appears to be binary. Automatically switching to
    #> family = 'binomial'. To fit a Linear Probability Model, explicitly specify
    #> family = 'gaussian'.
    #> Binary outcome 'Obese' recoded to 0/1: 'No' = 0 (reference), 'Yes' = 1 (modeled event). Set the factor levels (or supply a 0/1 outcome) to control which level is the event.

The warning is a feature, not a problem. To be explicit (and to silence the warning), pass `family = "binomial"` yourself – this is the recommended form for scripts:

    model_null <- fit_maihda(
      Obese ~ 1 + (1 | Gender:Race:Education),
      data   = maihda_health_data,
      family = "binomial"
    )

> If you actually want a linear probability model on a 0/1 outcome, ask for it explicitly with `family = "gaussian"`; the auto-switch only fires for the default family.

## The VPC is on the latent scale

`summary()` reports the VPC/ICC for the logistic model. There is no observed-scale residual variance for a Bernoulli outcome, so MAIHDA uses the standard latent-scale approximation: the level-1 (within-stratum) variance is fixed at \(\pi^2/3 \approx 3.29\) for the logit link (and the corresponding constant for a probit link). The VPC is therefore the between-stratum share of variation on that underlying latent scale, not on the probability scale.

    summary_null <- summary(model_null)
    print(summary_null)
    #> MAIHDA Model Summary
    #> ====================
    #>
    #> Variance Partition Coefficient (VPC/ICC):
    #>   Estimate: 0.0634
    #>
    #> Variance Components:
    #>                  component variance     sd proportion
    #>   Between-stratum (random)   0.2227 0.4719    0.06339
    #>  Within-stratum (residual)   3.2899 1.8138    0.93661
    #>                      Total   3.5125 1.8742    1.00000
    #>
    #> Discriminatory accuracy (binomial MAIHDA)
    #>   AUC (C-statistic): 0.626
    #>   Median Odds Ratio: 1.568
    #>   Cases / controls:  1077 / 1923
    #>
    #> Fixed Effects:
    #>         term estimate
    #>  (Intercept)   -0.616
    #>
    #> Stratum Estimates (first 10):
    #>  stratum stratum_id                           label random_effect     se
    #>        1          1  male × Hispanic × Some College     -0.075834 0.3155
    #>        2          2     male × Black × College Grad     -0.001524 0.3325
    #>        3          3   female × White × College Grad     -0.360949 0.1186
    #>        4          4     male × Hispanic × 8th Grade      0.218443 0.3808
    #>        5          5    female × Mexican × 8th Grade      0.437816 0.2809
    #>        6          6     male × White × College Grad     -0.293351 0.1187
    #>        7          7    female × White × High School     -0.006285 0.1322
    #>        8          8     male × White × Some College      0.502977 0.1077
    #>        9          9 female × White × 9 - 11th Grade      0.259733 0.1835
    #>       10         10 female × Hispanic × High School     -0.006773 0.3208
    #>  lower_95 upper_95
    #>   -0.6941  0.54246
    #>   -0.6533  0.65025
    #>   -0.5935 -0.12843
    #>   -0.5280  0.96489
    #>   -0.1127  0.98833
    #>   -0.5259 -0.06076
    #>   -0.2654  0.25283
    #>    0.2919  0.71405
    #>   -0.1000  0.61948
    #>   -0.6355  0.62197
    #>   ... and 40 more strata

Read from this null model, the VPC is the total between-stratum share of latent variation in the odds of obesity. As in the Gaussian case, the stratum random effects capture the combined additive + interaction differences across strata; they isolate the pure intersectional (interaction) component only once the additive main effects of the strata variables are in the model.

> For a binomial model `summary()` now reports the discriminatory accuracy (AUC / Median Odds Ratio) automatically, so the printed summary above already carries that block – the Discriminatory accuracy section below explains how to read it and how to obtain the pieces on their own.

For an interpretable probability-scale complement to the latent-scale VPC, the package provides `maihda_vpc_response()`, which estimates the response-scale VPC by simulation (Goldstein, Browne & Rasbash 2002):

    maihda_vpc_response(model_null, seed = 1)
    #> Response-scale VPC (simulation method)
    #>   VPC: 0.0478
    #>   10000 simulated stratum effects; between-stratum variance 0.2227 (latent scale).

Report it alongside – not instead of – the latent-scale VPC: it depends on the overall prevalence, and for adjusted models it is evaluated at the mean covariate profile, so it is best read from the null model.

## Adjusted model and PCV

A “Model 2” adds individual-level covariates to ask how much of the between-stratum variation they account for. Here we adjust for age, fit on the same analytic sample and strata as the null model, and compare with `calculate_pcv()`. PCV compares variances across models, so both must use the same complete-case sample:

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

    model_null2 <- fit_maihda(
      Obese ~ 1 + (1 | Gender:Race:Education),
      data = health_complete, family = "binomial"
    )
    #> Binary outcome 'Obese' recoded to 0/1: 'No' = 0 (reference), 'Yes' = 1 (modeled event). Set the factor levels (or supply a 0/1 outcome) to control which level is the event.

    # Model 2: adjust for an individual-level covariate (age)
    model_adj <- fit_maihda(
      Obese ~ Age + (1 | Gender:Race:Education),
      data = health_complete, family = "binomial"
    )
    #> Binary outcome 'Obese' recoded to 0/1: 'No' = 0 (reference), 'Yes' = 1 (modeled event). Set the factor levels (or supply a 0/1 outcome) to control which level is the event.

    pcv <- calculate_pcv(model_null2, model_adj)
    print(pcv)
    #> Proportional Change in Variance (PCV)
    #> =====================================
    #>
    #> PCV: 0.0167
    #>
    #> Between-stratum variance:
    #>   Model 1: 0.222667
    #>   Model 2: 0.218941
    #>   Change:  0.003726 (1.67%)
    #>
    #> Interpretation (PCV is the proportional change in between-stratum
    #> variance between the models):
    #>   Between-stratum variance is 1.7% lower in Model 2 than in Model 1.

The same caveats as in the continuous case apply, with one extra wrinkle: for non-Gaussian models the latent residual variance is fixed by the link, so a change in the between-stratum variance is partly a rescaling of the latent scale, not only “variance explained”. Interpret the PCV as a model-dependent, descriptive change, not a causal decomposition.

> A note on adjusting for the strata’s own categories. If instead you add the categorical main effects that define the strata (e.g. `Obese ~ Gender + Race + Education + ...`) to recover the additive-vs-interaction split shown in the introduction, be aware that in a logistic model those fixed effects are nearly collinear with the stratum random intercept, so `glmer()` often reports a convergence or “nearly unidentifiable” note. Scaling covariates and using `control = lme4::glmerControl(optimizer = "bobyqa")` usually helps.

## Discriminatory accuracy (AUC and Median Odds Ratio)

The VPC summarises variation; discriminatory accuracy summarises prediction. For a binomial model this is reported automatically: `summary()` of a binomial `maihda_model` carries a `discriminatory_accuracy` slot, and `maihda(..., family = "binomial")` surfaces it on its summaries and headline `print()`. The explicit `maihda_discriminatory_accuracy()` below is the same quantity on its own – it bundles the two individual-level summaries for a logistic MAIHDA model: the AUC / C-statistic (how well the predicted probabilities separate cases from non-cases) and the Median Odds Ratio (MOR) (the between-stratum heterogeneity expressed on the odds-ratio scale). The strata-only model’s AUC is the discriminatory accuracy of the intersectional strata themselves – Merlo’s central quantity. Comparing it with the adjusted model shows whether individual information beyond stratum membership sharpens classification:

    da_null <- maihda_discriminatory_accuracy(model_null2)
    da_adj  <- maihda_discriminatory_accuracy(model_adj)

    da_null
    #> Discriminatory accuracy (binomial MAIHDA)
    #>   AUC (C-statistic): 0.626
    #>   Median Odds Ratio: 1.568
    #>   Cases / controls:  1077 / 1923
    da_adj
    #> Discriminatory accuracy (binomial MAIHDA)
    #>   AUC (C-statistic): 0.628
    #>   Median Odds Ratio: 1.563
    #>   Cases / controls:  1077 / 1923

You can also call the pieces directly: `maihda_auc(prob, y)` on any vector of predicted probabilities and 0/1 outcomes (it equals the Mann-Whitney U statistic, so it needs no extra package), and `maihda_mor(model)` for the Median Odds Ratio.

    prob_null <- predict_maihda(model_null2, type = "individual", scale = "response")
    y_obs     <- as.numeric(lme4::getME(model_null2$model, "y"))

    maihda_auc(prob_null, y_obs)
    #> [1] 0.6261898

An AUC of 0.5 is chance. Here both models sit around 0.6: even a non-trivial between-stratum VPC translates into only modest accuracy at the individual level – exactly the cautionary message that motivates discriminatory-accuracy reporting in MAIHDA. Note the adjusted model barely moves the AUC: the categorical covariates that define the strata are already captured by stratum membership, so only the continuous covariate (`Age`) adds genuinely new individual-level information.

> The MOR needs the logit link. The Median Odds Ratio is an odds-ratio-scale quantity, so it is defined only for `binomial(link = "logit")` (the default). For a `binomial(link = "probit")` fit, `maihda_mor()` errors and `maihda_discriminatory_accuracy()` reports the AUC with `mor = NA` – the AUC, being rank-based, is link-agnostic.

## Plots adapt to the binomial family

The standard plots recognise the binomial family and switch to the probability scale and to deviance-based diagnostics automatically:

    # Predicted probabilities per stratum with intervals
    plot(model_adj, type = "predicted")

    # Latent-scale variance partition
    plot(model_adj, type = "vpc")

    # For binomial fits the dashboard highlights the largest absolute
    # deviance residuals rather than raw deviations from the mean.
    plot(model_adj, type = "prediction_deviation")

See the plot interpretation vignette for how to read each of these.

## Count outcomes work the same way

A Poisson (count) outcome follows the identical pattern – pass `family = "poisson"` to `fit_maihda()`. The VPC then uses the log-link latent-scale residual variance, and the summary, PCV, and plotting helpers all behave as above.

## References

- Merlo, J. (2018). Multilevel analysis of individual heterogeneity and discriminatory accuracy (MAIHDA) within an intersectional framework. Social Science & Medicine, 203, 74-80.

- Evans, C. R., Williams, D. R., Onnela, J. P., & Subramanian, S. V. (2018). A multilevel approach to modeling health inequalities at the intersection of multiple social identities. Social Science & Medicine, 203, 64-73.

- Merlo, J., Wagner, P., Ghith, N., & Leckie, G. (2016). An original stepwise multilevel logistic regression analysis of discriminatory accuracy: the case of neighbourhoods and health. PLOS ONE, 11(4), e0153778.