Author: Hamid Bulut
Date: 2026-06-18

# Planning a MAIHDA analysis

#### Hamid Bulut

#### 2026-06-18

## Before you fit

through those design decisions, with small runnable checks you can do on your own data first to evaluate dimensions, the number of strata, and the analytic sample

    library(MAIHDA)
    data("maihda_health_data")

## Is MAIHDA the right tool?

MAIHDA is for questions of the form “how much of the variation in an outcome lies between people’s intersectional social positions, and how much of that is more than the sum of its parts?” It is well suited when:

- you have several categorical social dimensions (gender, race/ethnicity, education, class, …) whose joint categories define the strata;
- the outcome is measured at the individual level;
- you have enough individuals to populate the cells (see below).

## The central tradeoff: more dimensions means emptier cells

Strata are the cross-product of the dimensions, so cell counts fall off fast as you add dimensions. `make_strata()` builds the strata and returns a `strata_info` table of counts you can inspect before modelling:

    s2 <- make_strata(maihda_health_data, vars = c("Gender", "Race"))
    nrow(s2$strata_info)                       # number of strata
    #> [1] 10
    summary(s2$strata_info$n)              # cell-size distribution
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>      75     102     127     300     175    1044

Add education and the same sample splits into many more, smaller cells:

    s3 <- make_strata(maihda_health_data, vars = c("Gender", "Race", "Education"))
    nrow(s3$strata_info)
    #> [1] 50
    summary(s3$strata_info$n)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>    1.00   13.25   25.50   60.00   45.50  349.00
    sum(s3$strata_info$n < 10)             # how many strata have < 10 people
    #> [1] 5

Each extra dimension multiplies the number of strata and divides the people among them. Small cells are not fatal, (partial pooling shrinkage is exactly what protects MAIHDA against noisy small strata) but they have consequences (next section). A useful rule: choose the fewest dimensions that answer your question, and look at the cell-size distribution before committing.

## What sparse cells do: singular fits

When cells get very small the maximum-likelihood (`lme4`) estimate of the between-stratum variance can collapse to the boundary ( a singular fit) and report a VPC of (near) zero with no uncertainty. The package records this and surfaces it in a “Fit diagnostics” note rather than letting it pass silently:

    over <- fit_maihda(
      BMI ~ 1 + (1 | Gender:Race:Education),
      data = maihda_health_data[1:60, ]       # deliberately too few people per stratum
    )
    #> boundary (singular) fit: see help('isSingular')
    over
    #> MAIHDA Model
    #> ============
    #>
    #> Engine: lme4
    #> Family: gaussian
    #> Formula: BMI ~ (1 | stratum)
    #>
    #> Fit diagnostics:
    #>   Singular fit: at least one variance component is estimated at (or near) zero.
    #>     The between-stratum variance and any VPC/PCV derived from it may be unreliable.
    #>   Convergence warnings reported by lme4:
    #>     - boundary (singular) fit: see help('isSingular')
    #>
    #>
    #> Underlying model:
    #> Linear mixed model fit by REML ['lmerMod']
    #> Formula: BMI ~ (1 | stratum)
    #>    Data: data
    #> REML criterion at convergence: 386.8857
    #> Random effects:
    #>  Groups   Name        Std.Dev.
    #>  stratum  (Intercept) 0.000
    #>  Residual             6.203
    #> Number of obs: 60, groups:  stratum, 24
    #> Fixed Effects:
    #> (Intercept)
    #>        28.8
    #> optimizer (nloptwrap) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings

If you see a singular-fit note, do not read the VPC as a clean zero. The solution is to collapse dimensions or categories (fewer, larger cells), or to use `engine = "brms"`, whose weakly-informative priors regularise the variance off the boundary and return a posterior interval, the subject of the Bayesian sparse vignette.

## Continuous variables and the analytic sample

- Keep continuous variables out of the strata. A continuous variable in the grouping term gives one stratum per value. `make_strata()` will auto-bin a numeric dimension into tertiles (with a `message()`), but a continuous covariate belongs in the fixed part of the formula, not the strata.

## What the summaries can and cannot tell you

Quantity | Answers | Does not answer
--- | --- | ---
VPC/ICC | share of variance between strata | the amount of between-stratum variation (a share can rise just because the residual fell)
PCV | additive share of the between-stratum variance | a causal decomposition; a negative PCV is not proof of hidden inequality
Discriminatory accuracy (AUC/MOR) | how well strata predict the individual outcome | how large the group differences are (a high VPC can go with modest AUC)

## Which engine, which design?

- `lme4` (default) – fast frequentist fits for adequately-sized cells.
- `brms` – Bayesian; preferred when cells are sparse or dimensions have few levels (regularising priors, posterior intervals).

For extensions beyond the cross-sectional case, see the crossed random effects (dimensions/contexts) and longitudinal vignettes.

## A suggested learning path

1. Introduction to MAIHDA – the end-to-end workflow.
2. Interpreting MAIHDA plots and diagnostics.
3. Finding interaction patterns.
4. Reporting MAIHDA results – tidy output and tables.
5. Specialised designs: binary outcomes, group comparison, survey weights, and Bayesian / sparse.

## References

- Evans, C. R., Leckie, G., Subramanian, S. V., Bell, A., & Merlo, J. (2024). A tutorial for conducting intersectional multilevel analysis of individual heterogeneity and discriminatory accuracy (MAIHDA). SSM - Population Health, 26,