Why Joint Distributions Matter in Synthetic Patient Data

§ · Marginals are not the population

A marginal distribution is one column on its own: the histogram of age, the share of each state, the spread of BMI. A joint distribution is how those columns relate: whether the 70-year-olds are also the ones with lower kidney function, whether the people on insulin are the ones who have diabetes.

When a tool says its data is "realistic," it almost always means the marginals look right. That is necessary but not sufficient. Age can have a perfect distribution, diabetes can have the right prevalence, and insulin use can hit the correct national rate — and the data can still put insulin in the hands of people who have no diabetes, because the two columns were filled in without reference to each other.

§ · What the structure actually looks like

Here is the joint structure in the numeric fields, measured across 8,000 records. Each cell is a Pearson correlation. The strong couplings are the ones a real population has: BMI tracks weight (r = +0.89, because BMI is computed from height and weight) and waist (+0.86); A1c tracks how long someone has had diabetes (+0.72); kidney function falls as age rises (eGFR against age, −0.55); systolic pressure climbs with age (+0.45).

Categorical fields carry the same kind of structure, measured with Cramér's V — 0 when two fields are independent, 1 when one fully determines the other. Blood-pressure medication is tied to a hypertension diagnosis (0.74), insulin to diabetes (0.66), pregnancy to sex (0.62). None of these are coincidences in the data; each is a rule the generator enforces.

§ · The test: shuffle the columns

There is a clean way to see what joint structure is worth. Take the same 8,000 real records and shuffle each column independently — permute the age column, permute the diabetes column, permute the insulin column, each on its own. This is exactly what independent-column generation produces: every column keeps its exact distribution, but the links between columns are destroyed.

The marginals come through untouched — the count of each diabetes category is identical to the row before and after. And yet:

Nine in ten records became impossible, and not one histogram changed. That is the whole argument in one number: a generator can pass every single-column check and still produce a population where almost nobody could exist. Matching marginals is the easy 80%; the joints are the part that makes the data usable.

§ · What the joints encode

The relationships are not decoration. They are the gradients any downstream analysis relies on. Mean A1c climbs cleanly across diabetes status and lands inside the correct diagnostic bands; insulin use rises from 0% in people without diabetes to 25% in diagnosed type-2 to 95% in type-1.

Age drives its own set of gradients — systolic pressure up, kidney function down — the kind of covariation a risk model or a cohort filter reads directly. Shuffle the columns and every one of these lines goes flat.

§ · Why most tools skip it

Not out of oversight — by design. Faker, Mockaroo, and most random-data tools generate each field independently from a type or a pick-list: a name here, an age there, a diagnosis from a weighted list. That is fast, it needs no reference data, and it is perfectly fine when all you want is well-formed values to fill a form or a UI.

It stops being fine the moment anything reads across columns. A risk score, a cohort query, a clinical rule, a join between tables, a model that learns from feature interactions — all of them see the shuffled version: correct margins, meaningless structure. This is the gap the comparison guides cover for Faker and Mockaroo.

§ · How SimpleIDGen keeps them

The generator draws fields in a fixed dependency order, and each field is conditioned on the ones already set. Age is drawn first. Conditions depend on age. Medications depend on conditions. BMI is computed from height and weight rather than sampled. Insulin is gated on a diabetes diagnosis. Because the draw order is fixed and seeded, the same seed reproduces the same people byte-for-byte, and the relationships hold on every row.

This is dependency-ordered conditional sampling, not a learned model — every rule is explicit and auditable. You can see the seed-invariance side of it in the determinism audit, and the joint structure at work in the HbA1c analysis.

§ · The honest limit

Generation only reproduces the joints it models. Look back at the matrix: age against BMI is r = +0.00, because BMI is drawn without conditioning on age, whereas real cohorts show a mild age–BMI link. The structure that is built in is faithful; the structure nobody built in is absent. If a conclusion hinges on one specific real-world correlation, validate it against real data first — see what synthetic data is for where that line falls.

§ · Reproduce the matrix

Both figures are a few lines. Generate a sample, read it into pandas, and the correlation matrix and the shuffle test fall out directly:

import pandas as pd

df = pd.read_json("people.jsonl", lines=True)

# the joint structure: the correlation matrix above
df.corr(numeric_only=True).round(2)

# the counterfactual: shuffle every column independently
shuffled = df.apply(lambda col: col.sample(frac=1).to_numpy())

# marginals are identical, joints are gone
(df["diabetes_status"].value_counts()
    == shuffled["diabetes_status"].value_counts()).all()   # True
((shuffled["on_insulin"]) & (shuffled["diabetes_status"] == "none")).sum()  # > 0

What is synthetic data? →What drives HbA1c (worked analysis) →Faker alternative →Person Profile generator →