An odds ratio of 0.43 for a risk ratio of 0.58

Fit it, and read what it prints

import pandas as pd
import numpy as np
import statsmodels.formula.api as smf

referrals = pd.read_csv("protection-referrals-2024.v1.csv")
referrals["disability"] = referrals["disability_reported"].map(
    {"true": 1, "Yes": 1, "false": 0, "No": 0})

consenting = referrals[referrals["consent_to_refer"]].copy()
consenting["completed"] = (consenting["referral_accepted"]
                           & consenting["days_to_first_service"].notna()).astype(int)
d = consenting.dropna(subset=["disability", "case_category", "age_band",
                              "sex", "service_requested", "admin1"])

crude = smf.logit("completed ~ disability", data=d).fit(disp=0)
print(np.exp(crude.params).round(3))
print(np.exp(crude.conf_int()).round(3))

library(dplyr)

crude <- glm(completed ~ disability, data = d, family = binomial())
exp(cbind(OR = coef(crude), confint(crude)))

Odds ratio 0.430, 95% CI 0.308 to 0.601.

Now compute what the data plainly say.

rates = d.groupby("disability")["completed"].agg(["sum", "size", "mean"])
print(rates.round(4))
p1, p0 = rates.loc[1, "mean"], rates.loc[0, "mean"]
print(f"risk ratio        {p1 / p0:.3f}")
print(f"risk difference   {p1 - p0:+.4f}")
print(f"odds ratio        {(p1/(1-p1)) / (p0/(1-p0)):.3f}")

d |> summarise(k = sum(completed), n = n(), rate = mean(completed), .by = disability)

Group	Completed	n	Rate
Disability reported	52	197	26.4%
Not reported	629	1,384	45.5%

Measure	Value	Reads as
Risk ratio	0.581	58% as likely to complete
Risk difference	−19.1 points	19 fewer completions per 100 cases
Odds ratio	0.430	—

The model printed 0.430 and the answer is 0.581. Both are correct; they are answers to different questions, and only one of them is the question anyone asked.

Why they differ, and when it gets worse

An odds is p / (1 − p). When p is small the denominator is near 1 and odds are close to risks, so the two ratios nearly agree. When p is large they do not.

for p0 in (0.02, 0.10, 0.25, 0.45, 0.70):
    p1 = 0.6 * p0                      # a true risk ratio of 0.6 throughout
    odds = (p1 / (1 - p1)) / (p0 / (1 - p0))
    print(f"baseline {p0:5.0%}   risk ratio 0.60   odds ratio {odds:.3f}")

# One true risk ratio, five baselines, five different odds ratios.

Baseline risk	True risk ratio	Odds ratio
2%	0.60	0.59
10%	0.60	0.57
25%	0.60	0.53
45%	0.60	0.45
70%	0.60	0.31

The odds ratio is not a fixed distortion of the risk ratio — it depends on the baseline. At a 2% outcome the two are interchangeable, which is why the odds ratio survives in epidemiology, where outcomes are rare.

Programme outcomes are not rare. Referral completion is 43%, attendance is 88%, enrolment is 97%, and at those baselines the odds ratio is a long way from what a reader will take it to mean. This is not a subtlety — it is the ordinary case in this work.

The trap that catches analysts, not just readers

Adding covariates changes an odds ratio even when nothing is confounded, and this surprises people who have only worked with linear models.

adjusted = smf.logit(
    "completed ~ disability + case_category + age_band + sex"
    " + service_requested + admin1", data=d).fit(disp=0)
print(f"crude OR    {np.exp(crude.params['disability']):.3f}")
print(f"adjusted OR {np.exp(adjusted.params['disability']):.3f}")

adjusted <- glm(completed ~ disability + case_category + age_band + sex +
                  service_requested + admin1, data = d, family = binomial())
exp(coef(adjusted))["disability"]

	Odds ratio	Risk difference
Crude	0.430	−19.05 points
Adjusted	0.388	−19.41 points

The odds ratio moved by 10% and the risk difference moved by a third of a point. The usual reading — “adjustment revealed a stronger effect” — is wrong here. Nothing was confounded away; the odds ratio simply is not collapsible.

A collapsible measure equals the average of the subgroup measures. Risk differences and risk ratios are; odds ratios are not. Add a covariate that predicts the outcome and the conditional odds ratio moves away from 1 even when the covariate is unrelated to the exposure.

So an adjusted odds ratio and a crude odds ratio cannot be compared to judge confounding. That comparison is the standard move in every regression tutorial and it does not work for logistic models. Compare risk differences instead, which the next lesson computes.

When the odds ratio is the right number

Three cases, and it is worth being precise because the answer is not “never”.

A case-control study. Cases and controls are sampled separately, so risks are not estimable at all and the odds ratio is the only ratio the design supports. It is what the measure was invented for.

A rare outcome. Below about 10%, the odds ratio approximates the risk ratio closely enough that the distinction stops mattering. Say which one you computed anyway.

Comparing to published literature that reports odds ratios. Then report both, and lead with the risk difference.

def report(label, p1, p0):
    return (f"{label}: {p1:.1%} vs {p0:.1%} — "
            f"risk difference {p1-p0:+.1%}, risk ratio {p1/p0:.2f}, "
            f"odds ratio {(p1/(1-p1))/(p0/(1-p0)):.2f}")

print(report("Referral completion, disability reported", p1, p0))

# Print all three. The reader picks; you do not pick for them by omission.

The sentence to write, and three not to

Not this: “Cases reporting a disability were 57% less likely to complete a referral (OR 0.43).” Two errors in one sentence — an odds ratio is not a likelihood, and 57% is not the reduction.

Nor this: “Disability halved the odds of completion.” Correct arithmetic, and “halved” will be read as risk by everyone who is not a statistician.

Nor this: “OR 0.43 (95% CI 0.31–0.60, p < 0.001).” Complete, checkable, and it tells a programme manager nothing they can act on.

This: “Cases reporting a disability completed a referral at 26.4% against 45.5% for cases with no disability reported — 19.1 percentage points lower (95% CI −25.7 to −12.4), or 58% of the completion rate. Adjusted for case category, age, sex, service requested and department, the gap is 19.4 points. The adjusted odds ratio is 0.39; it is reported here for comparability with published studies and should not be read as a risk.”

The last clause is the one that does the work. An odds ratio in a programme report with no translation beside it will be read as a risk by almost everyone who sees it, including the people who commissioned the analysis.

Report it whole

Referral completion by disability status, 1,581 consenting cases

  Disability reported     26.4%   52 of 197
  Not reported            45.5%   629 of 1,384

  Risk difference   -19.1 points   95% CI -25.7 to -12.4
  Risk ratio          0.58
  Odds ratio          0.43 (crude), 0.39 (adjusted)

  The odds ratio is reported for comparability only. Completion is a common
  outcome (43% overall), so the odds ratio overstates the relative
  difference: 0.43 in odds is 0.58 in risk.

  The adjusted odds ratio differs from the crude one partly because the odds
  ratio is not collapsible, not only because of confounding. The adjusted
  risk difference is -19.4 points, essentially unchanged from crude.

  Fitted on the 1,581 consenting cases with complete covariates. The
  statistics course reported this gap on all 1,638 consenting cases and got
  -19.4 points; the 57-case difference is the complete-case restriction.

The last paragraph is two lines and prevents the single most common over-claim — that adjustment “strengthened” a finding when the measure simply moved for arithmetic reasons.

What comes next

A logistic model prints coefficients on a scale nobody thinks in. The next lesson converts them back into probabilities — the number a programme manager can multiply by a caseload — and shows what the model says about the two gates the referral pathway actually has.