Binary Outcome Models for Experimental Research

When the outcome is zero or one, its conditional mean is a probability, and two models compete to describe it. The linear probability model reports differences in probability directly and can predict outside the zero-to-one range; logistic regression keeps probabilities in range and reports odds ratios, which are neither risk ratios nor probability differences. The two differ in the functional form assumed for $E [Y ∣ X]$ , in whether fitted values stay inside $[0, 1]$ , in the effect heterogeneity the link implies, in estimation, and in behaviour under extrapolation. The reporting scale is one consequence of that choice rather than the whole of it, and whichever scale is used must be stated.

Definition

For $Y \in {0, 1}$ , the conditional mean is a probability: $E [Y ∣ X] = P (Y = 1 ∣ X)$ . The linear probability model specifies $P (Y = 1 ∣ X) = X β$ , so each coefficient is a change in probability per unit change in the predictor, holding the others fixed; fitted values may fall outside $[0, 1]$ and the errors are inherently heteroskedastic, so robust standard errors are usual. Logistic regression specifies

\begin{aligned} P (Y = 1 ∣ X) & = \frac{1}{1 + e^{- X β}}, \\ \log \frac{p}{1 - p} & = X β, \end{aligned}

keeping fitted probabilities in range and modelling the log-odds linearly. A one-unit increase in $X_{j}$ multiplies the odds by $O R = e^{β_{j}}$ . An odds ratio is not a risk ratio and not a probability difference. Reporting on the probability scale requires transforming the coefficients, or using them to compute an appropriate probability-scale estimand: fitted probabilities, a risk difference, a discrete change over a stated contrast, a standardized contrast, or an average marginal effect, according to the quantity wanted. In a randomized two-arm experiment the unadjusted difference in sample proportions is the difference in mean outcomes, $\hat{τ} = {\bar{Y}}_{1} - {\bar{Y}}_{0} = {\hat{p}}_{1} - {\hat{p}}_{0}$ , the estimated risk difference.

Formal statement

$E [Y ∣ X] = P (Y = 1 ∣ X)$ ; LPM $P (Y = 1 ∣ X) = X β$ ; logistic $P (Y = 1 ∣ X) = (1 + e^{- X β})^{- 1}$ with $\log \frac{p}{1 - p} = X β$ and $O R = e^{β_{j}}$ ; randomized risk difference $\hat{τ} = {\hat{p}}_{1} - {\hat{p}}_{0}$ .

Assumptions and scope

An odds ratio is not a risk ratio and not a probability difference. Reporting one as though it were another misstates the magnitude, and the discrepancy grows as the outcome becomes more common.
Expressing logistic results on the probability scale requires transforming the coefficients or computing an appropriate probability-scale estimand. Which one depends on the question: fitted probabilities, a risk difference, a discrete change over a stated contrast, a standardized contrast, or an average marginal effect. The coefficients alone supply none of them.
The linear probability model can produce fitted probabilities outside $[0, 1]$ , which is a real defect when predictions near the boundaries matter and often harmless when the interest is an average difference in the middle of the range.
Errors in the linear probability model are inherently heteroskedastic because the variance of a Bernoulli outcome depends on its mean, so heteroskedasticity-robust standard errors are the default.
In a randomized two-arm experiment the difference in sample proportions is already the estimated risk difference; no model is required to obtain it.
Logistic coefficients are not comparable across models with different covariate sets, even when the added covariates are unrelated to treatment, because the scale itself shifts.
Logistic regression appears in this subject for two distinct jobs: modelling the outcome, and modelling treatment assignment for a propensity score. The estimand and the criteria for judging the fit differ, and the two must not be conflated.

Worked material

Example

One odds ratio, two very different effects

Two trials, each reporting an odds ratio of 2.0 for treatment.

Trial A: a rare outcome. Control event rate 1%. Odds in control: $0.01 / 0.99 \approx 0.0101$ . Doubling gives $0.0202$ , so $p_{1} = 0.0202 / 1.0202 \approx 1.98 %$ .

Risk difference: about 0.98 percentage points
Risk ratio: about 1.98

The odds ratio of 2.0 and the risk ratio of 1.98 are nearly identical. Reading "twice as likely" is approximately right.

Trial B: a common outcome. Control event rate 33%. Odds in control: $0.33 / 0.67 \approx 0.493$ . Doubling gives $0.985$ , so $p_{1} = 0.985 / 1.985 \approx 49.6 %$ .

Risk difference: about 16.6 percentage points
Risk ratio: about 1.50

Here the odds ratio of 2.0 corresponds to a risk ratio of 1.5. Reading "twice as likely" overstates the effect by a third.

Why this matters for decisions. Trial A's treatment moves one person per hundred; Trial B's moves seventeen. Both report $O R = 2.0$ . A reader given only the odds ratio cannot tell these apart, and the rare-disease case is where the habit of reading odds as risk is formed and then carried into settings where it fails.

What to report. The baseline rate alongside any odds ratio, or better, absolute risks in both arms and their difference.

Non-example

Statements these models do not support

"An odds ratio of 2 means twice as likely." True only when the outcome is rare. At a 33% baseline it corresponds to a risk ratio of about 1.5.

Reporting a logistic coefficient as a change in probability. $β_{j}$ is on the log-odds scale. Getting to probabilities requires marginal effects, which depend on where in the covariate space they are evaluated.

Comparing logistic coefficients across models with different covariates. The scale itself shifts when covariates are added, even covariates unrelated to treatment, so the coefficients are not comparable in the way OLS coefficients are.

Rejecting the linear probability model solely because it can predict outside $[0, 1]$ . When the interest is an average difference in the middle of the range, out-of-range fitted values at the extremes may be irrelevant to the estimand.

Using classical standard errors with the linear probability model. The error variance depends on the fitted probability by construction, so heteroskedasticity is guaranteed, not merely possible.

Confusing an outcome model with a propensity model. Logistic regression appears in this subject for both jobs. One estimates $P (Y = 1 ∣ X)$ ; the other estimates $P (W = 1 ∣ X)$ and is judged by balance and overlap.

Contrast

Choosing the reporting scale

	Linear probability model	Logistic regression
Models	$P (Y = 1 ∣ X) = X β$	$\log \frac{p}{1 - p} = X β$
Coefficient reads as	Change in probability	Change in log-odds; $e^{β_{j}}$ is an odds ratio
Fitted values	May leave $[0, 1]$	Always in $(0, 1)$
Standard errors	Robust, always — heteroskedasticity is structural	Model-based or robust
Directly decision-ready	Yes, a probability difference	No, needs marginal effects
Comparable across specifications	Yes	No — the scale shifts

Why the odds scale causes so much trouble. Odds are unfamiliar outside betting, and an odds ratio sounds like it should be a ratio of chances. It is a ratio of $p / (1 - p)$ , and that denominator is what makes it diverge from a risk ratio as $p$ grows.

The baseline rate resolves it. Given the control-arm probability, any odds ratio converts to a risk difference and a risk ratio. Reporting an odds ratio without the baseline leaves a reader unable to recover the magnitude.

Which to prefer. If the estimand is an average difference in probability, usually the case in a randomized trial, the linear probability model or plain proportions report it directly. If fitted probabilities near the boundaries matter, or the design calls for it, logistic regression plus marginal effects gets to the same scale by a longer route.

What both share. Neither supplies a causal reading. That comes from the design, exactly as in the continuous-outcome case.

Common errors

Common misconception

An odds ratio of 2 means the outcome is twice as likely, so odds ratios can be reported as though they were risk ratios or probability differences.

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Linear Regression for Experimental Research

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Propensity Scores (contrasts with)

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Binary Outcome Models for Experimental Research

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Inferential Foundations for Experimental Research

Sources

All of Statistics: A Concise Course in Statistical Inference (2004)
Causal Inference: What If (2020)

Binary Outcome Models for Experimental Research

Assumptions and scope

Worked material

One odds ratio, two very different effects

Statements these models do not support

Choosing the reporting scale

Common errors

Related units

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Connected

Learn this topic

Used in

Sources

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