Getting a Risk Ratio for a Common Outcome: The Step-Down When Log-Binomial Hits the Wall

อ่านฉบับภาษาไทย (Thai version)

Abstract

For a binary outcome reported as a risk ratio, the models form a fixed step-down. Fit log-binomial regression first. If it does not converge, use modified Poisson regression with a robust standard error, then a Gaussian working model with a robust standard error. Switch to logistic regression only when an odds ratio is acceptable. The order makes sense once three questions inside the word "model" are separated: which effect to report, which link, and which variance. The log link is what makes a coefficient a risk ratio; the family only sets the variance. Log-binomial matches binary data, but its fitted risk must stay at or below 1 for every patient, so its optimizer searches against a wall and can stall. Modified Poisson keeps the same mean equation, removes the wall and repairs the wrong Poisson variance with a robust standard error. This article walks through the step-down and then the reasons, with small hand calculations.

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Complications after surgery, and a model that will not converge

A surgical team studies factors linked to complications after an operation. The outcome is binary: a patient either has a complication or does not. The team wants to report an adjusted risk ratio, so it starts with a log-binomial regression.

The model does not converge: its fitting algorithm never settles on stable estimates. What should the team fit next? The answer follows a fixed step-down, and the reasons rest on three ideas: the link, the variance and a wall at a risk of 1.

The step-down for a risk ratio

When the outcome is binary, such as cured or not, dead or alive, diseased or not, and the result to report is a risk ratio (RR), the choice of model follows four steps in a fixed order [1, 2]. Each step is used only when the one above it fails or does not fit the question.

1. Log-binomial regression: start here

Goal: the risk ratio, directly.

The rule is short: binary outcome + a risk ratio wanted → start with log-binomial. The log-binomial model pairs the binomial distribution, the natural one for a yes-or-no outcome, with a log link, which models the log of the risk. It matches binary data and gives the risk ratio directly.

Its main weakness is that it sometimes does not converge, especially when the model is complex or has many covariates [3]. If the log-binomial model works, keep it. If the software reports a convergence failure, go down to step 2.

2. Modified Poisson regression + robust variance

This is the main fallback when log-binomial does not converge: log-binomial fails → modified Poisson [4].

Poisson regression was built for count data. Here the analysis only borrows Poisson's way of computing, to estimate a risk ratio. The data, though, are binary, not Poisson, so the ordinary Poisson variance does not suit them.

So a robust variance, or robust standard error (SE), is added: Poisson + robust SE = modified Poisson. The result can still be reported as a risk ratio.

3. Gaussian working model + robust variance

If modified Poisson still has problems, the next step in the order is a Gaussian working model + robust variance. The key words are working model. The Gaussian (normal) distribution is used as a tool to estimate the model; it does not mean the binary outcome has become normal data [5].

As with modified Poisson, a robust variance makes the standard error appropriate. The model keeps the log link, so its coefficient is still a risk ratio. The order so far: log-binomial → modified Poisson → Gaussian robust.

4. Logistic regression: when an odds ratio is accepted instead

If the earlier models do not suit the data, or the study may report an odds ratio (OR) in place of the risk ratio, use logistic regression. Logistic regression gives an odds ratio, not a risk ratio. This is the step where the effect measure itself changes: logistic → OR.

When the outcome is uncommon, the odds ratio and the risk ratio are close. As the outcome becomes more common, they can drift further apart [1]. A risk of 10% is often used as a memory aid for "rare", but treat it as a rule of thumb, not a fixed mathematical wall.

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Back to the surgical team

The team wants an adjusted risk ratio for a binary complication outcome, and the log-binomial fit did not converge.

1. What is wanted?

   A risk ratio, adjusted for covariates, for a binary outcome. That points to the risk-ratio chain, not to logistic regression.

2. Where did it fail?

   At step 1: log-binomial regression did not converge.

3. Step down once

   The main fallback for a failed log-binomial fit is modified Poisson regression with a robust variance. It still estimates a risk ratio.

Result: Next model: modified Poisson regression with robust variance. The reason in one line: a risk ratio is wanted but log-binomial did not converge, so modified Poisson with robust variance is the next step.

Why the step-down looks like this: three questions in one word

The word model mixes three separate questions: which effect to report, which link to use, and which variance to assume. Separate them, and the step-down becomes simple. There are several models because they answer different effect measures, or compute the same one in different ways.

Start with the effect. Take a 0/1 outcome such as neonatal death, with a risk of 10% in the unexposed group and 20% in the exposed group. The same data give three answers.

Same data, three answers

Logistic regression gives an odds ratio

Logistic regression models the log odds:

\[ \operatorname{logit}(p) = \log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X \]

so

\[ e^{\beta_1} = \mathrm{OR} \]

That is the whole reason logistic regression gives an odds ratio. It is not wrong; it answers a different question. If OR = 3.9, do not write "the risk is 3.9 times higher". It means the odds are 3.9 times higher, and when the outcome is common the odds ratio can sit far from the risk ratio.

Why a log link gives a risk ratio

To get a risk ratio directly, model the log of the risk, not the log odds.

1. The model

   \[ \log(p) = \beta_0 + \beta_1 X \;\Rightarrow\; p = e^{\beta_0 + \beta_1 X} \]

   Exponentiating turns the log of the risk back into the risk.

2. Unexposed, X = 0

   \[ p_0 = e^{\beta_0} \]

   Only the intercept is left.

3. Exposed, X = 1

   \[ p_1 = e^{\beta_0 + \beta_1} \]

   The intercept plus the exposure coefficient.

4. Divide

   \[ \frac{p_1}{p_0} = \frac{e^{\beta_0 + \beta_1}}{e^{\beta_0}} = e^{\beta_1} \]

   The intercept cancels.

Result: RR = exp(β1): the exponentiated exposure coefficient is the risk ratio.

Why both log-binomial and modified Poisson exist

Both models use the same log link. Log-binomial assumes $Y \sim \text{Binomial}$ with $\log(p) = X\beta$. Modified Poisson uses the working model $Y \sim \text{Poisson}$ but keeps the same $\log(p) = X\beta$. Here $X\beta$ is shorthand for $\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots$.

So in both, $e^{\beta}$ is read as a risk ratio. Why build two models if they give the same risk ratio?

Log-binomial is built to match a binary outcome directly, but it is harder to compute. Modified Poisson accepts a variance model that does not match a binary outcome to make fitting easier, then uses a robust SE to repair the variance.

Separate the link from the family

This is the root of the whole topic. Think of a model as two pieces: family + link [6].

The link sets what a coefficient means. With the logit link, $\log\{p/(1-p)\} = X\beta$, a coefficient gives an odds ratio. With the log link, $\log(p) = X\beta$, it gives a risk ratio. The link decides the main effect measure a coefficient shows.

The family sets how the variance is computed. A binary outcome truly has $\mathrm{Var}(Y) = p(1-p)$, the binomial variance. Poisson assumes $\mathrm{Var}(Y) = p$, which is wrong for a binary outcome.

Two variances at a risk of 0.30

1. Binomial, the true variance

   \[ p(1-p) = 0.3(0.7) = 0.21 \]

   The variance a binary outcome really has at p = 0.30.

2. Poisson, the assumed variance

   \[ \mathrm{Var}(Y) = p = 0.30 \]

   Poisson sets the variance equal to the mean.

Result: Modified Poisson says: borrow Poisson to estimate the coefficients, because it fits easily, but do not believe its variance; compute the SE with a robust sandwich instead. That is what "modified" means.

What the robust SE is for

First separate the coefficient from its standard error. The estimate $\hat\beta$ and its standard error $SE(\hat\beta)$ are different things.

The coefficient tells the effect, for example $e^{\beta_1} = 1.89$. The SE tells the uncertainty around that coefficient, and it builds the confidence interval and the P value.

In modified Poisson, the Poisson variance is wrong for a binary outcome, so the ordinary SE may be wrong too. The robust sandwich SE, computed from the observed spread of the data rather than from the family's formula, keeps inference usable even when the assumed variance is wrong [4]. Part 4 shows how the sandwich works.

Why log-binomial needs no "modification", and where it breaks

A binary outcome really is binomial, so the binomial variance $p(1-p)$ is already right. That makes the model conceptually natural: binary outcome → binomial family → log link → risk ratio [7].

The problem is the limit $0 \le p \le 1$. Log-binomial sets $p = e^{X\beta}$, and whenever $X\beta > 0$, $e^{X\beta} > 1$: an impossible probability.

The optimizer, the algorithm that searches for the best coefficients, must therefore search inside a narrow region where every observation has $p_i \le 1$. In some datasets that search ends in a convergence failure [3].

Why modified Poisson does not hit this problem

The Poisson model does not force a fitted mean to stay at or below 1. It will accept, for example, $\hat p = 1.03$. Read as a probability, that is impossible, but the optimization becomes much easier. That is the trade-off.

Log-binomial versus modified Poisson

Crude model: why both give exactly the same risk ratio

With one binary exposure, $X = 0$ or $X = 1$, the model has just enough parameters to produce two risks, $p_0$ and $p_1$. It therefore fits both groups' observed risks exactly.

1. Observed risks

   \[ p_0 = 0.0067241, \quad p_1 = 0.0139787 \]

   The risk in the unexposed and in the exposed group.

2. Risk ratio

   \[ \mathrm{RR} = \frac{0.0139787}{0.0067241} = 2.0789 \]

   Log-binomial and modified Poisson both reproduce these two risks, so both give this ratio.

Result: The crude model only asks: what is the exposed risk divided by the unexposed risk? Both models reach the same answer, 2.0789.

Adjusted model: why they can differ slightly

Add covariates, here antenatal care (ANC), age and socioeconomic status (SES), and each patient gets a risk of their own:

\[ \log(p_i) = \beta_0 + \beta_1\,\mathrm{ANC}_i + \beta_2\,\mathrm{Age}_i + \beta_3\,\mathrm{SES}_i + \cdots \]

Now $p_i$ differs from patient to patient. Binomial and Poisson use different variances, and so different weights, to estimate $\beta$. The two models may then give, for example, RR = 1.886733 and RR = 1.886706: a small difference.

Rare outcome: why they agree even more

The binomial variance is $p(1-p)$; the Poisson variance is $p$.

1. A rare outcome

   \[ p = 0.0088, \quad 1 - p = 0.9912 \]

   Almost every patient is free of the outcome.

2. Binomial variance

   \[ p(1-p) = 0.0088 \times 0.9912 = 0.00872 \]

   Hardly different from the Poisson variance, p = 0.0088.

Result: When the outcome is rare, binomial and Poisson see the variance almost the same way, so their results come out nearly identical.

A clear difference is not automatically interaction

One claim is too strong: "if the two adjusted models differ clearly, suspect interaction". Interaction is one possible cause, not the only one. A clear gap between log-binomial and modified Poisson can come from several kinds of model misspecification [8]:

• an interaction left out of the model;
• a continuous covariate that is really nonlinear but entered as linear;
• a wrong functional form;
• omitted covariates;
• a model that forces one common risk ratio when the risk ratio really varies with the covariates;
• extreme fitted risks.

If the two methods differ a lot, especially when the outcome is common, check the model specification rather than concluding interaction automatically. Also, modified Poisson usually converges very easily, but it should never be said to converge 100% of the time.

One tree for every binary outcome

With a binary outcome, start by asking: what do I want to report? The answer picks the link, and the link picks the model.

So there are not "many models" to memorise. There are only two questions. The first: which effect, OR, RR or RD? The second: if RR, estimate it with log-binomial or with modified Poisson + robust SE?

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Where the p in modified Poisson comes from

Writing $\log(p) = X\beta$ for modified Poisson can look odd, because ordinary Poisson regression has no p. Real Poisson regression is:

\[ Y_i \sim \text{Poisson}(\mu_i), \quad \log(\mu_i) = X_i\beta, \quad \mu_i = e^{X_i\beta} \]

Here $\mu_i$ is an expected count, a mean count, not a probability. Take the number of hospital admissions in a year, $Y_i = 0, 1, 2, 3, \ldots$, modelled as $\log(\mu_i) = \beta_0 + \beta_1\,\mathrm{Treatment}_i$. Then $e^{\beta_1} = \mu_1/\mu_0$, a count ratio or rate ratio depending on the setting (see Poisson regression for rates).

For a binary outcome, the mean is the risk

Modified Poisson puts a binary outcome, $Y_i \in \{0, 1\}$, into that model. A binary variable has one key property:

\[ E(Y) = 0(1-p) + 1(p) = p \]

So the Poisson mean model $\log(E[Y_i]) = X_i\beta$ can be rewritten as $\log(p_i) = X_i\beta$. That is why it looks so much like log-binomial.

Three models side by side

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A risk above 1 never comes from the data

In reality a probability can never exceed 1. A value of $p > 1$ is not produced by the data. It is a temporary value the model tries while it searches for $\beta$. So separate the real data from the equation used to estimate.

Real data, trial coefficients

Unexposed group: 20/100 have the outcome, so $p_0 = 0.20$. Exposed group: 40/100, so $p_1 = 0.40$. Everything lies between 0 and 1, as it should.

1. What the regression is told

   \[ \log(p) = \beta_0 + \beta_1 X \;\Rightarrow\; p = e^{\beta_0 + \beta_1 X} \]

   It does not know the answer in advance; the computer has to search for the best β0 and β1.

2. A trial that works

   \[ \beta_0 = -1.5,\ \beta_1 = 0.7: \quad p = e^{-1.5 + 0.7} = e^{-0.8} = 0.45 \]

   For X = 1 this risk is possible.

3. A trial that fails

   \[ \beta_0 = -0.2,\ \beta_1 = 0.5: \quad X\beta = -0.2 + 0.5 = 0.3, \quad p = e^{0.3} = 1.35 \]

   For X = 1 this is a probability of 135%, which is impossible.

Result: The 1.35 does not mean the data say the risk is 135%. It means the coefficient set the optimizer just tried cannot be used. That is the root of the problem.

Why the log link allows such numbers

Write $\eta = X\beta$. The symbol η (eta) is the linear predictor: the intercept plus each coefficient times its predictor (the last section takes it in full). Log-binomial uses $p = e^{\eta}$, and $e^{\eta}$ can be anything in $0 < e^{\eta} < \infty$:

\[ \eta = -2 \Rightarrow p = 0.135, \quad \eta = -1 \Rightarrow p = 0.368, \quad \eta = 0 \Rightarrow p = 1 \]

\[ \eta = 0.1 \Rightarrow p = 1.105, \quad \eta = 1 \Rightarrow p = 2.718 \]

The exponential function does not know it is standing in for a probability. It only knows that $e^x > 0$. To make it a probability, a constraint must be added: $X_i\beta \le 0$ for every patient, because then $e^{X_i\beta} \le 1$.

Why logistic regression fits easily

The logistic inverse link is $p = e^{X\beta}/(1 + e^{X\beta})$. Whatever $X\beta$ is, $p$ stays inside 0 and 1. $X\beta = -10$ gives $p \approx 0$, $X\beta = 0$ gives $p = 0.5$, and $X\beta = 10$ gives $p \approx 1$, but never above it.

\[ -\infty < X\beta < \infty \;\Rightarrow\; 0 < p < 1 \]

That is a huge advantage of the logit link. Set the two links side by side.

Log link: p = exp(η)

Logit link: p = exp(η) / (1 + exp(η))

The wall: one patient can block a step

The correct log-binomial model has a restricted parameter space:

\[ \{\beta : X_i\beta \le 0 \text{ for every } i\} \]

The computer must search for $\beta$ only inside this space. With many predictors, $\beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \cdots \le 0$ must hold for every observation.

One patient may be old, exposed and severely ill, so the linear predictor sits very close to zero: $X_i\beta = -0.0001$ and $p_i = e^{-0.0001} \approx 0.9999$. The optimizer wants to move a coefficient a little to fit other patients better. If that makes $X_i\beta = +0.001$, then $p_i = 1.001$: not allowed.

So the optimizer walks hugging the wall, and the best estimate may sit right against it. This is a main reason log-binomial reports "convergence not achieved". It is not because the data are wrong, nor because any true probability exceeds 1. It is because the log-link parameterization forces the optimizer to search under the constraint $p \le 1$.

Poisson stretches the search space, not the probability

The picture to see: Poisson does not stretch the probability. It stretches the space where the optimizer is allowed to look for coefficients. Log-binomial and Poisson both use $\mu = e^{\eta}$ with $\eta = X\beta$. On a graph of $y = e^x$, the line $y = 1$ is the ceiling of a probability.

Log-binomial reads $y$ as $p$, so it may use only the left side, $\eta \le 0$. There $\eta = -2$ gives 0.135, $\eta = -0.5$ gives 0.607 and $\eta = 0$ gives 1. But $\eta = 0.2$ gives 1.22, which is not allowed.

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How the search stalls at the wall

Suppose the search starts at $\eta = -0.5$, where $p = e^{-0.5} = 0.61$. The data suggest that moving the coefficient up a little should fit better, so the optimizer wants to walk $-0.5 \to -0.3 \to -0.1 \to +0.1$.

1. The walk reaches the wall

   \[ \eta = +0.1 \;\Rightarrow\; p = e^{0.1} = 1.105 \]

   Log-binomial stops at once: the step is forbidden.

2. Back off and shrink the step

   Try +0.1: not allowed, so go back. Then -0.05 and -0.01 are allowed. Try +0.005: not allowed, so go back again; -0.002 is allowed, and so on.

3. Bouncing off the wall

   The search behaves like someone looking for the best spot who keeps hitting a wall and bouncing back and forth.

Result: That bouncing is the convergence problem.

What Poisson does differently

Poisson does not say $\mu$ is a probability; it says $\mu$ is a mean count. So $\mu = e^{\eta}$ may be 0.5, 1, 2, 10 or 100, and there is no ceiling at 1. On the Poisson axis, $\eta$ from -3 to +2 gives 0.05, 0.14, 0.37, 1, 2.72 and 7.39, all usable.

If the optimizer walks $-0.5 \to -0.3 \to -0.1 \to +0.1 \to +0.2$ and back to $-0.05$, Poisson says: go ahead, the likelihood and the estimating equation can be computed at every point. The search can pass through the region that would mean a probability above 1 on its way to the best coefficients. That is what "fits more easily" means.

It does not mean $p > 1$ has become real. It means the optimizer has no wall in its way during the search.

Many patients make it worse

Real data have many patients. Take four, with $\eta_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i}$. Log-binomial needs $\eta_i \le 0$ for every one of them. The optimizer wants to raise $\beta_1$ by 0.1 to fit patients 1 to 3 better.

Four patients, one blocked step

Modified Poisson removes the wall

In log-binomial, the feasible coefficients form a fenced region, and the best estimate often sits against its edge. A step outside means a probability above 1 and is forbidden. Modified Poisson has no $p \le 1$ boundary, so the optimizer can move freely in every direction.

Numerical algorithms such as Newton-Raphson or IRLS (iteratively reweighted least squares) then never have to ask whether a step pushes anyone's predicted risk above 1. So they converge much more easily.

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Why it still works for a binary outcome

For a binary $Y = 0, 1$, $E(Y) = p$. Modified Poisson uses the mean equation $\log E(Y \mid X) = X\beta$, which equals $\log p = X\beta$, so $e^{\beta_1} = \mathrm{RR}$.

What is borrowed from Poisson is a way of estimating the mean with no boundary. The Poisson variance, $\mathrm{Var}(Y) = \mu$, is wrong for binary data, whose true variance is $p(1-p)$. That is fixed with a robust SE.

What η is

η is not a new variable from the data. It is just a name for the right-hand side of the regression equation. In $\log(p) = \beta_0 + \beta_1 X$, set $\eta = \beta_0 + \beta_1 X$; the equation shortens to $\log(p) = \eta$ and solves back to $p = e^{\eta}$. η is called the linear predictor.

One patient's η

1. Coefficients and the patient

   \[ \beta_0 = -2, \quad \beta_1 = 0.7, \quad X = 1 \]

   A patient with the exposure.

2. The linear predictor

   \[ \eta = \beta_0 + \beta_1 X = -2 + 0.7(1) = -1.3 \]

   The score before any conversion.

3. Back to a risk, log-binomial

   \[ \log(p) = \eta = -1.3 \;\Rightarrow\; p = e^{-1.3} \approx 0.27 \]

   Inverting the log link turns the score into a risk.

Result: The order is always: X → β0 + β1X → η → inverse link → p.

η for every model

With more variables it is the same, for example $\eta = \beta_0 + \beta_1(\text{ANC}) + \beta_2(\text{Age}) + \beta_3(\text{SES})$. η is the total score the regression computes for each patient before it is converted back to a probability or risk.

η is not a probability. It can be negative, positive or any number. The link function is what turns η into the quantity wanted:

• logistic: $\eta = \log\{p/(1-p)\}$;
• log-binomial: $\eta = \log(p)$;
• Poisson: $\eta = \log(\mu)$.

So η is simply $\eta = X\beta$, shorthand for the intercept plus each coefficient times its predictor.

Common misreadings and their fixes

• Reading OR = 3.9 as "the risk is 3.9 times higher".

  Logistic regression gives an odds ratio, and for a common outcome it can sit far from the risk ratio.

  Fix: Say the odds are 3.9 times higher, or use the risk-ratio chain when a risk ratio is wanted.

• Treating a risk of 10% as a hard line for "rare".

  The odds ratio and the risk ratio drift apart gradually as the outcome becomes common.

  Fix: Use 10% only as a rule of thumb, not as a mathematical wall.

• "Rare outcome → use modified Poisson."

  Modified Poisson estimates a risk ratio even when the outcome is common.

  Fix: Rarity matters for whether OR ≈ RR, not for whether modified Poisson can be used.

• "The two adjusted models differ clearly, so there is interaction."

  Interaction is one possible cause among several kinds of misspecification.

  Fix: Check the model specification: a missing interaction, nonlinearity, the functional form, omitted covariates, a forced common risk ratio and extreme fitted risks.

• "Modified Poisson always converges."

  It usually converges very easily, but not 100% of the time.

  Fix: Say it usually converges easily, never that it always does.

• Reading a fitted value of 1.35 as "the data say the risk is 135%".

  Values above 1 are trial values the optimizer tries while it searches for β.

  Fix: Read such a value as a coefficient set that cannot be used, not as a risk from the data.

• "Poisson fits easily, so it is the truer distribution for binary data."

  Its variance, p, is wrong for a binary outcome, whose true variance is p(1 − p).

  Fix: Easier fitting means easier optimization; the robust SE is what repairs the wrong variance.

• "Modified Poisson gives a risk ratio because it is Poisson."

  The risk ratio comes from the log link, which log-binomial shares.

  Fix: The link decides the effect measure; the family decides the variance.

Glossary

convergence

The point where a fitting algorithm's estimates stop changing; a fit that never gets there has not converged.

log-binomial model (แบบจำลอง log-binomial)

A binomial model with a log link; it gives a risk ratio directly but can fail to converge when fitted risks approach 1.

modified Poisson regression (Poisson ดัดแปลง)

Poisson regression with a log link and a robust SE, used with a binary outcome to estimate a risk ratio.

robust (sandwich) standard error (ค่าคลาดเคลื่อนมาตรฐานแบบแซนด์วิช)

A standard error computed from the observed spread of the data, which stays usable when the assumed variance is wrong.

working model

A distribution used as a tool for estimating, not as a claim about how the outcome is really distributed.

link function (ฟังก์ชันเชื่อมโยง)

The function that connects the mean of the outcome to the linear predictor and so fixes the effect measure.

family (family (การแจกแจงของผลลัพธ์))

The distributional assumption of a model, which sets how the variance is computed and so the standard error.

optimizer

The algorithm that searches for the coefficients that fit the data best.

linear predictor (η) (ตัวทำนายเชิงเส้น)

The intercept plus each coefficient times its predictor, before the link is inverted.

References

1. McNutt LA, Wu C, Xue X, Hafner JP. Estimating the relative risk in cohort studies and clinical trials of common outcomes. Am J Epidemiol. 2003;157(10):940-943. doi:10.1093/aje/kwg074 https://doi.org/10.1093/aje/kwg074
2. Spiegelman D, Hertzmark E. Easy SAS calculations for risk or prevalence ratios and differences. Am J Epidemiol. 2005;162(3):199-200. doi:10.1093/aje/kwi188 https://doi.org/10.1093/aje/kwi188
3. Williamson T, Eliasziw M, Fick GH. Log-binomial models: exploring failed convergence. Emerg Themes Epidemiol. 2013;10:14. doi:10.1186/1742-7622-10-14 https://doi.org/10.1186/1742-7622-10-14
4. Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol. 2004;159(7):702-706. doi:10.1093/aje/kwh090 https://doi.org/10.1093/aje/kwh090
5. Cheung YB. A modified least-squares regression approach to the estimation of risk difference. Am J Epidemiol. 2007;166(11):1337-1344. doi:10.1093/aje/kwm223 https://doi.org/10.1093/aje/kwm223
6. McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed. London: Chapman & Hall; 1989. doi:10.1007/978-1-4899-3242-6 https://doi.org/10.1007/978-1-4899-3242-6
7. Wacholder S. Binomial regression in GLIM: estimating risk ratios and risk differences. Am J Epidemiol. 1986;123(1):174-184. doi:10.1093/oxfordjournals.aje.a114212 https://doi.org/10.1093/oxfordjournals.aje.a114212
8. Chen W, Qian L, Shi J, Franklin M. Comparing performance between log-binomial and robust Poisson regression models for estimating risk ratios under model misspecification. BMC Med Res Methodol. 2018;18:63. doi:10.1186/s12874-018-0519-5 https://doi.org/10.1186/s12874-018-0519-5

Key takeaways

• For a binary outcome and a risk ratio, step down in order: log-binomial, modified Poisson with a robust SE, a Gaussian working model with a robust SE, and logistic regression only when an odds ratio is accepted.
• Ask first which effect you want: the same risks of 10% and 20% give an RD of 10%, an RR of 2.0 and an OR of 2.25.
• The risk ratio comes from the log link; the family only sets the variance, and Poisson's variance is wrong for binary data, so modified Poisson needs a robust SE.
• Log-binomial stalls because exp(η) must stay at or below 1 for every patient, so one patient near the wall can block a whole step.
• Modified Poisson removes the wall during the search: it fits more easily, not more truly, and a clear gap between the two models calls for a check of the model specification, not an automatic claim of interaction.

Related in the wiki: [[glm-link-family-guide]] [[heteroskedasticity-robust-standard-errors]]