GLM Families and Effect Measures: Choosing the Right Model for Y

When you're building models in clinical research, the starting point is always this question:

What is the outcome variable (Y), and what is its nature?

That’s because your entire analysis strategy in Generalized Linear Models (GLMs) hinges on the Family of Y. Each “family” corresponds to a different type of outcome — continuous, binary, count, time-to-event — and brings its own logic for link function and interpretation.

Let’s walk through the 6 major GLM families, linking each one to:

• the type of Y,
• the model you’d use,
• the link function under the hood,
• and the effect measure it gives you.

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🧮 1. Gaussian Family → For Continuous Outcomes

✅ When to use:

• Y is a numeric continuous variable (e.g., blood pressure, serum sodium).
• You're modeling differences in means.

🔗 Link functions and models:

• Linear regression: identity link — estimates mean difference.
• Log-linear regression: log link — for positively skewed data like cost or length of stay.

🎯 Effect measures:

• Linear → Mean difference
• Log-linear → Mean ratio

🔍 Example:

"Does Drug A reduce average systolic blood pressure compared to Drug B?" Use linear regression → output interprets as mmHg difference in means.

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⚫ 2. Binomial Family → For Binary Outcomes

✅ When to use:

• Y is binary (yes/no, event/no event, dead/alive).
• You're comparing probabilities or odds.

🔗 Link functions and models:

• Logistic regression (logit link) → gives odds ratio (OR)
• Log-binomial regression (log link) → gives risk ratio (RR)
• Linear probability model (identity link) → gives risk difference (RD)

🎯 Effect measures:

• Logistic → Odds ratio
• Log-binomial → Risk ratio
• Linear probability → Risk difference

⚠️ Pitfall alert:

Odds ratio ≠ risk ratio. Use RR when prevalence is high to avoid OR inflation.

🔍 Example:

"Is smoking associated with MI?" Use log-binomial if feasible for RR, or logistic if convergence issues occur.

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🔢 3. Poisson Family → For Rate Data (Count Outcomes)

✅ When to use:

• Y is a count (number of asthma attacks, infections).
• You're modeling events per time or person-unit.
• Assumes variance ≈ mean (no overdispersion).

🔗 Model:

• Poisson regression with log link.

🎯 Effect measure:

• Incidence rate ratio (IRR)

🔍 Example:

"Does ICU stay increase infection rates per 1000 catheter days?"

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🧨 4. Negative Binomial Family → For Overdispersed Count Data

✅ When to use:

• Y is a count with overdispersion (variance > mean), aka (σ2>μ).
• Common in real-world epidemiology, where counts vary more than Poisson expects.

🔗 Model:

• Negative binomial regression — similar link, better variance handling.

🎯 Effect measure:

• IRR, more robust to dispersion.

🔍 Example:

"How does COPD affect number of ER visits?" Overdispersion is likely — negative binomial > Poisson.

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💰 5. Gamma Family → For Skewed Continuous Outcomes

✅ When to use:

• Y is continuous, positive, and highly skewed (e.g., cost, length of stay).
• Avoids bias from using linear regression on non-normal Y.

🔗 Model:

• Log-gamma regression

🎯 Effect measure:

• Mean ratio (interpreted multiplicatively)

🔍 Example:

"What’s the effect of surgical approach on hospital cost?"

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⏱️ 6. Survival Family → For Time-to-Event Data

✅ When to use:

• Y is time until an event (e.g., death, discharge, relapse).
• Can handle censoring (people who don't reach the event).

🔗 Models:

• Cox proportional hazards model (semi-parametric, no Y distribution assumed)
• Parametric models: Exponential, Weibull, Gompertz (model shape of hazard)

🎯 Effect measure:

• Hazard ratio (HR)

🔍 Example:

"What’s the effect of chemotherapy on time to progression?"

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✅ Final Takeaway: The GLM "Match Game"

Always choose your model based on Y. Here's the golden shortcut: