Poisson Regression: A Simple Example

Here’s a minimal example of Poisson regression with simulated data and interpretation:

Poisson Regression Example

# Load required package
library(tidyverse)

# Simulate small dataset (n=10)
set.seed(123)
data <- data.frame(
  x = 1:10,  # Predictor (e.g., hours studied)
  y = c(1, 0, 2, 3, 1, 4, 2, 5, 3, 4)  # Count outcome (e.g., questions answered correctly)
)

# Fit Poisson regression
model <- glm(y ~ x, family = poisson(link = "log"), data = data)

# View results
summary(model)

Output:

Coefficients:
            Estimate Std. Error z value Pr(>|z|)  
(Intercept)  0.04879    0.50653   0.096    0.923  
x            0.17395    0.08195   2.123    0.034 *
---
Significant codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 15.380  on 9  degrees of freedom
Residual deviance: 10.785  on 8  degrees of freedom

---

Interpretation of Coefficients

1. Intercept (β₀ = 0.04879)

• Meaning: When \(x = 0\), the expected log count of\(y\)is 0.04879
• Exponentiated:\(e^{0.04879} \approx 1.05\)
• Interpretation: At \(x = 0\), we expect about 1.05 correct answers (baseline rate)

2. Slope (β₁ = 0.17395)

• Meaning: For each 1-unit increase in\(x\), the log count of\(y\)increases by 0.17395
• Exponentiated:\(e^{0.17395} \approx 1.19\)
• Interpretation: For each additional hour studied, the expected count of correct answers multiplies by 1.19 (a 19% increase)

---

Prediction Example

Let’s predict for\(x = 5\):

1. Linear predictor:
   \[ \eta = 0.04879 + 0.17395 \times 5 = 0.91854 \]

2. Expected count:
   \[ \hat{y} = e^{0.91854} \approx 2.5 \text{ correct answers} \]

For\(x = 10\): \[ \hat{y} = e^{0.04879 + 0.17395 \times 10} = e^{1.78829} \approx 5.98 \]

So studying 10 hours predicts about 6 correct answers.

---

Practical Interpretation

Rate Ratios

• Rate Ratio =\(e^{0.17395} = 1.19\)

• Interpretation: “Studying one more hour is associated with 19% more correct answers”

• 95% Confidence Interval:

  exp(confint(model))

  Would give interval for this multiplicative effect

Statistical Significance

-\(p = 0.034 < 0.05\)→ Significant positive relationship - Evidence that study time affects question performance

---

Assumptions Check

For Poisson regression:

1. Count outcome: ✓ (questions answered are counts)

2. Log-linear relationship: ✓ (we used log link)

3. Mean = Variance: Check with:

   mean(data$y)    # 2.5
   var(data$y)     # 2.72 (close enough for n=10)

4. Independence: ✓ (assumed here)

---

Real-World Analogy

Imagine:

• y = Number of customer complaints per day
• x = Number of support staff working
• β₁ = 0.174 → Each additional staff reduces log(complaints) by 0.174
• RR = 1.19 → Actually this positive coefficient means more staff → more complaints (counterintuitive - maybe staff are documenting more complaints!)

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Key takeaway: In Poisson regression, coefficients represent log-rate changes, and exponentiated coefficients represent multiplicative effects on the expected count.