Tutorial 4: Handling Overdispersion

Step 2: Choose the Link Function

✓

Same Link Function as Poisson

The link function choice is the same for Negative Binomial as for Poisson. Both model counts, so both need a link that ensures positive predictions. The key difference between the models is the distribution, not the link.

Step 2: Select the Link Function

Which function should map the linear predictor to expected counts?

Hint: We need g() such that g^-1(Xβ) is always positive.

Log Link

g(μ) = ln(μ)

Linear predictor maps to log of expected count. Ensures μ > 0.

Square Root Link

g(μ) = √μ

Historically used for count data. Less interpretable coefficients.

Identity Link

g(μ) = μ

Direct mapping - linear predictor equals expected value.

Logit Link

g(μ) = ln(μ/(1-μ))

Maps to log-odds. Used for probabilities.

Link Function Selected: Log

With the log link, your model equation is:

ln(E[cnt]) = β₀ + β₁·temp + β₂·hum + β₃·windspeed + ...

The log link is identical for Poisson and Negative Binomial. The difference will be in the distribution (variance function).

← Back to Systematic

✔ Correct! Log link is the standard choice.

The log link is the canonical link for both Poisson and Negative Binomial regression:

Positive predictions: Since μ = exp(Xβ), predictions are always > 0
Multiplicative interpretation: exp(β) gives rate ratios
Natural for count processes: Counts often grow multiplicatively with predictors

        Key insight: The link function is the same for Poisson and Negative Binomial.
        Both model counts, so both need the log link to ensure positive predictions.
        The difference is in the variance function, not the link.
      

⚠ Historically used, but not recommended today.

The square root link was sometimes used for count data:

Variance stabilising: For Poisson data, sqrt transform stabilises variance
Historical usage: Common before GLMs were well-understood
Less interpretable: Coefficients don't have a natural rate ratio interpretation

        Modern practice: The log link is strongly preferred because exp(β) gives
        interpretable rate ratios. Try selecting Log Link instead.
      

❌ Not appropriate for count data!

The identity link is problematic for counts:

Negative predictions: If Xβ < 0, we predict negative counts!
No natural lower bound: Counts must be ≥ 0, but μ = Xβ can be anything
Works for Gaussian: Identity is appropriate when the response can be any real number

        Example: With β0 = 100 and a large negative coefficient on temperature,
        cold days could predict -50 bike rentals. That's impossible!
      

❌ Wrong data type - logit is for probabilities!

The logit link is designed for binary/proportion outcomes:

Bounded output: Logit^-1(x) is always between 0 and 1
For probabilities: Perfect for Tutorial 2 (heart disease risk)
Not for counts: We need outputs > 1 (thousands of bike rentals!)

        Data type mismatch: Logit constrains output to [0,1].
        Our bike rentals range from 22 to 8,714 - way outside this range!
      