ST411 Weeks 7-8: Models for count responses

POISSON AND NEGATIVE BINOMIAL MODELS FOR SINGLE COUNTS

What are the key properties of the Poisson distribution?

It models count data where Y = 0, 1, 2, \dots with parameter \mu

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Probability: P(Y = y) = \frac{e^{-\mu} \mu^y}{y!}

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Mean and variance are both equal to \mu

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It belongs to the two-parameter exponential family (but without a dispersion parameter \phi)

What is the probability formula for Poisson distribution

Probability: P(Y = y) = \frac{e^{-\mu} \mu^y}{y!}

What is the mean and variance of a Poisson distribution?

Mean and variance are both equal to \mu

What is a Poisson log-linear model for counts?

This is a regression model using the Poisson distribution. It’s a GLM where Y_i \sim \text{Poisson}(\mu_i) and

\log(\mu_i) = x_i' \beta — the log link is the canonical link

\log(\lambda_i) = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_p x_{pi}

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Implies \mu_i = \exp(x_i' \beta)

\lambda_i = \exp(\beta_0 + \beta_1 x_{1i} + \cdots + \beta_p x_{pi})

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Estimated using maximum likelihood, with likelihood-based inference

Interpretation of regression coefficients in Poisson model

\exp(\beta_k) is the rate ratio — it gives the multiplicative change in the expected count when x_k increases by 1 unit, holding all other variables constant

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If \beta_k > 0, expected count increases; if \beta_k < 0, it decreases

How do you compute fitted counts in a Poisson regression model?

Use \hat{\mu} = \exp(x' \hat{\beta})

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You can compare fitted counts across scenarios by plugging in different values of x

How do you compute fitted probabilities in a Poisson regression model?

Use the Poisson formula:

\hat{P}(Y = y \mid x) = \frac{e^{-\hat{\mu}} \hat{\mu}^y}{y!}

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Where \hat{\mu} = \exp(x' \hat{\beta}) , obtained from the fitted count.

How do you compute the change in fitted counts when covariates change?

Change in fitted counts:

\hat{\mu}_b - \hat{\mu}_a = \exp(x_b' \hat{\beta}) - \exp(x_a' \hat{\beta})

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Compare fitted means for two different covariate values x_b and x_a

What is overdispersion in count data models?

When the observed variance is greater than the mean — violating the Poisson assumption that \text{E}(Y_i) = \text{Var}(Y_i)

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A common symptom is too many zeros in the data compared to what the Poisson model expects

Why might overdispersion persist even after fitting a Poisson regression model?

Because of unaccounted heterogeneity — the Poisson model assumes that all units with the same x share the same mean \mu

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But some variability (e.g. between individuals) may remain even after adjusting for observed covariates

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This can lead to continued overdispersion in the data

How does Negative Binomial regression handle overdispersion?

It adds a dispersion parameter (often \alpha = 1/k) so that:

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\text{E}(Y) = \mu (same as Poisson)

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\text{Var}(Y) = \mu(1 + \alpha \mu), allowing \text{Var}(Y) > \text{E}(Y)

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This extra flexibility helps model count data with overdispersion

What is the probability mass function of the Negative Binomial distribution?

P(Y = y) = \frac{\Gamma(y + k)}{\Gamma(k)\, \Gamma(y + 1)} \left( \frac{k}{\mu + k} \right)^k \left( \frac{\mu}{\mu + k} \right)^y

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Used for count data with overdispersion, where \mu is the mean and k is the dispersion parameter

Mean of the negative binomial distribution

E(Y) = \mu =

Variance of the negative binomial distribution, and how does it compare to the mean.

\text{Var}(Y) = \mu + \frac{\mu^2}{k} \quad \text{or} \quad \mu (1 + \alpha \mu)

Where \alpha = \frac{1}{k}

Var(Y) > E(Y)

What does R refer to k as?

theta = 1/alpha

Is negative binomial part of the two-param exponential family?

No, except when k is known and fixed.

How are fitted probabilities computed in Negative Binomial regression?

They are computed using the Negative Binomial probability function

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The mean is modeled as \mu_i = \exp(x_i' \beta) just like in Poisson regression

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The interpretation of \beta is the same, but the fitted \hat{P}(Y = y \mid x) uses the NB distribution to allow for overdispersion

How do you test for overdispersion in a Poisson model?

Use a likelihood ratio test comparing the Poisson model to a Negative Binomial model

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Test H_0: \alpha = 0 (no overdispersion) vs. H_1: \alpha > 0

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Because \alpha = 0 is on the boundary of the parameter space, divide the usual chi-squared p-value by 2

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Alternatively, use the LR test p-value as a conservative check

What happens when overdispersion is present and a Negative Binomial model is used?

If \alpha > 0 (e.g., \hat{\alpha} = 1.48 with significant p-value), there is strong overdispersion

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Estimated \hat{\beta} and fitted means \hat{\mu} may still look similar to Poisson

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But Poisson underestimates standard errors of \hat{\beta}

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Fitted probabilities from the NB model differ and better match observed data

Why might count outcomes y_i not be directly comparable across subjects?

Because subjects may have different levels or durations of exposure to risk

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E.g., longer follow-up time (person-years) or more vehicles on a road increases the chance of events occurring

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Without accounting for exposure, we may misinterpret the effect of explanatory variables

How do we account for differing exposure times in Poisson or Negative Binomial models?

Include \log(t_i) as an offset term in the log-linear model

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We model the rate \mu_i / t_i so that:

\log(\mu_i) = \log(t_i) + x_i' \beta

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The offset \log(t_i) has no coefficient and adjusts for different durations or exposure levels

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If t_i is constant across units, the offset isn’t needed