Poisson regression

A Poisson regression models independent count responses byIts likelihood is ${\prod }_i{e}^{-{\mu }_i}{\mu }_i^{y_i}/y_i!$.

For independent counts with means $p_{i1}{\lambda }_i$ and $p_{i2}{\lambda }_{i}r$, let $t_i=y_{i1}+y_{i2}$. Maximizing the Poisson regression likelihood in each baseline rate gives the displayed estimate. The remaining log-rate-ratio score isEquivalently, conditional on $t_i$, the second count has a binomial distribution with success probability $r{p}_{i2}/(p_{i1}+r{p}_{i2})$. Each positive-total term increases strictly with $r$; if both treatment totals are positive, the score has a unique positive root. This reduces a many-parameter fit to a scalar equation.

For a full-column-rank design $X$ and positive fitted means ${\mu }_i=\exp (x_i^T\beta )$, the log-likelihood has Hessian $-X^T\mathrm{diag}({\mu }_i)X$, which is negative definite. Thus any finite stationary point is the unique maximum-likelihood estimator. Full rank does not guarantee existence: with an intercept and all counts zero, the supremum is approached as the intercept tends to $-\infty$. If every observed count is positive, a full-rank design does give a finite maximum: each function $y_i{\eta }_i-e^{{\eta }_i}$ tends to $-\infty$ in either tail and is bounded above, and $\Vert X\beta \Vert \to \infty$ as $\Vert \beta \Vert \to \infty$, so the negative log-likelihood is coercive.

In Poisson regression with log means $a+\beta x_i$, the Fisher information matrix is $\left(\begin{array}{cc}{S}_0& S_1\\ S_1& S_2\end{array}\right)$, where $S_j={\sum }_i{x}_i^j{\mu }_i$. Inverting this matrix gives slope variance $S_0/(S_0{S}_2-S_1^2)$. Writing ${\bar{x}}_{\mu }=S_1/S_0$ reduces it to the displayed reciprocal weighted sum of squares. Thus a common intercept consumes information about the overall rate, while slope information comes from the spread of the covariates under the mean-count weights. Constant covariates make the two parameters unidentifiable.

For independent Poisson distributions with log means $Z\beta +o$, where $o$ is a known offset, differentiation of the log-likelihood gives the score function $Z^T(y-\mu )$. An interior maximum-likelihood estimator therefore satisfies the displayed equations. An intercept and categorical indicator columns make fitted count totals equal observed count totals in each represented factor level. These equations match counts, rather than unweighted averages of rates when exposures differ.

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For independent aggregate Poisson distributions with means $n\lambda$ and $n\lambda e^{\beta }$, conditioning on the total eliminates $\lambda$. Independent Poisson conditioning gives the displayed binomial distribution. Its conditional likelihood is proportional to $e^{\beta Y_1}(1+e^{\beta }{)}^{-m}$, also the profile likelihood after maximizing over the unrestricted positive baseline rate. Unequal exposures replace the binomial success probability by $n_1{e}^{\beta }/(n_0+n_1{e}^{\beta })$.

For conditionally independent paired counts with means ${\lambda }_i$ and ${\lambda }_i{e}^{\beta }$, condition on every pair total $N_i=Y_{i0}+Y_{i1}$. The joint conditional likelihood removes all pair-specific rates, leaving the displayed independent binomial distributions. A zero-total pair contributes no information. With unrestricted fixed baseline rates, this conditional likelihood has the same parameter-dependent part as the ordinary profile likelihood; a specified Gamma–Poisson hierarchical model can retain additional information by imposing structure on those rates.

Consider an approximate conditional likelihood in which a paired total $t_i$ leads to a structural zero with probability $\theta$, and otherwise a binomial distribution with success probability $p$. For an observed zero, the posterior cure weight is $q_i=\theta /[\theta +(1-\theta )(1-p{)}^{t_i}]$; for a positive count it is zero. Differentiating the log-likelihood gives the displayed equations at an interior maximum-likelihood estimator, with the weights evaluated at that estimator. Recomputing the weights and then updating the parameters is an expectation-maximization algorithm. Boundary fits must also be considered. These equations describe a conditional mixture: if cure is instead specified before two Poisson distributions are conditioned on their total, its conditional probability generally depends on the total and on the baseline nuisance rate. A zero observed in a short interval is not by itself proof of permanent biological cure.

For a continuous predictor $x$ and a categorical factor, an additive Poisson regression with log link has $\log {\mu }_j(x)=a_j+bx$. The log-mean curves are parallel and ${\mu }_j(x)/{\mu }_k(x)=e^{a_j-a_k}$ is constant in $x$. Adding a predictor-by-factor interaction gives $\log {\mu }_j(x)=a_j+b_{j}x$, so this ratio becomes $e^{a_j-a_k+(b_j-b_k)x}$. With $J$ factor levels, the common-slope restriction has $J-1$ independent constraints. A likelihood-ratio test compares the two deviances with an asymptotic chi-squared distribution on $J-1$ degrees of freedom. Evidence against common slopes establishes varying effects on the log scale; it does not by itself determine their signs or the ordering of the mean curves.

If $C_i\sim \mathrm{Poisson}(T_i{\lambda }_i)$ counts events over $T_i$ days, then the daily average $C_i/T_i$ has mean ${\lambda }_i$ and variance ${\lambda }_i/T_i$. It is not itself a Poisson variable unless $T_i=1$. Known observation periods allow count modelling with a log-exposure offset; otherwise Quasi-Poisson regression may be used as a working mean-variance model, with uncertainty qualified by the missing exposure information.

For independent samples of $n$ observations from Poisson distributions with positive means $a$ and $b$, the maximum-likelihood estimator of $\nu =\log (b/a)$ is the displayed difference when both totals are positive. Its asymptotic variance is $(a^{-1}+b^{-1})/n$, obtained by the delta method or by inverting the two-parameter Fisher information matrix. A zero group total makes its log-mean estimate a boundary value; finite normal intervals require nonzero totals.

For three distinct times, the columns $1,t,t^2$ are linearly independent by the Vandermonde determinant. Thus a quadratic Poisson regression for the log mean can reproduce any three positive time-group means. A grouped goodness-of-fit test of that mean form has zero residual degrees of freedom. Replicated observations still allow checks of the Poisson distribution and overdispersion within each time group; saturation of the mean does not establish the response distribution.

A categorical covariate can be grouped in an interaction term while retaining its original main effect. This shares selected slopes but keeps separate intercepts. Nested Poisson deviance differences test slope pooling and intercept pooling separately.

In a Poisson regression with $\log \mu ={\beta }_0+{\beta }_{T}T+{\beta }_{B}b+{\beta }_{TB}Tb$, the treatment logarithm of the mean ratio at baseline $b$ is ${\beta }_T+b{\beta }_{TB}$. The main treatment coefficient tests the contrast at $b=0$. An overall no-treatment test requires both ${\beta }_T=0$ and ${\beta }_{TB}=0$, with their joint covariance matrix for a Wald test.

For independent Poisson observations with fitted means ${\hat{\mu }}_i$, the deviance from the saturated model iswith $0\log 0=0$.

Write $y_i=e_i+r_i$. For $|r_i|/e_i$ small, the cell contribution to the Poisson deviance expands asSumming gives the Pearson chi-squared statistic as the quadratic approximation. The approximation can be inaccurate for sparse expected counts or cells with relative residuals near minus one.

The full Poisson deviance is $2{\sum }_i[y_i\log (y_i/e_i)-y_i+e_i]$. An unpenalized log-mean intercept has likelihood score ${\sum }_i(y_i-e_i)$, which vanishes at an interior fit. The linear terms then cancel, giving the displayed expression. Without that intercept score equation, discarding the linear terms is not justified. Use the limiting convention $0\log (0/e_i)=0$.