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The negative binomial regression assumes independent random variables conditional on their recorded days, withThe logarithmic link function applies to the conditional expectation, not to the observed count. The displayed maximum-likelihood estimates areand the dispersion parameter for the fitted family is one. Thus the fitted variance is ${\hat{\lambda }}_i+{\hat{\lambda }}_i^2/\hat{r}$, practically the Poisson distribution variance at these means.

Increasing the day by one multiplies the conditional expectation by $e^{{\hat{\beta }}_1}\simeq 0.996836$, a decrease of about $0.3164\%$. Comparing days 365 and 1 gives $e^{364{\hat{\beta }}_1}\simeq 0.3155$. The negative slope has Wald statistic approximately $-35.79$, so under the stated model assumptions there is very strong evidence of a decreasing conditional mean:This is an association with calendar day, not a causal effect established by the regression. The printed negative-binomial residual deviance is $7544.3$ on $1298$ residual degrees of freedom, which is a reason to investigate the adequacy of the assumed count model and dependence structure. In particular, the small reported slope standard error cannot make a misspecified model reliable.

The quasibinomial regression retains the logit link and mean function $p_i=(1+e^{-x_i^T\beta }{)}^{-1}$ but uses the working variance function $\mathrm{Var}(Y_i)=\varphi p_i(1-p_i)$. Its quasi-score equation isso multiplying by the common dispersion parameter does not change the coefficient estimates. Iteratively reweighted least squares therefore gives the same fitted means and coefficients as the binomial model. The usual Pearson dispersion estimator isThe estimated coefficient covariance matrix is $\hat{\varphi }(X^{T}WX{)}^{-1}$, with $W_{ii}={\hat{p}}_i(1-{\hat{p}}_i)$. From the intercept standard errors, $(1.08921/1.13031{)}^2\simeq 0.92860$; the rounded slope errors give approximately the same factor. The display uses approximate Student's t-tests with 118 residual degrees of freedom instead of fixed-dispersion normal tests.

For a genuinely individual binary response, the Bernoulli distribution identity $Y_i^2=Y_i$ forces $\mathrm{Var}(Y_i)=p_i(1-p_i)$. Thus $\varphi \ne 1$ is a working quasi-likelihood specification, not a different independent binary distribution with that mean. The mild estimated underdispersion does not establish an improved model, and the mean predictions are unchanged. In particular, a scalar rescaling does not model correlation among users, and usual likelihood-based Akaike information criterion comparisons are unavailable for a family without a specified probability likelihood. The output gives no convincing reason to prefer model3 to model2.

Write $\ell (\beta )$ for the log-likelihood and $U=\nabla \ell$ for the score function. Newton method uses the Observed Fisher information $M=-{\nabla }^2\ell$; Fisher scoring uses the Fisher information $M={\mathbb{E}}_{\beta }[-{\nabla }^2\ell ]$.

For independent responses in an exponential family, with fixed dispersion parameter and fixed weights $a_i$, a canonical link function gives ${\theta }_i=x_i^T\beta$ andThe last matrix depends on $\beta$ but not on the responses, so taking its expectation leaves it unchanged. Hence the two updates are identical, wherever this Fisher information matrix is invertible. Equivalently both solve the same iteratively reweighted least squares problem. This statement concerns the regression coefficients at a fixed dispersion parameter; a joint update of an unknown dispersion parameter need not have identical observed and expected full Fisher information matrices.

An exponential dispersion family has densities of the formwhere $\theta$ is the canonical parameter, $\varphi >0$ is the dispersion parameter, and the support does not depend on $\theta$. Its mean and variance are

For the given Gamma distribution, setThengivesThus the gamma laws form the Gamma exponential dispersion family.

Directly integrating, or using the cumulant-generating function of a gamma distribution,soDifferentiation at zero yields

For independent observations from an exponential dispersion family with common known dispersion parameter ${\sigma }^2$, the generalized linear model log-likelihood iswhere ${\mu }_i=b^{\prime }({\theta }_i)$ and the link function satisfies $g({\mu }_i)={\eta }_i=x_i^T\beta$.

For the Poisson variance function $V(\mu )=\mu$, the code computes the Pearson chi-squared statisticand the Pearson dispersion estimatorThe first quantity measures goodness of fit; the second estimates the dispersion parameter, which equals one in a correctly specified Poisson regression. A standard rough calculation substitutes the residual deviance for the Pearson statistic and givesIf the reported upper-tail probability $4.908651\times 10^{-11}$ is inverted numerically, the actual Pearson statistic used by the code is about $82.93$, giving $\hat{\varphi }\simeq 5.18$. Either calculation reveals severe overdispersion.

The second fit is a Quasi-Poisson regression. It retains the logarithmic link function and mean modelbut assumes only $\mathrm{Var}(Y_i)=\varphi {\mu }_i$ for an unknown dispersion parameter $\varphi$, rather than a complete Poisson distribution.

Quasi-likelihood specifies a mean and a variance function without necessarily specifying a full response distribution. For independent responses, the estimating equation is ${\sum }_i(\partial {\mu }_i/\partial \beta )(y_i-{\mu }_i)/(\varphi V({\mu }_i))=0$. A common scalar dispersion parameter rescales coefficient covariance while leaving the roots of the quasi-score equation unchanged.

The unscaled residual deviance of an exponential dispersion family compares its fitted mean with the saturated statistical model at a common dispersion parameter. Its normalization removes the factor $1/\varphi$ in the log-likelihood. In Poisson regression and binomial regression, $\varphi =1$, so it is twice the saturated-minus-fitted log-likelihood. Under suitable regularity, $D/\varphi$ has approximately a chi-squared distribution with the residual degrees of freedom; this is not an automatic accurate approximation for sparse responses. Nested deviance reductions with estimated dispersion lead to approximate F-tests after scaling by the dispersion estimate.

For an exponential dispersion family with a common dispersion parameter $\varphi$, scaled deviance compares a fitted model with the saturated statistical model at the same dispersion:For a normal distribution with variance ${\sigma }^2$, unscaled deviance is the residual sum of squares and scaled deviance is that sum divided by ${\sigma }^2$. For the binomial distribution, $\varphi =1$ and the two coincide. Unknown Gaussian dispersion requires variance estimation; raw deviance differences should not be treated as chi-squared statistics without the scale.

Responses observed on the same process or individual at different times may have nonzero correlation coefficients. Such serial correlation can arise from a shared random effect or from history-dependent conditional means. A scalar dispersion parameter adjustment alone does not represent this temporal dependence.

Underdispersion means less variation than prescribed by a reference statistical model, corresponding to a working dispersion parameter below one. It is not, by itself, evidence that the reference mean function is improved.