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The complementary log-log link maps $p\in (0,1)$ to $\ln [-\ln (1-p)]$, with inverse $p=1-\exp (-e^{\eta })$. For constant infection hazard $\lambda$ and exposure $e$, $p=1-e^{-\lambda e}$ implies $\mathrm{cloglog}(p)=\ln \lambda +\ln e$. Thus a binomial regression with a log-exposure offset estimates the log hazard as its intercept.

Maximum likelihood estimation is a method of statistical estimation: within a specified statistical model, choose the statistical parameter values that maximize the likelihood function of the observed data. For a parameter set $\Theta$ and observations $x$, its exact requirement isThe equality requires an attained, finite maximum; a supremum approached only at an excluded boundary is not enough. The resulting rule is a maximum-likelihood estimator. The likelihood function uses a probability mass function for discrete data and a probability density function for continuous data. See MIT 18.05, Reading 10b, sections 3 and 3.2.

The direction of the comparison is the interesting part: hold the observations fixed and vary the statistical parameter. Ten independent Bernoulli trials with eight successes and two failures give the maximum-likelihood estimate $\hat{p}=0.8$, as derived in log-likelihood. This chooses the member of the statistical model that assigns the greatest mass to that outcome. It does not assert an $80\%$ posterior probability that this parameter value is true. Bayesian statistics combines the same likelihood function with a prior distribution to obtain a posterior distribution; maximum a posteriori estimation maximizes that posterior mass or density instead.

The method turns many fitting tasks into one maximization problem. In the normal distribution model with known variance, it recovers the sample mean and ordinary least squares, while binomial regression and Poisson regression use different sampling assumptions to build their own likelihood functions. The log-likelihood turns independent contributions into a sum, exposing score equations and observed information. A peak's location gives an estimate; its shape records how much the observed data distinguish nearby parameter values.

Existence, uniqueness, bias, and statistical consistency are separate questions. Normal mean and variance maximum-likelihood estimators include a variance estimate with bias, likelihood supremum at an excluded boundary gives nonexistence, and the Neyman-Scott incidental parameter problem gives inconsistency. The usual asymptotic normality of a maximum likelihood estimator requires assumptions: a consistent estimator at an interior true parameter in a fixed finite-dimensional model with identifiability, sufficient smoothness and control of log-likelihood derivatives, and finite nonsingular Fisher information. A common parameter-independent support of a probability distribution is part of a standard sufficient framework; see Rigollet, MIT 18.650, Maximum Likelihood Estimation, slides 22-23. This is a description of the regular setting, not a complete theorem's hypotheses. The exact endpoint limit for a shifted exponential distribution shows why changing support can produce a different rate and a nonnormal limit. These properties do not follow just from maximizing a function.

In his 1922 paper in the Royal Society's Philosophical Transactions A, R. A. Fisher explicitly distinguished likelihood from probability and described choosing the parameter values with greatest likelihood. Section 6 also discusses comparing likelihoods after giving the maximum value one, the convention used in the figures here. See Fisher, On the Mathematical Foundations of Theoretical Statistics, section 6, pp. 326-328, DOI 10.1098/rsta.1922.0009.

Assume a constant annual force of infection $\lambda >0$ across ages and calendar years. For independent sampled people with ages $a_i>0$ and binary results $y_i$, fitApplying the complementary log-log link givesTherefore fit a binomial regression with the complementary log-log link, an intercept, and $\ln a_i$ as an offset with coefficient fixed at one. Age is measured in years. An ordinary freely fitted coefficient of age, or even a freely fitted coefficient of $\ln a_i$, would describe a different exposure model. The annual force of infection estimate isAn infection-naive person's one-year infection probability is $1-e^{-\hat{\lambda }}$, which is approximately $\hat{\lambda }$ only when that rate is small.

The log-likelihood being maximized is, apart from constants,A confidence interval for ${\beta }_0$ can be exponentiated to give a positive interval for $\lambda$. For groups of $n_j$ people at a common age $a_j$, use $Y_j\sim \mathrm{Binomial}(n_j,1-e^{-\lambda a_j})$ with the same offset. If an age band is broad, use individual ages or average $1-e^{-\lambda a}$ over the band's age distribution instead of treating its midpoint as exact. No numerical estimate is possible without the survey observations.

The biological interpretation assumes that immunoglobulin G positivity records previous infection, persists over the relevant ages, and is not substantially produced by maternal antibodies or vaccination. These are conditions to check, rather than conclusions from the fitted curve. Use known-positive and known-negative serum controls to assess the assay threshold, and validate specificity against relevant other infections. Cross-reactive antibodies to other flaviviruses can produce positive dengue virus serology, so a positive result need not be a marker specific to dengue virus exposure.^[ref]

If independently established assay diagnostic sensitivity and diagnostic specificity are $Se$ and $Sp$, the observed positive probability isUse that observation model in the binomial likelihood; it is generally no longer the simple intercept-plus-offset complementary log-log link fit. With only the stated survey, one cannot freely estimate $Se$, $Sp$, antibody loss and an arbitrary infection history simultaneously. A single binary result also does not identify how many subsequent infections a person has experienced.

Before introduction there is no local infection exposure. Let $H=2024-2005=19$ years, using the actual elapsed time between introduction and sampling if exact dates are available. A lifelong resident aged $a_i$ has therefore had onlyyears of possible exposure. With a constant post-introduction force of infection $\lambda$, replace age by this exposure duration:Fit the same binomial regression with $\ln e_i$ as its offset. The predicted seroprevalence increases with age up to 19 and then reaches a plateau, since older residents all began local exposure at introduction.

This is equivalent to using a calendar-time force of infection of zero before 2005 and $\lambda$ afterwards. With a changing post-introduction rate, the more general exposure is ${\int }_{\max (2005,2024-a_i)}^{2024}\lambda (u)du$. Account separately for migration or infection acquired elsewhere, since current residence alone need not specify lifetime exposure. Subjects with exactly zero exposure have model probability zero and cannot be included by taking $\ln 0$ as an ordinary finite offset; handle them through the original probability model.

read.table(...,header=T) reads four grouped rows with column names. The displayed outcome counts give the observed risks $0.1519,0.2145,0.2611,0.5111$ in the indicated factor order. rbind in the first test pools over predisposition: the two rows represent no cannabis use and some use, with event/non-event counts $(341,1775)$ and $(82,238)$. Thus the pooled risks are $341/2116=0.1612$ and $82/320=0.2563$. This test concerns the marginal statistical association between use and outcome, not an association adjusted for predisposition.

Under independence, expected counts in the two-by-two contingency table are $E_{ij}=O_{i+}{O}_{+j}/N$. The ordinary Pearson chi-squared statistic is $17.5183$. The printed $16.8618$ is reproduced by the default two-by-two Yates continuity correction:All absolute differences here exceed one half. Comparison with a chi-squared distribution on one statistical degree of freedom gives $p=4.02\times 10^{-5}$. There is strong evidence of marginal association. The simplified output omits the usual continuity-correction wording, but its numerical value shows which statistic was used.

attach makes the data columns available by name, and tot=with+without obtains group sizes $1936,275,180,45$. The grouped binomial regression fits $Y_g\sim \mathrm{Bin}(n_g,{\pi }_g)$, with response $Y_g/n_g$ and prior weights $n_g$. These weights tell R the binomial denominator; treating the four proportions as four equally precise individual observations would be wrong. The default logit link isLet $A$ indicate some use and $B$ indicate predisposition. The additive logistic regression specifies $\eta =a+bA+cB$. Its log-likelihood, including the binomial constants, isDifferentiation gives score equations $X^T(Y-n\pi )=0$ and Fisher information matrix $X^{T}WX$, where $W_{gg}=n_g{\pi }_g(1-{\pi }_g)$. Fisher scoring solves these equations, and the inverse fitted Fisher information matrix gives the reported approximate coefficient variances. The number of scoring iterations records numerical convergence, not degrees of freedom or statistical evidence.

The intercept corresponds to no use and no predisposition and gives baseline fitted risk ${\mathrm{logit}}^{-1}(-1.73881)=0.1495$. The additive coefficients imply common adjusted odds ratios:They multiply the odds, not the risks. The Wald tests divide the estimates by their standard errors and use a standard normal distribution; both predictors have small two-sided p-values. Approximate $95\%$ confidence intervals for these odds ratios are $(1.296,2.266)$ and $(1.685,3.110)$, obtained by exponentiating the log-scale intervals. The intercept test against zero asks whether the reference risk equals one half, and is not a test of either exposure association. The binomial dispersion is assumed to be one because $\mathrm{Var}(Y_g)=n_g{\pi }_g(1-{\pi }_g)$; the output does not demonstrate independence or rule out extra-binomial variation.

There are four grouped probabilities and three fitted coefficients, leaving one residual degree of freedom. The fitted risks are approximately $(0.1495,0.2314,0.2869,0.4080)$. The binomial deviance compares this fit with the four unrestricted observed proportions:The deviance goodness-of-fit test against ${\chi }_1^2$ has p-value $0.0796$. The common-odds-ratio fit is not rejected at $5\%$, although the doubly exposed group's observed risk exceeds its fitted risk appreciably and suggests possible interaction.

The second formula adds $\delta AB$. Its four coefficients fit all four probabilities, making a saturated statistical model with zero grouped residual degrees of freedom. The near-zero deviance is roundoff from exact interpolation; it supplies no remaining grouped lack-of-fit test. Unlike Q1's unreplicated Gaussian fit, however, these grouped responses represent many independent Bernoulli trials, and their binomial variances are specified by their sizes and probabilities. Coefficient standard errors remain available from binomial information without estimating dispersion from the zero residual degrees of freedom.

Indeed, let ${\eta }_{ab}=\log [Y_{ab}/(n_{ab}-Y_{ab})]$ be the fitted cell logit link value. The saturated coefficients areThese calculations yield the printed estimates. The delta method gives $\mathrm{Var}({\hat{\eta }}_{ab})\approx 1/Y_{ab}+1/(n_{ab}-Y_{ab})$, and independent groups make the variance of $\hat{\delta }$ the sum of all eight reciprocal counts. Its square root is $0.37857$, reproducing the printed standard error.

The saturated logistic regression describes the conditional odds ratios as follows:Thus the main coefficients now concern effects at the other factor's reference level. Logistic interaction as a ratio of odds ratios gives $e^{\delta }=1.939$, with approximate $95\%$ confidence interval $(0.923,4.073)$. The interaction Wald test has $p=0.0802$, while the nested likelihood-ratio test uses deviance difference $3.0733$ on one statistical degree of freedom and has $p=0.0796$. The data suggest a larger odds association in predisposed subjects, but do not establish this interaction at the conventional $5\%$ level. This conclusion is specific to the odds scale; absolute-risk differences can vary even with no logit-scale interaction.

The Akaike information criterion is $-2\hat{\ell }+2k$. The additive fit has $k=3$ and AIC $31.766$; the interaction fit has $k=4$ and AIC $30.692$. The extra coefficient improves twice the log-likelihood by $3.0733$ but incurs penalty $2$, so AIC falls by only $1.0733$. It weakly favors the larger fit rather than decisively establishing an interaction, and AIC is not a p-value.

The marginal odds ratio $1.793$ and the fitted conditional odds ratios are different quantities. Predisposition is associated with both use frequency and outcome, so confounding is a substantive possibility; noncollapsibility of the odds ratio also means marginal and conditional odds ratios need not agree. This is an observational association analysis with only one adjustment factor. It does not by itself establish a causal effect, exclude other confounding, or validate the assumed independent binomial sampling.

A generalized linear model has independent responses whose distributions belong to an exponential family:Here the expected value and variance are ${\mu }_i=b^{\prime }({\theta }_i)$ and $\mathrm{Var}(Y_i)=a_i(\varphi )b^{\prime \prime }({\theta }_i)$. A specified one-to-one link function $g$ connects the mean to the linear predictor: $g({\mu }_i)=x_i^T\beta$. The canonical link function is $g({\mu }_i)={\theta }_i$.

In binomial regression, $Y_i\sim \mathrm{Bin}(m_i,p_i)$ independently, with known trial counts $m_i$ and probabilities related to the predictor. The logistic regression and probit regression specifications are respectivelyHere $\Phi$ is the standard normal distribution function. The binomial log-density contains $y_i\log (p_i/(1-p_i))+m_i\log (1-p_i)$, so its natural parameter is the log-odds: the logistic link is canonical. If the response mean is written as ${\mu }_i=m_i{p}_i$, the canonical link is $g_i({\mu }_i)=\log ({\mu }_i/(m_i-{\mu }_i))$.

The first model's residual deviance is $671.27$ on $197$ residual degrees of freedom, a ratio of about $3.41$. This is far above the scale expected from a well-fitting dispersion-one grouped binomial regression. It suggests an inadequate mean function, overdispersion, or both. The linear age and exposure restrictions may miss nonlinear associations; the generalized additive model allows those shapes to be estimated through penalized cubic regression splines rather than assumed.

The very small exposure Wald p-value in the first fit suggests an association, but does not establish that a linear logit effect is adequate. Nor does the nonsignificant linear age term exclude every possible nonlinear age effect. Allowing estimated scale also addresses the excessive residual variation, which can arise from clustered data because several births belong to the same woman. The second model's estimated scale $3.3506$ confirms that permitting smooth means has not removed all extra variation. The reason to proceed is poor initial fit and possible nonlinear effects, with dispersion-aware inference, not merely the wish to obtain more significant tests. These observational fits alone do not establish causation by the disaster.

A generalized linear model assumes that the independent responses $Y_i$ have distributions in an exponential family, with means ${\mu }_i$, and thatThe invertible function $g$ is the link function; it connects the mean to the linear predictor.

For binomial regression, write $Y_i\sim \mathrm{Bin}(m_i,p_i)$ independently, so ${\mu }_i=m_i{p}_i$. The logistic regression link iswhile probit regression useswhere $\Phi$ is the standard-normal distribution function. The logit is the canonical link function for the binomial family.

Given numerical values of $\mu$ and ${\sigma }^2$, maximize the joint log-likelihood over the response rates $0<p_{jk}<1$:This is a penalized binomial regression problem and can be solved by Newton method or another numerical optimizer. One may alternate this maximization with the closed-form updates for $\mu$ and ${\sigma }^2$ from part i until convergence. The selected solution should have a negative-definite Hessian in the fitted log-odds parameters.

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.