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For a group with sample size $n_s$, let $Y_s$ be its overweight count. The natural sampling model is a binomial distribution, $Y_s\sim \mathrm{Bin}(n_s,{\pi }_s)$, with observed proportion $p_s=Y_s/n_s$. A grouped-binomial logistic regression specifiesFitting event/nonevent counts, or fitting the proportions with weights $n_s$, gives the same binomial likelihood. The weights represent trial counts; unweighted regression of the proportions would give small and large groups the same influence. If printed percentages are rounded, exact original event counts are preferable to manufacturing fractional counts from $n_s{p}_s$.

Year should ordinarily be a factor with three levels unless a linear trend over time is specifically intended. Country, sex and age group are also possible factors. The actual subset, reference levels and formula are absent from this PDF. For a year-by-age interaction term, writeWith treatment contrasts, $\alpha$ is the baseline logit, $u_a$ and $v_t$ are effects at the reference level of the interacting factor, and $w_{at}$ is an additional difference of log-odds differences. Exponentiating a main-effect contrast gives an odds ratio; exponentiating an interaction contrast gives a ratio of odds ratios. It is incorrect to interpret a main effect as a constant effect across the other factor when the interaction term is present.

A coefficient table should be read as estimate, standard error, a Wald estimate/error ratio and its reference-distribution probability. A whole multi-level interaction term is better tested by a nested likelihood-ratio test, using the decrease in deviance and the number of added independent coefficients. Null and residual deviances, residual degrees of freedom, convergence and fitted probabilities should all be checked. Large residual dispersion suggests lack of fit or dependence within sampled schools; a quasibinomial regression or an appropriate clustered analysis may then be preferable. Residual deviance is not automatically a valid chi-squared goodness-of-fit statistic for sparse groups or after estimating a free dispersion.

An interaction plot should display the group means or fitted probabilities against year, with separate lines for the interacting factor, and should also inspect the link scale. Nonparallel probability curves alone do not establish a logit interaction: even an additive logit model transforms nonlinearly to probabilities. As an independent descriptive illustration, the original study's Table 2 reports English boys' age-specific percentages. The following original plot shows their widening age differences after 1984 on both scales. It uses published descriptive values, not a reconstruction of the absent examination fit. The data are from pmc.ncbi.nlm.nih.gov/articles/PMC26603/ .

For example, on the probability scale the 1984–1994 increase for the oldest group exceeds that for the youngest by $6.9-0.8=6.1$ percentage points using the rounded table values. On the logit scale the corresponding difference of differences isThis describes the interaction pattern, but its sampling uncertainty requires the group counts. The absent output prevents identifying the exam fit's coefficient estimates and significance tests.

There are two different uses of Poisson regression. A rare-event approximation models $Y_s\sim \mathrm{Pois}(n_s{\pi }_s)$ withThe fixed offset $\log n_s$ accounts for exposure, and exponentiated slopes are risk ratios, not odds ratios. The approximation is most appropriate when probabilities are small; it is not exact for appreciable prevalence because binomial variance is $n_s{\pi }_s(1-{\pi }_s)$ rather than $n_s{\pi }_s$.

An exact alternative stacks event and nonevent counts in each group and fits a log-linear model with a separate group intercept and event-by-predictor effects:Conditioning on their total $n_s$ gives the binomial event probability ${\mu }_{s1}/({\mu }_{s0}+{\mu }_{s1})$, whose logit is exactly $x_s^T\beta$. Profiling the group intercepts enforces the observed totals and reproduces the grouped-binomial logistic regression. This Poisson surrogate for a conditional multinomial model is exact and should not be confused with the rare-event approximation.

The null model uses one parameter, the day model uses nine, and there are $27$ grouped observations. The four missing entries, in their table order, are thereforeThe test of no day effect is $H_0:{\alpha }_2=\cdots ={\alpha }_9=0$. The nominal binomial likelihood-ratio test statistic is $\Delta D=462.8363$, compared asymptotically with ${\chi }_8^2$. Its nominal $p$-value is about $6.6\times 10^{-95}$, so the common-probability model is decisively inadequate relative to the day model. The fitted probabilities in (i) show substantial differences between days, with days 1 and 7 much higher than days 3 and 8.

However, relative improvement does not establish absolute fit. The day model's deviance goodness-of-fit test compares $D=32.7945$ with ${\chi }_{18}^2$, giving $p\simeq 0.0177$. The counts are large enough for this approximation to be reasonable. Thus at $5\%$ the day-specific binomial model still shows lack of fit. Among the two printed models it is preferred, but it should not be regarded as a fully satisfactory final model.

The residual scale estimate $D/18\simeq 1.822$ suggests overdispersion. Directly computing the Pearson statistic from the day means givesThis corroborates extra dish variation rather than the nominal binomial variance. Dish heterogeneity or dependence among cells in a dish can produce such variation. Residual checks and a quasibinomial regression or an explicit model for dish heterogeneity are sensible refinements. Allowing dispersion of this moderate magnitude would enlarge standard errors but would not plausibly remove the very large day effect. Retain day effects, while accounting for the remaining extra-binomial variation.

Use grouped-binomial logistic regression, $S_i\sim \mathrm{Bin}(C_i,p_i)$ and $\mathrm{logit}(p_i)={\eta }_i$. Center the year as $u_i=i-7.5$. Maximizing $\ell ={\sum }_i[S_i{\eta }_i-C_i\ln (1+e^{{\eta }_i})]$ gives score $X^T(S-Cp)=0$ and information $X^T\mathrm{diag}\{C_i{p}_i(1-p_i)\}X$; these determine the estimates and working covariance matrix.

A straight-line logit fit is $\hat{\eta }=-1.95965+0.045384u$, with naive binomial standard errors $0.005382$ and $0.001352$. Its yearly odds multiplier is $e^{0.045384}=1.04643$, with naive 95% Wald confidence interval $[1.04366,1.04921]$. However, its residual deviance is $58.20$ on 12 degrees of freedom ($p=4.8\times 10^{-8}$), so those small standard errors do not describe an adequate independent-binomial fit.

Comparing polynomial mean structures gives residual deviances $1191.73,58.20,55.70,27.08$ for degrees zero through three, with respective residual degrees of freedom $13,12,11,10$. A quadratic term alone adds little; the cubic structure captures the middle-period acceleration and later leveling. Even the cubic independent-binomial fit has $p=0.00253$ for its residual deviance goodness-of-fit test. A useful working model is therefore cubic quasibinomial regression, retaining the logistic mean and estimating overdispersion rather than claiming an exact binomial law:The Pearson estimate is $\hat{\varphi }=27.182/10=2.718$. Using standard errors multiplied by $\sqrt{\hat{\varphi }}$ and a $t_{10}$ reference, the approximate 95% coefficient intervals are $[-1.98294,-1.92429]$, $[0.04982,0.07445]$, $[-0.001697,0.001131]$, and $[-0.000980,-0.000182]$. An approximate quasi-deviance comparison of cubic against linear gives $F_{2,10}=5.72$ and $p=0.022$, supporting the added shape while acknowledging that only fourteen annual groups inform dispersion.

In this preferred mean model the odds change is not constant:The fitted annual change is larger around the middle than at the ends; the straight-line 4.6% increase is a useful overall summary, not an exact yearly effect in the cubic model. The fitted curve, uncertainty band and residual plot appear in the figure. Singleton success increased overall, but a constant log-odds trend with unit binomial dispersion fits poorly. Quasibinomial intervals remain working-model intervals: serial dependence, changing patient mix and treatment practices cannot be separated from annual aggregates alone.

For woman $i$, write $m_i$ for the number of births, $Y_i$ for the number with defects, $a_i$ for baseline age, and $e_i$ for exposure. Let $p_i$ be the common marginal defect probability for her births. The first fit is a grouped grouped-binomial logistic regression:It assumes independent women and, conditionally on their covariates, independent births with the same probability within each woman. The supplied birth totals are the binomial denominators; fitting the proportions with weights $m_i$ is equivalent to this grouped-binomial specification. The fitted linear predictor is $-1.74272+0.02030a_i+2.33028e_i$.

The second fit is a generalized additive model with the same logit link function and the mean specificationThe functions $f_a,f_e$ are penalized cubic regression splines with natural boundary conditions. Centering constraints such as ${\sum }_i{f}_a(a_i)={\sum }_i{f}_e(e_i)=0$ separate them from the intercept; the fitted centered intercept is $-0.2031$. Their second derivative roughness penalties control complexity.

The call estimates a common scale rather than fixing it at one. Its working variance specification for the counts isEquivalently the variance of $Y_i/m_i$ is $\varphi p_i(1-p_i)/m_i$. This is the working moment interpretation of an overdispersed binomial generalized additive model: the code uses binomial deviance for fitting and estimated scale for inference. With $\varphi \ne 1$, it is not an exact independent-binomial sampling model. Women are still treated as independent groups, but within-woman dependence or unobserved heterogeneity can motivate the extra dispersion. The quasibinomial regression interpretation states what the scaled analysis assumes without inventing a full probability distribution for its counts.

The original PDF's age smooth is nearly flat, its uncertainty band includes zero across the age range, and its approximate p-value is $0.859$. The exposure smooth is increasing and convex: it is fairly flat at low exposure and rises more steeply at high exposure. Its effective degrees of freedom $1.895$ suggest a modest departure from a straight line, rather than a highly oscillatory relationship.

A suitable simpler model is therefore the grouped quasibinomial regressionwith age omitted and dispersion estimated. A positive quadratic coefficient captures the convex exposure relationship; retaining the linear term respects the usual polynomial hierarchy. This model should be refitted, rather than assigning coefficients from the smooth plot. Compare its residual pattern and dispersion-adjusted fit with the additive model to check that the quadratic approximation is adequate. An exposure-only quadratic logit model is a justified parsimonious candidate, while the graph does not justify a sharp threshold or a zero-dispersion model.

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.