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The first fit is a Bernoulli logistic-regression model for the probability of a positive response. Its fitted linear predictor isand its fitted probability is $\hat{\pi }=(1+e^{-\hat{\eta }}{)}^{-1}$. There are eight coefficients. The null degrees of freedom identify $n=200$, and the residual degrees of freedom are $200-8=192$.

Each slope is a change in the log odds per unit of its covariate, holding the others fixed. Exponentiating gives conditional odds ratios; for example a ten-unit glucose increase multiplies fitted odds by $e^{10(0.032115958)}=1.379$, and a ten-year age increase by $e^{10(0.041182353)}=1.510$. These are changes in odds, not multiplicative changes in probability, and the observational associations are not automatically causal effects. The intercept corresponds to all numerical covariates being zero, generally an extrapolation beyond meaningful subjects.

The standard errors come from the inverse Fisher information matrix of the logistic regression. With binomial dispersion parameter fixed at one, the displayed t ratios are asymptotic normal Wald statistics. In the full fit, glucose and the pedigree measure have clear individual evidence of nonzero slopes, with two-sided $p\simeq 2.09\times 10^{-6}$ and $0.00609$. The body-mass and age terms are borderline, with $p\simeq 0.0504$ and $0.0618$; pregnancy count is weaker, $p\simeq 0.110$. Blood pressure and skinfold have little individual evidence after adjustment. These labels should not be converted into a claim that the weaker predictors have no effect, or used as automatic model-selection rules.

A deviance residual here isIts square is the observation's contribution to binomial deviance, so the squared residuals sum to the printed residual deviance. The five-number summaries describe the distribution of these signed residuals. The null deviance $256.4142$ belongs to an intercept-only logistic regression; the full fitted deviance is $178.3907$. Four Fisher scoring iterations record convergence of the numerical likelihood fit.

The second model uses only glucose:Its two coefficients leave $198$ residual degrees of freedom. A ten-unit glucose increase multiplies its fitted odds by $e^{0.3778371}\simeq 1.459$. This differs from the adjusted multiplier $1.379$, because the models condition on different covariates; both adjustment and noncollapsibility of the odds ratio can change logistic coefficients. The glucose-only model has deviance $207.3727$, so its maximized log-likelihood is lower than the full model's.

For these fixed-dimensional nested models, analysis of deviance and Wilks theorem giveThus the additional six predictors collectively improve the glucose-only model, even though several individual Wald tests are not significant. These comparisons assume independent subjects, an appropriate logistic mean structure, full rank, and regular finite parameter estimates without separation.

It would be incorrect to infer that absolute fit is good merely because $178.39<192$. Individual Bernoulli deviance need not have a chi-squared calibration: there is just one binary observation per saturated probability, and the saturated dimension increases with sample size. A simple counterexample is an intercept-only model with true probability $1/2$, for which $D/n\to 2\log 2$, not one. For absolute adequacy, inspect residual patterns and predicted-risk calibration, use meaningful grouped comparisons when appropriate, or calibrate a chosen fit statistic by parametric bootstrap. The regular nested-model tests above are not subject to that saturated-model dimensionality problem. Only the displayed summaries and a few example rows are supplied, so patient-level diagnostics or complete numerical refitting cannot be reconstructed from this page alone.

The population-averaged logistic regression describes $P(Y_{ij}=1\mid z_i,x_i)$ after averaging over unmeasured individual heterogeneity. Its intercept is a marginal log odds at reference baseline values and time zero, its treatment coefficient is a population conditional-on-baseline log odds ratio, and its time coefficient is the change in that marginal log odds per month. Time zero is an extrapolation here because the recorded visits begin later; centring time at the first visit would give a more interpretable intercept.

In the random-intercept logistic model, the intercept is instead the conditional log odds for a subject with $B_i=0$. The treatment and time coefficients apply at fixed $B_i$; the subject's actual intercept is ${\alpha }_C+B_i$. Even with a treatment-independent random intercept, the conditional and marginal coefficients generally differ through noncollapsibility of the odds ratio. Indeed, if $\eta ={\alpha }_C+{\varphi }_{C}z+{\beta }_C^{T}x+{\delta }_{C}t$, marginalization giveswhich is not generally a logistic curve with the same linear coefficients. To quantify random-intercept attenuation of marginal logistic slopes, let $p_B={\mathrm{logit}}^{-1}(\eta +B)$. ThenFor a nondegenerate finite random intercept this derivative lies strictly between zero and one and usually varies with $\eta$. Thus the marginal treatment contrast is an integrated, attenuated version of the conditional contrast, and the marginal time slope need not be constant. One should not equate the two sets of coefficients, even in a randomized controlled trial without treatment confounding.

For missing completely at random, an unweighted observed-response GEE with cluster-robust standard errors can consistently fit the marginal mean, and a correctly specified integrated GLMM likelihood is valid as well. Under missing at random depending on previous observed outcomes, ordinary unweighted GEE is generally biased because the remaining responses are selected by informative observed history. Correct inverse-observation-weighted estimating equations for longitudinal dropout recover the marginal target under the sequential MAR and positivity assumptions already stated. Covariate-only observation mechanisms are a simpler special case in which correctly conditioned unweighted mean equations can remain valid.

A correctly specified joint response GLMM, fitted by observed-data likelihood, is valid under ignorable missing at random with distinct parameters, including dependence of missingness on observed response history. This is a likelihood property, not a claim that every mixed model automatically fixes missingness. Under missing not at random, neither ordinary GEE, MAR-based weights nor a response-only GLMM is generally valid; the unseen-outcome dependence requires further modelling and sensitivity assumptions.

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.

Let $Y_i=1$ indicate a depressed outcome and $C_i=1$ indicate the control arm. A logistic regression could specifywith $x_i$ containing prespecified baseline characteristics, including the stratification factor. Fit the coefficients by maximizing the likelihood for independent Bernoulli random variables ${\prod }_i{p}_i^{Y_i}(1-p_i{)}^{1-Y_i}$. For equal covariate values the control-to-intervention odds ratio is $e^{{\beta }_C}$. Thus an estimate ${\hat{\beta }}_C=\log (2.1)$ produces the quoted adjusted ratio. An ordinary Wald confidence interval is obtained by exponentiating the endpoints of the coefficient's confidence interval.

The raw control-to-intervention odds ratio, using the actual counts, isUsing only the quoted percentages gives about $2.05$, so both versions are approximately $2.1$. The printed $14\%$ is not the usual whole-percentage rounding of $40/297\simeq 13.47\%$; using the counts avoids that small numerical inconsistency. The reversed intervention-to-control ratio is about $0.475$.

The apparent surprise is a coding issue: with depression as the adverse outcome and intervention relative to control, protection would give an odds ratio below one. A ratio above one is favourable if the comparison is control relative to intervention, as above, or if the outcome has been recoded as absence of depression and intervention is compared with control. Both comparison and outcome must be specified. The adjusted ratio need not equal the crude one, even in a randomized trial: covariate adjustment and noncollapsibility of the odds ratio can change an odds ratio. Their close numerical values here do not prove that the regression adjustment was unnecessary.

There is an apparent reversal, but not a mathematical contradiction. The crude odds ratio averages over the actual background distribution in each group, whereas the adjusted logistic regression compares groups at the same fitted background values. Strong confounding could make those distributions very different: for example, elective patients might disproportionately come from low-risk backgrounds or clinics, while the reference patients come from higher-risk backgrounds. The Simpson paradox can then reverse the marginal comparison even when every conditional comparison favours the same direction.

The noncollapsibility of the odds ratio can also make marginal and conditional odds ratios differ in magnitude. It does not, by itself, explain a sign reversal if both groups have the same covariate distribution and the common conditional effect has one sign: if $p(1,x)>p(0,x)$ for every $x$, averaging over the same distribution of $x$ still gives a larger exposed risk. A genuine reversal therefore requires differing background distributions, heterogeneous effects or another change in what is being compared. The observed reversal is a reason to investigate these issues, not proof of a programming error.

Put $p_B={\mathrm{logit}}^{-1}(\eta +B)$ and $M(\eta )=\mathbb{E}(p_B)$. Differentiation gives $M^{\prime }=\mathbb{E}[p_B(1-p_B)]=M(1-M)-\mathrm{Var}(p_B)$. Hence the displayed derivative lies between zero and one, strictly below one for a nondegenerate random intercept. A marginal logit is therefore generally nonlinear and its slope differs from the conditional slope even without confounding. This is a precise instance of noncollapsibility of the odds ratio.