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The cumulative-response logistic model uses $C_{it}={\sum }_{s<t}{Y}_{is}$ instead of the two individual lag predictors. It and the two-lag history-dependent logistic regression are nonnested: the entire accumulated history generally cannot be represented using only the two recent outcomes. Their binomial deviance difference therefore has no ordinary nested chi-squared distribution calibration.

Use the Akaike information criterion, which up to the same saturated-model constant equals $D+2k$ for these Bernoulli distribution conditional likelihoods. The two-lag fit has six coefficients, while the cumulative fit has five:Thus prefer the cumulative-history model, whose Akaike information criterion is lower by 35 despite its smaller number of statistical parameters. This is a model-selection comparison of conditional histories, not a nested likelihood-ratio test.

The preferred cumulative-response logistic model estimatesIts logit link describes conditional smoking-free probability given baseline predictors and prior successful weeks. Holding the other predictors fixed, males have $e^{-0.364092}=0.695$ times the female success odds; the reported p-value is $0.0152$, and the approximate 95% odds ratio confidence interval is $(0.518,0.932)$. Age has estimated odds ratio $e^{0.001224}=1.0012$ per additional year, with p-value $0.918$, giving little evidence for an age association in this fit.

The combined treatment has conditional success odds ratio $e^{0.327118}=1.387$ versus the reference treatment, with approximate 95% confidence interval $(1.035,1.859)$ and p-value $0.0285$. The fitted conditional odds are about 39% higher for the combined treatment. The trial randomization supports treatment comparisons, but conditioning on accumulated post-treatment outcomes means this coefficient is not directly the marginal total treatment effect.

Each previous successful week multiplies current success odds by $e^{0.399296}=1.491$, with approximate 95% confidence interval $(1.329,1.673)$ and very small p-value $1.06\times 10^{-11}$. This is strong fitted persistence. It can reflect true state dependence, unobserved heterogeneity, or an omitted calendar-time trend; this fit alone cannot distinguish them. The baseline intercept implies success probability ${\mathrm{logit}}^{-1}(-1.417630)\simeq 0.195$ for a reference-treatment female aged zero with no previous success. That age is outside the study's useful interpretation range, so the intercept chiefly anchors the regression. The Bernoulli distribution dispersion parameter is fixed at one, and the residual binomial deviance is 1122 on 995 statistical degrees of freedom. With individual binary outcomes, comparing that binomial deviance mechanically to a chi-squared distribution is not a reliable general goodness-of-fit test.

Take the event to mean three smoking-free weeks in succession, $Y_{i1}=Y_{i2}=Y_{i3}=1$. The initial cumulative count is zero. Along this path it equals $0,1,2$ in weeks one, two, three, respectively. In the cumulative-response logistic model, defineThe chain rule for probabilities then givesThe three conditional probabilities are approximately $0.19891,0.27015,0.35559$. If “stop during the first three weeks” instead means at least one smoking-free week by week three, its different event has probability $1-(1-{\pi }_0{)}^3\simeq 0.48590$: along the all-failure path the cumulative count stays zero. Stating the event resolves this wording ambiguity.