= Solution
Let $Y_{it}\in\{0,1\}$ indicate a smoking-free week, $S_i$ the recorded sex indicator, $A_i$ age in years and $T_i$ assigned treatment. The screening history supplies $Y_{i0}=0$; earlier lag initializations are also zero. A <history-dependent logistic regression> for the first model is
$$
Y_{it}\mid\mathcal H_{i,t-1},S_i,A_i,T_i\sim\operatorname{Bernoulli}(p_{it}),\qquad
\log\frac{p_{it}}{1-p_{it}}=\beta_0+\beta_SS_i+\beta_AA_i+\beta_TT_i+\gamma Y_{i,t-1}.
$$
Here $t=1,\ldots,10$, $\mathcal H_{i,t-1}$ is the observed past, and the five unknown <regression coefficients> have time-invariant values. Conditional on baseline <covariates>, this model has the first-order <Markov property>: only the immediately preceding outcome enters the current conditional <probability>. Distinct subjects have independent histories. There is no subject <random effect> or additional time trend in this fit.
Using the <chain rule for probabilities>, the individual conditional <likelihood> is
$$
\boxed{L_i(\beta,\gamma)=\prod_{t=1}^{10}p_{it}^{y_{it}}(1-p_{it})^{1-y_{it}},\qquad
p_{it}=\frac{e^{\eta_{it}}}{1+e^{\eta_{it}}}.}
$$
The lagged values in $\eta_{it}$ are the individual's actual preceding outcomes. This product is a sequential conditional <likelihood>, not an assertion of unconditional independence of the ten readings. The full conditional <likelihood> is $\prod_iL_i$; no extra <Bernoulli distribution> factor is attached to the fixed screening history.
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