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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 isHere $t=1,\dots ,10$, ${\mathcal{H}}_{i,t-1}$ 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 isThe 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 ${\prod }_i{L}_i$; no extra Bernoulli distribution factor is attached to the fixed screening history.

Let $q_{rs}$ be the transition intensity from state $r$ to $s$, and let $Q$ be the transition intensity matrix, with $q_{rr}=-{\sum }_{s\ne r}{q}_{rs}$. For a continuous-time multi-state model with the time-homogeneous Markov property, the transition probability matrix is $P(u)=e^{uQ}$, with entry $p_{rs}(u)$. Condition on each recorded initial state. Panel visits contribute transition probabilities; an exact entry into the absorbing death state contributes a statistical probability densityThis mixed panel and exact-death likelihood sums over the living state just before death. It accounts for survival until the event; replacing its final factor by $p_{r3}(u)$ would count deaths throughout the interval.

Using the actual visit times recovered from the PDF, the three individual likelihood contributions areHere $g_{23}$ means $g_{r3}$ with $r=2$, not a transition probability. Subjects 8 and 9 supply no event-density factor after their last panel observation. Assume independent subjects and noninformative examination and censoring times; conditional on their observation schedule, its distribution supplies no additional statistical parameter-dependent factor. The patients' covariates can be incorporated by using their own $Q_i$ in these same expressions.

For the progressive structure used in the subsequent output, put $a=q_{12}$, $b=q_{13}$, $c=q_{23}$ and $\lambda =a+b$. The progressive illness-death model permits no recovery, soWhen $\lambda =c$, the continuous limit is $p_{12}(u)=au{e}^{-cu}$. Thus the likelihood contributions simplify toIntermediate unobserved disease transitions remain integrated into each panel transition probability.

Let $z_i=(S_i,E_i,A_i{)}^T$ contain sex, the education indicator, and age at diagnosis. Write $\overline{z}$ for the sample means. The output's baseline transition intensities are evaluated at those means, so use the centred log-linear transition intensity modelThe full transition intensity matrix isThe fitted centred baselines are $(q_{12},q_{13},q_{23})(\overline{z})=(0.1821,0.0126,0.0450)$, and fitted slope vectors in sex–education–age order areThe two sex coefficients displayed as zero are fixed by the specified constraints; they are not estimated to be exactly zero. There are three free baseline transition intensities and seven free covariate slopes. The sample means are not printed, so uncentred intercepts at $z=0$ cannot be recovered numerically from this output.

Conditional on fixed covariates, subjects follow independent, correctly classified continuous-time multi-state models obeying the time-homogeneous Markov property, with $P_i(u)=e^{u{Q}_i}$. The progression is irreversible and death absorbing, with constant transition intensities during follow-up for each subject. In particular, the model uses fixed age at diagnosis rather than attained age. It assumes noninformative observation and censoring, and treats death times as exact through the mixed panel and exact-death likelihood. The Markov property rules out an additional effect of elapsed time in the current state after conditioning on it and the recorded predictors.

Age changes during follow-up, so the generator matrix $Q(s)$ changes with time. The transition matrix must solve ${\partial }_{t_2}P(t_1,t_2)=P(t_1,t_2)Q(t_2)$, with $P(t_1,t_1)=I$. It is generally not the matrix exponential of one constant matrix; even $\exp (\int Q(s)ds)$ is not generally valid because the matrices at different times need not commute.

For example, write the age-dependent onset rate as $\lambda (s)$ and keep progression rate $\nu$ constant. ThenEven when the integrated onset rate is elementary, this remaining integral need not have an elementary closed form. Special cases can have closed forms; time dependence does not by itself prove universal impossibility.

For a tractable likelihood function, freeze age at its left-endpoint value, or a midpoint value, on each observation interval $[t_j,t_{j+1})$. The genetic covariate is already constant. This piecewise-constant covariate approximation gives a constant matrix $Q_j$ on each interval and a product of closed-form interval probabilities:Here $x_j$ is the observed state at $t_j$, and the likelihood function is conditional on the initial state. The approximation improves when intervals are short relative to the variation in the age effect.

Put $U=H(T)$ and $V=H(C)$, so $Y=\min (U,V)$ and $0\le \mathbb{E}Y\le \mathbb{E}U=1$. Its expectation is not generally one and depends on censoring. Under independent censoring, condition on $V=v$: the exponential distribution givesWriting $\Delta =1_{T\le C}$ for the observed event indicator, it follows that $\mathbb{E}Y=\mathbb{E}\Delta$. The exponential mean imputation under independent censoring therefore gives a modified Cox–Snell residual with known mean:On censoring, the added one is the expected remaining transformed lifetime, by memorylessness of the exponential distribution. This gives a mean-one variable, not necessarily another exponential variable. Equivalently, $M=\Delta -Y=1-Y^{\ast }$ is a martingale residual with mean zero. Independent censoring, or its appropriate conditional version, is essential for these identities; arbitrary informative censoring does not justify the correction.

Fit a model using the relevant covariates, calculate ${\hat{M}}_i=v_i-{\hat{H}}_i(x_i)$, and plot these martingale residuals against a candidate explanatory variable or against an included variable's value. A smooth systematic trend away from zero can indicate an omitted effect or an unsuitable functional form, such as a nonlinear age effect. After fitting an appropriate effect, the residual trend should diminish. The equivalent plot of ${\hat{Y}}_i^{\ast }$ has mean-one reference rather than zero.

This is administrative censoring. Under the usual assumption that entry calendar time is independent of the subsequent event time, it is an example of independent censoring, not informative censoring: the planned study closure, rather than a patient's prognosis, determines the available follow-up. Uniform entry by itself does not prove that assumption; if entry date is associated with prognosis, independence must instead be justified conditionally on the relevant covariates. The calculations below use independence.

In the counting-process intensity in survival analysis, the counting process $N_i(t)$ jumps once if an observed event occurs; the at-risk process $Y_i(t)$ is one immediately before $t$ only while individual $i$ is still observed and event-free. Conditional on that information, the chance of an event in a short interval is $h_i(t)dt+o(dt)$ for an at-risk individual and zero otherwise. This is the meaning ofThe history ${\mathcal{H}}_{t-}$ is the filtration generated by the observed event and censoring histories, entry information, and available covariates strictly before $t$. In particular, it determines the current risk set but does not reveal future event times. The right-hand side is the infinitesimal compensator of a counting process; the equality is interpreted as an intensity statement to first order in $dt$.

A proportional hazards model assigns individual $i$ the hazard function $h_i(t)=h_0(t){\varphi }_i$, where $h_0$ is the baseline hazard for multiplier one and ${\varphi }_i>0$ is a time-constant hazard multiplier. In a Cox proportional-hazards model, ${\varphi }_i=e^{x_i^T\beta }$ for a covariate vector $x_i$ and coefficient vector $\beta$; ratios $h_i(t)/h_k(t)={\varphi }_i/{\varphi }_k$ are constant whenever the baseline is nonzero.

Assume independent individuals, independent censoring conditional on the covariates, and no tied events. Immediately before an event at $t$, condition on its observed risk set $R(t)$ and one event in a small time interval. The individual event probability is $h_0(t){\varphi }_{i}dt+o(dt)$, while the total is $h_0(t){\sum }_{k\in R(t)}{\varphi }_{k}dt+o(dt)$. Their limiting ratio is ${\varphi }_i/{\sum }_{k\in R(t)}{\varphi }_k$. Multiplication over observed events gives the Cox partial likelihoodThe unspecified baseline hazard cancels. This is the usual successive conditional event contribution defining the partial likelihood, not the full event-time likelihood conditioned simultaneously on all event times.

For the four individuals the successive event risk sets are $\{a,b,c,d\}$, $\{c,d\}$ and $\{d\}$. Individual $b$ leaves at its right censoring time and contributes no event numerator. ConsequentlyThe last event contributes one and carries no further relative-risk information.

Because $t^{\ast }>t_b>t_a$, the three possible complete event orders are $t_a<t^{\ast }<t_c<t_d$, $t_a<t_c<t^{\ast }<t_d$, and $t_a<t_c<t_d<t^{\ast }$. Write $K={\varphi }_a/({\varphi }_a+{\varphi }_b+{\varphi }_c+{\varphi }_d)$ and $V={\varphi }_b+{\varphi }_c+{\varphi }_d$. Their partial likelihoods, with the final singleton factor omitted, are respectivelyAdding first $L_2$ and $L_3$ gives $K{\varphi }_c/V$, and thereforeThusThis is Cox rank-likelihood deletion consistency: summing out the unobserved position of $b$ leaves the relative order information in the observed events. It is an algebraic marginalization of complete-order probabilities under time-constant hazard multipliers. It does not include the probability density function of the actual censoring time, establish the distribution of $t^{\ast }$ given all observed times, or justify informative censoring. Ignoring the censoring mechanism in the survival analysis still requires independent censoring given the modeled covariates.

Paul R. Rosenbaum develops the design and analysis of observational studies. His work asks how a study can support a causal inference when investigators cannot assign the exposure by randomization. His research includes matching, sensitivity analysis for unmeasured confounding and ways to design studies whose conclusions withstand larger hidden biases.^[ref]

With Donald B. Rubin, he established the central role of the propensity score $e(X)=P(A=1\mid X)$: conditioning on this score balances measured baseline covariates between treatment groups. Under the required treatment-assignment assumptions and positivity in causal inference, the score permits adjustment for a many-dimensional $X$ through a scalar. Measured balance alone leaves the possibility of unmeasured confounding.^[ref]

Approximate changing covariates by constant values on observation intervals. The generator matrix is then constant on each interval, with transition matrix $\exp (Q_j\Delta t_j)$. Multiplying the relevant entries gives a conditional likelihood function for discretely observed states.

Positivity requires every compared treatment to have positive probability at each target covariate value.

The propensity score for binary treatment $A$ and pretreatment covariates $X$ is $e(X)=P(A=1\mid X)$. It is a balancing score:This distributional fact does not require conditional exchangeability of the potential outcomes. It follows from the definition of the true assignment probability.^[ref]

For a bounded function $g$, condition successively on $X$ and on $e=e(X)$:Also $P(A=1\mid e)=e$, so division by $e$ gives the same conditional covariate distribution among treated units as in the full score stratum; the analogous argument uses $1-e$ for controls. Comparison of both arms requires $0<e<1$.

Identifying a causal effect is an additional step. If $(Y(0),Y(1))\perp A\mid X$, then the balancing property lets the scalar propensity score replace the full measured $X$ in that exchangeability statement. Without this causal assumption, balancing measured covariates leaves possible unmeasured confounding. An estimated score also need not deliver exact finite-sample covariate balance; its estimation and the subsequent matching in causal inference or weighting procedure must be checked.

A randomized controlled trial assigns experimental units to competing treatments by a specified random mechanism and compares their outcomes. Under complete randomization of $N$ units, exactly $n_1$ receive assignment one and $n_0=N-n_1$ assignment zero. With fixed potential outcomes, each unit has assignment probability $n_1/N$. ThereforeThis proves design-unbiasedness of the mean difference for the finite-population average treatment effect, assuming consistency in causal inference and no interference in causal inference.

Randomization controls the assignment distribution; it does not force every realized baseline covariate difference to be zero. Equal assignment probabilities yield unconditional exchangeability in the design, whereas covariate-dependent randomization can require conditional exchangeability and an analysis respecting that design.^[ref]

When assigned and received treatments differ, an intention-to-treat analysis estimates an assignment-policy effect. A treatment-receipt effect needs additional assumptions. Allocation concealment, blinded outcome assessment and handling of missing follow-up address other sources of distortion beyond random assignment itself.

For each new eligible participant, compare the imbalance that would result from each treatment assignment across prespecified baseline covariate margins. Assign with a probability favoring the less imbalanced choice, while retaining a random component and allocation concealment. This targets marginal balance when a large number of joint strata makes stratified randomization with blocks inefficient. It is a sequential allocation rule, and inference should respect its design.

A regression model specifies features of the conditional distribution of a response given predictor covariates, often a conditional mean or a full parametric density. Linear regression specifies a linear conditional mean; logistic regression specifies a logit-linear conditional probability for a binary response. The conditional comparison represented by a coefficient depends on which covariates are included.

Model the joint distribution by multiplying a response model by a sequential dropout model. For example, the logit of dropout before visit $j$, conditional on remaining observed, can depend on the currently unobserved $Y_{ij}$ as well as previous responses and baseline covariates. The observed-data likelihood sums over the missing binary responses. Such a dependence parameter is usually weakly identified or unidentified from observed responses alone; sensitivity analysis must vary it over scientifically justified values.

Let $p_{gs}$ be a group-specific conditional probability in stratum $s$, and let $w_s\ge 0$ with ${\sum }_s{w}_s=1$. Average each group's conditional probabilities using the same weights. ThenThis is a descriptive comparison under a common covariate composition. Its difference from a crude contrast need not be a causal effect. Causal outcome standardization additionally requires its identification assumptions.

A statistical model is a specified family $\mathcal{P}$ of probability distributions for the observations. A statistical parameter often indexes it as $\mathcal{P}=\{P_{\theta }:\theta \in \Theta \}$, where $\Theta$ is the parameter space. The choice states which patterns of variation are considered possible and supplies a basis for statistical inference; it does not guarantee that any member exactly describes the data-generating mechanism.

For example, suppose $n$ binary observations are independent and identically distributed with a common unknown success probability $p$. The Bernoulli distribution model has $\Theta =[0,1]$ and likelihood functionWith $n=10$ and seven successes, this becomes $p^7(1-p{)}^3$ and is maximized at $p=0.7$. The binary observation type alone does not supply independence or a common $p$: those are additional model assumptions. A regression model can instead let the response law depend on observed covariates, while a mixed model represents a specified grouping structure through random effects.

Survival data describe the time $T$ from a specified origin to an event, together with covariates and the observation mechanism. Under right censoring, the observed exit time is $Y=\min (T,C)$ and the event indicator is $\delta =1_{\{T\le C\}}$. With left truncation, an entry time $E$ is also recorded and subjects are observed only if $T>E$. Recording the time origin, entry, exit and event indicator is essential for constructing valid risk sets.