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I have been studying counting process theory for time to recurrent event processes and am interested in the explicit use of the conditioning set in the model notation; $$E[dN(t)|\mathcal{F}_{t^{-}}]=\lambda(t)dt$$ where $\lambda(t)$ is called the intensity of the risk of an event at time $t$. Where $\mathcal{F}_{t^{-}}$ is a sigma-field representing all relevant information about the counting process up to "just before" time $t$. Typically the sigma-field $\mathcal{F}_{t^{-}}$ is thought of as being genrated by a random covariate process $\mathcal{F}_{t^{-}}=\sigma(X(s):s<t)$ that contains the counting process history itself, and a Cox PHM type model is employed

$E[dN(t)|\mathcal{X}_{t^{-}}]=\lambda(t)dt=Y(t)\lambda_{0}(t)\exp(X(t)\beta)$

and given the "total information" in the conditioning information the recurrent events within individual subjects (people in a medical study or clinical trial say) are independent. This model is known as the Anderson-Gill model (1982, Annals of Statistics: Cox's regression model for counting processes). Given this conditional independence within subjects and assumed independence between subjects, my understanding is that the entire sample of event times can be modelled as per the standard Cox model and inference is valid. The key step is that the "compensator" $\Lambda(t)=\int_{0}^{t}\lambda(u)du$ renders the residuals $M_{\lambda}(t)$

$M_{\lambda}(t):=N(t)-\Lambda(t)$

a zero-mean martingale with respect to a filtration $(\mathcal{F}_{s})_{0\leq s <t}$ allowing the above model to be derived.

In constrast omitting some "dynamic" covariates such as event history leads to marginal models in the sense that lack of predictive information in the regression component means dependence within subjects will remain in the residuals and conditioning on this set (say $Z(t)\subset X(t)$ with sigma field $\mathcal{Z}_{t}\subset \mathcal{X}_{t}$) leads to the use of the model

$E[dN(t)|\mathcal{Z}_{t^{-}}]=\psi(t)dt=Y(t)\psi_{0}(t)\exp(Z(t)\beta)$

where for $\Psi(t)=\int_{0}^{t}\psi(u)du$ the residuals

$M_{\psi}(t):=N(t)-\Psi(t)$

are generally not a zero mean martingale. These models have been studied by Lin et al (2000, J. R. Statist. Soc: Semiparametric regression for the mean and rate functions of recurrent events) and inference proceeds by "naively" fitting a Cox PHM model as per the Anderson-Gill model and then correcting the variance of the resulting parameter estimator.

This approach of Lin for marginal models is redolent of Generalized estimating equations(GEE) for longitudinal data and this finally brings me to my question. The concept of conditioning sets and dependence of model residuals will surely hold too in the longitudinal setting? That is if total information was available and if one was permitted to use it (i.e. not in a randomised controlled trial for risk of bias) then model residuals would be conditionally independent within subjects and OLS regression could be used as opposed to (say) mixed effects models that model the covariance within subjects?

Mathematically I am wondering why the counting process notation is not used in presenting regression models. For example the standard for OLS regression is

$Y_{i} = X_{i}\beta + e$

where $e_{i}\sim N(0,\sigma^{2})$ and where the $X_{i}$ are modelled as fixed. This implies of course $E[Y_{i}|X_{i}]=X_{i}\beta$ which is the same [?] as assuming $X_{i}$ is random, independent of $e_{i}$ and that $e_{i}|X_{i}\sim N(0,\sigma^{2})$. The point is that the combination of the model, the covariate set and the residuals/errors are all linked: if $Y_{i}$ is cholesterol levels and $X_{i}$ intake of food and/or nutritional information etc., then $e_{i}\sim N(0,\sigma^{2})$ is plausible but if $X_{i}$ is the price of coffee in Brazil then clearly not. What in essence is being stated is surely something akin to counting process models? For example

$E[Y_{i}|\mathcal{X}_{t}]=X_{i}\beta$

where

$M_{i}:=Y_{i}-X_{i}\beta$

is a zero mean process with respect to a sigma-field $\mathcal{X}_{t}$;

$E[M_{i}|\mathcal{X}_{t}]=0$.

In this notation for repeated measures $Y_{i,j}$ for $i=1,...N$ subjects at times $j=1,...,n_{i}$, and if $X_{i}$ represents total information then conditional on $\mathcal{X}_{t}$ the entire set of observations $\{Y_{i,j}\}_{i,j}$ can be modelled as per OLS regression (in the spirit of Anderson-Gill models) but typically (due to lack of total information, or a preference for marginal models - they are easier to interpret) we use $\mathcal{Z}_{t}$ and model within-subject dependence.

Is the lack of sigma-field/conditioning set type notation for regression models because it is for some reason not relevant, or is it for convinience?

Update

Given the helpful response to this question I can now refine my question it a little and offer some of my own thoughts;

The time-series regression book provided develops the theory of Generalized Linear Models (GLMs) for the stochastic processes $(Y_{t})_{t\geq 0}$ and $(X_{t})_{t\geq 0}$ where the X's are a covariate process that incude past response history. Thus the theory appears to be precisely the same as the theory for counting process models (albeit with different models). For example $Y_{t}$ is assumed distributed from the exponential family;

$Y_{t}\sim f(y_{t}|\theta_{t},\phi,\mathcal{F}_{t})=\exp\left\{\frac{y_{t}\theta_{t}-b(\theta_{t})}{\alpha_{t}}+c(y_{t}|\phi)\right\}$

as per standard GLM notation - but obviously here there is a family of distributions indexed by $t$. The standard assumptions are made but conditional on the sigma field representing full covariate history (including response history);

$\mu_{t}=E[Y_{t}|\mathcal{F}_{t}]$

where $\mathcal{F}_{t}=\sigma\left\{Y_{t-1},Y_{t-2},...,X_{t-1},Y_{t-2,...}\right\}$. The difference with the counting process theory is that the times here are discrete instead of continuous. The mean $\mu_{t}$ is mapped onto the real line through the link function $g_{t}$;

$g(\mu_{t})=\eta_{t}=X_{t-1}\beta $

The mean and variance functions are related as per usual;

$\mu_{t}=b^{'}(\theta_{t})$

and

$Var[Y_{t}|\mathcal{F}_{t}]=\alpha_{t}(\phi)b^{''}(\theta_{t})$

In terms of inference the book focuses on partial likelihood which starts from the position of treating the pair $(Y_{t},X_{t})$ as random and factoring the joint density as follows

$f_{\theta}(x_{1},y_{N},...,x_{N},y_{N})=f_{\theta}(x_{1})\left[\prod_{t=2}^{N}f_{\theta}(x_{t}|d_{t})\right]\left[\prod_{t=1}^{N}f_{\theta}(y_{t}|c_{t})\right]$

where here the $N$ variables are on the same subject (subject index $i$ say dropped for clarity) and where $d_{t}=(y_{1},x_{1},...,y_{t-1},x_{t-1})$ and $c_{t}=(y_{1},x_{1},...,y_{t-1},x_{t-1},x_{t})$. The main idea is to drop the first two terms on the RHS the $X_{t}$ and hope that not too much information is lost about $\theta$. Obviously the story is that conditions are imposed to ensure this and of much interest is how reasonable or not these are. The partial likelihood is then defined as

$PL(\theta|y_{1},...,y_{N})=\left[\prod_{t=1}^{N}f_{\theta}(y_{t}|c_{t})\right]$

but since the $c_{t}$ include the past values of both $X_{t}$ and $Y_{t}$ (information that generates $\mathcal{F}_{t}$ then the densities in the right hand product are (presumably) defined to be conditional densities w.r.t the filtration $\mathfrak{F}:=(\mathcal{F}_{t})_{t\geq 0}$ - i.e. $\mathcal{F}_{t}\subset \mathcal{F}_{t+1}$ for all $t\geq 0$ and each $Y_{t}$ is measurable $\mathcal{F}_{t}/\mathbb{R}$

$PL(\theta|y_{1},...,y_{N})=\left[\prod_{t=1}^{N}f_{\theta}(y_{t}|\mathcal{F}_{t})\right]$

The martingale property that is central to the $M(t)$ in the counting process models appears in the score function $S_{N}(\beta)$ of the partial-likelihood which is a product involving the term $(Y_{t+1}-\mu_{t+1}(\beta))$ [... to be completed shortly]

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  • $\begingroup$ As a note - $X$ doesn't have to be random in a regression setting, and often isn't. $\endgroup$ – jbowman Sep 2 '18 at 18:16
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In regression, the predictors $x$ is most often taken as given constants, not as random. The name design matrix comes from that, the statistician has chosen (designed) the $x$ values where the observations is taken. This viewpoint is especially relevant for design of experiments.

In counting process models, time dependence is important, and the sigma-algebras is used there. So you could expect, maybe, that the same notation could be useful in regression settings where time dependence is an issue. Sometimes, one uses lagged values of the response $y$ in modeling time dependence. Now, the response $y$ becomes part of the predictors $x$, and the supposition that $y$ is random while $x$ is deterministic becomes "problematic". And there is indeed a book, https://www.amazon.com/Regression-Models-Analysis-Benjamin-Kedem/dp/0471363553/ref=sr_1_1?ie=UTF8&qid=1535912891&sr=8-1&keywords=regression+models+for+time+series+analysis&dpID=41l1u-nkkGL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch "Regression Models for Time Series Analysis" (Wiley) that uses measure-theoretic notation.

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    $\begingroup$ Thank you for your reply, and reference. I see your point but many "design" matrices in clinical trials involve baseline response measurements (height, weight, response itself) which, although planned, are really quite random. Planned times may also be missed and so appear random in behaviour (and really we choose discrete times to capture a continuous time processes). The fact covariates are treated as fixed is a modelling choice (for convienience ?) and so there must be an argument for residuals to posess conditional distributions depending on random covariates? $\endgroup$ – dandar Sep 2 '18 at 18:45
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    $\begingroup$ Usually, the argument is that the analysis is conditional on the observed values of the random covariates, and for inference, that the (random) covariates by itself not is informative about the regression parameters, that is, is ancillary. So the argument is the usual argument of conditioning on ancillaries. But it might well be that this argument is used automatically to often? $\endgroup$ – kjetil b halvorsen Sep 2 '18 at 19:08

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