A simple model for a stationary sequence of integer-valued random variables with lag-one dependence is given and is referred to as the integer-valued autoregressive of order one (INAR(1)) process. The model is suitable for counting processes in which an element of the process at time t can be either the survival of an element of the process at time t -1 or the outcome of an innovation process. The correlation structure and the distributional properties of the INAR(1) model are similar to those of the continuousvalued AR(1) process. Several methods for estimating the parameters of the model are discussed, and the results of a simulation study for these estimation methods are presented.

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional rnarginals of the process (X,,) generated according to the scheme X , , =~O X , , -~ +c,,, where aoX,,-l denotes a sum of X,,-l independent 0-1 random variables fin-'), independent of X,,-l with Pr(fi"-')=l)=l -P r ( V -' ) = O ) = a .The distribution of the innovation pro cess (c,,) is obtained when the marginal distribution of the process [&) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.

An extension of the INAR(1) process which is useful for modelling discrete-time dependent counting processes is considered. The model investigated here has a form similar to that of the Gaussian AR(p) process, and is called the integer-valued pth-order autoregressive structure (INAR(p)) process. Despite the similarity in form, the two processes differ in many aspects such as the behaviour of the correlation, Markovian property and regression. Among other aspects of the INAR(p) process investigated here are the limiting as well as the joint distributions of the process. Also, some detailed discussion is given for the case in which the marginal distribution of the process is Poisson.

An extension of the INAR(1) process which is useful for modelling discrete-time dependent counting processes is considered. The model investigated here has a form similar to that of the Gaussian AR(p) process, and is called the integer-valued pth-order autoregressive structure (INAR(p)) process. Despite the similarity in form, the two processes differ in many aspects such as the behaviour of the correlation, Markovian property and regression. Among other aspects of the INAR(p) process investigated here are the limiting as well as the joint distributions of the process. Also, some detailed discussion is given for the case in which the marginal distribution of the process is Poisson.

Clinical trials target patients who are expected to benefit from a new treatment under investigation. However, the magnitude of the treatment benefit, if it exists, often depends on the patient baseline characteristics. It is therefore important to investigate the consistency of the treatment effect across subgroups to ensure a proper interpretation of positive study findings in the overall population. Such assessments can provide guidance on how the treatment should be used. However, great care has to be taken when interpreting consistency results. An observed heterogeneity in treatment effect across subgroups can arise because of chance alone, whereas true heterogeneity may be difficult to detect by standard statistical tests because of their low power. This tutorial considers issues related to subgroup analyses and their impact on the interpretation of findings of completed trials that met their main objectives. In addition, we provide guidance on the design and analysis of clinical trials that account for the expected heterogeneity of treatment effects across subgroups by establishing treatment benefit in a pre-defined targeted subgroup and/or the overall population. Copyright © 2016 John Wiley & Sons, Ltd.