762notes.pdfCHAPTER 13 ST 762, M. DAVIDIAN13 Approaches to modeling multivariate response13.1 IntroductionIn the previous chapters, we have focused on the situation in which the responses may be viewed asindependent. We thought about this in different ways.• In some circumstances, units may be (randomly) sampled from some population of interest, andunit j gives rise to (Yj, xj). Thus, we have i.i.d. pairs (Yj, xj), j = 1, . . . , n, and interest focuseson characterizing the association between Yjand xjby modeling E(Yj|xj). Here, we may viewthe Yjconditional on the xjas ind ependent, as is customary in the “classical” regression set-upand more generally.• In other situations, an experiment may be carried out in which the xjare fixed by the investigator.For each setting xj, Yjis observed independently of the observations for the other x settings; e.g.,each j involves observing a certain chemical reaction at temp erature xj, or xjis the dose of a druggiven to animal j in a random s ample of su ch animals. Here, again, th e assumption that the Yjconditional on the xjare independent is reasonable.• We also discussed examples (e.g Examples 1.1, 1.2, and 1.3) in which rep eated measurements aretaken on a single unit. In Examples 1.1 and 1.2, for instance, a single subject is given a doseof a drug at time 0, and drug concentrations Yjat times xjfollowing the dose are derived fr omblood samples taken at those times, and the objective is to characterize the concentration-timerelationship for this particular subject.In this setting, we noted the possibility that observations taken very close together in time mightbe expected to be “more alike” than those far apart in time; that is, we might expect such repeatedmeasurements to be serially correlated. In Section 13.2, we will discuss this in more detail. Often,the blood samples are taken sufficiently far apart in time that concern over this issue is negligible,so it is routine to assume that the Yj, j = 1, . . . , n, are independent over time.We noted in Chapter 1 that, in this situation, thinking of “time” as a “covariate” in the usualsense may not be quite appropriate; it may be more reasonable to consider a stochastic processover time, realizations of which we obs erve at times xj. We will f or the most part downplay thisdistinction but discuss it further shortly.There are many circumstances, however, in which assuming independence among all observations in aregression setting is not reasonable.PAGE 320CHAPTER 13 ST 762, M. DAVIDIAN• Example 1.7 is an example of a situation in which the data fall into natural “clusters” due to theway in which the observations arise.In this example, pregnant rats indexed by i = 1, . . . , m receive a dose xiof a toxic agent; eachrat gives birth to nipups, where the jth pup has birthweight Yij, j = 1, . . . , ni. As discuss edpreviously, it is natural to think that the birthweights of pups f rom the same mother might be“more alike” than those from different mothers; a “high” birthweight m other’s pups will tend tobe heavier than the average pup across all such mothers at dose xi, so might all tend to be ab ovethis average together. Of course, we would expect that the way birthweights turn out for pupsfrom different mothers would h ave nothing to do with one another.Under these circumstances, it is natural to think of the observations from mother i together as agroup or “cluster.” Letting Yi= (Yi1, . . . , Yini)Tbe the vector of b ir thweights for mother i, weobserve pairs (Yi, xi), i = 1, . . . , m. The Yigiven the ximay be reasonably viewed as independentfrom above; however, the observations within Yifor any i are correlated.So, clearly, if we consider the entire data vector Y = (YT1, . . . , YTm)T, it would be unreasonable toassume in dependence among all elements of Y . Thus, trying to view this problem from the per-spective we have considered previously, as N =Pni=1niindependent pairs (Yij, xi), i = 1, . . . , m,j = 1 . . . , ni, would be erroneous.• Example 1.8 reflects the same issue, with some additional complication. Here, the ith of m subjectsreceives dose of a drug Di, and drug concentrations Yijare taken at times tijpost-dose for subjecti, j = 1, . . . , ni. Thus, letting xij= (Di, tij)T, subject i has nipairs of observations (Yij, xij),j = 1, . . . , ni. Here, we include time as part of the covariate xijfor notational convenience.Again, it is natural to think of the observations in “clusters,” by individuals: Letting Yi=(Yi1, . . . , Yini)T, it is clearly reasonable to consider the m Yias independent, as each arises fr oma different individual. In particular, as ph armacokinetic behavior is a w ithin -individual process,there is no reason to expect that the way in which drug concentrations arise for one person wouldbe related to that for another.However, it is natural to be concerned about correlation of the observations within a given datavector Yifrom two sources:(a) As discussed above, the time-ordered nature of the observations raises concern over serialcorrelation. This concern is an “individu al” one – even if the available data were only from asingle subject and interest focused on that subject, s uch correlation would still be an issue.PAGE 321CHAPTER 13 ST 762, M. DAVIDIAN(b) As in the developmental toxicology example, from the perspective of th e entire population ofindividuals, it is natural to be concerned that concentrations on the same ind ividual wouldbe “more alike” than concentrations from different individuals.e.g., a certain subject may be a “high” concentration s ubject, whose concentrations tend tobe inherently higher than the average concentration across all subjects if they were all toreceive the s ame dose and be measured at a particular time. Thus two concentrations fromsuch a subject might be expected to be “high” in this way together.We will discuss these ideas further in Section 13.2. It is clear, however, that regardin g the elementsof Y = (YT1, . . . , YTm)Tas independent would be erroneous.WHY WORRY? We have made the case above that, under these more complex data s tr uctures, it isnot legitimate to view all N observations as independent.• As regression methods are available under the independence assum ption whose properties are wellestablished and for which implementation is widely available, it is natural to wonder about theconsequences of using these methods anyway.• On the other hand, it