ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Multivariate Responses In the general mean-variance specification E ( Y j | x ) = f ( x j , β ) , var ( Y j | x j ) = σ 2 g ( β , θ , x j ) 2 , we have assumed that the responses Y 1 , Y 2 , . . . , Y n are conditionally independent , conditioning on x 1 , x 2 , . . . , x n . In many situations, this assumption may fail: clusters of observations, such as pups born to mother rats; serial correlation in repeated measurements on each experimental unit. 1 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Recall Example 1.7: Developmental toxicology studies Developmental toxicology studies in rodents are used in testing and regulation of potentially toxic substances that may pose danger to developing fetuses. A total of m pregnant rats are exposed to different doses of a toxic agent, and each mother rat gives birth to n i pups. The response Y i , j , i = 1 , 2 , . . . , n i is birthweight, and the objective is to characterize the effect on birthweight of different doses of the agent across the population of all exposed mothers and their pups. 2 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Recall Example 1.8: Pharmacokinetics of theophylline Subject i receives an oral dose D i of theophylline. Response Y i , j is the subject’s level of the drug at time t i , j after administration, j = 1 , 2 , . . . , n i . For a given subject, a pharmacokinetic model may explain the time-variation in the response. A broader objective is to understand pharmacokinetic behavior in the entire population of subjects. 3 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Why worry? If we ignore dependence: parameter estimates are generally inefficient; standard errors are generally wrong, hence inferences (confidence intervals, hypothesis tests) do not have nominal properties (coverage probability, size); statistical framework may be inappropriate for scientific objectives. Inefficiency may not be important, invalidity is always important, but relevance to the science is paramount. 4 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response We limit discussion to situations where groups of observations may unambiguously be assumed to be independent: m response vectors Y i , i = 1 , 2 , . . . , m ; n i observations on subject i Y i , 1 Y i , 2 Y i = . . . . Y i , n i 5 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Covariates Within-individual covariates: describe conditions under which Y i , j was observed; needed even if inference were restricted to individual i ; e.g., t i , j = time of j th observation on individual i . Among-individual covariates: same value for all observations on individual i ; e.g., treatment assigned to this individual, or individual characteristics such as gender. 6 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Covariate notation Within-individual covariate vector z i , j ; Stacked: z i , 1 z i , 2 z i = . . . z i , n i Among-individual covariate vector a i . Combined: � z i � x i = . a i 7 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Sources of Dependence Dependence simply means that n i � f i ( Y i | x i ) � = f i , j ( Y i , j | x i ) . j =1 Very general, hence difficult to specify. It is helpful to distinguish: “individual-level” sources; “population-level” sources. 8 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Individual-Level Sources of Dependence For example, suppose we model repeated measurements of a subject’s blood pressure using a within-individual linear regression: Y i , j = β 0 , i + β 1 , i t i , j + e i , j , where t i , j is the time of the j th measurement on the i th subject. The linear trend β 0 , i + β 1 , i t represents the mean response for that subject. 9 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response The subject’s actual blood pressure at time t at the time of testing is β 0 , i + β 1 , i t + e P ( t ) where e P ( t ) is random variation around that mean response, perhaps a stationary stochastic process. If measurement error e M , i , j is non-negligible, then e i , j = e P ( t i , j ) + e M , i , j . 10 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Then var ( Y i , j ) = var ( e P , i , j ) + var ( e M , i , j ) and, for j ′ � = j , cov ( Y i , j , Y i , j ′ ) = cov ( e P , i , j , e P , i , j ′ ) . We would need to specify a model for these variances and covariances in order to make inferences about β 0 , i and β 1 , i . 11 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response The conceptual representation says that the response vector for a single individual is intermittent observations on a stochastic process, whose realizations fluctuate to some extent about a smooth inherent trend, possibly subject to additional measurement error. Here the frame of inference is the individual subject. 12 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Population-Level Sources of Dependence If the subjects are themselves a random sample from some population, then the “parameters” � β 0 , i � β i = β 1 , i associated with the i th subject are a random sample from the corresponding population of parameter vectors. We shall be interested in the mean and dispersion in this population, which describe the average across subjects and the variation among subjects. Here the frame of inference is the population of subjects. 13 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Subject-Specific Modeling Example: Theophylline concentration-time profiles 14 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Pharmacokinetics suggests the subject-specific model � β 1 , i ti , j � − − e − β 3 , i t i , j D i β 3 , i e β 2 , i E ( Y i , j | z i , j , β i ) = . ( β 2 , i β 3 , i − β 1 , i ) where z i , j contains D i = dose for i th subject, and t i , j = time of j th measurement for i th subject ( D i is the same for all measurements, but is included in z i , j instead of a i for convenience); the vector β 1 , i β i = β 2 , i β 3 , i consists of parameters specific to subject i . 15 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response This may be written E ( Y i , j | z i , j , β i ) = f ( z i , j , β i ) . Because β i is associated with the randomly selected i th subject, it is a random variable, and the model for that subject is conditional on the value of β i . We also need some assumptions about how β i varies from subject to subject. We might assume β i ∼ N ( β , D ), or equivalently β i = β + b i , where b i ∼ N ( 0 , D ). 16 / 37 Multivariate Responses Here β is the mean parameter vector across the population of
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response If the subjects were in two groups, e.g. smokers vs non-smokers, the mean might depend on the group to which the subject belongs: β i = β (0) + δ i β (1) + b i � � β (0) � � = I δ i I + b i β (1) where δ i is an indicator variable for smokers. More generally: β i = A i β + b i , where A i is a subject-specific design matrix , which is a function of the individual-level (or among-individual) covariate vector a i , and β is the vector of all relevant parameters. 17 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response For instance, for the theophylline data, β Cl , 0 β Cl , w β 1 , i 1 w i c i 0 0 0 0 β Cl , c = β i = 0 0 0 1 0 0 + b i . β 2 , i w i β V , 0 β 3 , i 0 0 0 0 0 1 w i β V , w β ka , 0 β ka , w Here w i = i th subject’s body weight and c i = i th subject’s creatinine clearance rate (a measure of kidney function), both of which are components of a i . The 7 β s have pharmacokinetic interpretation. 18 / 37 Multivariate Responses
ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Summary of two-stage modeling Stage 1: Individual model . E ( Y i , j | z i , j , β i ) = f ( z i , j , β i ) . Stage 2: Population model . β i = A i β + b i . 19 / 37 Multivariate Responses
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