Metodología de las unidades experimentales

Considernindependent observations(y1,x1), . . . ,(yn,xn), whereyiis theM-valued

response variable andxi is adx×1vector of covariates. Our objective is to introduce an intrinsic regression model for RSS responses and covariates of interest fromn subjects.

The specification of the intrinsic regression model involves three key steps including (i) a link function mapping from the space of covariates to M, (ii) the definition of a residual, and (iii) the action of transporting all residuals to a common space. First, we explicitly formalize the link function. From now on, all covariates have been centered to have mean zero. We consider a single-center link function given by

µ(x,q,β) :Rdx_{× M ×R}dβ _→M, (2.1)

where µ(xi,q,β) is a known link function, q∈ Mcan be regarded as the intercept or

center, and β= (β1, . . . , βdβ)

0 _{is a d}

β×1 vector of regression coefficients. Moreover, it

is assumed that µ(x,q,β) satisfies a single-center property as follows:

µ(0,q,β) =µ(x,q,0) = q. (2.2)

When the regression coefficient vector β equals 0, the link function is independent of

the covariates and thus, it reduces to the single center (or ’mean’) q ∈ M. When all the covariates are equal to zero, the link function is independent of the regression coefficients and reduces to the centerq∈ M.

More generally, we may consider a multicenter link function to account for the pres- ence of discrete covariates, such as gender and diagnostic group. Letxi = (xi,C,xi,D),

a dx,C×1 vector of all the continuous covariates and their potential interactions with xi,D. We may introduce a center for each covariate class based onxi,D (McCullagh and

A.Nelder 1989). In this case, we may define the multicenter link function as follows:

µ(x,q(xD),β) :Rdx_{× M}dD _×Rdβ _→M, (2.3)

wheredD is an integer associated with the number of covariate classes andβis primarily

associated with continuous covariates. Moreover, it is assumed thatµ(x,q,β)satisfies a multicenter property as follows:

µ((0,xD),q(xD),β) =µ(x,q(xD),0) = q(xD). (2.4)

When the regression coefficients vector β equals 0, the link function is independent of

continuous covariates and reduces to q(xD) in M. When all continuous covariates are

equal to zero, the link function is independent of the regression coefficients and reduces to the center q(xD) in M. For notational simplicity, we focus on (2.1) from now on and as the extension to (2.3) is trivial.

Secondly, we introduce a definition of “residual” to ensure that µ(xi,q,β) is the

proper “conditional mean” of yi given xi, which is the key concept of many regression

models (McCullagh and A.Nelder 1989, Fahrmeir and Tutz 2001). For instance, in the classical linear regression model, the response can be written as the sum of the regression function and a residual term and the regression function is the conditional mean of the response only when the conditional mean of the residual is equal to zero. Given the points yi and µ(xi,q,β) on the RSS M, we need to define the residual as

“a difference” between yi and µ(xi,q,β). Assume that yi and µ(xi,q,β) are “close

such that for all i= 1, . . . , n,

yi ∈Expµ(xi,q,β)(B(0, ρ)) or Logµ(xi,q,β)(yi)⊂B(0, ρ).

Thus, Log_µ(x

i,q,β)(yi) may make it a good candidate to play the role of a ‘residual’. These residuals, however, lie on different tangent spaces toM, so it is difficult to carry out a multivariate analysis of these residuals.

Thirdly, sinceMis a RSS, this enables us to “transport” all the residuals, separately, to a common space, say TpM, by exploiting that the parallel transport along the

geodesics can be expressed in terms of the action of G on M. Indeed, since M is a symmetric space, the base point p and the point µ(xi,q,β) can be joined in M by a geodesic, denoted by γi(t; q,β) = γ(t;xi,q,β), satisfying γi(0; q,β) = p and

γi(1; q,β) = µ(xi,q,β). Moreover, γ can be seen as the action of a one-parameter

subgroup ofGand γi(t; q,β) =ci(t; q,β)·pfor t∈R, where ci(t; q,β) = c(t;xi,q,β) :

R→G. Thus, since L−1 a∗ =La−1_∗ onTM fora ∈G, we have L−_c(1;1_x i,q,β)∗(Logµ(xi,q,β)(yi)) =Lc(1;xi,q,β)−1∗(Logµ(xi,q,β)(yi)) = Log_p Lc(1;xi,q,β)−1(yi) = Log_p c(1;xi,q,β)−1·yi ∈TpM.

We define the rotated residual E(yi,xi; q,β) of yi ∈ M with respect to µ(xi,q,β)

as the parallel transport of the actual residual,Log_µ(x

i,q,β)(yi), along the geodesic from the conditional mean,µ(xi,q,β), to the base pointp. That is,

E(yi,xi; q,β) =Ei(q,β) := Logp c(1;xi,q,β)−1·yi

∈TpM (2.5)

Mis defined by

E[E(yi,xi; q∗,β∗)|xi] = 0, (2.6)

where (q∗,β∗) denotes the true value of (q,β) and the expectation is taken with respect to the conditional distribution of yi given xi. Model (2.6) is equivalent to

E[Log_{µ(xi,q∗,β∗)}(yi)|xi] = 0 for i = 1, . . . , n, since Lc(1;xi,q∗,β∗)−1∗ is an isomorphism of linear spaces (invariant under the metric m) between the fibers of TM. This model does not assume any parametric distribution for yi given xi, and thus it allows for a

large class of distributions. The model is essentially semi-parametric, since we do not restrict the joint distribution of(y,x)except by the conditional moment restriction in (2.6).