2.3 Intrinsic Regression Model
2.3.4 Hypotheses Testing
Many scientific questions involve in the comparison of the M-valued data across groups and subjects and the detection of the change in the M-valued data over time. Such questions usually can be formulated as testing the hypotheses of q and β. We consider two types of hypotheses as follows:
H0(1) : C0β=b0 vs. H (1) 1 : C0β6=b0, (2.20) H0(2) : q = q0 vs. H (2) 1 : q6= q0, (2.21)
where C0 is a r×dβ matrix of full row rank and q0 and b0 are specified in M and
Rr, respectively. Further extensions of these hypotheses are definitely interesting and
possible. For instance, for the multicenter link function, we may be interested in testing whether allq(xD)are independent of xD.
We develop several test statistics for testing the hypotheses given in (2.20) and (2.21). Firstly, we consider the Wald test statistic for testing H0(1) against H1(1) in
(2.20), which is given by Wn,φ(1) = (C0β˜E −b0)> h (0C0) ˆΣE,φ(0C0)> i−1 (C0β˜E −b0) = (C0β˜E −b0)> h C0ΣˆE,φ;22C>0 i−1 (C0β˜E−b0),
where ΣˆE,φ is given in Theorem 2.3.3 or Theorem 2.3.4 , and ΣˆE,φ;22 is its lower-right dβ×dβ submatrix. Since β˜E and its asymptotic covariance matrix are independent of
the chart on M, the test statistic Wn,φ(1) is independent of the chart.
Secondly, we consider the Wald test statistic for testing the hypotheses given in (2.21) when there is a local chart (U, φ)onMcontaining bothˆqE and q0. Specifically,
the Wald test statistic for testing (2.21) is defined by
Wn,φ(2) = (φ(˜qE)−φ(q0))> h
(IdM 0) ˆΣE,φ(IdM 0)> i−1
(φ(˜qE)−φ(q0)).
Thirdly, we develop an intrinsic Wald test statistic, that is independent of the chart, for testing the hypotheses given in (2.21). We consider the asymptotic covariance estimator ΣbE,φ based on ˜qE and its upper-left dM × dM submatrix ΣbE,φ;11. Since
both are compatible with the manifold structure of M, ΣbE,φ;11 defines a unique non-
degenerate linear map ΣbE;11(·) from the tangent space Tq˜EM of M at ˜qE onto itself, which is independent of the chart (U, φ). In a maximal normal chart centered at q˜E,
then in any such normal chart, the Wald test statistic for testing (2.21) is given by
WM(2),n = m˜qE((ΣbE;11)
−1(Log ˜
qEq0),Log˜qEq0).
We obtain the asymptotic null distributions ofWn,φ(1), Wn,φ(2), and WM(2),n as follows.
Theorem 2.3.5. Let (U, φ) be a local chart on M so that ˜qE,q∗ ∈ U. Assume that all conditions in Theorem 2.3.3 hold. Under the corresponding null hypothesis, we have the following results:
(i) Wn,φ(1) and Wn,φ(2) are asymptotically distributed as χ2r and χ2dM, respectively; (ii) Wn,φ(1) is independent of the chart (U, φ);
(iii) for any other local chart (U, φ0) with ˜qE and q0 in U,
Wn,φ(2)0 =W (2)
(iv) For any normal chart (U, φ) centered at q˜E, Wn,φ(2) =WM(2),n.
Theorem 2.3.5 has several important implications. Theorem 2.3.5 (i) characterizes the asymptotic null distributions ofWn,φ(1) andWn,φ(2). Theorem 2.3.5 (ii) shows thatWn,φ(1)
does not depend the choice of the chart (U, φ) on M. Theorem 2.3.5 (iii) shows that
Wn,φ(2)0 and W (2)
n,φ are asymptotically equivalent for any two local charts. Theorem 2.3.5
(iv) shows that Wn,φ(2)0 can be used to construct an intrinsic test statistic.
We consider a local alternative framework for (2.20) and (2.21) as follows:
H0(1) : C0β =b0 vs. H1,n(1) : C0β=b0+δ/ √ n+o(1/√n), (2.22) H0(2) : q = q0 vs. H (2) 1,n : q = Expq0(v/ √ n+o(1/√n)), (2.23)
whereδ andv are specified (and fixed) inRr andT
q0M, respectively, and we establish the asymptotic distributions of Wn,φ(1), Wn,φ(2), and WM(2),n under these local alternatives.
Theorem 2.3.6. Let (U, φ) be a local chart on Mso that ˜qE,q∗ ∈U. Assume that all conditions in Theorem 2.3.3 hold. Under the local alternatives (2.22) and (2.23), we have the following results:
(i) UnderH1,n(1), Wn,φ(1) is asymptotically distributed as noncentralχ2r with noncentral- ity parameter δ>
h
C0ΣˆE,φ;22C>0 i−1
δ.
(ii) Under H1,n(2), Wn,φ(2) is asymptotically distributed as noncentral χ2
dM, with noncen- trality parameter J(φ ◦ Expq0)0(v)> h ˆ ΣE,φ;11 i−1 J(φ ◦ Expq0)0(v).
The noncentrality parameter does not depend on the choice of the coordinate system at q0. Here, J(f)a denotes the Jacobian matrix of map f at a.
(iii) Under H1,n(2), WM(2),n is asymptotically distributed as noncentral χ2dM, with non- centrality parameter m˜qE((ΣbE;11) −1(J(Log ˜ qE)q0(v)),(J(Log˜qE)q0(v))).
The noncentrality parameter does not depend on the choice of the coordinate systems at ˜qE and q0, respectively.
We consider another scenario that there are no local charts on Mcontaining both ˜
qE and q0. In this case, we restate the hypotheses H (2)
0 and H
(2)
1 as follows:
H0(2) :distM(q,q0) = 0 vs. H1(2) :distM(q,q0)6= 0. (2.24)
We propose a geodesic test statistic given by
Wdist =distM(˜qE,q0)2, (2.25)
which is independent of the chart(U, φ). Theoretically, we can establish the asymptotic distribution ofWdist under both the null and alternative hypotheses as follows.
Theorem 2.3.7. Assume that all conditions in Theorem 2.3.5 hold.
(a) Under H0(2), nWdist is asymptotically weighted chi-square χ2(λ1, . . . , λdM) dis-
tributed, where the weights λ1, . . . , λdM are the eigenvalues of the matrix ΣE,Logq0,11,
which is the upper-left dM×dM submatrix of the asymptotic covariance matrixΣE,Logq0
of ˜qE in a normal chart centered at q0. Moreover, the weights are independent, up to
a permutation, of the choice of the normal chart centered at q∗.
(b) Under the alternative hypothesis, Wdist is asymptotically normal distributed and
we have
√
n(Wdist −distM(q∗,q0)2) d
where Ddist is the column vector representation of gradq∗(dist(·,q0)2) with respect to
the orthonormal basis of Tq∗M associated with the normal chart used to represent the
asymptotic covariance of ˜qE as the matrix ΣE,Logq∗. In particular, when q0 is close to
q∗, then √
n(Wdist−distM(q∗,q0)2) d
→NdM(0,4[Logq∗q0]>ΣE,Logq∗,11[Logq∗q0]).
Theorem 2.3.7 establishes the asymptotic distribution of Wdist when q˜E and q0 do
not belong to the same chart ofM. In practice, the covariance matrixΣE,Logq∗,11is not
available, sinceΣE,Logq∗ is not known; it also depends on the unknown true valueβ∗, so
we may use the estimate ΣbE,Logq∗ as defined in Theorems 2.3.3 and 2.3.4 . Therefore,
under the null hypothesis, the asymptotic distribution ofWdist can be approximated by
the weighted chi-square distributionχ2(ˆλ
1, . . . ,ˆλdM), in which the weights ˆλ1, . . . ,λˆdM
are the eigenvalues of the covariance matrix(ΣbE,Log
q0)11/n.
Finally, we develop a score test statistic for testingH0(2) againstH1(2). An advantage
of using the score test statistic is that it avoids the calculation of an estimator under the alternative hypothesis H1(2). For notational simplicity, we only consider the ILSE
estimator of (q,β), denoted by(q0,β˜I), under the null hypothesis H0(2). For any chart
(U, φ)on Mwith q0 ∈U, we define Fφi = (Fφi,1> , F > φi,2) > = ∂(t,β)distM(f(xi, φ−1(t),β), yi)2 t=φ(q0),β˜I, Uφ= Utt Utβ Uβt Uββ = n X i=1 ∂(2t,β)distM(f(xi, φ−1(t),β), yi)2 t=φ(q0),β˜I ,
shown that the score testWSC,φ reduces to WSC,φ = ( n X i=1 Fφi,1)>Σ˜−φ,q1( n X i=1 Fφi,1), (2.26) where Σ˜ φ,q = (IdM,−UtβU−ββ1)[ Pn i=1(Fφi − Fφ)⊗2](IdM,−UtβU−ββ1) >, in which F φ = n−1Pn
i=1Fφi. Theoretically, we can establish the asymptotic distribution of WSC,φ
under the null hypothesis.
Theorem 2.3.8. Assume that all conditions in Theorem 2.3.5 hold. We have the following results:
(i) For any suitable local chart(U, φ), the score test statisticWSC,φ is asymptotically
distributed as χ2
dM under the null hypothesis H (2) 0 .
(ii) Under H0(2), for any other local chart (U, φ0) with q0 ∈U, we have
WSC,φ0 =WSC,φ.
Theorem 2.3.8 establishes the asymptotic distribution ofWSC,φforM−valued data.