ABSTRACT
We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.
Acknowledgments
We would like to thank the Editor and the referees for their detailed reading and very valuable comments on the manuscript.
Funding
M. Ahkim's research was supported by the Special Research Fund (BOF) of Universiteit Antwerpen [grant number 42FA070300FFB5994]. A. Verhasselt gratefully acknowledge support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy) and the FWO [grant number 1.5.137.13N]. The infrastructure of the VSC—Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government—department EWI, was used for the simulations.
Appendix A. Notation
1. | For a real-valued function h on | ||||
2. | Let |
Appendix B. Assumptions
Assumption 1.
1. | The observation times tij, j = 1, …, Ni, i = 1, …, n, are chosen independently according to a distribution function FT(t) on | ||||
2. | The eigenvalues η0(t), …, ηd(t) of | ||||
3. | There exists a positive constant M5 such that |Xp(t)| ⩽ M5 for | ||||
4. | There exists a positive constant M6 such that | ||||
5. |
|
These conditions are commonly used (e.g., Huang, Wu and Zhou Citation2004) and are satisfied in many practical examples. As for Assumption 1.1, when dealing with deterministic time points we can replace this assumption by
for some distribution function FT having a Lebesgue density function fT which is bounded away from zero and infinity, uniformly over
, where
and
is the indicator function (Huang, Wu and Zhou Citation2004). Note that we do not assume zero modeling bias, since we allow the knots to increase to infinity.
Appendix C. Theorem of Tan (Citation1977)
In the proof of Theorem 3 and 4 we need the following Lemma, based on Theorem 3.1 of Tan (Citation1977).
Lemma 1.
Let with V invertible and Q = Z′AZ, where A is a real symmetric matrix. Then Q = ∑ki = 1λiχ2(ri, θ2i) where χ2(ri, θ2i) are independent non-central chi-square variables, λ1, …, λk are the non-zero distinct eigenvalues of VA with algebraic multiplicities r1, …, rk, respectively, and
where VA has the spectral decomposition VA = ∑kj = 1λjEj. Moreover, we have that
Appendix D. Proof of Theorem 1
Proof.
Under hypothesis H1 we have that βp(t) = ∑lαplBpl(t; qp) and αpl = cp for l = 1, …, mp; p = 0, …, d. Therefore and
Hence, we obtain that
The specified distribution of Q1 ∼ ∑ki = 1λiχ2(ri, θ2i) follows from Lemma 1 in Appendix C with 0 = ∑iλiθ2i. We now show that ∑ki = 1ri = N − dim and that all θi = 0. Note that the idempotent matrix has eigenvalues 0 and 1. Therefore we have the decomposition
, where
is the eigenspace corresponding to the eigenvalue λ = b of the matrix
. Moreover,
has dimension
. Denote by
the eigenspace of the eigenvalue λ = 0 of the matrix
. One can verify that
. Hence, in order to find the eigenvectors corresponding to a non-zero eigenvalue we can restrict to the space
. This also means that the λi are eigenvalues of
. Since
is positive definite and the fact 0 = ∑iλiθ2i, we obtain that all θi = 0. The eigenspace of
has dimension N, and therefore we have
It remains to show that Q1 and Q2 are independent. By Theorem 3.2 of Tan (Citation1977) Q1 and Q2 are independent if and only if
(A1) It takes a small effort to verify the equation above by noting that
.
Appendix E. Proof of Theorem 2
Proof.
The proof of this theorem is along the same lines as the proof of Theorem 3 in Li, Xu and Liu (Citation2011), some of the details are however different due to our longitudinal setting. Recall the definition of (see Appendix A). Set
, then
. We can also write
, so that under hypothesis H0 we obtain
Note that
under H0. Hence
so
Denote
. We define
Using Lemma 1, we obtain that
where γ2 and θ2i are specified in Lemma 1. Denote
and
. To prove Theorem 2, we need to show that
(A2)
Some mathematical preparation is needed to prove (EquationA2(A2) ). The Takagi factorization of
leads to a matrix G ∈ IR(N − dim) × N such that
Throughout ‖A‖ (‖c‖) denotes the Frobenius (Euclidean) norm of a matrix A (vector c), and ⟨a, b⟩ denotes the standard in-product of vectors a, b. Let
, then
where
Let
. Note that if
, then there is nothing to prove since in that case ξ0 = ξ1 and η0 = η1, so we proceed with the case
. We also have that
from which it follows that
. Define an orthogonal transformation T ∈ IR(N − dim) × (N − dim) with first row equal to
and let
We obtain the expressions
Therefore
(A3) since for a mean zero normal variable Z we have the property
. Now
and TGG′T′ = IN − dim. We want to bound
. Let b = (b1, b2, …, bN) denote the first row of the orthogonal matrix TG, then we know ‖b‖ = 1, also denote by c1, …, cN the columns of
. Using the fact
which is obtained by the Cauchy–Schwarz inequality, and the symmetric property of
, we have that
Using the previous inequality, we can continue from equation (EquationA3
(A3) ) to obtain
(A4) Let
, then
and
. Analogously as in (EquationA4
(A4) ) we obtain
(A5) since for any orthogonal transformation
, the variance of the first component of
, where
is obtained by the entry with index (1, 1) of the matrix
Note that
and
are independent multivariate normal random vectors, because on the one hand
on the other hand, by the same argument as in (EquationA1
(A1) )
from which we find that
Hence
Fix a t > 0, then
(A6) For the last inequality, since η1 and ξ1 are independent, and η1 and ξ0 are independent, we have that
where f is the density function of the multivariate normal distribution
Continuing from equation (EquationA6(A6) ) with k a positive real number
(A7) where
is the maximum of the density function of ξ0 (the Markov inequality is applied in (EquationA7
(A7) )). Substitute
in (EquationA7
(A7) ) to find that
and by (EquationA6
(A6) ) we obtain that for all t ⩾ 0
On the other hand, we obtain in a similar fashion
(A8) where
is the maximum of the density function of the random variable η0. Substitute in (EquationA8
(A8) )
to finally establish
(A9)
Note that
since H′H and G′G are idempotent matrices, thus 0 and 1 are the only eigenvalues. Then by (EquationA3
(A3) ),(EquationA5
(A5) ) and (EquationA9
(A9) ), it follows that
Appendix F. Rate of convergence
In Theorem 2 we assume (Equation9(9) ). We shed more light on this rate by assuming that
is bounded (Nmax = max i = 1, …, nNi and Nmin = min i = 1, …, nNi),
and
. Suppose that subjects with equal number of repeated measurements have the same time points, we do not need this assumption if the correlation structure does not depend on time, as is the case with any time independent correlation structure.
For the first part we use the fact that (Li, Xu and Liu Citation2011). Thus the first part is bounded by
Bounding ![](//:0)
For the second part, we note that there is no closed-form expression of the density function of a linear combination of chi-square variables (see Bausch (Citation2013) among others). However, we obtain a reasonable bound on which is the maximum of the density of ∑ki = 1λiχ2(ri).
First, it does not hold that ri = 1 for all i. To prove this, suppose otherwise, i.e., ri = 1 for all i. Then, by Theorem 1, we have k = ∑ki = 1ri = N − dim. Next, we obtain a bound on k. We argue, as in the proof of Theorem 1, that to find a bound on k we restrict to the eigenspace of eigenvalue 1 of
. Thus, restricting to
, we only look at the number of positive eigenvalues of W1/2VW1/2 which is a block diagonal matrix. By the restriction on the time points (see above), W1/2VW1/2 contains at most Nmax − Nmin + 1 different block matrices with dimensions not exceeding Nmax . Hence, the number of different positive eigenvalues does not exceed Nmax (Nmax − Nmin + 1), i.e., k ⩽ Nmax (Nmax − Nmin + 1). By assumption all ri = 1, and thus it should hold
(A10) Divide (EquationA10
(A10) ) by N. Since Nmax /Nmin is bounded by C > 0 and Nmax /n → 0, we obtain from the previous inequality using also the fact N ⩾ nNmin that the left-hand side is 1 + o(1), while the right-hand side is o(1). This is a contradiction. Hence, there is a 1 ⩽ j ⩽ k such that rj > 1.
Also, we can write ∑ki = 1λiχ2(ri) as a sum of a scaled chi-square distribution and the remaining part, where λmax ≔ max iλi is assumed to be an eigenvalue of a vector in
. Moreover, we assume that
. The density of this sum is a convolution which is bounded by
(after a small calculation). Moreover, by Theorem 2.1 of Wolkowicz and Styan (Citation1980) we know that
since V contains only ones on its diagonal. Hence, we derived