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Keywords:

  • F-type test;
  • L2-norm-based test;
  • one-way anova for functional data;
  • pointwise F-test;
  • Welch–Satterthwaite χ2-approximation

ABSTRACT

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

In this paper, we propose and study a new global test, namely, GPF test, for the one-way anova problem for functional data, obtained via globalizing the usual pointwise F-test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root-n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one-way anova for functional data. Via some simulation studies, it is found that in terms of size-controlling and power, the GPF test is comparable with two existing tests adopted for the one-way anova problem for functional data. A real data example illustrates the GPF test.

Introduction

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

Let inline image denote k groups of random functions defined over a given finite interval inline image. Let SP(μ,γ) denote a stochastic process with mean function inline image and covariance function inline image. Assuming that

  • display math(1)

it is often interesting to test the equality of the k mean functions

  • display math(2)

against the usual alternative that at least two mean functions are not equal. The aforementioned problem is known as the k-sample testing problem or the one-way anova problem for functional data. For the aforementioned one-way anova problem, the k mean functions are often decomposed as μi(t) = μ0(t) + αi(t),i = 1,2, ⋯ ,k, where μ0(t) is the grand mean function, and αi(t),i = 1,2, ⋯ ,k are the k main-effect functions so that (2) is often equivalently written as the problem for testing if the main-effect functions are all 0, that is, inline image.

For testing (2), some interesting work has been performed in the literature. The L2-norm-based test and the F-type test proposed and studied by Faraway (1997), Shen & Faraway (2004), Zhang & Chen (2007), and Zhang (2011) may be adopted; see Section 2.2 for some details. The most related literature is because of Cuevas et al. (2004) who proposed and studied a L2-norm-based test directly for (2), derived the random expression of their test statistic, and approximated the null distribution by a parametric bootstrap (PB) method via re-sampling the Gaussian processes involved in the limit random expression of the test statistic under the null hypothesis. Although their method worked reasonably well in their numerical implementation, Cuevas et al. (2004) noted that their PB test is time-consuming. In this paper, we aim to develop a new global test, namely, GPF test, obtained via globalizing the pointwise F-test. The GPF test is much less computationally intensive than the aforementioned PB test because no bootstrapping is needed in conducting the GPF test.

The pointwise F-test was proposed by Ramsay & Silverman (2005, p.227), naturally extending the classical F-test to the context of functional data analysis. The test statistic of the pointwise F-test for (2) is defined as

  • display math

where and throughout, inline image denotes the total sample size, inline image and inline image denote the pointwise between-subject and within-subject variations, respectively, inline image and inline image are the sample grand mean function and the sample group mean functions, respectively. There are a few advantages for using the pointwise F-test. When the functional data (1) are realizations of Gaussian processes, for any given inline image, Fn(t) ∼ Fk − 1,n − k for all inline image, where Fk − 1,n − k denotes the F-distribution with k − 1 and n − k DOF. Therefore, we can test (2) at all points of inline image using the same critical value Fk − 1,n − k(α) for any predetermined significant level α, where Fk − 1,n − k(α) denotes the upper 100 α percentile of Fk − 1,n − k. However, the aforementioned pointwise F-test has some limitations too. For example, it is time-consuming to conduct the pointwise F-test at all inline image and it is not guaranteed that the one-way anova problem (2) is overall significant for a given significance level even when the pointwise F-test is significant for all inline image at the same significance level. To overcome this difficulty, in this paper, we propose the aforementioned GPF test via globalizing the pointwise F-test using its integral over inline image

  • display math(3)

The main contributions of the paper are as follows. First of all, we derived several asymptotic random expressions of Tn under the null hypothesis (2) and very general conditions. This allows describing several methods for approximating the null distribution of Tn. We recommend the so-called Welch–Satterthwaite χ2-approximation method. This method allows to conduct the GPF test quickly without involving bootstrapping. Second, we derived the asymptotic power of the GPF test under some local alternatives and show that the GPF test is root-n consistent. On the contrary, Cuevas et al. (2004) did not study the asymptotic power of their testing procedure. Third, via some simulation studies, we found that in terms of size-controlling and power, the GPF test is generally comparable with the aforementioned L2-norm-based and F-type tests adopted for (2). The computational effort needed for the three testing procedures are also comparable although the GPF test is constructed via directly summarizing the pointwise F-test as in (3), whereas the L2-norm-based and F-type tests are defined in different ways. Finally, we described how to implement the GPF test when the k functional samples (1) are sampled densely and noisily, irrespective of whether the sampling design is regular or irregular across subjects.

Alternatively, we may globalize the pointwise F-test using its maximum value over inline image

  • display math(4)

For any given significance level α, we can simulate the critical value of the resulting Fmax-test via bootstrapping. This Fmax-test is somewhat similar to the one suggested in Ramsay & Silverman (2005, p.234), where they used the square root of Fn(t) as their test statistic and used a permutation-based critical value. A further study on the Fmax-test is important but is skipped here for space saving because the Fmax-test is not the focus of this paper although we shall illustrate its use in our real data example presented in Section 4.

The rest of the paper is organized as follows. The main results are presented in Section 2. Simulation studies and a real data example are given in Sections 3 and 4, respectively. Some concluding remarks are given in Section .

Main results

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

Asymptotic random expression under the null hypothesis

Notice that for any inline image, the pointwise between-subject variation SSRn(t) can be expressed as

  • display math(5)

where Ik is the k × k identity matrix,

  • display math(6)

Because inline image, it is easy to verify that inline image is an idempotent matrix with rank (k − 1). In addition, as n[RIGHTWARDS ARROW] ∞ , we have

  • display math(7)

where inline image as given in Condition A3. Note that Ik − bbT is also an idempotent matrix of rank (k − 1).

Let inline image denote the set of all square-integrable functions over inline image and let inline image denote the trace of γ(s,t). For further study, we list the following conditions:

Condition A

  1. The k samples (1) are with inline image and tr(γ) < ∞ .

  2. The subject–effect functions vij(t) = yij(t) − μi(t),j = 1,2, ⋯ ,ni;i = 1, ⋯ ,k are i.i.d..

  3. As n[RIGHTWARDS ARROW] ∞ , the k sample sizes satisfy ni ∕ n[RIGHTWARDS ARROW]τi ∈ (0,1), i = 1,2, ⋯ ,k.

  4. The subject–effect function v11(t) satisfies inline image.

  5. For any inline image, γ(t,t) > 0. In addition, the maximum variance inline image.

  6. The expectation inline image is uniformly bounded. That is, for any inline image, we have inline image, where C is some constant independent of (s,t).

The first two conditions are regular. Condition A3 requires that the k sample sizes n1,n2, ⋯ ,nk tend to ∞ proportionally. This guarantees that as n[RIGHTWARDS ARROW] ∞ , the sample group mean functions inline image will converge to Gaussian processes weakly. The last three conditions are imposed so that the pointwise F-statistic Fn(t) is well-defined at any inline image, and the pooled sample covariance function inline image converges to γ(s,t) uniformly over inline image where on the basis of the k functional samples (1), the pooled sample covariance function inline image, as an unbiased estimator of γ(s,t), is given by

  • display math(8)

Let GPk(μ,Γ) denote ‘a k-dimensional Gaussian process with vector of mean functions inline image and matrix of covariance functions inline image’. We write Γ(s,t) = γ(s,t)Ik for simplicity when Γ(s,t) = diag[γ(s,t),γ(s,t), ⋯ ,γ(s,t)]. In particular, GP(η,γ) denotes ‘a Gaussian process with mean function η(t) and covariance function γ(s,t)’. Further, let ‘ inline image’ denote ‘converge in distribution’ in the sense of Laha & Rohatgi (1979, p.474) and van der Vaart & Wellner (1996, p.50–51). To investigate the asymptotic distribution of Tn, we need the following lemma.

Lemma 1. Under Condition A, as n[RIGHTWARDS ARROW] ∞ , we have

  • display math(9)

where ϖ{(s1,t1),(s2,t2)} = E{v11(s1)v11(t1)v11(s2)v11(t2)} − γ(s1,t1)γ(s2,t2). In addition, we have

  • display math(10)

where OUP means ‘bounded in probability uniformly’.

The proof of Lemma 1 is outlined in the Appendix. Lemma 1 shows that zn(t) is asymptotically a k-dimensional Gaussian process and inline image is asymptotically Gaussian and uniformly consistent over inline image. These properties are useful for deriving the asymptotical distribution of Tn under the null and some local alternative. When the k functional samples (1) are Gaussian, we can show that zn(t) ∼ GPk(0,γIk) and inline image even when n is finite, where WP(f,γ) denotes a Wishart process with f degrees of freedom and covariance function γ(s,t). Let ‘ inline image’ denote that ‘X and Y have the same distribution’. By Lemma 1, we can derive the asymptotical random expression of Tn under the null hypothesis (2).

Proposition 1. Under Condition A and the null hypothesis (2), as n[RIGHTWARDS ARROW] ∞ , we have inline image with

  • display math(11)

where w(t) = [w1(t), ⋯ ,wk(t)]T ∼ GPk(0,γwIk) with inline image, Ik − bbT is given in (7), and λr,r = 1,2, ⋯ , ∞ are the decreasing-ordered eigenvalues of γw(s,t).

The proof of Proposition 1 is deferred to the Appendix. Proposition 1 shows that the asymptotical distribution of Tn under the null hypothesis (2) is the same as that of a central χ2-type mixture (Zhang, 2005), which is known except the unknown eigenvalues λr,r = 1,2, ⋯ , ∞ of γw(s,t). Proposition 1 is a key for approximating the null distribution of the test statistic Tn. This issue will be discussed in the next subsection.

Approximating the null distribution

By Proposition 1, inline image which are known except γw(s,t). The covariance function γw(s,t) can be estimated as

  • display math(12)

where inline image is given in (8). In this case, we can adopt the PB method of Cuevas et al. (2004) to obtain an approximate critical value of Tn for any given significance level α. Its key idea is to re-sample the Gaussian processes wi(t),i = 1,2, ⋯ ,k from inline image a large number of times so that based on the first expression of (11), a large sample of Tn under the null hypothesis can be obtained. It is obvious that the PB method is time-consuming, as noticed by Cuevas et al. (2004).

To obtain an approximate critical value for Tn, however, we can actually avoid re-sampling any Gaussian processes. On the basis of the second expression of (11), as mentioned earlier, inline image is known except the unknown eigenvalues λr,r = 1,2, ⋯ of γw(s,t). These unknown eigenvalues can be estimated by the eigenvalues inline image of inline image. Furthermore, it is often sufficient to use only the positive eigenvalues of inline image. That is, the direct simulation method proposed in Zhang & Chen (2007) can be adopted here to generate a large sample of inline image, where inline image denotes the number of positive eigenvalues of inline image. This direct simulation method is less intensive than the PB method mentioned earlier because no Gaussian processes need to be generated. It generally works well but it is still time-consuming because it needs to estimate the eigenvalues of inline image and to re-sample inline image from inline image a large number of times. To overcome this difficulty, we propose here to adopt the Welch–Satterthwaite χ2-approximation for approximating the null distribution of Tn.

The key idea of the Welch–Satterthwaite χ2-approximation is to approximate the null distribution of Tn by that of a χ2-random variable multiplied by a constant, namely, inline image via matching the means and variances of Tn and Rw to determine the parameters βw and dw. First of all, we have

  • display math(13)

On the basis of the second expression of (11), under the null hypothesis (2), we have

  • display math(14)

where inline image. On the basis of (13) and (14), matching the means and variances of Rw and Tn under the null hypothesis (2) results in

  • display math(15)

ignoring the higher-order terms. On the basis of the data, the natural estimators of βw and dw are then given by

  • display math(16)

obtained via replacing the covariance function γw(s,t) in (15) by its estimator inline image given in (12). Then, the proposed GPF test is conducted by computing the p-value using the following approximate distribution: under the null hypothesis,

  • display math(17)

or for any given significance level α, the estimated critical value of Tn is specified as

  • display math(18)

where inline image denotes the upper 100 α percentile of inline image for any ν > 0. The following proposition shows that as n[RIGHTWARDS ARROW] ∞ , the estimated critical value (18) will tend to its theoretical critical value

  • display math(19)

when assuming γw(s,t) is known. Let ‘ inline image’ denote ‘converge in probability’.

Proposition 2. Under Condition A, as n[RIGHTWARDS ARROW] ∞ , we have

  • display math(20)

The proof of Proposition 2 will be given in the Appendix. Proposition 2 says that under Condition A and when n[RIGHTWARDS ARROW] ∞ , the estimated parameters inline image and inline image converge in probability to their theoretical values βw and dw, which are obtained when γ(s,t) is assumed to be known. This result is expected. However, how well the distribution (17) approximates the underlying null distribution of Tn is usually not determined by n but by the value of dw. According to our experience, the larger the value of dw, the better the distribution of (17) approximates the underlying null distribution of Tn.

Remark 1. When the k functional samples (1) are Gaussian, it is straightforward to show that the exact value of E(Tn) is given by (n − k)(b − a) ∕ (n − k − 2). In this case, the estimators given in (16) can be replaced with

  • display math(21)

Before concluding this subsection, let us briefly discuss how to adopt the L2-norm-based test (Faraway, 1997; Zhang & Chen, 2007) and the F-type test (Shen & Faraway, 2004; Zhang, 2011) in the current context. The L2-norm-based test for (2) uses the test statistic inline image. Under the null hypothesis (2), we can show that

  • display math(22)

where inline image, inline image, and inline image. The approximate distribution (22) can be used to compute the p-value of Sn or its critical value. The F-type test for (2) uses the test statistic inline image. Under the null hypothesis (2), we can show that

  • display math(23)

where inline image is the same as the one for the L2-norm-based test given earlier and inline image. Similarly, the approximate null distribution (23) of the F-type test can be used to compute the p-value of Fn or its critical value. Further details about the aforementioned L2-norm-based and F-type tests for one-way anova for functional data can be found in Zhang (2013, ch.5).

The asymptotic power

To study the asymptotic power of the proposed GPF test, we specify the following local alternative

  • display math(24)

where μ0(t) is the grand mean function, and d1(t), ⋯ ,dk(t) are any real functions, independent of n. By (24), we have inline image, where 1k denotes the k × 1 vector of ones. It follows that μn(t) = μ0(t)bn + d(t) where μn(t) is defined in (6) and d(t) = [d1(t), ⋯ ,dk(t)]T. Under Condition A3, as n tends to ∞ , the local alternative (24) will tend to the null with the root-n rate. In this sense, when we can show that the GPF test can detect the local alternative (24) with probability 1 as long as the information provided by d(t) diverges to ∞ , the GPF test is called to be root-n consistent. A good test usually should admit the root-n consistency. The main purpose of this subsection is to show that the GPF test is root-n consistent.

Because inline image, under (24), we have

  • display math

where zn(t) is defined in (6). Under the conditions of Lemma 1, we have inline image and inline image, an idempotent matrix of rank (k − 1), which has the following singular value decomposition

  • display math(25)

where the columns of U are the eigenvectors of Ik − bbT. Let λ1, ⋯ ,λm be all the positive decreasing-ordered eigenvalues of γw(s,t), and ϕ1(t),ϕ2(t), ⋯ ,ϕm(t) are the associated eigenfunctions. Notice that we allow m = ∞ when all the eigenvalues of γw(s,t) are positive. In this case, the asymptotical random expression (11) can be further written as inline image. We now have the following proposition about the asymptotic distribution of Tn under the local alternative (24).

Proposition 3. Under Condition A and the local alternative (24), as n[RIGHTWARDS ARROW] ∞ , we have inline image with

  • display math(26)

where w(t) is as defined in Proposition 1, inline image, inline image, r = 1,2, ⋯ ,m, and inline image.

We shall give the detailed proof of Proposition 3 in the Appendix. From Proposition 3, it is seen that the asymptotic distribution of Tn under the local alternative (24) is the same as the distribution of non-central χ2-mixture plus a constant. The noncentral parameters depend on γw(s,t) and d(t) through the positive eigenvalues λr,r = 1,2, ⋯ ,m of γw(s,t) and the squared L2-norm-based of the projections of (Ik − 1,0)UTh(t) onto the eigen-subspaces spanned by the eigenfunctions ϕr(t),r = 1,2, ⋯ ,m of γw(s,t). Proposition 3 is the key for our deriving the asymptotic power of the GPF test under the local alternative (24).

Remark 2. It is clear that via introducing m, the number of positive eigenvalues of γw(s,t), we aim to express the noncentral parameters of Br,r = 1,2, ⋯ ,m properly.

For further study, set inline image. It is easy to see that

  • display math(27)

where δ2 is defined in Proposition 3 and inline image. Recall that inline image and inline image are the estimated and theoretical critical values of Tn as defined in (18) and (19), respectively.

Proposition 4. Let Z ∼ N(0,1). Then, under Condition A and the local alternative (24), as n[RIGHTWARDS ARROW] ∞ , the power of Tn is given by

  • display math(28)

which will tend to 1 as δ[RIGHTWARDS ARROW] ∞ .

We defer the detailed proof of Proposition 4 to the Appendix. From Proposition 4, it is seen that the asymptotic power of Tn under H1n can asymptotically be expressed as the right-hand side of (28). The inequality (27) shows that δλ can be finite or infinite. When δλ is finite, it is easy to see that the right-hand side of (28) tends to 1 as δ[RIGHTWARDS ARROW] ∞ and when δλ[RIGHTWARDS ARROW] ∞ , the asymptotic power of Tn under H1n can be further expressed as P(Z ≥ − δ2 ∕ [2δλ]) + o(1), which tends to 1 as δ[RIGHTWARDS ARROW] ∞ . From the aforementioned, we claim that the proposed GPF test is root n-consistent.

Numerical implementation of the GPF test

In practice, it is not possible to observe the k functional samples (1) continuously. Rather, each individual function may be observed at a grid of design time points, often with errors. When all the individual functions are observed at a common grid of design time points, the GPF test can be directly applied to the observed functional data as mentioned in Remark 5 and see the real data example presented in Section 4. In many situations, however, the design time points may be different for different individual functions. In this case, to numerically implement the GPF test, one has to first reconstruct the k functional samples from the k observed discrete functional samples using some smoothing technique, then discretize each individual function of the k reconstructed functional samples at a common grid of time points and finally apply the GPF test accordingly.

We first briefly describe how to reconstruct the k functional samples from the k observed discrete functional samples. Let inline image denote the j-th underlying function of the i-th functional sample, let tijl,l = 1,2, ⋯ ,nij be the associated design time points in inline image, and let εijl,l = 1,2, ⋯ ,nij be the associated measurement errors. Assume that fij(t),j = 1,2, ⋯ ,ni;i = 1,2, ⋯ ,k are smooth. Then, the standard nonparametric regression model yij(tijl) = fij(tijl) + εijl, l = 1, ⋯ ,nij allows one to reconstruct fij(t) via one of the existing smoothing techniques such as regression splines (Eubank, 1999), smoothing splines (Wahba, 1990; Green & Silverman, 1994), P-splines (Ruppert et al., 2003), local polynomial smoothing (Wand & Jones, 1995; Fan & Gijbels, 1996), and reproducing kernel Hilbert space decomposition (Wahba, 1990) among others. Details about smoothing-based function reconstruction can be found in Zhang (2013, ch.3). The resulting k reconstructed functional samples can be expressed as

  • display math(29)

Notice that nonparametric smoothing allows removing as much noise as possible from the raw functional data, evaluating the reconstructed individual functions at any desired resolution and generally improving the power of a testing procedure for functional data; see Section  for an illustrative example. Following the arguments in Zhang & Chen (2007), we can show that if the k reconstructed functional samples (29) are obtained using local polynomial smoothing, then under some mild conditions including that the number of measurements for an individual function is slightly larger than the number of individual functions, the effects of substitutions of the k functional samples (1) with the k reconstructed functional samples (29) can be ignored asymptotically. That is why, in the previous discussions, we assume that the k functional samples (1) can be observed continuously for ease of presentation, whereas in numerical implementation, the GPF test is actually applied to the k reconstructed functional samples (29) provided that the measurements of all the subjects are sufficiently densely sampled, irrespective of whether the sampling design is regular or irregular across subjects.

We now briefly discuss how to apply the GPF test to the k reconstructed functional samples (29) numerically. For this end, we have to first discretize the k reconstructed functional samples (29) at a common grid of resolution time points. Let t1,t2, ⋯ ,tM be M resolution time points which are equally spaced in inline image. Then, the k reconstructed functional samples (29) are discretized as

  • display math(30)

and the one-way anova problem (2) is then discretized accordingly as

  • display math(31)

where μi = [ μi(t1), ⋯ ,μi(tM)]T, i = 1,2, ⋯ ,k denote the vectors consisting of the values of μi(t),i = 1,2, ⋯ ,k evaluated at the resolution time points, respectively. Let Fn(t),Tn, and inline image be calculated on the basis of the k reconstructed functional samples (29). Let Fn(t1), ⋯ ,Fn(tM) denote the values of Fn(t) evaluated at the M resolution time points and let inline image denote the covariance matrix obtained via evaluating inline image at the M resolution time points so that its (l,s)th entry is inline image for l,s = 1,2, ⋯ ,M. Then, we have

  • display math

where inline image denotes the volume of inline image and inline image. Similarly, we have

  • display math

Then, the estimated parameters inline image and inline image defined in (16) for the Welch–Satterthwaite χ2-approximation can be approximately expressed as

  • display math

where inline image and inline image denote the estimated βw and dw when the Welch–Satterthwaite χ2-approximation method is applied to the discretized one-way anova problem (31) based on the k discretized samples (30). It follows that

  • display math

We have the following remarks.

Remark 3. In numerical implementation, the resolution M should be large enough so that each individual function of the k reconstructed functional samples (29) can be well-represented by the resulting vector. According to our experiences, M = 50 ∼ 1000 is often sufficient for various purposes in functional data analysis. A simple simulation study about this can be found in Zhang (2013, ch.4).

Remark 4. When we conduct the GPF test, the constant factor inline image in Tn and inline image can be omitted at the same time in computation. This will not affect the test result. Alternatively speaking, the GPF test can actually be applied directly to the discretized one-way anova problem (31) based on the k discretized samples (30).

Remark 5. From the aforementioned description, it is seen that in numerical implementation, we have to discretize the individual functions at a common grid of resolution time points so that the GPF test is actually applied to the k discretized samples (30) via computing inline image, and so on. Therefore, when the functional data can be observed simultaneously over a common grid of (possibly unequally spaced) time points so that k samples of vectors such as (30) can be obtained, the GPF test can be applied directly to the k observed samples of vectors. However, when the smoothness assumption about the underlying individual functions is valid, the GPF test using the k reconstructed functional samples (29) is expected to have higher power than the GPF test using the raw functional data directly. See an illustrative example in Section 4.

Simulation studies

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

In this section, we present some simulation studies, aiming to check if the GPF test is comparable, in terms of size controlling and powers, with the L2-norm-based test (Faraway, 1997; Zhang & Chen, 2007) and the F-type test (Shen & Faraway, 2004; Zhang, 2011) adopted for the one-way anova problem (2) as briefly described in Section 2.

We use the following model to generate k discrete functional samples:

  • display math(32)

where the k coefficient vectors ci = [ci1,ci2,ci3,ci4]T are used to flexibly specify the k group mean functions μi(t) = ci1 + ci2t + ci3t2 + ci4t3; the i.i.d. random variables zijr (having mean 0 and variance 1), the orthonormal basis vector Ψ(t) = [ψ1(t), ⋯ ,ψq(t)]T, and the q decreasing-ordered variance components λr are used to flexibly specify the common covariance function inline image so that the subject–effect functions inline image; and the measurement errors εijl are i.i.d normal with mean 0 and variance σ2. The unequal design time points tijl are obtained via randomly removing some design time points at a rate rmiss from a common grid of time points

  • display math(33)

where M is some positive integer and rmiss is some small number, measuring the missing rate of the design time points.

The tuning parameters are specified as follows. For specifying the k group mean functions μi(t) = ci1 + ci2t + ci3t2 + ci4t3, we set c1 = [1,2.3,3.4,1.5]T and ci = c1 + (i − 1)δu,i = 2, ⋯ ,k where δ controls the differences μi(t) − μ1(t),i = 2, ⋯ ,k, and u controls the direction of these differences. We set δ properly as listed in Table 1 so that the null hypothesis (when δ = 0) and the three alternatives (when δ > 0) are considered. In addition, we set inline image. For specifying the common covariance function inline image, we set λr = r,r = 1,2, ⋯ ,q with a = 1.5, ρ = 0.10,0.50 or 0.90, and q = 7 and set the basis functions as inline image r = 1, ⋯ ,(q − 1) ∕ 2. Notice that the values 0.10, 0.50, and 0.90 of ρ represent the three cases when the simulated functional data have high, moderate, and low correlations. We set rmiss = 10% and set the noise variance σ2 =  ∕ 16, ∕ 4, ∕ 2, representing the three cases of small, moderate, and large measurement errors. We set k = 3 and specify three cases of n = [n1,n2,n3] as n1 = [20,30,30],n2 = [40,60,60], and n3 = [60,90,90], representing the small, moderate, and large sample size cases. We specify two cases of the distribution of the i.i.d. random variables zijr's as follows: inline image and inline image, allowing to generate Gaussian and non-Gaussian functional data, respectively. Notice that the inline image distribution is chosen because it has nearly the heaviest tails among the t-distributions with finite first two moments. Finally, we specify two cases of the number of distinct design time points, M = 80 and M = 150.

Table 1. Empirical sizes and powers (in percentages) of the L2-norm-based, F-type, and GPF tests for the one-way anova problem (2) when inline image and M = 80. The associated standard deviations (in percentages) for the GPF test are given in parentheses
ρnσ2L2FGPFL2FGPFL2FGPFL2FGPF
  1. n1 = [20,30,30], n2 = [40,60,60], n3 = [60,90,90].

   δ = 0δ = 0.07δ = 0.15δ = 0.22
0.10n10.15/165.424.945.30(0.31)15.3514.2815.52(0.51)59.7457.7060.02(0.69)92.1691.4892.22(0.37)
  0.15/45.304.925.26(0.31)15.2814.0415.12(0.50)61.2659.8161.50(0.68)92.8892.0892.80(0.36)
  0.15/25.404.925.40(0.32)15.0414.0915.20(0.51)61.3660.0861.16(0.69)92.7692.0292.60(0.37)
 
 n20.15/165.525.305.40(0.31)26.1425.1826.14(0.62)90.9690.5891.18(0.40)99.6699.6699.66(0.08)
  0.15/45.445.225.42(0.32)27.7227.0227.74(0.63)91.0290.6290.92(0.40)99.6499.6499.66(0.08)
  0.15/25.525.345.60(0.33)28.0427.1327.90(0.63)91.0290.6490.80(0.41)99.6699.6499.64(0.08)
 
 n30.15/165.465.325.46(0.32)38.3237.7638.40(0.68)98.7098.6498.78(0.15)100.0100.0100.0(0.00)
  0.15/45.225.005.28(0.31)38.7438.2038.82(0.68)98.8498.7498.82(0.15)99.9699.9699.96(0.02)
  0.15/25.305.185.30(0.32)38.8238.2738.47(0.69)98.8298.7298.82(0.15)99.9699.9699.96(0.02)
 
   δ = 0δ = 0.10δ = 0.20δ = 0.30
0.50n10.75/165.084.685.42(0.32)8.888.389.50(0.41)20.1019.1821.20(0.57)44.7643.2246.52(0.70)
  0.75/45.004.765.28(0.31)8.547.968.80(0.40)21.0220.0421.86(0.58)43.0641.6644.72(0.70)
  0.75/25.124.805.16(0.31)8.467.968.88(0.40)21.0219.9422.08(0.59)42.7841.3444.40(0.70)
 
 n20.75/164.924.805.04(0.30)11.0410.7811.56(0.45)39.1638.6440.26(0.69)76.3475.7877.96(0.58)
  0.75/44.544.294.86(0.30)12.3811.9212.30(0.46)39.6639.0440.48(0.69)78.4277.7879.42(0.57)
  0.75/24.424.204.84(0.30)12.4412.0212.26(0.46)39.5339.0040.34(0.69)78.2877.5479.00(0.57)
 
 n30.75/165.205.095.30(0.31)16.6116.2016.94(0.53)57.4456.9259.04(0.69)92.5492.4093.10(0.35)
  0.75/45.165.085.40(0.31)16.6016.3417.14(0.53)58.0657.4659.38(0.69)91.9891.8692.50(0.37)
  0.75/25.325.205.44(0.32)16.5216.2617.08(0.53)57.5456.9458.64(0.70)91.8091.6692.16(0.38)
 
   δ = 0δ = 0.23δ = 0.46δ = 0.69
0.90n11.35/164.043.704.56(0.29)9.008.269.88(0.42)34.3633.2635.84(0.67)72.1171.1873.74(0.62)
  1.35/43.803.544.28(0.28)9.448.8410.19(0.42)32.5431.4234.54(0.67)72.3071.3173.34(0.62)
  1.35/24.503.944.94(0.31)9.268.8610.08(0.43)34.0232.8435.24(0.68)71.6870.6072.72(0.63)
 n21.35/164.664.484.82(0.30)16.8816.3418.10(0.54)67.1666.5067.64(0.66)97.4097.2497.70(0.21)
  1.35/44.884.785.24(0.31)17.7417.4218.50(0.54)67.6267.0668.58(0.65)97.3097.2297.64(0.21)
  1.35/25.165.065.44(0.32)17.3016.8617.90(0.54)66.4066.0067.70(0.66)96.8096.7697.00(0.24)
 n31.35/164.974.825.28(0.31)24.8824.5025.46(0.61)88.0487.8288.66(0.44)99.8299.8299.84(0.05)
  1.35/45.124.965.20(0.31)25.5025.2426.12(0.62)86.7886.5287.02(0.47)99.8499.7899.84(0.05)
  1.35/24.684.584.90(0.31)26.6826.4627.20(0.63)87.2487.0687.38(0.47)99.8699.8699.86(0.05)

For a given set of tuning parameters, the k discrete functional samples (32) are generated. Because the design time points tijl for different functions are different, as suggested in Section 2.4, some smoothing technique should be used to reconstruct the individual functions before the L2-norm-based, F-type, and GPF tests can be applied. For simplicity, the regression spline reconstruction method (Zhang, 2013, ch.3) is applied to the simulated data of each individual function with a common set of equally spaced interior knots. The number of interior knots is selected by the GCV (generalized cross-validation) rule. As described in Section 2.4, the three tests are then applied to the k reconstructed functional samples evaluated at the common grid of the distinct design time points (33). Their p-values are then recorded. When the p-values are smaller than the nominal significance level α (5% here), the null hypothesis (2) is rejected. The aforementioned process is repeated N = 5000 times. For each case, the empirical size or power of a test is then computed as the proportion of the number of rejections (out of N = 5000 replications) based on the calculated p-value.

Table 1 presents the empirical sizes and powers (in percentages) of the three tests under various tuning parameters and when inline image and M = 80. The associated standard deviations (in percentages) of the GPF-test's empirical sizes or powers are given in the parentheses. Under the same tuning parameters, the associated standard deviations of the other two tests’ empirical sizes or powers are approximately the same as that of the GPF-test's empirical size or power and are hence omitted for space saving. In fact, they can be easily calculated on the basis of the associated empirical sizes or powers. From the columns associated with δ = 0, we see that in terms of size controlling, when we take the associated standard deviations of the empirical sizes into account, the three tests are overall comparable. This is because when the functional data are highly and moderately correlated (when ρ = 0.10,0.50), the L2-norm-based and GPF tests are slightly more liberal than the F-type test, but when the functional data are less correlated (when ρ = 0.90), they are also less conservative than the F-type test. In addition, the differences between the empirical sizes of the three tests are generally not significant at 5% level and with the sample sizes increasing, these differences generally become smaller and smaller. From the columns associated with δ > 0, we also see that in terms of power, when we take the associated standard deviations of the powers into account, the three tests are also overall comparable. This is because the differences between the powers of the three tests are generally not significant even at 5% level although we do observe, in numerical values, that the GPF test generally have the highest powers, followed by the L2-norm-based test and then by the F-type test. In addition, we also see that (i) with increasing the sample sizes, the empirical sizes of the three tests generally become better in size controlling and the empirical powers of the three tests generally become higher; and (ii) for different noise variance σ2, the empirical sizes and powers of the three tests are generally comparable. Observation (i) is expected, whereas observation (ii) is probably because the effect of the measurement errors are largely removed by regression spline smoothing.

In the aforementioned, we only present the simulation results when inline image and M = 80 because the main conclusions based on the simulation results in other cases are similar. From all these simulation results, we conclude that in terms of size controlling and powers, the GPF test is generally comparable with the L2-norm-based and F-type tests adopted for the one-way anova problem (2).

An illustrative example

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

An orthotic is an orthopedic device applied externally to limb or body to provide support, stability, and prevention of deformity from getting worse or replacement of lost function. Depending on the diagnosis and physical needs of an individual, a large variety of orthosis is available. According to Abramovich et al. (2004), the orthosis data were acquired and computed in an experiment by Dr. Amarantini David & Dr. Martin Luc (Laboratoire Sport et Performance Motrice, EA 597, UFRAPS, Grenoble University, France). The aim of the experiment was to analyze how muscle copes with an external perturbation. The experiment recruited seven young male volunteers who wore a spring-loaded orthosis of adjustable stiffness under the following four experimental conditions: a control condition (without orthosis), an orthosis condition (with orthosis), and two spring conditions (with spring 1 or with spring 2) in which stepping-in-place was perturbed by fitting a spring-loaded orthosis onto the right knee joint. All the seven volunteers tried all four conditions 10 times for 20 seconds each, whereas only the central 10 seconds were used in the study in order to avoid possible perturbations in the initial and final parts of the experiment. The resultant moment of force at the knee was derived by means of body segment kinematics recorded with a sampling frequency of 200 Hz. For each stepping-in-place replication, the resultant moment was computed at 256 time points equally spaced and scaled to [0,1] so that a time interval corresponds to an individual gait cycle.

In this section, for illustrative purpose, we use the first volunteer's orthosis data under the four experimental conditions only. Panels (A)–(D) of Figure 1 present the 40 raw orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. Of interest is to test if the mean orthosis curves of the first volunteer are different under the four experimental conditions. This is equivalent to testing the null hypothesis H0 : μ1(t) ≡ μ2(t) ≡ μ3(t) ≡ μ4(t), t ∈ [a,b], where μi(t),i = 1,2,3,4 denote the underlying mean curves of the first volunteer's orthosis curves under the four experimental conditions, and [a,b] is a finite interval of interest.

image

Figure 1. Panels (a)–(d): noisy raw orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. Panel (e): pointwise F-value line (solid) with 95% pointwise F-test critical line (dashed) and 95% Fmax-test critical line (dot-dashed).

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Because the 256 design time points are common for all the orthosis curves, we can apply the pointwise F-test, the L2-norm-based, F-type, GPF, and Fmax tests directly to the raw orthosis data of the first volunteer. The resulting F-value line (solid) of the pointwise F-test over the whole range [0,1], together with the associated 95% pointwise F-test critical line (dashed) and the 95% Fmax-test critical line (dot-dashed), is displayed in Figure 1(E). Both the pointwise F-test and Fmax-test suggest that the mean orthosis curves of the first volunteer under the four experimental conditions are unlikely the same, but they may be the same at the last stage of the experiment over the interval [0.8,1]. Table 2 displays the test results of the L2-norm-based, F-type, GPF, and Fmax tests. All the four global tests suggest that the mean orthosis curves of the first volunteer under the four experimental conditions are unlikely the same over [0,1] but they are very likely to be the same at the last stage of the experiment over [0.8,1]. This is in agreement with what we observed from Figure 1(E). Notice that throughout this section, the critical values and p-values of the Fmax-test were obtained on the basis of 10,000 bootstrap replicates and we use 7.6e5 to denote 7.6 × 105 for simplicity.

Table 2. One-way anova of the first volunteer's raw orthosis data by the L2-norm-based, F-type, GPF, and Fmax tests
[a,b]L2-norm-based testF-type testGPF testFmax test
Sn(inline image)p-valueFn(inline image)p-valueTn(inline image)p-valueFmaxp-value
[0,1]7.6e5(2404, 1.58)019.9(1.58,189)017.1(0.046,23.1)050.60
[0.80,1]4.5e3( 483, 8.40)0.3471.12( 8.40,100)0.3581.15(0.114,9.51)0.3753.730.308

Figure 1 shows that the raw orthosis curves of the first volunteer are very noisy and so is the F-value line of the pointwise F-test. This indicates that the measurement errors from the individual raw orthosis curves may have some impact on the pointwise F-test and the Fmax-test as well, possibly resulting in misleading results in some situations. To overcome this problem, we may apply some smoothing technique as described in Section 2.4 to largely remove these measurement errors. For the first volunteer's orthosis data, for simplicity, we applied the regression spline smoothing method to each of the 40 raw orthosis curves, respectively, but with a common set of interior knots equally spaced over [0,1]. The optimal number of interior knots is 20, selected by the GCV rule (Zhang, 2013, ch.3). The regression spline smoothing technique also allows to reconstruct the orthosis curves at any desired resolution. For comparison purpose, the reconstructed orthosis curves of the first volunteer were evaluated at the 256 design time points of the first volunteer's raw orthosis data. Panels (A)–(D) of Figure 2 presents the 40 reconstructed orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. It appears that the measurement errors of the raw orthosis curves are largely removed.

image

Figure 2. Panels (a)–(d): reconstructed orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. Panel (e): pointwise F-value line (solid) with 95% pointwise F-test critical line (dashed) and 95% Fmax-test critical line (dot-dashed).

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We now apply the pointwise F-test and L2-norm-based, F-type, GPF, and Fmax tests to the reconstructed orthosis data of the first volunteer. The resulting F-value line (solid) of the pointwise F-test, together with the associated 95% pointwise F-test critical line (dashed) and the 95% Fmax-test critical line (dot-dashed), is displayed in Figure 2(E), and the test results of the L2-norm-based, F-type, GPF, and Fmax tests are displayed in Table 3. The main conclusions based on the reconstructed orthosis data of the first volunteer are similar to those based on his raw orthosis data described earlier. However, the impact of the measurement errors and regression spline smoothing is still spotted. The after-smoothing p-values (0.333, 0.346, 0.372) of the first three tests over [0.80,1] are comparable with but are generally smaller than their before-smoothing p-values ( 0.347,0.358,0.375), probably indicating an improvement of the powers of the three tests. The after-smoothing p-value (0.378) of the Fmax-test over [0.80,1] is comparable with the before and after-smoothing p-values of the first three tests but is much larger than its before-smoothing p-value (0.308), probably indicating a reduction of the impact of the measurement errors on the Fmax-test. It appears that the first three tests are relatively robust against the measurement errors but this is not the case for the Fmax-test. This is probably because the first three tests are the integrals of some pointwise test statistics, whereas the Fmax-test is the maximum value of the pointwise F-test statistics so that the impact of the measurement errors on the first three tests may have been integrated out partially before smoothing is applied while it is not the case for the Fmax-test. Therefore, smoothing may be more preferred for noisy functional data when the Fmax-test is conducted than when the first three tests are conducted.

Table 3. One-way anova of the first volunteer's reconstructed orthosis data by the L2-norm-based, F-type, GPF, and Fmax tests
[a,b]L2-norm-based testF-type testGPF testFmax test
Sn(inline image)p-valueFn(inline image)p-valueTn(inline image)p-valueFmaxp-value
[0,1]7.5e5(2761,11.4)023.8(11.4,137)019.6(0.065,16.3)048.70
[0.80,1]3.2e3( 675, 4.08)0.3331.15( 4.08, 49)0.3461.14(0.232,4.57)0.3721.890.378

Concluding remarks

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

In this paper, we proposed and studied the GPF test for the one-way anova problem (2) for functional data under very general conditions. With help of the Welch–Satterthwaite χ2-approximation, the GPF test can be easily conducted. Via intensive simulation studies, we found that in terms of size controlling and power, the GPF test is comparable with the L2-norm-based test (Faraway, 1997; Zhang & Chen, 2007) and the F-type test (Shen & Faraway, 2004; Zhang, 2011) adopted for the one-way anova problem for functional data.

Notice that the GPF test is widely applicable. Actually, it is rather easy to extend the GPF test for various hypothesis testing problems for functional data because for these hypothesis testing problems, it is very easy to construct the associated pointwise F-test (Ramsay & Silverman, 2005; Zhang, 2013). These hypothesis testing problems include two-way anova problems for functional data (Zhang, 2013, ch.5) and functional linear models with functional responses (Zhang, 2013, ch.6), among others.

In the introduction section, we briefly introduced the Fmax-test. It is an alternative for globalizing the pointwise F-test. Further studies about the Fmax-test is interesting and warranted.

Acknowledgements

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

The work was supported by the National University of Singapore Academic Research Grant R-155-000-128-112. The authors thank the editor, associate editor, and two referees for their insightful comments and helpful suggestions.

References

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results
  • Abramovich, F., Antoniadis, A., Sapatinas, T. & Vidakovic, B. (2004). Optimal testing in a fixed-effects functional analysis of variance model. Int. J. Wavelets Multiresolut. Inf. Process. 2, 323349.
  • Cuevas, A., Febrero, M. & Fraiman, R. (2004). An anova test for functional data. Comput. Statist. Data Anal. 47, 111122.
  • Eubank, R. L. (1999). Nonparametric Regression and Spline Smoothing, Marcel Dekker, New York.
  • Fan, J. & Gijbels, I. (1996). Local Polynomial Modeling and its Applications, Chapman & Hall, London.
  • Faraway, J. (1997). Regression analysis for a functional response. Technometrics 39, 254261.
  • Green, P. J. & Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models, Chapman & Hall, London.
  • Laha, R. G. & Rohatgi, V. K. (1979). Probability Theory, Wiley, New York.
  • Ramsay, J. O. & Silverman, B. W. (2005). Functional Data Analysis, (2nd edn)., Springer Series in Statistics, Springer, New York.
  • Ruppert, D., Wand, M. P. & Carroll, R. J. (2003). Semiparametric Regression, Cambridge University Press, Cambridge, United Kingdom.
  • Shen, Q. & Faraway, J. (2004). An F test for linear models with functional responses. Statist. Sinica 14, 12391257.
  • van der Vaart, A. W. & Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer, New York.
  • Wahba, G. (1990). Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, SIAM, Philadelphia.
  • Wand, M. P. & Jones, M. C. (1995). Kernel Smoothing, Chapman & Hall, London.
  • Zhang, J.-T. (2005). Approximate and asymptotic distributions of chi-squared-type mixtures with applications. J. Amer. Statist. Assoc. 100, 273285.
  • Zhang, J.-T. (2011). Statistical inferences for linear models with functional responses. Statist. Sinica. 21, 14311451.
  • Zhang, J.-T. (2013). Analysis of Variance for Functional Data, Chapman & Hall, London.
  • Zhang, J.-T. & Chen, J. W. (2007). Statistical inferences for functional data. Ann. Statist. 35, 10521079.

Appendix: proofs of the main results

  1. Top of page
  2. ABSTRACT
  3. Introduction
  4. Main results
  5. Simulation studies
  6. An illustrative example
  7. Concluding remarks
  8. Acknowledgements
  9. References
  10. Appendix: proofs of the main results

Proof of Lemma 1. By (6), it is obvious that the k components of zn(t) are independent. Under Conditions A1–A3 and by the central limit theorem for i.i.d. stochastic processes (random elements taking values in a Hilbert space) (Laha & Rohatgi, 1979, p.474 and van der Vaart & Wellner, 1996, p.50-51), as n[RIGHTWARDS ARROW] ∞ , we have inline image. The first expression of (9) then follows immediately.

We now show the second expression of (9). By (8), with inline image, we have inline image. Under Conditions A1–A5 and by the central limit proposition for i.i.d. stochastic processes, as n[RIGHTWARDS ARROW] ∞ , it is standard to show that inline image, where ϖ{(s1,t1),(s2,t2)} = E{v11(s1)v11(t1)v11(s2)v11(t2)} − γ(s1,t1)γ(s2,t2). The second expression of (9) then follows immediately from the fact that the k samples (1) are independent and inline image .

We now show (10). It follows from the second expression of (9) that as n[RIGHTWARDS ARROW] ∞ , we have inline image. The expression (10) follows immediately from the fact that inline image, where we use the fact γ2(s,t) ≤ γ(s,s)γ(t,t) ≤ ρ2, and the constants ρ and C are given in Conditions A5 and A6, respectively. The lemma is then proved.□

Proof of Proposition 1. Under the given conditions, by Lemma 1, as n[RIGHTWARDS ARROW] ∞ , we have inline image uniformly for all inline image, inline image and by (7), inline image. Then, the first expression of (11) follows from the well-known Slutsky's theorem, with inline image. Since inline image is a finite interval, we have inline image. Then the proof of the second expression of (11) follows about the same lines as those in the proof of Theorem 1 of Zhang (2011). This completes the proof of Proposition 1.□

Proof of Proposition 2. Under the given conditions, by Lemma 1,

  • display math

uniformly over all inline image. It follows that as n[RIGHTWARDS ARROW] ∞ , we have inline image. Therefore, as n[RIGHTWARDS ARROW] ∞ , we have

  • display math

By Slutsky's theorem, we have inline image as desired. The theorem is proved.□

Proof of Proposition 3. Under the given conditions, by Lemma 1, as n[RIGHTWARDS ARROW] ∞ , the first expression of (26) is obvious; see also the proof of the first expression of (11). On the basis of the singular value decomposition (25) of Ik − bbT, we set u(t) = (Ik − 1,0)UT[w(t) + h(t)]. We have u(t) ∼ GPk − 1(μu,γwIk − 1), where μu(t) = (Ik − 1,0)UTh(t) and inline image. Then the proof of the second expression of (26) follows about the same lines as those in the proof of Theorem 1 of Zhang (2011). This completes the proof of Proposition 3.□

Proof of Proposition 4. Under Condition A and the local alternative (24), by Proposition 3, we have inline image where inline image is given in (26). By the second expression of (26), we have

  • display math

where inline image and inline image. Thus,

  • display math

where inline image is as given in (11), inline image and inline image with inline image. Under the given conditions, by Proposition 2, we have inline image. It follows that under the given conditions, as n[RIGHTWARDS ARROW] ∞ , we have inline image. The expression (28) follows immediately. We now show that the power (28) tends to 1 as δ2[RIGHTWARDS ARROW] ∞ . This is obvious when inline image. We now assume δλ[RIGHTWARDS ARROW] ∞ . This implies that inline image. By (27), we further have inline image Proposition 4 is then proved.□