A note on repeated measures analysis for functional data
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Abstract
In this paper, the repeated measures analysis for functional data is considered. The known testing procedures for this problem are based on test statistic being the integral of the difference between sample mean functions, which takes into account only “between group variability”. We modify this test statistic to use also information about “within group variability”. More precisely, we construct the new test statistics being integral and supremum of pointwise test statistic obtained by adapting the classical paired ttest statistic to functional data framework. The testing procedures are based on different methods of approximating the null distribution of the test statistics, namely the Boxtype approximation, nonparametric and parametric bootstrap and permutation approaches. These approximations do not perform equally well under finite samples, which is established in simulation experiments, indicating the best new tests. The simulations and an application to mortality data suggest that some of the new procedures outperform the known tests in terms of size control and power.
Keywords
Bootstrap Boxtype approximation Functional data analysis Hypothesis testing Permutation Repeated measures analysis1 Introduction
In the recent past two decades, the functional data analysis (FDA) dealing with samples of random curves or functions has increased its popularity. The first reason for this is the technological progress allowing to continuously monitor and record data for many processes in biostatistics, chemometrics, econometrics, geophysics, medicine, meteorology etc. Moreover, the increasing computer power allows to conduct an effective analysis of such (highdimensional) data in a reasonable time. Secondly, the monographs by Ramsay and Silverman (1997, 2002, 2005) offering a broad range of the FDA methods and presenting their applications to real data problems have helped in learning the basics of the FDA in an orderly and accessible way for a wide audience. The nonparametric perspective of the FDA is proposed by Ferraty and Vieu (2006). The books by Horváth and Kokoszka (2012) and Zhang (2013) complement the previous monographs on new theoretical and practical aspects of the FDA.
Testing statistical hypothesis about functions or curves is one of the most commonly investigated problems in the FDA. The testing procedures for functional analysis of variance problem, where one wants to check the equality of mean functions in a number of groups of independent functional observations, are very popular. In Cuevas et al. (2004), Górecki and Smaga (2015), Ramsay and Silverman (2005), Zhang (2013), Zhang et al. (2018) and Zhang and Liang (2014), the parametric functional ANOVA was studied by a broad range of different methods. On the other hand, Delicado (2007) considered nonparametric functional ANOVA, while Górecki and Smaga (2017) investigated the multivariate analysis of variance for functional data. Some other hypothesis testing problems are as follows: Gabrys and Kokoszka (2007) proposed a simple portmanteau test of independence for functional observations based on their principal component expansion. In functional regression analysis, the statistical significance of (fixed or functional) coefficients was investigated, for example, by Collazos et al. (2016), Kokoszka et al. (2008) and Maslova et al. (2010). Testing the linearity of a scalaronfunction regression relationship is considered by McLean et al. (2015). Kosiorowski et al. (2017) proposed a novel statistic for detecting a structural change in functional time series based on a local Wilcoxon statistic. The interval testing procedure for functional data in different frameworks (e.g., one or twopopulation frameworks) was proposed by Pini and Vantini (2017). Their procedure is based on means of different basis expansions. In the case of spatial functional data, Giraldo et al. (2018) proposed the test based on the Mantel statistic for checking spatial autocorrelation, which is an important step in the statistical analysis of spatial data.
In this paper, we consider the repeated measures analysis for functional data, where we have the repeated functional data for the same subjects probably submitted to different conditions. MartínezCamblor and Corral (2011) proposed first testing procedure for this problem. Their tests are based on test statistic being the integral of the difference between sample mean functions and bootstrap and permutation approaches. Smaga (2017) proposed the test based on the same test statistic, but its distribution was approximated by the Boxtype approximation, which resulted in less timeconsuming procedure than resampling and permutation methods. Nevertheless, all these tests perform equally well in terms of size control and power under finite samples. Note that since the tests considered in these papers are constructed mainly on asymptotic distribution of the test statistic, it is based only on “between group variability”. In this work, we show how to modify this test statistic to take into account also “within group variability”. Namely, we adapt the usual test statistic of the classical paired twosample test to functional data framework obtaining an appropriate pointwise test statistic. We construct two test statistics being integral and supremum of the pointwise test statistic. Their distributions are approximated by parametric methods based on derived asymptotic distributions as well as nonparametric bootstrap and permutation approaches. Simulation results indicate that the new testing procedures may not perform equally well under finite samples (in contrast to methods proposed by MartínezCamblor and Corral 2011, and Smaga 2017), and some of them are not recommended when the sample size is small. However, the best new permutation tests usually work better than the tests by MartínezCamblor and Corral (2011) and Smaga (2017), which is seen in simulations and real data application.
The rest of the paper is organized as follows: In Sect. 2, we first formulate the repeated measures analysis for functional data. Then, we propose new test statistics for this problem and consider different methods of approximating their null distribution to construct testing procedures. Section 3 is devoted to Monte Carlo simulation studies providing an idea of finite sample behavior of the new tests and comparing them with the known tests. In Sect. 4, a real data example based on the mortality data is presented. Section 5 concludes the paper. In Appendix, the technical assumptions and the proof of theoretical result are given.
2 Repeated measures analysis for functional data
In this section, we present new testing procedures for the repeated measures analysis for functional data. They can be seen as modifications of the tests proposed in MartínezCamblor and Corral (2011) and Smaga (2017) by taking into account also “within group variability”.
2.1 Test statistics
In fact, the test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\) can use more information from the data than the test statistic \({\mathcal {C}}_n\), which may result in more powerful testing procedures for (2), as we will see in simulation experiments and real data example (Sects. 3 and 4).
2.2 Approximating the null distribution
To construct the tests based on the test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\), we use different methods to approximate their null distributions. Similarly to MartínezCamblor and Corral (2011), we start with the use of the asymptotic null distributions of the test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\). They are presented in the following result. Its proof and technical assumptions are given and discussed in Appendix. Throughout the paper, \({\mathop {\rightarrow }\limits ^{d}}\) denotes convergence in distribution (Laha and Rohatgi 1979, p. 474), and \(X{\mathop {=}\limits ^{d}}Y\) denotes that the random variables X and Y have the same distribution.
Theorem 1
 1.If Assumptions A1–A5 given in Appendix are satisfied, thenas \(n\rightarrow \infty \), where z(t), \(t\in [0,2]\) is a Gaussian process with zero mean and covariance function \({\mathbb {C}}(s,t)\), \(s,t\in [0,2]\), \(\lambda _1^*,\lambda _2^*,\dots \) are the decreasingordered eigenvalues of \({\mathbb {K}}_*(s,t)={\mathbb {K}}(s,t)/({\mathbb {K}}(s,s){\mathbb {K}}(t,t))^{1/2}\), and \(A_1,A_2,\dots \) are the independent random variables following a central chisquared distribution with one degree of freedom.$$\begin{aligned} {\mathcal {D}}_n{\mathop {\rightarrow }\limits ^{d}}\int _0^1\frac{(z(t)z(t+1))^2}{{\mathbb {K}}(t,t)}\mathrm{d}t{\mathop {=}\limits ^{d}}\sum _{k=1}^\infty \lambda _k^*A_k, \end{aligned}$$(7)
 2.If Assumptions A1–A6 given in Appendix are satisfied, thenas \(n\rightarrow \infty \), where z(t), \(t\in [0,2]\) is the same Gaussian process as in 1.$$\begin{aligned} {\mathcal {E}}_n{\mathop {\rightarrow }\limits ^{d}}\sup _{t\in [0,1]}\left\{ \frac{(z(t)z(t+1))^2}{{\mathbb {K}}(t,t)}\right\} , \end{aligned}$$(8)
The distribution of a \(\chi ^2\)type mixture can be also approximated by exact methods as, for example, the Imhof’s algorithm (Imhof 1961). However, the simulation results (not shown) indicate that such algorithms work similarly to the Boxtype approximation, but are usually more timeconsuming. Thus, we do not consider the exact methods in the paper.
Since the tests mentioned above are constructed based on the asymptotic null distributions of the test statistics, they may need larger number of observations to perform satisfactorily, as we will see in simulations. Thus, we also investigate nonparametric approaches, namely nonparametric bootstrap and permutation procedures, which were considered by MartínezCamblor and Corral (2011) and Konietschke and Pauly (2014). These procedures do not require the assumptions as the tests based on the asymptotic null distributions of the test statistics.
Note that the nonparametric bootstrap tests take into account the null hypothesis at the moment of computing the value of the test statistics instead of at the resampling moment, as it is suggested, for example, in MartínezCamblor and Corral (2011, 2012).
Simulation and real data example results in MartínezCamblor and Corral (2011) and Smaga (2017) indicate that the all tests (namely, using parametric and nonparametric bootstrap, permutation approach and Boxtype approximation) based on the test statistic \({\mathcal {C}}_n\) perform very similarly under finite samples in terms of size control and power. However, this is not true for the testing procedures based on the new test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\) as we will see in the next sections.
3 Simulation studies
In this section, the simulation studies are performed to investigate the behavior of the tests proposed in Sect. 2 with regard to maintaining the preassigned typeI error level under the null hypothesis, and the power under alternatives. Since all testing procedures based on the test statistic \({\mathcal {C}}_n\) behave very similarly under finite samples, we considered only the least timeconsuming BT test by Smaga (2017), as a competitor to the new tests.
3.1 Simulation setup
The paired functional samples are generated according to the formulas: \(X_i(t) =m_{1}(t)+e_{i1}(t)\) and \(X_i(t+1)=m_{2}(t)+e_{i2}(t)\) for \(t\in [0,1]\), \(i=1,\dots ,n\). The number of observations n is set to 15, 25, 35.

M1 \(m_1(t)=m_2(t)=(\sin (2\pi t^2))^5I_{[0,1]}(t)\),

M2 \(m_1(t)=(\sin (2\pi t^2))^5I_{[0,1]}(t)\), \(m_2(t)=(\sin (2\pi t^2))^7I_{[0,1]}(t)\),

M3 \(m_1(t)=(\sin (2\pi t^2))^5I_{[0,1]}(t)\), \(m_2(t)=(\sin (2\pi t^2))^3I_{[0,1]}(t)\).

Normal setting: \(e_{i1}(t)=0.5B_{i1}(t)\), \(e_{i2}(t)=\rho e_{i1}(t)+0.5\sqrt{1\rho ^2}B_{i2}(t)\),

Lognormal setting: \(e_{i1}(t)=\exp (0.5B_{i1}(t))\), \(e_{i2}(t)=\exp (\rho e_{i1}(t)+0.5 \sqrt{1\rho ^2}B_{i2}(t))\),

Mixed setting: \(e_{i1}(t)=0.5B_{i1}(t)\), \(e_{i2}(t)=\exp (\rho e_{i1}(t)+0.5\sqrt{1\rho ^2}B_{i2}(t))\),
Since the functional data are not usually continuously observed in practice, the trajectories of \(X_1(t),\dots ,X_n(t)\), \(t\in [0,2]\) are discretized at design time points \(t_1,\dots ,t_I,t_1+1,\dots ,t_I+1\), where \(t_i\), \(i=1,\dots ,I\) are equispaced in [0, 1]. We consider \(I=26,101\).
S  n  \(\rho \)  \(I=26\)  \(I=101\)  

\({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  \({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  
BT  BT  PB  NB(I)  NB(II)  P  PB  NB(I)  NB(II)  P  BT  BT  PB  NB(I)  NB(II)  P  PB  NB(I)  NB(II)  P  
N  15  0.00  5.6  7.2  7.0  3.2  3.6  4.7  14.0  2.0  2.7  4.5  6.5  8.6  8.4  2.8  3.8  4.6  20.4  1.2  3.0  4.6 
0.25  5.7  7.5  7.5  3.0  3.7  4.7  14.6  1.6  3.3  4.6  6.6  8.7  8.6  2.1  3.7  5.1  19.1  1.5  3.7  4.6  
0.50  6.5  7.8  7.9  2.8  3.8  5.0  14.5  1.3  3.9  4.9  6.5  8.6  8.8  2.6  3.6  5.1  18.9  1.2  3.7  4.8  
0.75  6.4  8.0  7.9  3.2  4.8  5.4  15.2  1.4  4.8  5.2  6.2  8.0  7.9  2.7  3.7  4.9  20.3  1.0  3.7  4.2  
25  0.00  5.1  5.9  6.1  3.3  3.6  4.7  9.9  3.1  3.9  5.2  5.0  6.8  6.4  3.3  3.9  4.1  11.0  3.0  3.3  4.7  
0.25  4.2  5.4  5.5  3.6  3.8  4.2  9.2  3.2  3.7  4.6  5.2  6.0  5.7  3.3  3.6  4.4  10.2  2.6  3.5  4.5  
0.50  4.4  5.1  5.3  3.7  3.7  4.2  8.7  2.8  3.9  4.6  5.3  5.8  5.7  3.5  3.7  4.5  9.7  2.6  3.9  4.4  
0.75  5.2  5.8  6.0  3.5  4.0  4.2  9.8  2.5  3.9  4.8  4.6  5.7  5.8  3.1  3.8  4.2  10.9  3.0  4.3  5.2  
35  0.00  5.3  5.9  6.0  4.3  4.5  4.9  8.8  3.4  3.9  4.5  5.3  5.9  6.3  4.4  4.9  4.9  8.0  2.7  3.5  4.5  
0.25  4.7  5.7  5.8  4.5  4.3  4.9  8.2  3.8  4.4  4.9  4.9  5.8  6.0  4.2  4.8  5.1  8.5  2.8  4.2  4.1  
0.50  4.5  5.6  5.9  3.9  4.1  4.3  8.7  3.7  4.4  5.3  5.1  5.9  6.4  4.1  4.6  5.0  8.6  3.1  4.1  4.3  
0.75  4.8  5.6  5.7  3.8  4.7  4.6  9.4  3.9  5.5  5.6  4.9  5.9  6.5  3.7  4.5  4.8  8.2  2.4  3.9  4.4  
L  15  0.00  3.4  7.0  6.9  1.1  2.0  4.3  16.7  2.1  2.9  4.9  4.6  8.0  8.0  2.6  3.7  5.0  21.2  1.2  2.5  4.7 
0.25  3.4  6.6  7.4  1.3  2.6  3.9  16.7  2.0  3.4  5.2  4.5  8.6  8.6  2.4  3.4  4.8  22.3  1.0  3.4  5.3  
0.50  3.5  7.0  7.2  1.2  2.9  4.5  17.3  1.8  3.8  5.1  4.4  8.7  8.5  2.4  3.4  5.4  22.6  1.3  3.8  5.4  
0.75  4.5  8.3  7.8  1.2  3.3  5.0  15.7  1.8  4.4  5.8  5.0  8.3  8.7  2.0  3.5  5.4  21.2  1.4  4.5  5.8  
25  0.00  3.5  5.2  5.0  2.9  2.8  4.0  10.6  2.1  2.7  4.2  5.2  6.4  6.7  4.1  4.3  5.2  12.0  2.0  3.1  4.0  
0.25  3.4  5.8  5.7  2.8  2.9  4.0  10.4  2.6  3.4  4.6  4.9  6.0  6.0  3.5  4.1  5.2  12.0  1.7  2.8  4.3  
0.50  3.4  4.9  4.6  2.5  2.9  4.1  10.4  2.4  3.7  4.8  4.4  6.1  5.8  3.0  3.5  4.4  11.4  1.9  3.7  4.1  
0.75  3.9  5.7  6.3  2.9  3.3  4.5  9.6  2.1  4.2  4.9  4.6  5.9  5.8  3.5  3.6  4.5  11.7  2.1  3.5  4.2  
35  0.00  4.1  5.2  5.0  2.5  2.9  4.2  8.5  3.2  3.9  4.5  5.1  6.4  6.2  4.4  4.6  5.6  10.1  3.4  4.2  5.2  
0.25  3.8  5.1  5.1  2.9  3.3  4.3  9.1  2.6  3.3  4.3  4.7  5.8  6.1  4.1  4.6  5.3  9.2  2.7  3.5  4.7  
0.50  4.3  5.6  5.5  3.3  3.6  4.8  9.3  2.8  3.5  4.4  4.9  6.0  5.6  4.1  4.7  5.0  8.9  2.9  3.4  3.9  
0.75  4.4  5.8  5.9  3.2  3.8  4.4  8.6  2.9  3.3  4.1  4.5  5.5  5.6  4.0  4.4  4.8  9.9  3.1  4.5  4.4  
M  15  0.00  4.6  7.6  7.5  1.8  2.9  4.2  15.1  1.7  2.7  4.6  5.4  8.7  9.1  2.7  4.0  5.7  21.5  0.9  3.1  5.9 
0.25  4.9  8.3  7.6  2.5  3.9  4.7  15.3  1.6  3.2  5.0  5.8  9.0  9.3  2.7  4.5  5.6  20.9  1.1  3.0  5.9  
0.50  5.6  8.1  8.7  3.3  5.5  5.6  15.0  2.0  2.8  4.7  6.1  8.7  8.5  2.7  5.9  5.1  18.2  1.7  3.2  5.5  
0.75  6.4  7.6  8.0  3.6  6.8  5.0  13.6  1.9  2.3  4.7  6.3  8.8  8.6  3.2  7.6  5.2  17.2  1.4  2.4  5.7  
25  0.00  4.9  7.2  6.6  3.4  4.3  5.2  11.6  3.5  4.3  5.8  7.3  9.7  9.5  5.5  6.2  5.7  14.9  3.9  4.5  5.9  
0.25  5.1  6.6  6.9  3.7  4.6  5.2  10.6  3.7  4.3  5.3  6.8  9.0  8.8  5.3  6.8  6.1  13.6  4.1  4.8  6.3  
0.50  5.0  6.9  7.0  3.6  5.3  5.0  10.0  3.6  3.8  5.2  6.6  8.9  9.0  5.6  8.0  6.2  14.4  4.4  5.0  6.0  
0.75  5.1  6.5  6.9  3.4  8.1  4.6  9.9  3.4  3.0  5.6  7.0  9.0  9.0  5.5  10.4  6.3  13.9  5.4  4.1  6.0  
35  0.00  6.3  8.0  7.5  5.3  5.9  5.9  12.6  4.7  5.3  6.1  6.3  7.9  8.3  4.9  5.4  6.2  12.9  4.4  4.8  5.8  
0.25  6.2  7.5  7.9  5.1  6.7  6.2  11.8  4.6  5.1  5.8  6.2  7.5  7.2  4.6  6.4  5.9  12.5  4.8  4.9  6.1  
0.50  6.4  7.5  7.9  5.5  7.5  6.1  11.2  5.9  5.2  5.6  6.0  6.9  6.9  4.5  6.9  5.5  12.2  4.3  4.2  6.1  
0.75  6.7  7.5  7.8  5.9  9.1  6.4  11.4  5.9  4.7  6.3  5.9  6.1  6.4  4.6  9.0  5.5  11.8  5.2  3.2  6.1 
3.2 Simulation results
S  n  \(\rho \)  \(I=26\)  \(I=101\)  

\({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  \({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  
BT  NB(I)  NB(II)  P  NB(I)  NB(II)  P  BT  NB(I)  NB(II)  P  NB(I)  NB(II)  P  
N  15  0.00  14.5  8.4  10.9  14.3  17.3  24.7  30.3  16.9  9.7  12.4  16.7  11.6  21.6  30.1 
0.25  19.0  11.3  15.0  21.1  24.0  35.4  41.9  21.4  13.4  15.9  21.9  18.5  32.3  39.3  
0.50  28.3  20.9  27.7  35.9  40.9  58.4  62.5  30.8  21.3  27.5  36.1  29.9  51.0  56.6  
0.75  62.0  55.9  67.4  75.4  83.9  93.1  94.4  63.2  57.1  69.1  74.9  73.4  90.1  91.7  
25  0.00  21.6  20.8  22.7  29.0  52.4  56.7  60.7  21.8  20.2  21.9  28.0  47.5  53.6  57.9  
0.25  32.2  32.9  36.0  43.2  71.6  75.7  79.4  31.7  32.6  35.4  41.9  65.6  71.2  74.7  
0.50  52.9  57.2  62.8  68.0  92.3  94.6  95.3  53.8  58.4  63.3  69.2  87.9  91.7  92.2  
0.75  94.3  97.3  98.9  98.9  100  100  100  94.0  96.2  97.6  98.1  100  100  100  
35  0.00  30.3  36.5  37.9  43.3  78.6  79.2  81.0  31.7  35.8  37.9  42.7  74.2  76.9  79.2  
0.25  46.1  56.1  58.0  63.2  91.7  92.8  93.4  46.2  55.7  57.2  64.0  90.2  91.9  93.3  
0.50  74.2  84.1  87.2  88.6  98.9  99.3  99.3  73.9  85.3  87.4  89.6  99.0  99.4  99.3  
0.75  99.8  99.8  99.9  100  100  100  100  99.5  100  100  100  100  100  100  
L  15  0.00  32.6  21.7  28.6  37.4  23.0  31.4  41.6  35.3  22.7  27.2  36.4  16.4  30.5  41.7 
0.25  45.3  31.0  40.8  49.5  32.2  45.9  55.8  46.8  32.1  41.6  50.5  25.5  44.0  55.4  
0.50  67.5  53.1  63.9  72.4  52.4  69.8  77.3  69.3  52.5  64.2  73.0  44.5  68.3  76.7  
0.75  96.1  90.4  95.1  97.3  89.5  97.2  98.0  97.0  90.7  95.9  97.7  86.0  97.0  98.1  
25  0.00  60.8  54.4  56.1  65.0  64.2  68.2  72.0  65.6  57.7  61.7  69.4  64.6  70.5  76.2  
0.25  78.5  72.7  75.2  80.5  78.8  83.8  86.7  81.1  74.9  78.0  82.5  80.5  85.2  88.4  
0.50  94.3  91.4  93.5  95.3  95.3  97.2  97.9  95.8  92.3  94.4  96.4  94.6  97.1  97.8  
0.75  100  99.9  100  100  99.9  100  100  100  99.9  99.9  100  100  100  100  
35  0.00  81.4  78.2  81.3  83.9  87.1  88.7  90.2  83.6  79.1  81.0  84.6  87.3  88.8  91.6  
0.25  93.5  90.6  92.8  94.1  95.9  96.8  97.5  94.7  91.6  93.5  94.8  95.5  96.5  97.1  
0.50  99.5  99.4  99.5  99.6  99.6  99.7  99.6  99.5  99.3  99.6  99.7  99.9  99.9  99.9  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
M  15  0.00  16.6  9.1  13.2  20.7  13.4  19.4  27.3  20.0  11.3  15.5  21.1  8.1  18.1  26.7 
0.25  20.0  11.9  17.6  23.7  17.1  24.7  32.4  22.2  13.5  18.8  23.5  13.3  22.6  32.8  
0.50  23.8  14.7  24.1  26.2  23.5  29.4  39.3  25.6  16.3  24.0  27.5  18.6  27.8  41.8  
0.75  27.4  17.6  32.9  28.2  39.3  40.4  56.7  30.7  18.7  35.0  30.7  36.5  38.4  59.1  
25  0.00  33.3  30.2  33.0  38.7  44.6  48.8  54.0  33.9  28.5  30.8  38.4  41.6  48.3  56.0  
0.25  41.1  35.1  39.9  44.6  55.3  58.7  64.0  41.9  33.1  40.8  45.2  53.1  59.1  67.2  
0.50  49.0  40.7  52.2  50.6  69.5  70.3  75.8  52.0  40.9  55.1  53.8  71.3  72.1  80.9  
0.75  56.9  46.8  69.0  57.5  86.6  83.3  91.4  59.9  47.9  73.8  58.8  89.2  84.7  92.9  
35  0.00  49.1  49.4  51.0  56.0  71.0  72.7  75.7  49.7  48.6  51.0  56.4  68.9  72.5  75.8  
0.25  59.0  56.9  61.3  64.4  79.8  81.3  84.4  60.6  56.3  62.9  64.1  79.8  80.3  84.7  
0.50  70.6  65.6  76.6  72.9  92.3  91.3  93.2  71.7  65.2  77.1  72.8  89.9  89.2  93.1  
0.75  80.7  75.0  91.9  81.4  98.5  97.7  99.1  81.3  73.9  92.1  80.4  98.8  97.3  99.3 
S  n  \(\rho \)  \(I=26\)  \(I=101\)  

\({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  \({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  
BT  NB(I)  NB(II)  P  NB(I)  NB(II)  P  BT  NB(I)  NB(II)  P  NB(I)  NB(II)  P  
N  15  0.00  40.1  34.0  42.1  52.6  52.5  65.5  71.6  40.7  37.3  45.7  57.5  49.4  67.7  76.8 
0.25  55.6  52.1  61.9  71.7  71.3  82.8  86.2  59.4  55.9  64.9  75.7  68.7  83.6  89.8  
0.50  82.4  79.8  89.3  93.2  90.2  96.6  98.0  84.9  82.0  89.9  93.3  90.1  97.1  98.7  
0.75  99.5  99.3  100  99.9  99.7  100  100  99.8  99.7  100  100  99.8  100  100  
25  0.00  68.0  80.2  83.9  87.9  96.3  97.3  97.9  71.7  82.1  84.4  88.6  97.5  98.1  98.5  
0.25  88.5  94.0  96.1  97.5  99.4  99.6  99.6  87.6  94.9  96.6  97.5  99.5  99.7  99.7  
0.50  98.9  99.6  99.9  99.9  99.9  100  100  99.4  99.9  100  100  100  100  100  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
35  0.00  93.1  97.8  98.2  99.1  99.9  99.8  99.9  91.9  97.6  98.5  98.9  100  100  100  
0.25  99.0  99.9  100  100  100  100  100  98.8  100  100  100  100  100  100  
0.50  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
L  15  0.00  86.3  72.6  80.5  87.8  66.7  79.2  87.0  89.5  76.0  83.1  90.3  59.9  79.8  88.3 
0.25  95.4  88.5  92.6  95.7  82.3  91.7  94.6  96.6  89.8  93.8  96.3  77.9  91.5  95.8  
0.50  99.4  97.2  99.4  99.7  96.1  99.1  99.6  99.8  98.3  99.3  99.7  93.0  98.4  99.5  
0.75  100  100  100  100  100  100  100  100  100  100  100  99.9  100  100  
25  0.00  99.5  98.8  99.1  99.8  99.3  99.9  100  99.8  99.2  99.7  99.6  98.6  99.1  99.6  
0.25  100  100  100  100  100  100  100  100  99.9  100  100  99.7  100  100  
0.50  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
35  0.00  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.25  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.50  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
M  15  0.00  71.9  58.2  66.5  78.9  56.0  68.6  79.9  74.3  59.9  68.4  78.8  48.8  67.1  80.0 
0.25  81.9  68.7  77.8  85.3  69.0  79.4  88.0  84.3  69.3  78.0  86.1  59.5  77.3  88.9  
0.50  90.4  76.7  88.2  91.2  84.2  90.0  95.9  91.7  78.6  90.3  92.3  77.2  87.3  96.8  
0.75  96.7  85.2  96.3  95.6  95.9  97.0  99.8  97.3  86.3  97.3  97.0  95.5  96.1  99.6  
25  0.00  96.7  95.5  96.2  98.3  97.9  98.7  99.4  97.0  96.4  97.8  98.9  98.3  99.0  99.4  
0.25  99.0  98.5  99.0  99.4  99.6  99.8  99.9  99.6  99.2  99.8  100  99.8  99.8  100  
0.50  99.8  99.4  99.9  99.8  100  100  100  100  99.9  100  100  100  100  100  
0.75  100  99.8  100  100  100  100  100  100  100  100  100  100  100  100  
35  0.00  99.9  100  100  100  100  100  100  99.8  99.9  99.9  100  100  100  100  
0.25  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.50  100  100  100  100  100  100  100  100  100  100  100  100  100  100  
0.75  100  100  100  100  100  100  100  100  100  100  100  100  100  100 
Now, we investigate the empirical power of tests, which do not have too liberal character in most cases, namely the BT test and the nonparametric bootstrap and permutation tests based on the test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\). (Note, however, that the power of the BT test may be overstated under normal and mixed settings. The same is true for the test based on the test statistics \({\mathcal {D}}_n\) and the nonparametric bootstrap method (II) under mixed setting.) The empirical power of the other tests is not comparable, as they do not maintain the typeI error rate. The empirical powers obtained under Models M2 and M3 are presented in Tables 2 and 3.
In general, the empirical power increases when the number of observations or the correlation between curves also increases. The estimated statistical powers are often comparable for different numbers of design time points. However, it may happen that they increase with I, which is indicated by simulation results not shown. Unfortunately, they may also decrease with I for the tests based on the test statistic \({\mathcal {E}}_n\) (especially nonparametric bootstrap method (I)) when the number of observations is small.
The empirical powers of the nonparametric bootstrap tests are usually smaller than these of the permutation procedures, which follows from conservativity of the first tests. From permutation tests, the test based on \({\mathcal {E}}_n\) outperforms that based on \({\mathcal {D}}_n\), but sometimes their powers are comparable. In the group of nonparametric bootstrap methods, the NB(II) approach usually has greater empirical power, as it is less conservative than the NB(I) method. Under normal and mixed settings, the permutation test based on \({\mathcal {D}}_n\) works better than the BT test, except under mixed setting and \(\rho =0.75\), where the empirical powers of these tests are comparable. Under the same settings, the nonparametric bootstrap tests based on \({\mathcal {E}}_n\) usually have greater power than the permutation procedure based on \({\mathcal {D}}_n\). Under lognormal setting, the empirical powers of the BT test, permutation test based on \({\mathcal {D}}_n\) and nonparametric bootstrap tests based on \({\mathcal {E}}_n\) are comparable for \(n=25,35\), but when \(n=15\), the last three tests perform usually better, worse than the others, respectively. The conservativity of the nonparametric bootstrap tests based on \({\mathcal {D}}_n\) implies that they often have smaller power than the BT test.
To summarize, the permutation test based on \({\mathcal {E}}_n\) usually outperforms the other tests, but the permutation procedure based on \({\mathcal {D}}_n\) also works quite well and has larger power than the BT test in most cases. The nonparametric bootstrap tests have conservative character especially in case of small sample size, which may result in the loss of power. On the other hand, the asymptotic tests based on the test statistics \({\mathcal {D}}_n\) and \({\mathcal {E}}_n\) are not recommended because of their liberality.
The finite sample behavior of the tests under the null and alternative hypotheses discussed here will be seen in the real data application of the next section.
4 Real data example
In this section, we present a real data example to illustrate the practical use of the tests proposed in Sect. 2 and of the tests of MartínezCamblor and Corral (2011) and Smaga (2017). For this purpose, we consider the human mortality data, which are often used to quantify health state of people living in different countries and to appraise biological limits of longevity (Oeppen and Vaupel 2002; Vaupel et al. 1998). The data were obtained from the Human Mortality Database (Wilmoth et al. 2017; http://www.mortality.org).
For illustrative purposes, we take into account the mortality rates for two decades 1980–1989 and 1990–1999 for the following 32 countries: Australia, Austria, Belarus, Belgium, Bulgaria, Canada, Czech Republic, Denmark, Estonia, Finland, France, Hungary, Iceland, Ireland, Italy, Japan, Latvia, Lithuania, Luxembourg, Netherlands, New Zealand, Norway, Poland, Portugal, Russia, Slovakia, Spain, Sweden, Switzerland, United Kingdom, Ukraine, USA. We focus on the mortality rates of older individuals, i.e., in age from 60 to 100 years. The death rates for these countries were also studied by using the principal component analysis for repeated functional observations in Chen and Müller (2012), but for longer time period.
For a given decade, the observed period mortality rates can be treated as realizations of some random process being a function of age (\(t\in [60,100]\)). Moreover, the observations of the mortality rates for two decades can be considered as two samples of repeated functional data, since they are measured repeatedly for different countries, which are the subjects in this example. Thus, we have \(n=32\) repeated functional observations measured at \(I=41\) design time points in each of two conditions (decades). These data as well as the corresponding sample mean functions are shown in the first row of Fig. 1.
n  \({\mathcal {C}}_n\)  \({\mathcal {D}}_n\)  \({\mathcal {E}}_n\)  

BT  PB  NB(I)  P  BT  PB  NB(I)  NB(II)  P  PB  NB(I)  NB(II)  P  
32  0.065  0.061  0.076  0.052  0.000  0.000  0.001  0.000  0.000  0.000  0.000  0.000  0.000 
5  0.000  0.000  0.000  0.000  0.000  0.000  0.120  0.009  0.000  0.000  0.111  0.008  0.000 
27  0.001  0.001  0.005  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
5 Conclusions
In this paper, we have studied the testing procedures for the repeated measures analysis for functional data, which are based on test statistics taking into account both “between and within group variabilities”, in contrast to tests by MartínezCamblor and Corral (2011) and Smaga (2017). By simulation studies and real data example, different proposed methods of approximating the null distribution of the test statistics do not work equally well under finite samples. The Boxtype approximation and parametric bootstrap are too liberal when the number of observations is small, and therefore they are not recommended in such situation. On the other hand, the nonparametric bootstrap method may result in conservative tests and the loss of power may be noted. In terms of size control and power, the permutation test based on the test statistic being the supremum of the pointwise test statistic seems to be the best testing procedure, which also outperforms the tests of MartínezCamblor and Corral (2011) and Smaga (2017). We also observed that the procedures proposed in these papers may be too anticonservative for very small sample size (\(n\le 15\)).
Notes
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