Nonparametric MANOVA in meaningful effects

Abstract

Multivariate analysis of variance (MANOVA) is a powerful and versatile method to infer and quantify main and interaction effects in metric multivariate multi-factor data. It is, however, neither robust against change in units nor meaningful for ordinal data. Thus, we propose a novel nonparametric MANOVA. Contrary to existing rank-based procedures, we infer hypotheses formulated in terms of meaningful Mann–Whitney-type effects in lieu of distribution functions. The tests are based on a quadratic form in multivariate rank effect estimators, and critical values are obtained by bootstrap techniques. The newly developed procedures provide asymptotically exact and consistent inference for general models such as the nonparametric Behrens–Fisher problem and multivariate one-, two-, and higher-way crossed layouts. Computer simulations in small samples confirm the reliability of the developed method for ordinal and metric data with covariance heterogeneity. Finally, an analysis of a real data example illustrates the applicability and correct interpretation of the results.

Appendices

Appendix

A Proofs

Throughout, let $\mathbb P_{1, n_1}, \ldots , \mathbb P_{a, n_a}$ be the empirical processes based on the samples $\mathcal X_1, \ldots , \mathcal X_a$, respectively, which are indexed by the class of functions $\mathcal G = \mathcal F \circ \Pi$, where

$$\begin{aligned} \mathcal F = \{{\mathbb {1}}_{(-\infty , x]}(\cdot ), {\mathbb {1}}_{(-\infty , x)}(\cdot ) : x \in \mathbb {R}\}, \end{aligned}$$

and $\Pi = \{\pi _j : j=1,\ldots , d \}$ is the class of all canonical coordinate projections $\pi _j : \mathbb {R}^d \rightarrow \mathbb {R}, (x_1, \ldots , x_d) \mapsto x_j$. Using this indexation, it is easily possible to derive the normalized empirical distribution functions $\widehat{F}_{ij}$ from $\mathbb P_{i, n_i}$. In particular, $\widehat{F}_{ij}(x)= \mathbb P_{i, n_i} [\frac{1}{2} ({\mathbb {1}}_{(-\infty , x]}(\cdot ) + {\mathbb {1}}_{(-\infty , x)})\circ \pi _j]$, where we used the definition $Pf = \int f \mathrm dP$ for a suitable function $f \in \mathcal G$ and a probability measure P. We also see that every group-specific empirical process $\mathbb P_{i, n_i}$ can be considered as an element of $\ell ^\infty (\mathcal G)$ which contains all bounded sequences with indices in $\mathcal G$: Based on the definition $\mathbb P_{i, n_i} \in \ell ^\infty (\mathcal G)$ if $\sup _{f \in \mathcal G} |Pf| < \infty$.

It is important to note that the class $\mathcal G$ is obtained from the Vapnik–C̆ervonenkis subgraph class $\mathcal F$ concatenated with the class of all canonical coordinate projections $\Pi$. This conserves the Vapnik–C̆ervonenkis subgraph property as argued in Lemma 2.6.17(iii) and 2.6.18(vii) of van der Vaart and Wellner (1996). Hence, $\mathcal G$ is a Donsker class.

Proof of Theorem 1

Clearly, $\widehat{\mathbf{p}}$ is a multivariate version $\phi$ of the Wilcoxon functional $\tilde{\phi }(f,g) = \int f(u) \mathrm dg(u)$ which is applied to all of the normalized empirical distribution functions ${\widehat{F}_{ij}(x)=}$$\frac{1}{n_i} \sum _{k=1}^{n_i} {c(x-X_{ijk})}$. The Hadamard differentiability of the Wilcoxon functional $\tilde{\phi }$ for normalized distribution functions has been argued in the proof of Theorem 2.1 in Dobler and Pauly (2018), and a similar result for the multivariate $\phi$ follows immediately. Hence, asymptotic normality follows from an application of the functional delta-method (Theorem 3.9.4 in van der Vaart and Wellner, 1996): it follows that $\sqrt{N}(\widehat{\mathbf{p}} - \mathbf{p})$ is asymptotically equal to $\phi '_{F_{11}, \ldots , F_{ad}}( \sqrt{N}(\widehat{F}_{ij} - F_{ij})_{i,j} )$, where $\phi '_{F_{11}, \ldots , F_{ad}}$ is a continuous and linear functional. Hence, the Donsker theorem yields that $\sqrt{N}(\widehat{F}_{ij} - F_{ij})_{i,j}$ converges in distribution to a Gaussian process as $N \rightarrow \infty$ and the application of $\phi '_{F_{11}, \ldots , F_{ad}}$ proves the asymptotic multivariate normality of $\sqrt{N}(\widehat{\mathbf{p}} - \mathbf{p})$.

The asymptotic covariance structure of the resulting multivariate normal distribution is derived in detail in Section 10 of in the supplement, where the asymptotic linear expansion of $\widehat{\mathbf{p}}$ in all empirical distribution functions is utilized. $\square$

Proof of Theorem 2

Similarly, as argued in the proof of Theorem 1, $\mathbf{p}^*$ is obtained as a Hadamard-differentiable functional of all bootstrapped (normalized) empirical distribution functions $F_{ij}^*(t) = \frac{1}{n_i} \sum _{k=1}^{n_i} c(t - X^*_{ijk})$, $j=1, \ldots , d$, $i=1, \ldots , a$. As the conditional central limit theorem holds in outer probability for each bootstrapped empirical distribution function, i.e. for each

$$\begin{aligned} \sqrt{n_i} (F_{ij}^*(t) - \widehat{F}_{ij}(t)) = \frac{1}{\sqrt{n_i}} \left( \sum _{k=1}^{n_i} c(t - X^*_{ijk}) - \sum _{k=1}^{n_i} c(t - X_{ijk}) \right) , \end{aligned}$$

cf. Theorem 3.6.1 in van der Vaart and Wellner (1996), the convergence is transferred to $\sqrt{N}(\mathbf{p}^* - \widehat{\mathbf{p}})$ by means of the functional delta-method for the bootstrap; cf. Theorem 3.9.11 in van der Vaart and Wellner (1996). $\square$

Proof of Theorem 3

First note that, given $\mathcal X$, we have conditional convergence in distribution of $\mathbf{F}_N^\star = \sqrt{N} (F_{11}^\star , F_{12}^\star , \ldots , F_{ad}^\star )'$ to an (ad)-variate Brownian bridge process $(\mathbf{U}_t)_{t \in \mathbb {R}}$ in outer probability: indeed, each $F_{ij}^\star$ can be written as

$$\begin{aligned} F_{ij}^\star (x)&= \frac{1}{n_i} \sum _{k=1}^{n_i} \widehat{\varepsilon }_{ijk} = \frac{1}{n_i} \sum _{k=1}^{n_i} D_{ik} \cdot [c(x - X_{ijk}) - \widehat{F}_{ij}(x)] \\&= \frac{1}{n_i} \sum _{k=1}^{n_i} (D_{ik} - \bar{D}_{i \cdot }) \cdot c(x - X_{ijk}) \end{aligned}$$

which is due to $\sum _{k=1}^{n_i} [c(x - X_{ijk}) - \widehat{F}_{ij}(x)] = 0$. Here we let $\bar{D}_{i\cdot } = \frac{1}{n_i} \sum _{k=1}^{n_i} D_{ik}$. If we further define $\delta _{\mathbf{X}_{ik}}$ to be the Dirac measure in $\mathbf{X}_{ik}$ and $f_{j,x} = \frac{1}{2} ({\mathbb {1}}_{(-\infty , x]}(\cdot ) + {\mathbb {1}}_{(-\infty , x)})\circ \pi _j$, we see that $c(x - X_{ijk}) = \delta _{\mathbf{X}_{ik}} f_{j,x}$. Consequently, $F_{ij}^\star (x)$ is a marginal distribution of $\frac{1}{n_i} \sum _{k=1}^{n_i} (D_{ik} - \bar{D}_{i \cdot }) \cdot \delta _{\mathbf{X}_{ik}}$. This proves that $F_{ij}^\star (x)$ has the required structure for an application of the conditional Donsker Theorem 3.6.13 in van der Vaart and Wellner (1996). Finally, Example 3.6.12 shows that this Donsker theorem still holds for our choice of wild bootstrap multipliers $D_{ik}$.

Next, recall the asymptotic linear representation (8) of

$$\begin{aligned} \sqrt{N}(\widehat{p}_{ij} - {p}_{ij}) = \sqrt{N} \int ( \widehat{G}_j - G_j) \mathrm dF_{ij} - \sqrt{N} \int (\widehat{F}_{ij} - F_{ij}) \mathrm dG_j + o_p(1) \end{aligned}$$

which followed from the functional delta-method and which motivated the wild bootstrap version (9). That presentation motivates that the Hadamard-derivative of the multivariate Wilcoxon-type functional $\phi$ which depends on unknown quantities should be estimated by

$$\begin{aligned} \phi _{ij; \widehat{F}}' : (\ell ^\infty (\mathcal G))^a \rightarrow \mathbb {R}, \quad (\text {P}_1, \ldots , \text {P}_a) \mapsto \int \left( \frac{1}{a} \sum _{\ell = 1}^a \text {F}_{\ell j} \right) \mathrm d\widehat{F}_{ij} - {\int \left( \frac{1}{a} \sum _{\ell = 1}^a \widehat{F}_{\ell j} \right) \mathrm d\text {F}_{i j}. } \end{aligned}$$

Here each $\text {P}_\ell \in \ell ^\infty (\mathcal G), \ell =1,\ldots , a,$ is a distribution on $\mathbb {R}^d$ with marginal normalized distribution functions $\text {F}_{\ell j}$.

We apply the extended continuous mapping theorem (Theorem 1.11.1 in van der Vaart and Wellner, 1996) to the (random) functional $\phi '_{\widehat{F}} = (\phi _{11; \widehat{F}}', \phi _{12; \widehat{F}}', \ldots , \phi _{ad; \widehat{F}}') : \ell ^\infty (\mathcal G) \rightarrow \mathbb {R}^{ad}$. Due to the subsequence principle (Lemma 1.9.2 in van der Vaart and Wellner, 1996) convergence in outer probability is equivalent to outer almost sure convergence along subsequences given a realization of $\mathcal X$.

The actual requirement for an application of the extended continuous mapping theorem is satisfied as well: note that $\phi '_{\widehat{F}}$ basically consists of integral mappings of the form

$$\begin{aligned} \psi : D(\mathbb {R}) \times BV_1(\mathbb {R}) \rightarrow \mathbb {R}, \quad (f,g) \mapsto \int f{(u)} \mathrm dg{(u)} \end{aligned}$$

where $D(\mathbb {R})$ is the space of right- (or left-)continuous functions on $\mathbb {R}$ with existing left- (or right-)sided limits and $BV_1(\mathbb {R})$ is the subspace of functions with total variation bounded by 1. Lemma 3.9.17 in van der Vaart and Wellner (1996) states that $\psi$ is Hadamard-differentiable, hence continuous. We conclude that for all sequences of functions $(f_n)_{n \in \mathbb {N}}$ and $(g_n)_{n \in \mathbb {N}}$, which converge to $f_0$ in $D(\mathbb {R})$ and to $g_0$ in $BV_1(\mathbb {R})$, respectively, the sequence of functionals $\psi _n : f \mapsto \int f \mathrm dg_n$ satisfies $\psi _n(f_n) \rightarrow \int f_0 \mathrm dg_0$ as $n \rightarrow \infty$.

All in all, the extended continuous mapping theorem, combined with the conditional central limit theorem for the wild bootstrapped empirical distribution functions as stated at the beginning of this proof, concludes this proof: $\phi '_{\widehat{F}}(\mathbf{F}_N^\star )$ converges in distribution to $\phi (\mathbf{U})$ for almost every sample $\mathcal X$ which is the same limit in distribution as in Theorem 1. Another application of the subsequence principle yields the desired convergence result in outer probability. $\square$