Two-sample density-based empirical likelihood ratio tests based on paired data, with application to a treatment study of attention-de cit/hyperactivity disorder and severe mood dysregulation

Abstract

It is a common practice to conduct medical trials to compare a new therapy with a standard-of-care based on paired data consisted of pre- and post-treatment measurements. In such cases, a great interest often lies in identifying treatment effects within each therapy group and detecting a between-group difference. In this article, we propose exact nonparametric tests for composite hypotheses related to treatment effects to provide efficient tools that compare study groups utilizing paired data. When correctly specified, parametric likelihood ratios can be applied, in an optimal manner, to detect a difference in distributions of two samples based on paired data. The recent statistical literature introduces density-based empirical likelihood methods to derive efficient nonparametric tests that approximate most powerful Neyman–Pearson decision rules. We adapt and extend these methods to deal with various testing scenarios involved in the two-sample comparisons based on paired data. We show that the proposed procedures outperform classical approaches. An extensive Monte Carlo study confirms that the proposed approach is powerful and can be easily applied to a variety of testing problems in practice. The proposed technique is applied for comparing two therapy strategies to treat children’s attention deficit/hyperactivity disorder and severe mood dysregulation.

1. Introduction and Technical Preliminaries

Investigators in various fields of medical studies often deal with paired data to compare different population groups. In this article, we propose a paired data-based methodology motivated by the following comparative study of attention-deficit/hyperactivity disorder (ADHD) and severe mood dysregulation (SMD). ADHD is a commonly diagnosed psychiatric disorder in children (e.g., [1, 2]). SMD is a diagnostic label recently created by the Leibenluft’s laboratory in the National Institute of Mental Health’s intramural program to refer to children with an abnormal baseline mood, hyperarousal, and increased reactivity to negative emotional stimuli (e.g., [3–6]). A novel group therapy study at University at Buffalo enrolled 32 children aged 7–12 with ADHD and SMD. These children were treated for 11 weeks. The study participants were randomized between two therapy groups: experimental group therapy program (case; new therapy group) and community psychosocial treatment (control; old therapy group). An objective of the study was to compare the feasibility and efficacy of these two treatments using the Children’s Depression Rating Scale — Revised total score (CDRS-Rts). The Children’s Depression Rating Scale, revised version (CDRS-R), is a clinician-rated instrument for the diagnosis of childhood depression and the assessment of the severity of depression in children 6–12 years of age [7, 8]. The CDRS-R consists of 17 clinician rated items, with 14 items based on the child’s self-report or reports from the parents or teachers and three items based on the child’s nonverbal behavior during the interviews. The CDRS-R provides more reliable depression ratings compared with the other children depression rating scales, because it collects information from more sources through interviewing the child, parents or school teachers, independently, and it considers the child’s behavior during the interview, and lengthens scales to capture slight differences of symptomatology. On the basis of clinical experience, a CDRS-Rts of below 40, 40–60, and above 60 corresponds to none to mild, moderate, and severe depression, respectively [7–9]. Thus, the fact that the CDRS-Rts drops significantly over the course of the study implies an effectiveness of a treatment. To record paired data of this study, two measurements were taken from the same subjects. The paired data were constituted by observed values of CDRS-Rts at week 0 (baseline) and week 11 (endpoint).

In this medical study, the main research problems are to test differences between distributions of the two therapy groups and to detect treatment effects within each group. Testing the hypothesis of no difference between distributions of two therapy groups using the paired data is only one aspect of the comparisons between treatments. For example, in the context of the treatment study of ADHD and SMD, previously, Waxmonsky et al. [6] carried out a study to examine the tolerability and efficacy of methylphenidate (MPH) and behavior modification therapy, where multiple comparisons with the Bonferroni technique using the independent sample t-tests and the pairwise t-tests were conducted. The former tests were implemented to evaluate between-group differences in baseline characteristics and measures of tolerability (the Pittsburgh Side Effect Rating Scale). The latter ones were undertaken in the SMD group to compare pre-differences and post-differences in CDRS-Rts and Pittsburgh Side Effect Rating Scale from low dose MPH to high dose MPH. In this article, we avoid considerations of combined p-values, proposing a simple and efficient way to create nonparametric tests that attends to special alternative hypotheses directly, in an analogy to parametric likelihood ratio tests’ developments. Note that, in controlling the type I error, the used t-tests are known to be inefficient, when utilized data are skewed, and the applied Bonferroni method tends to be conservative. The nonparametric statistical analyses of two populations described above require to consider more versatile testing methods than those well addressed in the classic literature (e.g., [10, 11]). In this article, we propose and examine distribution-free tests for multiple hypotheses to detect various differences related to treatment effects in study groups based on paired data.

To formalize the testing problems, let (X_ij, Y_ij) be independent identically distributed (i.i.d.) pairs of observations within a subject j from sample i, where i = 1, 2 are referred to as treatments; j = 1, …, n_i are referred to as subjects. In the nonparametric setting, the classic one-sample tests for paired data, for example, the paired t-test and the Wilcoxon signed rank test, are based on differences Z_ij = Y_ij − X_ij, where Z_ij denotes a within-pair difference of subject j from sample i, i = 1, 2; j = 1, …, n_i. Note that {Z₁₁, …, Z_1n1} and {Z₂₁, …, Z_2n2} consist of i.i.d. observations from populations Z₁ and Z₂ with distribution functions, say, F_Z1(.) and F_Z2(.), respectively. In contexts of treatment evaluations, Z_ij can be defined to be the difference of measurements between pre- and post-treatment. In this article, we consider different hypotheses simultaneously for the symmetry of F_Z1 and/or F_Z2 (detecting a treatment effect into groups) and for the equivalence F_Z1 = F_Z2. Here, we refer to the nonparametric literature to connect the term ‘treatment effect’ with tests for symmetry (e.g., [10]). Note that the Kolmogorov–Smirnov test is a known procedure to compare distributions of populations, whereas the standard testing procedures such as the paired t-test, the sign test, and the Wilcoxon signed rank test can be applied to the symmetric problem, that is, to test for H₀ : F_Z(Z) = 1 − F_Z(−Z). Comparisons between distributions of new therapy and control groups and detecting treatment effects may be based on multiple hypotheses tests. To this end, one can create relevant tests combining, for example, the Kolmogorov–Smirnov test and the Wilcoxon signed rank test. The use of the classical procedures commonly requires complex considerations to combine the known nonparametric tests. Alternatively, we will develop a direct distribution-free method for analyzing the two-sample problems. The proposed method can be easily applied to test nonparametrically for different composite hypotheses. The proposed approach approximates nonparametrically most powerful Neyman–Pearson test-rules, providing efficiency of the proposed procedures.

When parametric forms of the relevant distributions are known, corresponding parametric likelihood ratios can be easily applied to test for the problems mentioned above. According to the Neyman–Pearson lemma, the parametric likelihood ratio tests are optimal decision rules (e.g., [11–13]). We propose to approximate corresponding likelihood ratios using an empirical likelihood (EL) concept. The EL methodology has been addressed in the statistical literature as one of powerful nonparametric techniques (e.g., [14–23]). The EL methodology allows researchers to use distribution-free procedures with efficient characteristics that are asymptotically close to those of related parametric likelihood approaches (e.g., [14]). The EL approach is developed via terms of cumulative distribution functions (e.g., [17, 24, 25]). Vexler and Yu [25] demonstrated that the classical EL method based on distribution functions is very suitable for testing parameters; however, the EL technique based on density functions performs more efficiently to test for distributions. To approximate Neyman–Pearson test statistics, Vexler and Gurevich [26, 27] proposed to focus on the density-based EL, $L_f={\prod }_{i=1}^n{f}_i$, where f_i = f(T_(i)), f(·) is an unknown density function of the observations {T₁, …, T_n} and T_(i) denotes the i^th ordered statistic based on {T₁, …, T_n}. In this case, approximate values of f_i are obtained by maximizing L_f subject to an empirical version of the constraint ∫ f(u)du = 1.

We extend and adapt the density-based EL approach for the two-sample testing issues carrying out multiple testing problems in paired data settings. Despite the fact that many statistical inference procedures have been developed for two-sample problems, to our knowledge, relevant nonparametric likelihood techniques to deal with the presented two-sample issues based on paired data have not been well addressed in the literature. The proposed density-based EL tests are exact, which ensure accurate computations of relevant p-values based on data with small sample sizes.

The paper is organized as follows. In Section 2, we address the purpose of each testing hypothesis considered in this article, and then we develop corresponding density-based EL ratio test statistics. The theoretical results will be presented to show the asymptotic consistency of the proposed tests. To evaluate the proposed approaches, extensive Monte Carlo studies are carried out in Section 3. An application to analyze the CDRS-Rts data is presented in Section 4. In Section 5, we provide some concluding remarks.

2. Statement of problems and Methods

2.1. Hypotheses setting

To test for equality of the distribution of the new therapy group and the control therapy group based on paired observations {Z₁₁, …, Z_1n1} and {Z₂₁, …, Z_1n2}, one may consider the hypotheses,

$$H_{N0}:F_{Z_1}=F_{Z_2}=F_Z\text{vs}{H}_A:F_{Z_1}\ne F_{Z_2}.$$

To incorporate evaluation of the treatment effect on each therapy group, we point out three tests related to the null hypothesis, (1) the equality of the distributions of two therapy groups, and (2) no treatment effect in each group. This can be presented by H₀ : 1) F_Z1 = F_Z2 = F_Z, and 2) F_Z(Z) = 1 − F_Z(−Z), for all Z ∈ (−∞, ∞). Against H₀, we can set up three different alternative hypotheses, namely H_A1, H_A2, and H_A3, where

1. H_A1: not H₀, that is, F_Z1 ≠ F_Z2 or F_Z1(Z₁) ≠ 1 − F_Z1(−Z₁) or F_Z2(Z₂) = 1 − F_Z2(Z₂);

2. H_A2: There is a treatment effect in one therapy group while there is no treatment effect in the other;

3. H_A3: One asserts that both therapy groups have the same treatment effect. In this case, because the distributions of two groups are assumed to be identical under H₀ and H_A3, a one-sample test for symmetry can be applied.

The cases (i)–(iii) are formally noted in Table I.

Table I.

Hypotheses of interest to be tested based on paired data.

Null hypothesis	versus	Alternative hypothesis
HN0 : FZ1 = FZ2 = FZ		HA : FZ1 ≠ FZ2
H0 : FZ1 = FZ2 = FZ;FZ(Z) = 1 − FZ(−Z), for all Z ∈ (−∞, ∞)		HA1 : FZ1 ≠ FZ2 or fZi(Zi) ≠ 1 − fZi (−Zi), for i = 1 or 2 (i.e., not H0)
		HA2 : FZ1 ≠ FZ2; FZ1(Z1) ≠ 1 − FZ1(−Z1); FZ2(Z2) = 1 − FZ2(−Z2)
		HA3 : FZ1 = FZ2 = FHA3, Z; FHA3,Z(Z) ≠ 1 − FHA3,Z(−Z)

Let Test 1, Test 2, and Test 3, refer to the hypothesis tests for the composite hypotheses H₀ versus H_A1, H₀ versus H_A2, and H₀ versus H_A3, respectively.

2.2. Test statistics

In this section, we develop test statistics for Tests 1–3. The proposed three tests will be shown to be exact.

2.2.1. Test 1: H₀ versus H_A1

Consider the scenario where one is interested to test for

$$H_0:F_{Z_1}=F_{Z_2}=F_Z;F_Z(Z)=1-F_Z(-Z),\text{for}\text{all}Z\in (-\infty ,\infty )\text{versus}{H}_{A1}.$$

The likelihood ratio test statistic based on observations, Z_ij, i = 1, 2; j = 1, …, n_i, is given by

$${LR}_{H_{A1}}=\frac{{\prod }_{j=1}^{n_1}{f}_{Z_1}(Z_{1j}){\prod }_{j=1}^{n_2}{f}_{Z_2}(Z_{2j})}{{\prod }_{j=1}^{n_1}{f}_Z(Z_{1j}){\prod }_{j=1}^{n_2}{f}_Z(Z_{2j})}=\prod _{j=1}^{n_1}\frac{f_{Z_1,j}}{f_{{ZZ}_1,j}}\prod _{j=1}^{n_1}\frac{f_{Z_{2,j}}}{f_{{ZZ}_{2,j}}},$$

where f_Zi, i = 1, 2, are density functions related to f_Zi, i = 1, 2; f_Z is a density function related to a symmetric distribution F_Z; f_Z1,j = F_Z1 (Z_1(j)), f_Z2,j = F_Z2 (Z_2(j)), f_ZZ1,j = f_Z(Z_1(j)), and f_ZZ2,j = f_Z(Z_2(j)) and Z₁₍₁₎ ≤ Z₁₍₂₎ ≤ ··· ≤ Z_1(n1), Z₂₍₁₎ ≤ Z₂₍₂₎ ≤ ··· ≤ Z_2(n2) are the order statistics based on {Z₁₁, …, Z_1n1} and {Z₂₁, …, Z_2n2}, respectively. The main novelty of the proposed method for developing the nonparametric test statistic is that we modify the maximum EL concept to obtain directly estimated values of f_Z1j, j = 1, … n₁, maximizing ${\prod }_{j=1}^{n_1}{f}_{Z_1,j}$ subject to an empirical constraint. This constraint controls estimated values of f_Z1,j, j = 1, … n₁, preserving the main property of the density function F_Z1 under the complex structure of the tested hypothesis. To obtain the associated empirical constraint, we utilize the fact that the values of f_Z1,j should be restricted by the equation ∫ F_Z1(u)du = 1. By applying the Mean Value Theorem to approximate the constraint ∫ F_Z1(u)du = 1 (for details, see [24–27]), for all positive integer m ≤ n/2, we have

$$\begin{array}{l}{(2m)}^{-1}\sum _{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du={(2m)}^{-1}\sum _{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)\frac{f_Z(u)}{f_Z(u)}du\\ \cong {(2m)}^{-1}\sum _{j=1}^{n_1}\frac{f_{Z_1,j}}{f_{{ZZ}_1,j}}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_Z(u)du\\ \cong {(2m)}^{-1}\sum _{j=1}^{n_1}\frac{f_{Z_1,j}}{f_{{ZZ}_1,j}}(F_Z(Z_{1_{(j+m)}})-F_Z(Z_{1_{(j-m)}})).\end{array}$$ (1)

Because under the null hypothesis H₀, the distribution function F_Z = F_Z1 = F_Z2 is assumed to be symmetric, the idea presented by Schuster [28] can be adapted to estimate (F_Z(Z_{1(j + m)}) − F_Z(Z_{1(j − m)})) at (1) by using the following estimator, which is denoted as η_m,j,

$${\eta }_{m,j}=(F_{n_1+n_2}(Z_{1_{(j+m)}})-F_{n_1+n_2}(Z_{1_{(j-m)}})),$$ (2)

where $F_{n_1+n_2}(u)=\frac{1}{2(n_1+n_2)}{\sum }_{i=1}^2{\sum }_{j=1}^{n_i}[I(Z_{ij}\le u)+I(-Z_{ij}\le u)]$, I(.) is the indicator function.

By virtue of Lemma 2.1 in [24] and Proposition 2.1 in [26], we have that, for all integer m ≤ 0.5n₁,

$${(2m)}^{-1}\sum _{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du={\int }_{Z_{1_{(1)}}}^{Z_{1_{(n_1)}}}{f}_{Z_1}(u)du-\sum _{r=1}^{m-1}\frac{(m-r)}{2m}\times \left[{\int }_{Z_{1_{(n_1-r)}}}^{Z_{1_{(n_1-r+1)}}}{f}_{Z_1}(u)du+{\int }_{Z_{1_{(r)}}}^{Z_{1_{(r+1)}}}{f}_{Z_1}(u)du\right],$$ (3)

where Z_{1(j − m)} = Z₁₍₁₎, if j − m ≤ 1 and Z_{1(j + m)} = Z_1(n1), if j + m ≥ n₁.

Because

$${\int }_{Z_{1_{(1)}}}^{Z_{1_{(n_1)}}}{f}_{Z_1}(u)du\le {\int }_{-\infty }^{\infty }f(u)du=1,$$

Equation (3) demonstrates that ${(2m)}^{-1}{\sum }_{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du\le 1$, and ${(2m)}^{-1}{\sum }_{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du\approx 1$ when m/n₁ → 0 as m, n₁ → ∞.

By replacing the distribution functions in (3) by their empirical counterparts, $F_{n_1}(u)=n_1^{-1}{\sum }_{j=1}^{n_1}I(Z_{1j}\le u)$, the empirical version of the Equation (3) then has the form of

$${(2m)}^{-1}\sum _{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du\cong F_{n_1}\left(Z_{1_{(n_1)}}\right)-F_{n_1}(Z_{1_{(1)}})-\sum _{r=1}^{m-1}\frac{(m-r)}{2m}\left[F_{n_1}\left(Z_{1_{(n_1-r+1)}}\right)-F_{n_1}\left(Z_{1_{(n_1-r)}}\right)+F_{n_1}(Z_{1_{(r+1)}})-F_{n_1}(Z_{1_{(r)}})\right].$$ (4)

This leads to

$${(2m)}^{-1}{\sum }_{j=1}^{n_1}{\int }_{Z_{1_{(j-m)}}}^{Z_{1_{(j+m)}}}{f}_{Z_1}(u)du\cong 1-(m+1){(2n_1)}^{-1}.$$ (5)

Now, by the Equations (1), (2), and (5), the resulting empirical constraint for values of f_Z1,j is

$${(2m)}^{-1}{\sum }_{j=1}^{n_1}\frac{f_{Z_1,j}}{f_{{ZZ}_1,j}}{\eta }_{m,j}=1-(m+1){(2n_1)}^{-1}.$$ (6)

To find values of f_Z1,j that maximize the likelihood ${\prod }_{j=1}^{n_1}{f}_{Z_1,j}$ provided the condition (6), we formalize the Lagrange function as

$$\sum _{j=1}^{n_1}log{f}_{Z_1,j}+{\lambda }_1\left[1-(m+1){(2n_1)}^{-1}-{(2m)}^{-1}\sum _{j=1}^{n_1}\frac{f_{Z_1,j}}{F_{{ZZ}_1,j}}{\eta }_{m,j}\right],$$

where λ₁ is a Lagrange multiplier. Maximizing the equation above, the values of f_Z1,j, … f_Z1,n1 have the form of

$$f_{Z_1,j}=\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}}{f}_{{ZZ}_1,j},j=1,\dots ,n_1,$$

where Z_{1(j − m)} = Z₁₍₁₎, if j − m ≤ 1 and Z_{1(j + m)} = Z_1(n1), if j + m ≥ n₁.

As a consequence, the density-based EL estimator of the ratio ${\prod }_{j=1}^{n_1}{f}_{Z_1,j}/f_{{ZZ}_1,j}$ can be formulated by

$$V_{1,m,1}=\prod _{j=1}^{n_1}\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}}.$$

One can show that properties of the statistic V_1,m,1 strongly depend on the selection of values of the integer parameter m. A similar problem also arose in the well-known goodness-of-fit tests based on sample entropy (e.g., [24,26,29]). Attending to this issue, we eliminate the dependence on the integer parameter m. Towards this end, we utilize the maximum EL concept in a similar manner to arguments proposed in [24, 26, 27:Appendix A]. Thus, the modified test statistic can be written as

$$V_{1,1}=\underset{{n_1}^{0.5+\delta }\le m\le {n_1}^{1-\delta }}{min}\prod _{j=1}^{n_1}\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}},\delta \in (0,1/4),$$ (7)

Likewise, the approximation to the likelihood ratio ${\prod }_{j=1}^{n_2}{f}_{Z_2,j}/f_{{ZZ}_2,j}$ is

$$V_{2,2}=\underset{{n_2}^{0.5+\delta }\le k\le {n_2}^{1-\delta }}{min}\prod _{j=1}^{n_2}\frac{k(2n_2-k-1)}{n_2^2{\varphi }_{k,j}},\delta \in (0,1/4).$$ (8)

where

$${\varphi }_{k,j}=(F_{n_1+n_2}(Z_{2_{(j+k)}})-F_{n_1+n_2}(Z_{2_{(j-k)}})),F_{n_1+n_2}\text{is}\text{defined}\text{in}(2).$$

Finally, the proposed test statistic for Test 1 has the form of

$$V_{n_1{n}_2}^{H_{A1}}=\prod _{i=1}^2{V}_{i,i}=\underset{{n_1}^{0.5+\delta }\le m\le {n_1}^{1-\delta }}{min}\prod _{j=1}^{n_1}\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}}\underset{{n_2}^{0.5+\delta }\le k\le {n_2}^{1-\delta }}{min}\prod _{j=1}^{n_2}\frac{k(2n_2-k-1)}{n_2^2{\varphi }_{k,j}},$$

that approximates the likelihood ratio LR_HA1. Consequently, the decision rule is to reject H₀ if

$$log\left(V_{n_1{n}_2}^{H_{A1}}\right)>C^{H_{A1}},$$ (9)

where C^H_A1 is a test threshold. (Similarly to [30], we will arbitrarily define η_m,j = 1/(n₁ + n₂) or ϕ_k,j = 1/(n₁ + n₂), if η_m,j = 0 or ϕ_k,j = 0, respectively.) Proposition 1 in Section 2.3 will demonstrate that the proposed test-statistic $log\left(V_{n_1{n}_2}^{H_{A1}}\right)$ in (9) is asymptotically consistent. The upper and lower bounds for the integer parameters m and k in definitions of (7) and (8) were selected to provide the asymptotic consistency. Note that, to test the composite hypotheses H₀ versus H_A1, a complex consideration regarding a reasonable combination of the Kolmogorov–Smirnov test and the Wilcoxon signed rank test can be applied (see, for example, Section 3.2). Alternatively, the test (9) uses measurements from the therapy groups, in an approximate Neyman–Person manner, providing a simple procedure to evaluate the treatment effect on each therapy group. Section 3 shows, in various situations, the test (9) is superior to the combinations of the classic tests based on Kolmogorov–Smirnov and Wilcoxon procedures. It is also shown that the proposed nonparametric test has power comparable with that of correct parametric likelihood ratio tests. Thus, in contexts of the study described in Section 1, the direct application of the density-based EL ratio test (9) provides an efficient evaluation of treatment effects with ADHD and SMD in children.

2.2.2. Test 2: H₀ versus H_A2

Our goal is to test for

$$\begin{array}{c}{H}_0:F_{Z_1}=F_{Z_2}=F_Z,F_Z(Z)=1-F_Z(-Z),\text{for}\text{all}Z\in (-\infty ,\infty ),\\ \text{versus}{H}_{A2}:F_{Z_1}\ne F_{Z_2},F_{Z_1}(Z_1)\ne 1-F_{Z_1}(-Z_1),F_{Z_2}(Z_2)=1-F_{Z_2}(-Z_2).\end{array}$$

In a similar manner to the development of the density-based EL approximation to the ratio ${\prod }_{j=1}^{n_1}{f}_{Z_1,j}/f_{{ZZ}_1,j}$ mentioned in Section 2.2.1, the EL ratio related to test for H₀ versus H_A2 can be defined as

$$V_{1,1}=\underset{{n_1}^{0.5+\delta }\le m\le {n_1}^{1-\delta }}{min}\prod _{j=1}^{n_1}\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}},\delta \in (0,1/4),$$ (10)

where η_m,j are defined in (2). Consider the density-based EL approximation to the corresponding ratio ${\prod }_{j=1}^{n_2}{f}_{Z_2,j}/f_{{ZZ}_2,j}$. The empirical constraint for values of f_Z2,j can be constructed based on the symmetric property of F_Z2. By analogy with Equations (1)–(6), one can show the resulting empirical constraint on values of f_Z2,j in Test 2 has the form of

$${(2k)}^{-1}\sum _{j=1}^{n_2}\frac{f_{Z_2,j}}{f_{{ZZ}_2,j}}{\varphi }_{k,j}={\mathrm{\Lambda }}_{n_2}^k,$$ (11)

where ϕ_k,j are defined in (8) and

$$\begin{array}{l}{\mathrm{\Lambda }}_{n_2}^k={(2n_2)}^{-1}\{\prod _{j=1}^{n_2}\left[I(-Z_{2j}\le Z_{2_{(n_2)}})-I(-Z_{2j}\le Z_{2_{(1)}})\right]+n_2-1\\ -\sum _{r=1}^{k-1}\frac{(k-r)}{2k}\sum _{j=1}^{n_2}[I(-Z_{2j}\le Z_{2_{(n_2-r+1)}})-I(-Z_{2j}\le Z_{2_{(n_2-r)}})\\ +I(-Z_{2j}\le Z_{2_{(r+1)}})-I(-Z_{2j}\le Z_{2_{(r)}})]-\frac{(k-1)}{2}\}.\end{array}$$ (12)

The formal derivation of this constraint is given in Appendix A of this article. Then the corresponding Lagrange function can be formulated by

$$\sum _{j=1}^{n_2}log{f}_{Z_2,j}+{\lambda }_2\left[{\mathrm{\Lambda }}_{n_2}^k-{(2k)}^{-1}\sum _{j=1}^{n_2}\frac{f_{Z_2,j}}{f_{{ZZ}_2,j}}{\varphi }_{k,j}\right],$$ (13)

where λ₂ is a Lagrange multiplier. Thus, approximate values of f_{Z2, 1}, …, f_{Z2, n2} are

$$f_{Z_2,j}=\frac{2k{\mathrm{\Lambda }}_{n_2}^k}{n_2{{\varphi }_{k,}}_j}{f}_{{ZZ}_2,j},j=1,\dots ,n_2,$$

where Z_{2(j − m)} = Z₂₍₁₎, if j − k ≤ 1 and Z_{2(j + k)} = Z_2(n2), if j + k ≥ n₂.

Similarly to (7) and (8), the density-based EL estimator of the ratio ${\prod }_{j=1}^{n_2}{f}_{Z_2,j}/f_{{ZZ}_2,j}$ can be presented as

$${\stackrel{\sim }{V}}_{2,2}=\underset{{n_2}^{0.5+\delta }\le k\le {n_2}^{1-\delta }}{min}{\prod }_{j=1}^{n_2}\frac{2k{\mathrm{\Lambda }}_{n_2}^k}{n_2{\varphi }_{k,j}},\delta \in (0,1/4).$$ (14)

Finally, taking into account (10) and (14), the proposed test statistic for Test 2 can be constructed as

$$V_{n_1{n}_2}^{H_{A2}}=\underset{{n_1}^{0.5+\delta }\le m\le {n_1}^{1-\delta }}{min}\prod _{j=1}^{n_1}\frac{m(2n_1-m-1)}{n_1^2{\eta }_{m,j}}\underset{{n_2}^{0.5+\delta }\le k\le {n_2}^{1-\delta }}{min}\prod _{j=1}^{n_2}\frac{2k{\mathrm{\Lambda }}_{n_2}^k}{n_2{\varphi }_{k,j}}.$$

In this case, the decision rule developed for Test 2 is to reject the null hypothesis if

$$log(V_{n_1{n}_2}^{H_{A2}})>C^{H_{A}2},$$ (15)

where C^H_A2 is a test threshold.

2.2.3. Test 3: H₀ versus H_A3

Consider the following hypotheses of interest

$$\begin{array}{c}{H}_0:F_{Z_1}=F_{Z_2}=F_Z,F_Z(Z)=1-F_Z(-Z),\text{for}\text{all}Z\in (-\infty ,\infty ),\\ \text{versus}{H}_{A3}:F_{Z_1}=F_{Z_2},=F_{H_{A3},Z},F_{H_{A3},Z}(Z)\ne 1-F_{H_{A3},Z}(-Z).\end{array}$$

The corresponding likelihood ratio test statistic based on observations Z_ij, i = 1, 2; j = 1, , , n_i can be

$${LR}_{H_{A3}}=\frac{{\prod }_{i=1}^2{\prod }_{j=1}^{n_i}{f}_{H_{A3},Z}(Z_{ij})}{{\prod }_{i=1}^2{\prod }_{j=1}^{n_i}{f}_Z(Z_{ij})}=\frac{{\prod }_{s=1}^N{f}_{H_{A3},Z}(Z_{(s)})}{{\prod }_{s=1}^N{f}_Z(Z_{(s)})}=\prod _{s=1}^N\frac{f_{H_{A3},s}}{f_{H_0,s}},$$

where N = n₁ + n₂; f_{H0, s} = f_Z(Z_(s)) and f_{HA3, s} = f_HA3,Z(Z_(s)) denote the density functions of observations Z under H₀ and H_A3, respectively, and Z_(s)s = 1, …, N, are the order statistics based on the pooled sample of {Z_11, …, Z_1n1} and {Z_21, …, Z_2n2} that are denoted by Z_s,s = 1, …, N. Using the same technique as in Section 2.2.1, we derive values of f_HA3,s, s = 1, …, N, that maximize the log-likelihood ${\sum }_{s=1}^{N}log(f_{H_{A3},s})$ given a constraint, empirical form of ∫ f_HA3(u)du = 1. The proposed test statistic for Test 3 is

$$V_N^{H_{A3}}=\underset{N^{0.5+\delta }\le m\le N^{1-\delta }}{min}\prod _{s=1}^N\frac{m(2N-m-1)}{N^2{\omega }_{m,s}},$$ (16)

where ${\omega }_{m,s}={(2N)}^{-1}{\sum }_{s=1}^N[I(Z_s\le Z_{(s+m)})+I(-Z_s\le Z_{(s+m)})-I(Z_s\le Z_{(s-m)})-I(-Z_s\le Z_{(s-m)})]$ and δ ∈ (0, 1/4).

Thus, the decision rule for Test 3 is to reject the null hypothesis if

$$log(V_N^{H_{A3}})>C^{H_{A3}},$$ (17)

where C^H_A3 is a test threshold.

2.3. Asymptotic consistency of the tests

In this section, we present the following propositions to demonstrate the asymptotic consistency of the proposed tests:

Proposition 1

Let f_Zi(Z) be the density function with the expectations E(log f_Zi(Z_i1)) < ∞ and E(log f_Zi(Z_i1)) < ∞, i = 1, 2. Let f_i(u) = (f_Zi(u) + f_Zi(−u)/2. Then, under $H_0,{(n_1+n_2)}^{-1}log(V_{n_1{n}_2}^{H_{At}})\stackrel{p}{\to }0$, t = 1, 2, and under H_A1 and H_A2,

$${(n_1+n_2)}^{-1}log(V_{n_1{n}_2}^{H_{At}})\stackrel{p}{\to }-\frac{\gamma }{1+\gamma }{E}_{H_{At}}log\left\{\frac{\gamma }{1+\gamma }+\frac{1}{1+\gamma }\left(\frac{f_2(Z_{11})}{f_1(Z_{11})}\right)\right\}-\frac{1}{1+\gamma }{E}_{H_{At}}log\left\{\frac{1}{1+\gamma }+\frac{\gamma }{1+\gamma }\left(\frac{f_1(Z_{21})}{f_2(Z_{21})}\right)\right\}\ge 0,t=1,2,$$

as n₁ → ∞, n₂ → ∞, n₁/n₂ → γ > 0, where γ is a constant.

Proof

We outline the proof in Appendix A1 provided in Supporting material (S1)^‡ (ref. [31] is cited in the proof).

Consider the testing problem H₀ versus H_A3.

Proposition 2

Let the pooled sample Z_s,s = 1, …, N, have the density function, f_(Z), with the expectations E(log f(Z₁)) < ∞ and E(log f(−Z₁)) < ∞. Then under $H_0,N^{-1}log(V_N^{H_{A3}})\stackrel{p}{\to }0$, and under $H_{A3},N^{-1}log(V_N^{H_{A3}})\stackrel{p}{\to }-E_{H_{A3}}log\left\{0.5+0.5\left(\frac{f(-Z_1)}{f(Z_1)}\right)\right\}\ge 0$, as N → ∞

Proof

We omit the proof, since it is similar to the proof of Proposition 1.

2.4. Null distributions of the proposed test statistics

To obtain critical values of the proposed tests, we utilize the fact that the proposed test statistics are based on indicator functions I(.) and I(Z₁ < Z₂) = I(F_Z(Z₁) < F_Z(Z₂)) and I(Z₁ < −Z₂) = I(F_Z(Z₁) < F_Z(−Z₂)) = I(F_Z(Z₁) < 1 − F_Z(Z₂)), where the random variables F_Z(Z₁) and F_Z(Z₂) have the uniform distribution, Unif [0,1], under H₀. Thus, the distributions of the proposed test statistics are independent of the distributions of observations and hence, the critical values of the proposed tests can be exactly computed. For each proposed test, we conducted the following procedures to determine the critical values, C_α, of the null distributions. We first generated data of Z₁ and Z₂ from the standard normal distribution N(0, 1) and then calculated the test statistics corresponding to each proposed test. At each sample sizes n₁ and n₂, we obtained 50,000 generated values of the test statistics (9), (15), and (17), with δ = 0.1, tabulating the critical values for the null distributions of the test statistics at the significance levels α, α = 0.01; 0.05; 0.1 (see Table II).

Table II.

The critical values for Test 1 by (9) (Test 2 by (15)) [Test 3 by (17)] with δ = 0.1 for different sample sizes (n₁, n₂) and significance levels α.

n1	α	n2
		10	15	20	25	30	35
10	0.01	7.44 (5.75) [4.17]	7.18 (5.85) [4.18]	7.39 (6.02) [4.18]	7.33 (6.17) [4.16]	7.45 (6.33) [4.25]	7.47 (6.20) [4.42]
	0.05	5.22 (3.86) [2.68]	5.06 (3.93) [2.82]	5.25 (4.09) [2.84]	5.35 (4.28) [2.85]	5.38 (4.37) [2.97]	5.42 (4.43) [3.11]
	0.1	4.27 (3.11) [2.14]	4.19 (3.17) [2.27]	4.39 (3.35) [2.30]	4.48 (3.52) [2.36]	4.56 (3.60) [2.46]	4.59 (3.68) [2.60]
15	0.01		6.83 (5.39) [4.15]	6.96 (5.57) [4.16]	6.98 (5.68) [4.24]	6.88 (5.80) [4.33]	6.96 (5.81) [4.39]
	0.05		4.90 (3.73) [2.83]	5.02 (3.89) [2.85]	5.15 (4.04) [2.97]	5.14 (4.11) [3.12]	5.20 (4.18) [3.13]
	0.1		4.09 (3.07) [2.30]	4.25 (3.24) [2.34]	4.38 (3.38) [2.45]	4.36 (3.45) [2.59]	4.44 (3.52) [2.63]
20	0.01			6.86 (5.44) [4.24]	6.96 (5.55) [4.38]	6.99 (5.69) [4.41]	7.01 (5.73) [4.42]
	0.05			5.11 (3.94) [2.96]	5.24 (4.11) [3.11]	5.25 (4.19) [3.16]	5.27 (4.26) [3.19]
	0.1			4.35 (3.33) [2.45]	4.48 (3.47) [2.58]	4.51 (3.55) [2.64]	4.52 (3.62) [2.69]
25	0.01				7.08 (5.62) [4.40]	7.01 (5.76) [4.44]	7.10 (5.85) [4.56]
	0.05				5.33 (4.18) [3.15]	5.36 (4.25) [3.22]	5.37 (4.32) [3.34]
	0.1				4.56 (3.55) [2.64]	4.59 (3.63) [2.70]	4.62 (3.69) [2.82]
30	0.01					6.89 (5.63) [4.55]	6.99 (5.75) [4.68]
	0.05					5.31 (4.23) [3.32]	5.33 (4.27) [3.44]
	0.1					4.59 (3.65) [2.80]	4.61 (3.68) [2.93]
35	0.01						6.91 (5.68) [4.64]
	0.05						5.35 (4.28) [3.47]
	0.1						4.64 (3.69) [2.98]

Remark 1

The definitions (9), (15), and (17) of the proposed test statistic include δ ∈ (0, 1/4). We set up δ = 0.1. To investigate the test statistics with different values of δ, we conducted an extensive Monte Carlo study. The Monte Carlo powers of the proposed tests were not found to be significantly dependent on values of δ ∈ (0, 1/4). These experimental results are similar to those shown in [24,25] and [27]: Appendix A.

Remark 2

The computer codes related to the outputs of this section are provided in the Supplemental Section S2. These codes can be easily modified to obtain results of the next section or to perform the proposed tests based on real data.

3. Simulation study

In this section, we examine the power properties of the proposed tests in various cases using Monte Carlo simulations. The proposed tests based on (9), (15), and (17), with δ = 0,1, are compared with the common test procedures: the maximum likelihood ratio (MLR) tests, assuming parametric conditions on distributions of observations (for details of the constructions and definitions of the MLR tests, see Appendix A2 of the supporting information); combined classic nonparametric tests with a structure based on the Wilcoxon signed rank test or/and the Kolmogorov–Smirnov test. We fixed the significance level of the tests to be 0.05 in all considered cases.

3.1. Power comparison with the parametric method (the MLR tests)

To present the comparative power of the proposed tests versus the corresponding MLR tests, we performed the following Monte Carlo study. Critical values of the MLR test statistics were obtained based on 50,000 simulations under H₀ based on N(0, 1)-distributed observations Z. To study the powers of the tests, 10,000 samples for each size (n₁, n₂) were generated from a variety of distributions. Tables III–V depict the Monte Carlo powers of the proposed tests and those of the corresponding MLR tests.

Table III.

The Monte Carlo powers of Test 1 by (9) versus the MLR test for H₀ versus H_A1 with different sample sizes (n₁, n₂) at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FX<sub>2</sub>	FY<sub>2</sub>	n1	n2	Proposed test at (9)	MLR
N(0, 1)	N(0.2, 0.252)	N(0.1, 0.52)	N(0.5, 1)
				10	10	0.1541	0.1579
				50	50	0.6232	0.6921
N(2.5, 0.82)	N(1.5, 0.52)	N(1, 1.52)	N(1.5, 0.62)
				10	10	0.7267	0.7351
				25	25	0.9956	0.9978
N(0.3, 0.52)	N(0.5, 1)	N(0.25, 0.252)	N(0.5, 0.52)
				10	10	0.2818	0.4497
				50	50	0.9764	0.9993
N(0.5, 0.52)	N(1, 1)	N(0, 1)	N(0, 1)
				10	10	0.1723	0.1764
				50	50	0.7171	0.7942
N(0, 1)	N(0, 1)	N(1.5, 1.12)	N(1, 1.32)
				10	10	0.1125	0.1300
				50	50	0.4321	0.5531
N(0, 1)	N(0.5, 1)	N(0.5, 1.22)	N(1, 0.52)
				10	10	0.2208	0.2230
				50	50	0.8348	0.8785

Table V.

The Monte Carlo powers of Test 3 by (17) versus the MLR test for H₀ versus H_A3 with different sample sizes (n₁, n₂) at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FX<sub>2</sub>	FY<sub>2</sub>	n1	n2	Proposed test at (17)	MLR
N(1, 1)	N(1.5, 1.52)	N(1, 1)	N(1.5, 1.52)
				10	10	0.2058	0.2191
				50	50	0.6709	0.7919
N(0, 0.52)	N(0.6, 1)	N(0, 0.52)	N(0.6, 1)
				10	10	0.5922	0.6377
				50	50	0.9974	0.9996
N(0.5, 0.25)2	N(1, 1)	N(0.5, 0.252)	N(1, 1)
				10	10	0.5072	0.5444
				50	50	0.9869	0.9977
N(2.5, 1.252)	N(2, 0.52)	N(2.5, 1.252)	N(2, 0.52)
				10	10	0.3383	0.3509
				50	50	0.9004	0.9582

When observations are normally distributed, as anticipated, the MLR tests would be more powerful than the proposed nonparametric tests. The tables show that the powers of the proposed tests are very close to those of the MLR tests, demonstrating that the density-based EL tests are comparable to the parametric method that utilizes the correct information regarding distributions of observations. Table VI displays the actual type I errors of the MLR tests under the misspecification of underlying distributions, i.e., when observations were simulated from t distributions with different degrees of freedoms (DOFs). a logistic with parameters (0,1), a Laplace with parameters (0,1), and the Unif [0,1] under H₀. As can be seen from Table VI, the type I errors of the MLR tests for H₀ versus H_A1 and H₀ versus H_A2 are not under control until the DOFs of the t distribution < 200. For the cases of the logistic and the Laplace distributions, the type I errors of the MLR tests are not well controlled. When the observations are from Unif [0,1], the impact of the misspecification of the model on the type I errors of the MLR tests is more significant. This illustrates that the considered MLR tests are strongly dependent on assumptions regarding distributions of observations.

Table VI.

The Monte Carlo type I errors of the MLR tests.

FZ<sub>1</sub>	FZ<sub>2</sub>	n1	n2	MLR test for H0 versus HA1	MLR test for H0 versus HA2	MLR test for H0 versus HA3
t3	t3
		10	10	0.1599	0.1838	0.0428
		50	50	0.2934	0.3260	0.0465
t5	t5
		10	10	0.0955	0.1066	0.0458
		50	50	0.1447	0.1687	0.0488
t200	t200
		10	10	0.0507	0.0493	0.0499
		50	50	0.0503	0.0506	0.0502
Logistic(0,1)	Logistic(0,1)
		10	10	0.0717	0.0759	0.0463
		50	50	0.0855	0.0968	0.0508
Laplace(0,1)	Laplace(0,1)
		10	10	0.1108	0.1293	0.0438
		50	50	0.1446	0.1645	0.0488
Unif[0, 1]	Unif[0, 1]
		10	10	1	0.9993	1

3.2. Power comparison with classic nonparametric methods

In this section, we compare the power of the proposed tests to the power of procedures based on the classic nonparametric tests. Because Tests 1 and 2 are based on the composite hypotheses regarding between-group differences and treatment effects, the respective Kolmogorov–Smirnov test and Wilcoxon signed rank test cannot be directly applied to test for these hypotheses. In this case, one can perform combined tests based on the Kolmogorov–Smirnov test and the Wilcoxon signed rank test for H₀ versus H_A1 and H₀ versus H_A2. For the comparison, we used combined nonparametric tests with the Bonferroni method. To this end, the R procedure ‘p.adjust’ with the method ‘bonferroni’ was utilized. Let ‘W-test’ denote the Wilcoxon signed rank test and ‘K–S test’ denote the Kolmogorov–Smirnov test. The combined nonparametric test for H₀ versus H_A1 consists of two W-tests for symmetry and one K–S test based on Z₁, Z₂ for F_Z1 = F_Z2. The former tests are employed to assess a treatment effect of each therapy group, whereas the latter test is conducted to detect the group difference. Similarly, we performed the combined nonparametric test for H₀ versus H_A2 that includes one W-test and one K–S test. The classical procedure for H₀ versus H_A3 is the W-test for symmetry.

To test H₀ versus H_A1 and H₀ versus H_A3, we assigned different distributions for baseline measurements X and endpoint observations Y in each group under the alternative hypothesis (i.e., (F_X1, F_Y1) versus (F_X2, F_Y2)), whereas when testing for H₀ versus H_A2, we directly generated observations Z₂ from three cases of symmetric distributions under H_A2: N(0, 1); Unif [−1, 13; t₂ distribution. Tables VII–IX contain the results of the power comparisons of the two different testing procedures: the proposed procedures and the nonparametric testing procedures based on the W-test and/or K–S test using the Bonferroni approach.

Table VII.

The Monte Carlo powers of the proposed test (9) versus the combined nonparametric test (the two Wilcoxon signed rank tests and one Kolmogorov–Smirnov test) at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FX<sub>2</sub>	FY<sub>2</sub>	n1	n2	Proposed test at (9)	W and K–S test
Exp(1)	Lognorm(0, 22)	N(0, 1)	N(0.5, 1.52)
				10	10	0.2238	0.0946
				50	50	0.9616	0.6273
Lognorm(1, 1)	Lognorm(1, 0.52)	N(0,1)	N(1.5, 22)
				10	10	0.3646	0.2754
				50	50	0.9953	0.9849
Exp(3)	Lognorm(0, 22)	Gamma(5,1)	Gamma(1, 5)
				10	10	0.6321	0.5016
				50	50	1	1
${\chi }_{(6)}^2$	Gamma(1,10)	N(0,1)	N(0.5, 22)
				10	10	0.3815	0.1179
				50	50	1	0.8576
Exp(1)	Cauchy(1,1)	N(0.5, 1)	N(1.5, 22)
				10	10	0.1819	0.1255
				50	50	0.7928	0.7426
Exp(1)	Lognorm(0, 22)	Unif [−1, 1]	Unif [−1, 1]
				10	10	0.2325	0.0836
				50	50	0.9981	0.6939

Table IX.

The Monte Carlo powers of the proposed test (17) versus the Wilcoxon signed rank test at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FX<sub>2</sub>	FY<sub>2</sub>	n1	n2	Proposed test at (17)	W test
Exp(1)	Lognorm(0, 22)	Exp(1)	Lognorm(0, 22)
				10	10	0.4136	0.2933
				50	50	0.9992	0.9223
Lognorm(1, 1)	Lognorm(1, 0.52)	Lognorm(1, 1)	Lognorm(1, 0.52)
				10	10	0.1218	0.0731
				50	50	0.7906	0.2208
Gamma(5,1)	Gamma(1, 5)	Gamma(5,1)	Gamma(1, 5)
				10	10	0.1218	0.0886
				50	50	0.8074	0.2294
${\chi }_{(6)}^2$	Gamma(1,10)	${\chi }_{(6)}^2$	Gamma(1,10)
				10	10	0.3125	0.2344
				50	50	0.9629	0.8517
Beta(0, 0.8)	Exp(1.5)	Beta(0, 0.8)	Exp(1.5)
				10	10	0.2094	0.1518
				50	50	0.9928	0.6003

The Monte Carlo outputs shown in Tables VII–IX indicate that the new tests have higher powers against the combined nonparametric tests. In particular, for the cases of small sample sizes (e.g., (n₁, n₂) = (10, 10), (25, 25)), the proposed tests are significantly superior to the classic tests. In several cases, the powers of the proposed tests have values that are 3–4 times larger than those of the combined nonparametric tests.

4. Data analysis

In this section, we apply the proposed method to the study described in Section 1, which evaluates treatment effects of ADHD and SMD in children. Study subjects were randomized to receive either the experimental 11-week group therapy program (n₁ = 17) or community psychosocial treatment (n₂ = 15). We defined the former as group 1 and the latter as group 2. For each child enrolled in the study, CDRS-Rts was taken at the baseline (week 0) and endpoint (week 11). Specifically, we computed the differences of CDRS-Rts: Z_ij = Y_ij − X_ij, for i = 1, 2; j = 1, …, n_i, where X_ij stands for the CDRS-Rts assessed at baseline before subject j receives treatment i and Y_ij represents another CDRS-Rts at the endpoint after subject j receives treatment i.

The empirical histograms of the CDRS-Rts at baseline and endpoint for each group are shown in Figure 1. As can be seen from Figure 1, it appears that both therapy groups have a decline in the CDRS-Rts after the baseline but the decrease in the CDRS-Rts seems to be more significant in group 1.

Figure 1.

Histograms of CDRS-Rts related to the baseline and endpoint in group 1: X₁, Y₁ and in group 2: X₂, Y₂.

In the context of the study’s interest to test a claim that the distributions of the changes in CDRS-Rts are not equivalent with respect to the therapy groups or at least one therapy group has a treatment effect, we performed the proposed Test 1. In this case, the observed value of the test statistic by (9), with δ = 0.1, is 22.8217 and the corresponding p-value is 0.00002, indicating the null hypothesis of ‘no group differences and the lack of treatment effects in both groups’ is rejected. The combined non-parametric test (the two ‘W-tests’ and one ‘K–S test’) also rejects the null hypothesis with the p-value 0.000005. On the basis of these results, there is a strong evidence to reject the null hypothesis.

In addition, to demonstrate applicability of the proposed tests, we carried out Test 2, which might be appropriate to test an assertion that there is a treatment effect in one group and no such effect in the other besides a group difference. The observed value of the test statistic by (15), with δ = 0.1, is 11.9370 and the corresponding p-value is 4 × 10⁻⁵. The combined nonparametric test (the ‘W-test’ and one ‘K-S test’) with the Bonferroni method also supports the result to reject the null hypothesis with the p-value of 5 × 10⁻⁷.

These results show that the proposed procedures are in conjunction with the classic tests, demonstrating that our proposed tests can be utilized in the ADHD and SMD study.

In addition to the analysis above, we also conducted a bootstrap type study to evaluate the efficiency (power) of the considered tests based on small datasets randomly selected from the original data. To perform this study for Test 1, we executed the following procedure. We randomly selected samples with the sample sizes of (n₁, n₂) = (9, 6), (9, 7), (11, 9), (13, 10), (13, 11), (15, 13) from the original dataset. Then we calculated the corresponding test statistic $log(V_{n_1{n}_2}^{H_{A1}})$ by (9), where δ = 0.1. We repeated this strategy 10,000 times calculating the proportion of rejections at α = 0.05 of the null hypothesis; that is, we computed the percentage of times when $log\left(V_{n_1{n}_2}^{H_{A1}}\right)>C_{\alpha =0.05}$. The bootstrap type study for Test 2 was also carried out following the same procedures as described above. The results regarding the proportion of the rejections of the null hypothesis for each considered test are provided in Table X.

Table X.

The proportions of rejections^a based on the bootstrap method for each considered test.

Bootstrapped sample sizes	Test 1		Test 2
(n1, n2)	Proposed test (9)	Classic testb	Proposed test (15)	Classic testc
(9, 6)	0.9858	0.7135	0.9755	0.9134
(9, 7)	0.9870	0.7172	0.9800	0.9165
(11, 9)	0.9989	0.9795	0.9955	0.9844
(13, 10)	0.9998	0.9962	0.9993	0.9972
(13, 11)	0.9999	0.9965	0.9997	0.9975
(15, 13)	1	0.9997	0.9999	0.9994

^aThe proportion of rejections of each test from the bootstrap method was computed based on sample sizes (n₁, n₂) and 10,000 replications.

^bThe combined classic test for H₀ versus H_A1 is based on two W-tests and one K–S test.

^cThe combined classic test for H₀ versus H_A2 is based on one W-test and one K–S test.

Table X demonstrates that the proposed procedures have larger proportion of the rejections in comparison with the combined nonparametric tests. In particular, when the sample sizes are relatively small (e.g., (n₁, n₂) = (9, 6), (9, 7)), the differences in the proportions of the rejections between two approaches are strongly recognizable. For example, we selected a sample of size 9 from the group 1 and a sample of size 6 from the group 2. This subdataset was tested for the hypotheses H₀ versus H_A1 (Test 1). In contrast with the result that the nonparametric test based on the two ‘W-tests’ and one ‘K–S test’ for H₀ versus H_A1 is not statistically significant (the Bonferroni adjusted p-values of these classic tests are 0.0617, 0.1050, and 0.9873, respectively), the proposed Test 1, with δ = 0.1, is statistically significant (p-value= 0.0005). Figure 2 shows the empirical histograms of Z₁ and Z₂ from the subdataset. All these results indicate that the proposed methods for tests 1 and 2 are more sensitive to detect the difference between the null hypothesis and the alternative hypotheses involved in Tests 1 and 2 compared with the corresponding combined nonparametric tests.

Figure 2.

Histograms of the paired observations Z₁ and Z₂ based on the CDRS-Rts data, with sample sizes (n₁, n₂) = (9, 6), that were sampled from the original data set.

5. Concluding remarks

In this article, we proposed and examined the two-sample density-based EL ratio tests based on paired observations. While constructing the tests, we used approximations to the most powerful test statistics with respect to the stated problems, providing efficient nonparametric procedures. The proposed tests are shown to be exact and simple in performing. The extensive Monte Carlo studies confirmed powerful properties of the proposed tests. We showed that our tests outperform different tests with a structure based on the Wilcoxon signed rank test and/or the Kolmogorov–Smirnov test, and outperform the parametric likelihood ratio tests when the underlying distributions are misspecified. The data example illustrated that the proposed tests can be easily and efficiently used in practice.

Table IV.

The Monte Carlo powers of Test 2 by (15) versus the MLR test for H₀ versus H_A2 with different sample sizes (n₁, n₂) at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FZ<sub>2</sub>	n1	n2	Proposed test at (15)	MLR
N(0, 0.52)	N(0.5, 1)	N(0,1)
			10	10	0.2325	0.2351
			50	50	0.7989	0.8560
N(0, 0.52)	N(1.5, 1)	N(0,1)
			10	10	0.9564	0.9593
			15	15	0.9957	0.9967
N(0, 1)	N(1.5, 1)	N(0,1)
			10	10	0.877	0.8991
			25	25	0.9995	0.9997
N(1, 0.52)	N(2, 1.52)	N(0,1)
			10	10	0.5219	0.6194
			50	50	0.9975	0.9997

Table VIII.

The Monte Carlo powers of the proposed test (15) versus the combined nonparametric test (the one Wilcoxon signed rank test and one Kolmogorov–Smirnov test) at the significance level α = 0.05.

FX<sub>1</sub>	FY<sub>1</sub>	FZ<sub>2</sub>	n1	n2	Proposed test at (15)	W and K–S test
Exp(3)	N(1.5, 22)	N(0, 1)
			10	10	0.5638	0.2876
			50	50	0.9995	0.9816
Exp(1)	Beta(1,1)	N(0, 1)
			10	10	0.2256	0.1193
			50	50	0.9983	0.8046
Exp(1)	Cauchy(1,1)	N(0, 1)
			10	10	0.1613	0.0401
			50	50	0.9736	0.2323
Exp(1.5)	N(0.5,1)	Unif [−1, 1]
			10	10	0.1677	0.0384
			50	50	0.9984	0.3042
Exp(1.5)	Beta(3,1)	Unif [−1, 1]
			10	10	0.1328	0.0934
			50	50	0.8774	0.4128
Exp(1)	Cauchy(1,1)	Unif [−1, 1]
			10	10	0.4052	0.0608
			25	25	0.9952	0.9042
Lognorm(1, 1)	Lognorm(1.2, 1)	t2
			10	10	0.2892	0.0657
			50	50	0.8807	0.6635
${\chi }_{(6)}^2$	Gamma(10,1)	t2
			10	10	0.9044	0.6699
			25	25	0.9999	0.9914
Exp(1)	Cauchy(1,1)	t2
			10	10	0.0792	0.0316
			50	50	0.2891	0.0877

Appendix A. Computing the empirical constraint (11) in the development of Test 2

Similarly to the Equations (1)–(4), by virtue of the Mean Value Theorem and Lemma 2.1 in [24], we have

$${(2k)}^{-1}\sum _{j=1}^{n_2}{\int }_{Z_{2_{(j-k)}}}^{Z_{2_{(j+k)}}}{f}_{Z_2}(u)du\cong {(2k)}^{-1}\sum _{j=1}^{n_2}\frac{f_{Z_2,j}}{f_{{ZZ}_2,j}}{\varphi }_{k,j}$$

and

$$\begin{array}{l}{(2k)}^{-1}\sum _{j=1}^{n_2}{\int }_{Z_{2_{(j-k)}}}^{Z_{2_{(j+k)}}}{f}_{Z_2}(u)du\cong F_{Z_2}\left(Z_{2_{(n_2)}}\right)-F_{Z_2}(Z_{2_{(1)}})\\ -\sum _{r=1}^{k-1}\frac{(k-r)}{2k}\left[F_{Z_2}\left(Z_{2_{(n_2-r+1)}}\right)-F_{Z_2}\left(Z_{2_{(n_2-r)}}\right)+F_{Z_2}(Z_{2_{(r+1)}})-F_{Z_2}(Z_{2_{(r)}})\right],\end{array}$$ (A1)

where ϕ_k,j are defined in (8). Here, because of the symmetric property of F_Z2, under the alternative hypothesis H_A2, applying the estimation proposed by Schuster [28], F_Z2(r) − F_Z2(s) can be estimated by

$${\stackrel{\sim }{F}}_{Z_2}(r)-{\stackrel{\sim }{F}}_{Z_2}(s)={(2n_2)}^{-1}\sum _{j=1}^{n_2}[I(Z_{2j}\le r)+I(-Z_{2j}\le r)-I(Z_{2j}\le s)-I(-Z_{2j}\le s)].$$

Now, the right-hand side of Equation (A1) can be estimated by ${\mathrm{\Lambda }}_{n_2}^k$ defined in Equation (12). This concludes that the resulting empirical constraint on values of F_{Z2, j} in Test 2 has the form of

$${(2k)}^{-1}\sum _{j=1}^{n_2}\frac{f_{Z_2,j}}{f_{{ZZ}_2,j}}{\varphi }_{k,j}={\mathrm{\Lambda }}_{n_2}^k.$$