in terms of Type I error probability and penalized power. In the case of Type I error probability of the compared tests are different, the penalized power is used which allows fair power comparisons. In this way, we conclude the performance of the tests by taking into account two possible errors in hypothesis testing.
1 Introduction
Testing equality of normal distributed and independent groups’ means is a basic analysis in statistics and related fields. The Fisher’s F-test is a powerful test to do this analysis with the assumptions of variance homogeneity, normality, and statistical independency. Violation of the variance homogeneity assumption is a commonly encountered statistical problem in a variety of application areas such as agriculture, pharmacy, and biostatistics. There is number of methods improved because of the negative effect of the violation of variance homogeneity assumption on the performance of Classical F-test in terms of Type I error probability and power. These tests are, Cochran F (CF), Welch-Aspin (WA), Welch (WE), Box (BX), Scott-Smith (SS), Brown-Forsythe (BF), Johansen (JF), Approximate F (AF), Alexander-Govern (AG), Generalized F (GF), Modified Brown-Forsythe (MBF), Permutation F (PF), Adjusted Welch (AW), B2, Parametric Bootstrap (PB), Fiducial Approach (FA) and Alvandi et al. Generalized F (AGF) test, chronologically. The fact that the high number of methods in the literature raises the problem of choosing the most appropriate method for researchers.
There are many articles to investigate the performance of the tests for equality of normal distributed and independent group means under unequal variances in the literature. However, only some of the tests are included in these studies. The results of these studies help researchers to solve the problem of choosing the appropriate method for their work. (Gamage and Weerahandi 1998) compared the size performance of the GF test and four widely used procedures: CF, BF, and Welch test in case of deviation from normality. The highly skewed Gamma distributions and Gamma distributions with shapes close to being normal are considered. While the GF was found to have size not exceeding the intended level, as heteroscedasticity becomes severe the others were found to have poor size performance. (Hartung et al. 2002) compared the CF, C, W, BF, MBF, AF, and AW tests under normal populations, balanced-unbalanced sample sizes and an increasing number of populations. None of the tests considered is uniformly dominating the others. The BF and the W test perform well over a wide range of parameter configurations, the MBF test by Mehrotra keeps generally the level, but other tests may also perform well, depending on the constellation of the parameters under study. The W test becomes liberal when the sample sizes are small and the number of populations is large. They propose a modified version of Welch’s test that keeps the nominal level in these cases. With the understanding that methods are unacceptable if they have Type I error rates that are too high, only the testing procedure associated with the MBF test can be recommended, the modified Welch test can also be recommended. (Argac 2004) constructed a systematic pattern in simulations of the tests for equality of normal distributed and independent group means under unequal variances. Classical F, Cochran, Welch, modified Welch, Brown-Forsythe, modified Brown-Forsythe, and approximate F test considered are divided into two groups, Cochran-Welch type tests and the Brown-Forsythe type tests. There seems to be considerably higher variability in the power of C-W type tests in the balanced case. In the unbalanced case, there does not appear to be a huge difference between the power of the different tests. (Sadooghi-Alvandi et al. 2012) proposed a new GF test and compared it with GF, PB, Welch, and Cochran test in an extensive Monte-Carlo simulation study. According to results, it controls the Type I error probability better and its power closed to the others. (Gokpinar and Gokpinar 2012) compared the Type I error probability and power of CF, BF, GF, PB, and W test under different variance heterogeneities and effect sizes for three and five groups. Their results indicate that PB is the best control Type I error probability and has the highest power. In addition to these articles, the scope of the other articles are not comprehensive in the literature ((Hartung et al. 2002), (Lee and Ahn 2003), (Li et al. 2011), (Mutlu et al. 2017)). A comprehensive Monte-Carlo simulation study is conducted under normal distribution in this article in order to fill this gap. Especially, the penalized power is used which allows fair power comparisons when the Type I error probabilities are different. In this way, we conclude the performance of the tests by taking into account two possible errors in hypothesis testing.
Another problem experienced by the researchers is most of these tests are not available in any R package. However, some R packages contain the tests for equality of normal distributed and independent group means under unequal variances, asbio by (Aho 2018), coin by (Hothorn et al. 2008), lawstat by (Hui et al. 2008), onewaytests by (Dag et al. 2018), welchADF by (Villacorta 2017), WRS2 by (Mair and Wilcox 2018). These packages contain only the Brunner-Dette-Munk, Permutation F, Kruskal-Wallis, Brown-Forsythe, Alexander-Govern, James Second Order, Welch test. In particular, the performance of the tests such as the GF, PB, FA, and AGF test by Monte-Carlo simulations prevents the easy use of these tests. Clearly, a package should contain these tests. We propose the package doex provides the tests for equality of normal distributed group means under unequal variances which previously have not been implemented in any R package such as AF, AGF, B2, FA, JF, MBF, MW, PB, and PF. Also, it consists of the modified Generalized F-test (MGF) which is proposed by (Cavus et al. 2017) to test the equality of group means under heteroscedasticity and non-normality caused by outliers. It is a useful procedure for non-normal distributed groups and (Cavus et al. 2018) showed in a real data application.
The following sections detail the tests for equality of normal distributed and independent group means under unequal variances considered in doex. The performance of these tests is investigated in terms of penalized power and Type I error probability. Finally, we conclude with a brief summary and future works.
2 Tests for Testing Equality of Normal Distributed Groups’ Means under Unequal Variance
The linear model within the context of a one-way independent group design for testing the equality of groups’ means is given in ((1)) .
\[\label{eq:1} Y_{ij}=\mu_i+\epsilon_{ij} \tag{1}\]
where \(Y_{ij}\) is the dependent variable associated with the \(i\)th observation in the \(j\)th group for \(i= 1, 2, ..., n_i\) and \(j= 1, 2, ..., k\). \(\mu_i\) is the group mean for the \(i\)th group, and \(\epsilon_{ij}\) is the random error component associated with \(Y_{ij}\). The null hypothesis \(H_0: \mu_1=\mu_2= ...= \mu_k\) is tested as the Classical F-test assumed that the \(\epsilon_{ij}\)’s are independent, normally distributed, and have an equal variance \(\sigma^2\) for each group of \(k\). Type I error probability of Classical F-test inflates and its power decreases in case of the violation of variance homogeneity assumption. There are many procedures improved in the literature to solve this problem. In this section, the tests for equality of normal distributed and independent group means under unequal variances, considered in doex and discussed in the Monte-Carlo simulation study, are introduced. These tests are, Alexander-Govern, Alvandi et al. generalized F, Approximate F, Box F, Brown-Forsythe, \(B^2\), Cochran F, Fiducial Approach, Generalized F, Johansen, Modified Brown-Forsythe, Adjusted Welch, Parametric Bootstrap, Permutation F, Scott-Smith, Welch, Welch-Aspin test.
Alexander-Govern (AG) test
(Alexander and Govern 1994) improved a test using the Hill’s normality transformation to the Student’s t variables. Consider \(X_{i1}, X_{i2}, ..., X_{in_i} \sim N(\mu_i, \sigma_i^2)\) and the standard deviations of normal groups computed as in ((2)). \[\label{eq:AG1} S_{\bar{X}_i}= \frac{\sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij}-\bar{X}_i)^2}{n_i(n_i-1)} \tag{2}\] The weights are computed using the \(S_{\bar{X}_i}\) as in ((3)). \[\label{eq:AG2} wi=\frac{1/S_{\bar{X}_i}^2}{\sum_{i=1}^k 1/S_{\bar{X}_i}^2} \tag{3}\] The weight mean is computed using the \(w_i\) in ((4)). \[\label{eq:AG3} \bar{X}^*=\sum_{i=1}^k w_i\bar{X}_i \tag{4}\] The values of \(t_i=(\bar{X}_i-\bar{X}^*)/S_{\bar{X}_i} \sim t_{n_i-1}\) are transformed using the following transformation. \[z_i=c+\frac{c^3+3c}{b}+\frac{4c^7+33c^5+240c^3+855c}{10b^2+8bc^4+1000b}\] where \(a=v_i-0.5, c=\sqrt{a ln (1+\frac{t_i^2}{v_i})}\) and \(b=48a^2\). The test statistic of AG test is computed as in ((5)). \[\label{eq:AG4} T_{AG}=\sum_{i=1}^k z_i^2 \tag{5}\] The \(H_0\) is rejected when \(T_{AG}>\chi^2_{(k-1); \alpha}\).
Alvandi et. al. Generalized F (AGF) test
(Sadooghi-Alvandi et al. 2012) proposed the test statistic in (6) as an alternative of Weerahandi’s Generalized F-test. \[T_G(S_1^2, S_2^2, ..., S_k^2)=\sum_{i=1}^k \frac{n_i}{S_i^2}\bar{X}_i-\frac{[\sum_{i=1}^k n_i\bar{X}_i/S_i^2]^2}{\sum_{i=1}^k n_i/S_i^2}\]
\[\label{eq:AGF1} T_{AGF}=\sum_{i=1}^k \frac{n_i-1}{U_i} (\bar{X}_i-q_i\tilde{X})^2 \tag{6}\] where \(q_i=\sqrt{\frac{n_i/s_i^2}{\sum_{i=1}^k n_i/s_i^2}}\) and \(\tilde{X}=\sum_{i=1}^k q_i \bar{X}_i\). The p-value of AGF test computed using Monte-Carlo simulations with Algorithm 1.
Algorithm 1. Computation of Monte-Carlo estimate of the AGF test
Compute the vectors of \((\bar{x}_1, \bar{x}_2, ..., \bar{x}_k)\) and \((s_1^2, s_2^2, ..., s_k^2)\) for \(k\) groups
Compute the \(T_G\) using the vectors in Step 1
for \(j\) in \(\{1, ..., r\}\) do
Generate \(U_i \sim \chi^2_{n_i-1}\) random samples Compute the \(T_{AGF}\)
using generated random samples Set the counter \(Q_j=1\) when \(T_{AGF}>T_G\)
end forCompute the Monte-Carlo estimate of p-value as \(\sum_{i=1}^k Q_j/r\)
Approximate F (AF) test
(Asiribo and Gurland 1990) proposed a modification to the F-test as in ((7)). \[\label{eq:AF1} T_{AF}=N\frac{\sum_{i=1}^k n_i(\bar{X}_i-\bar{X}_{..})^2}{\sum_{i=1}^k (N-n_i)S_i^2} \tag{7}\] where \(\bar{X}_{..}=\sum_{i=1}^k \bar{X}_i\) and \(N=\sum_{i=1}^k n_i\). The \(H_0\) is rejected when \(T_{AF}>F_{v1, v2; \alpha}\). The degrees of freedom of the AF test statistic is computed in ((8)). \[\label{eq:AF2} v_1=\frac{[\sum_{i=1}^k (1-n_i/N)S_i^2]^2}{\sum_{i=1}^k S_i^4+[\sum_{i=1}^k n_i S_i^2/N]^2-2\sum_{i=1}^k n_i S_i^4/N}, v_2=\frac{[\sum_{i=1}^k (1-n_i/N)^2 S_i^2]^2}{\sum_{i=1}^k (n_i-1)S_i^4} \tag{8}\]
Box (BX) test
(Box 1954) proposed the test statistic in ((9)). \[\label{eq:BO1} T_{BO}=\frac{N_k}{N(k-1)\sum_{i=1}^k \frac{(N-n_i)S_i^2}{(n_i-1)S_i^2}} \tag{9}\] The \(H_0\) is rejected when \(T_{BO}>F_{v_1, v_2; \alpha}\) where \[v_1=\frac{[\sum_{i=1}^k (N-n_i)S_i^2]^2}{[\sum_{i=1}^k n_i S_i^2]^2+N\sum_{i=1}^k (N-2n_i)S_i^2}, v_2=\frac{[\sum_{i=1}^k (n_i-1 S_i^2)]^2}{\sum_{i=1}^k (n_i-1)S_i^+}\]
Brown-Forsythe (BF) test
(Brown and Forsythe 1974) proposes the following test statistic. \[T_{BF}=\frac{\sum_{i=1}^k n_i(\bar{X}_i-X_{..})^2}{\sum_{i=1}^k (1-n_i/N)S_i^2}\] where \(X_{..}=\sum_{i=1}^k \bar{X}_i\) and \(N=\sum_{i=1}^k n_i\). The \(H_0\) is rejected when \(T_{BF}>F_{(k-1), v; \alpha}\). The degrees of fredom of the test statistic computed as in ((10)). \[\label{eq:BF1} v=\frac{[\sum_{i=1}^k n_i(\bar{X}_i-X_{..})]^2}{\sum_{i=1}^k \frac{(1-n_i/N)^2 S_i^4}{(n_i-1)}} \tag{10}\]
The \(B^2\) test
(Ozdemir and Kurt 2006) proposed the following procedure using the Bailey’s normality transformation to the Student’s t variables. Consider \(X_{i1}, X_{i2}, ..., X_{in_i} \sim N(\mu_i, \sigma_i^2)\) and the standard deviations of normal groups computed as in ((11)). \[\label{eq:B21} S_{\bar{X}_i}=\frac{\sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij}-\bar{X}_i)^2}{n_i(n_i-1)} \tag{11}\] The weights computed using the \(S_{\bar{X}_i}\) as in ((12)). \[\label{eq:B22} w_i=\frac{1/S_{\bar{X}_i}^2}{\sum_{i=1}^k 1/S_{\bar{X}_i}^2} \tag{12}\] The weighed mean computed using the \(w_i\)’s as in ((13)). \[\label{eq:B23} \bar{X}^*=\sum_{i=1}^k w_i\bar{X}_i . \tag{13}\] The values of \(t_i=(\bar{X}_i-\bar{X}^*)/S_{\bar{X}_i} \sim t_{n_i-1}\) are transformed using Bailey’s (1980) normality transformation. \[z_i=\frac{4v_i^2+\frac{5(2z_c^2+3)}{24}}{4v_i^2+v_i+\frac{4z_c^2+9}{12}}\sqrt{v_i ln(1+\frac{t_i^2}{v_i})} \sim N(0, 1)\] where \(z_c=Z_{\alpha/2} \sim N(0, 1)\) and the test statistic of \(B^2\) test computed as in ((14)). \[\label{eq:B24} T_{BK}=\sum_{i=1}^k z_i^2=\sum_{i=1}^k (\frac{4v_i^2+\frac{5(2z_c^2+3)}{24}}{4v_i^2+v_i+\frac{4z_c^2+9}{12}})^2 . \tag{14}\] The \(H_0\) is rejected when \(T_{BK}>\chi^2_{(k-1); \alpha}\).
Cochran (CF) test
(Cochran 1937) proposes the test statistic in ((15)). \[\label{eq:CF1} T_C=\sum_{i=1}^k w_i (\bar{X}_i - \sum_{j=1}^k h_j\bar{X}_j)^2 \tag{15}\] where \(w_i=n_i/s_i^2\) and \(h_i=w_i/\sum_{i=1}^k w_i\). The \(H_0\) is rejected when \(T_C > \chi^2_{(k-1); \alpha}\).
Fiducial Approach (FA) test
(Li et al. 2011) proposed the test statistic in ((16)). \[\label{eq:FA1} T_{FA}=\sum_{i=1}^k t_i^2-\frac{(\sum_{i=1}^k \frac{\sqrt{n_i}}{S_i}t_i)^2}{\sum_{i=1}^k \frac{n_i}{S_i^2}} . \tag{16}\] The p-value of the FA test can be computed using Monte-Carlo simulations with Algorithm 2.
Algorithm 2. Computation of Monte-Carlo estimate of the FA test
Compute the vectors of \((\bar{x}_1, \bar{x}_2, ..., \bar{x}_k)\) and \((s_1^2, s_2^2, ..., s_k^2)\) for \(k\) groups
Compute the \(T_G\) using the vectors in Step 1
for \(j\) in \(\{1, ..., r\}\) do
Generate \(Z_i \sim N(0, 1)\) and \(U_i \sim \chi^2_{n_i-1}\) random samples
Compute the \(T_{FA}\) using generated random samples
Set the counter \(Q_j=1\) when \(T_{FA}>T_G\)
end forCompute the Monte-Carlo estimate of p-value as \(\sum_{i=1}^k Q_j/r\)
Generalized F (GF) test
(Weerahandi 1995) proposed the test statistic in ((17)) using the generalized p-value approach. \[\label{eq:GF1} T_{GF}=\sum_{i=1}^k (n_iU_i/v_i^2)\bar{x}_i^2-\frac{[\sum_{i=1}^k (n_iU_i/v_i^2)\bar{x}_i]^2}{\sum_{i=1}^k n_iU_i/v_i^2} \tag{17}\] where \(v_i^2=(n_i-1)S_i^2\). The p-value of GF test can be computed using Monte-Carlo simulations with Algorithm 3.
Algorithm 3. Computation of Monte-Carlo estimate of the GF test
Compute the vectors of \((\bar{x}_1, \bar{x}_2, ..., \bar{x}_k)\) and \((s_1^2, s_2^2, ..., s_k^2)\) for \(k\) groups
Compute the \(T_G\) using the vectors in Step 1
for \(j\) in \(\{1, ..., r\}\) do
Generate \(U_i \sim \chi^2_{n_i-1}\) random samples
Compute the \(T_{GF}\) using generated random samples
Set the counter \(Q_j=1\) when \(T_{GF}>T_G\)
end forCompute the Monte-Carlo estimate of p-value as \(\sum_{i=1}^k Q_j/r\)
Johansen (JF) test
(Johansen 1980) proposed an approximate solution to the W test as in ((18)).
\[\label{eq:tj} T_J=\frac{\sum_{i=1}^k \frac{\bar{X}_i^2}{S_i^2}-\frac{[\sum_{i=1}^k\bar{X}_i/S_i^2]^2}{\sum_{i=1}^k 1/S_i^2} }{c} \tag{18}\]
where \(c=(k-1)+2A-6A/(k+1)\), \(v=(k-1)(k+1)/3A\) and \(A=\sum_{i=1}^k (1-w_i/w)^2/(n_i-1)\) The \(H_0\) is rejected when \(T_J > F_{k-1, v; \alpha}\).
Modified Brown-Forsythe (MBF) test
(Mehrotra 1997) proposed the test statistic in ((19)), which is a modification of BF, to well-performing in case of small sample size. \[\label{eq:MBF1} T_{MBF}=\frac{\sum_{i=1}^k n_i(\bar{X}_i-\bar{X}_{..})^2}{\sum_{i=1}^k (1-n_i/N)S_i^2} . \tag{19}\] where \(\bar{X}_{..}=\sum_{i=1}^k \bar{X}_i\) and \(N=\sum_{i=1}^k n_i\). The \(H_0\) is rejected when \(T_{MBF>F_{v_1, v_2; \alpha}}\). The degrees of freedom of the MBF test statistics is computed as in ((20)). \[\label{eq:MBF2} v_1=\frac{[\sum_{i=1}^k (1-n_i/N)S_i^2]^2}{\sum_{i=1}^k S_i^4+(\sum_{i=1}^k n_iS_i^2/N)^2-2\sum_{i=1}^k n_iS_i^4/N} , v_2=\frac{[\sum_{i=1}^k (1-n_i/N)^2 S_i^2]^2}{\sum_{i=1}^k \frac{(1-n_i/N)^2S_i^4}{n_i-1}} . \tag{20}\]
Adjusted Welch (AW) test
(Hartung et al. 2002) proposed an adjustment to the Welch test. The test statistic of adjusted Welch test is computed as in ((21)). \[\label{eq:AW1} T_W=\frac{\sum_{i=1}^k w_i^* (\bar{x}_i-\sum_{j=1}^k h_j^*\bar{x}_j)^2}{(k-1)+2\frac{k-2}{k+1}\sum_{i=1}^k \frac{1}{n_i-1}(1-h_j^*)^2} . \tag{21}\] where \(w_i^*=[\frac{n_i}{(n_i-1/n_i-3)s_i^2}]\) and \(h_i^*=\frac{w_i^*}{\sum_{i=1}^k w_i^*}\). The \(H_0\) is rejected when \(T_W>F_{(k-1), v; \alpha}\). The degrees of freedom of the test statistic computed in ((22)). \[\label{eq:AW2} v=\frac{\frac{k^2-1}{3}}{\sum_{i=1}^k \frac{(1-h_i^*)^2}{n_i-1}} . \tag{22}\]
Parametric Bootstrap (PB) test
(Krishnamoorthy et al. 2007) proposed a procedure to test the equality of group means under heteroscedasticity. \[T_G(S_1^2, S_2^2, ..., S_k^2)=\sum_{i=1}^k \frac{n_i}{S_i^2}\bar{X}_i-\frac{[\sum_{i=1}^k n_i\bar{X}_i/S_i^2]^2}{\sum_{i=1}^k n_i/S_i^2}\] Assume \(Z_i \sim N(0, 1)\) and \(U_i \sim \chi^2_{n_i-1}\) random samples, the test statistic of the PB test is computed as in ((23)). \[\label{eq:PB2} T_{PB}(S_1^2, S_2^2, ..., S_k^2)=\sum_{i=1}^k \frac{Z_i^2 (n_i-1)}{U_i}-\frac{[\sum_{i=1}^k \sqrt{n_i}Z_i(n_i-1)/S_iU_i]^2}{\sum_{i=1}^k n_i(n_i-1)/S_i^2U_i} \tag{23}\] The \(H_0\) is rejected when \(T_{PB}>T_G\). The p-value of PB test is computed using Monte-Carlo simulations with Algorithm 4.
Algorithm 4. Computation of Monte-Carlo estimate of the PB test
Compute the vectors of \((\bar{x}_1, \bar{x}_2, ..., \bar{x}_k)\) and \((s_1^2, s_2^2, ..., s_k^2)\) for \(k\) groups
Compute the \(T_G\) using the vectors in Step 1
for \(j\) in \(\{1, ..., r\}\) do
Generate \(Z_i \sim N(0, 1)\) and \(U_i \sim \chi^2_{n_i-1}\) random samples
Compute the \(T_{PB}\) using generated random samples
Set the counter \(Q_j=1\) when \(T_{PB}>T_G\)
end forCompute the Monte-Carlo estimate of p-value as \(\sum_{i=1}^k Q_j/r\)
Permutation F (PF) test
(Berry and Mielke 2002) proposed the test statistic in ((24)) as the permutational alternative of F-test. \[\label{eq:PF1} T_{PF}=\frac{(T-N\bar{X}^*)/(k-1)}{(V-T)/(N-k)} \tag{24}\] where \(T=\sum_{i=1}^k n_i\sum{x}_i^2\), \(\bar{X}^*=1/N\sum{n_i\bar{x}_i}\) and \(V=\sum_{i=1}^k \sum_{j=1}^{n_i} X_{ij}^2\). The \(H_0\) is rejected when \(T_{PF}>F_{k-1, N-k; \alpha}\).
Scott-Smith (SS) test
(Scott and Smith 1971) proposed the test statistic in ((25)). \[\label{eq:SS1} T_{SC}=\sum_{i=1}^k \frac{n_i(\bar{X}_i-\bar{X}_{..})^2}{S_i^{*2}} \tag{25}\] where \(S_i^{*2}=\frac{n_i-1}{n_i-3}S_i^2\). The \(H_0\) is rejected when \(T_{SC}>\chi^2_{k; \alpha}\).
Welch (WE) test
(Welch 1951) improved the test statistic in (26) based on the weighted group variance as an alternative to the F-test under heteroscedasticity. \[\label{eq:W1} T_W=\frac{\sum_{i=1}^k w_i(\bar{x}_i-\sum_{i=1}^k h_j\bar{x}_j)^2}{(k-1)+2\frac{k-2}{k+1}\sum_{i=1}^k \frac{1}{n_i-1}(1-h_i)^2} \tag{26}\] where \(w_i=n_i/s_i^2\) and \(h_i=w_i/\sum_{i=1}^k w_i\). The \(H_0\) is rejected when \(T_W>F_{(k-1), v; \alpha}\). The degrees of freedom of the Welch test computed as in (27). \[\label{eq:W2} v=\frac{(k^2-1)/3}{\sum_{i=1}^k \frac{(1-h_i)^2}{n_i-1}} \tag{27}\]
Welch-Aspin (WA) test
(Aspin 1948) proposed the test statistic in ((28)) with a
modification to the degrees of freedom of Welch test.
\[\label{eq:WA1}
T_{WA}=\frac{\sum_{i=1}^k (\bar{X}_i-\bar{X})^2/S_i^2}{(k-1)[1+\frac{2k-2}{k^2-1}\lambda]} \tag{28}\]
where \(\lambda=\sum_{i=1}^k [(1-w_i)^2/w_i]\), \(v_1=k-1\) and
\(v_2=(k^2-1)/3\lambda\). The \(H_0\) is rejected when
\(T_{WA}>F_{v_1; v_2; \alpha}\).
3 Using doex package
The doex package provides to perform several tests for equality of
normal distributed and independent distributed group means under unequal
variances. These tests are called a function with the initials of their
name which are given in the previous sections. In particular, the
following tests are not included in any R package or statistical package
program: AF, AGF, B2, FA, JF, MBF, MW, PB, and PF. In this section, the
examples are given how to use these tests by using doex. After the
explanatory data analysis, the variance homogeneity assumption must be
checked to move on to the next stage (Noguchi and Gel 2010; Erps and Noguchi 2019). The
Levene Test is used to this, and we did not include it in the package is
because it is included in many R package such as
car by (Fox and Weisberg 2019),
rstatix by
(Kassambara 2020), lawstat by (Gastwirth et al. 2020),
inferr by (Hebbali 2018).
We want to stick with the idea of creating a package that includes tests
not included in the CRAN.
Example 1: The data are inputted to the functions with two parts:
observations and the group labels. As an example hybrid data from
Weerahandi (1995) is given in the package. It consists of two parts:
data are observations and species are the labels of species of the
corn hybrids.
# Call the doex package
> library(doex)
# print hybrid data of Weerahandi (1995)
> hybrid
data species
1 7.4 A
2 6.6 A
3 6.7 A
4 6.1 A
5 6.5 A
6 7.2 A
7 7.1 B
8 7.3 B
9 6.8 B
10 6.9 B
11 7.0 B
12 6.8 C
13 6.3 C
14 6.4 C
15 6.7 C
16 6.5 C
17 6.8 C
18 6.4 D
19 6.9 D
20 7.6 D
21 6.8 D
22 7.3 D
# observations of the hybrid data
> hybrid$data
[1] 7.4 6.6 6.7 6.1 6.5 7.2 7.1 7.3 6.8 6.9 7.0 6.8 6.3 6.4 6.7 6.5 6.8 6.4 6.9 7.6 6.8 7.3
# group labels of the hybrid data
> hybrid$species
[1] A A A A A A B B B B B C C C C C C D D D D D
Levels: A B C D
# The ggplot2 package can be used to plot the box plot of the data in Figure 1.
> ggplot(hybrid, aes(x = species, y = data)) +
> geom_boxplot() +
> ylab("Yield") +
> xlab("Corn Species")
# Look at the summary statistics of the data before using the tests.
# Use psych package to obtain the descriptive statistics of the hybrid data
> library(psych)
# Describe the hybrid data by species using describe.by(.) function
> describe.by(hybrid$data, hybrid$species)
#The output of the describe.by function as follows:
Descriptive statistics by group
group: A
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 6 6.75 0.48 6.65 6.75 0.52 6.1 7.4 1.3 0.11 -1.7 0.19
----------------------------------------------------------
group: B
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 5 7.02 0.19 7 7.02 0.15 6.8 7.3 0.5 0.28 -1.72 0.09
----------------------------------------------------------
group: C
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 6 6.58 0.21 6.6 6.58 0.3 6.3 6.8 0.5 -0.13 -2.02 0.09
----------------------------------------------------------
group: D
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 5 7 0.46 6.9 7 0.59 6.4 7.6 1.2 0.04 -1.84 0.21
# It is seen that the variances of the species are unequal
# Thus we need to use the tests for equality of the group means under unequal variances
#
# Examples of the use of the AF and GF tests on the hybrid data are given in the follows.
# The following code performs the Approximate F-test on the hybrid data.
> library(doex)
> AF(hybrid$data,hybrid$species)
# This function returns a result matrix consists of a test statistic, degrees of freedom,
and p-value of Approximate F-test as follows:
Test Statistic df1 df2 p-value
Approximate F 1.8538 2 12 0.1943# Following code performs the Generalized F-test.
> library(doex)
> GF(hybrid$data,hybrid$species)
# The p-value of the GF test is computed Monte-Carlo estimates and its size is
# controlled with the rept parameter in the function. It is implemented as
# default rept=10000
# This function returns the p-value of the Generalized F-test as follows:
p-value
Generalized F 0.0492
# The results of the AF and GF tests are different at the nominal level 0.05.
# It is needed to investigate the performance of these tests in
# a Monte-Carlo simulation study.Example 2: This example is provided an external data involves litter weights of mice born from mothers assigned to three different dosage groups and a control. For the low dose group the dose metameter is 5, for the medium dose group it is 50, and for the high dose group it is 500. In here, the problem is testing the equality of mean of litter weights of mice born according to the used dose. The dataset is available in the following repository: https://github.com/mcavs/doex_TheRJournal.
# Print born weight data using the data is given in GitHub repository.
weight_data dose
1 22.69 0
2 26.59 0
3 28.85 0
4 28.03 0
5 29.05 0
6 23.61 0
7 22.21 0
8 26.81 0
9 26.01 0
10 25.98 0
. . .
. . .
. . .
70 26.31 500
71 30.61 500
72 26.48 500
73 24.31 500
74 27.98 500
# The ggplot2 package can be used to plot the box plot of the data in Figure 2.
> ggplot(born_weight_data, aes(x = dose, y = weight_data)) +
> geom_boxplot() +
> ylab("Born Weight (gr)") +
> xlab("Dose Treatment")
# Describe the born weight data by species using describe.by(.) function
> describe.by(born_weight_data$weight_data, born_weight_data$dose)
#The output of the describe.by function as follows:
Descriptive statistics by group
group: 0
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 20 25.73 2.02 26.1 25.74 2.43 22.21 29.05 6.84 -0.1 -1.16 0.45
-------------------------------
group: 5
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 19 23.52 3.9 22.75 23.51 4.28 16.34 30.95 14.61 0.01 -0.97 0.89
-------------------------------
group: 50
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 18 23.79 2.83 24.11 23.84 1.92 17.54 29.21 11.67 -0.28 0.03 0.67
-------------------------------
group: 500
vars n mean sd median trimmed mad min max range skew kurtosis se
X1 1 17 23.72 4.08 24.31 23.76 3.84 16.13 30.61 14.48 -0.4 -0.91 0.99
# It is seen that the variances of the dose groups may be unequal
# To conclude whether the variance homogenity assumption is valid,
# Levene test is used.
> library(car)
> car::LeveneTest(weight_data ~ dose)
# LeveneTest(.) function returns the test statistic and
# p-value of Levene variance homogeneity test as follows:
Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 3 3.3819 0.0229 *
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’
# The p-value of Levene test is lower than the nominal level 0.05,
# so it is concluded that the variance homogeneity assumption is violated.
# Thus we need to use the tests for equality of the group means under
# unequal variances in doex.
# The GF, AF, and PB are used to conclude there is a significance difference between
# the mean born weight of mice according to used dose group.
> doex::GF(weight_data, dose)
p-value
Generalized F 0.0331
> doex::AF(weight_data, dose)
Test Statistic df1 df2 p-value
Approximate F 1.9408 3 57 0.1484
> doex::PB(weight_data, dose)
p-value
Parametric Bootstrap 0.0366
# The results of the GF and PB tests indicate that there is a significant difference,
# while the result of the AF indicates that there is no significant difference between
# the mean born weight of mice according to used dose group.
# It is also needed to investigate the performance of these tests in
# a Monte-Carlo simulation study.4 Monte-Carlo simulation study
In this section, the performance of the tests for equality of normal distributed and independent groups’ means under unequal variances are investigated in terms of Type I error probability and penalized power of the test. We used the penalized power instead of the classical power of the test, because any comparison of the powers is invalid when Type I error probabilities are different in Monte-Carlo simulation studies. (Zhang and Boos 1994) and (Lloyd 2005) proposed alternatives for the power of the tests have some deficiencies. To overcome this problem, (Cavus et al. 2019) proposed the penalized power of the test in ((29)) to compare the power of the test even if Type I error probabilities are different.
\[\label{eq:pp} \gamma=\frac{1-\beta}{\sqrt{1+\left|1-\frac{\alpha_i}{\alpha_0}\right|}} \tag{29}\]
where \(\beta\) is Type II error rate, \(\alpha_i\) is Type I error of the test, and \(\alpha_0\) is the nominal level. Penalized power adjusts the power function with the square root of the percentile deviation between type I error probability and the nominal level. Thus, penalized power is used to compare the power of the tests in the simulation studies. An extensive Monte-Carlo simulation study is conducted to investigate the performance of the tests in terms of Type I error probability and penalized power. Firstly, the ability of the tests to control the Type I error probability is examined. Then, the penalized power of the test which controls the Type I error probability in the (Bradley 1978)’s robustness limits are compared. In this way, we conclude the performance of the tests by taking into account two possible errors in hypothesis testing. The sample size, design type, variance heterogeneity, and effect sizes are used as configuration factors beyond this part of the study. The R code used in this simulation study is available in the following repository: https://github.com/mcavs/doex_TheRJournal.
The properties of the tests to control the Type I error probability
Type I error probabilities of the tests are investigated in an extensive Monte-Carlo simulation study under balanced and unbalanced design with small, moderate, and large sample sizes in this section. Also, the number of the groups is fixed as \(k=3, 5, 7\), and different heteroscedasticity setups are also used. Hereby, the properties of the tests to control the Type I error probability are revealed under various scenarios.
The boxplots in Figs. 3,4,5 are constructed for several heteroscedasticity scenarios. In this way, the ability of the tests to control the Type I error probability are obtained. According to the Fig.3, AF, AG, B2, GF, JF, MBF, PB, WA and WE test controls the Type I error probability in the (Bradley 1978) limits which are shown with dashed red lines. However, the AGF, CF, PF, and SS test could not control the Type I error probability for \(k=3\). The GF test controls the Type I error probability unlike in the case of \(k=3\) in Fig.4. The AF, AG, B2, MBF, PB, and WA test control Type I error probability for \(k=7\). When the results are summarized, it is concluded that the AGF, BF, BX, CF, MW, PF, and SS test could not control the Type I error probability for each of the \(k\)’s. Thus, the tests only which control Type I error probability in the limits are considered in the next section for power comparisons to avoid making a wrong decision.
The results of the penalized powers
In this section, the penalized power results are given under four configuration factors are sample size, design type, effect size, heteroscedasticity level for \(k=3, 5, 7\). The samples follows normal distribution with the parameters \((\mu_i, \sigma_i^2)\) as given in Tables 1, 2, 3. The mean parameter of the samples are shown as the effect size \(\Delta_i\) in each line. This means that the mean parameter of the samples are zero except the last sample is \(\Delta_i\).
When the effect of the configuration factors on the power are examined, it is observed that the larger sample size increase, the higher level of heteroscedasticity decrease, and the higher effect sizes increase the power of all tests as expected. Also, some interesting results are obtained such as the penalized power of all tests are higher in the unbalanced designs. The performances of the AF and MBF, the AG and B2, the JF and WE tests are very close to each other in terms of penalized power. Thus, these tests may be used interchangeably.
The AF and MBF test are superior than others in most of the scenarios. In the lower level of heteroscedasticity for all sample sizes, the penalized power of the tests are higher than 0.90 for \(k=3\). It is the same situation for \(k=5\) except for a small sample-unbalanced design. In this case, the penalized power of the AF and MGF is close to the 0.90, and the performance of the others is unacceptable. For \(k=7\), the penalized power of the tests is higher than 0.90 except for small sample-lower heteroscedasticity scenarios. In this case, the AF and MBF tests show acceptable performance in terms of penalized power in only small sample-lower level heteroscedasticity. As a result, it is clearly seen that the penalized power of the tests decreases dramatically in the higher level of heteroscedasticity for \(k=5, 7\).
| \(n_i\) | \(\sigma_i^2\) | \(\Delta_i\) | AG | AF | B2 | GF | JF | MBF | PB | WE | WA |
|---|---|---|---|---|---|---|---|---|---|---|---|
| (10, 10, 10) | (0.1, 0.2, 0.3) | 0.3 | 0.2232 | 0.2630 | 0.2236 | 0.2133 | 0.2314 | 0.2628 | 0.2306 | 0.2314 | 0.2128 |
| 0.8 | 0.9045 | 0.9391 | 0.9047 | 0.8915 | 0.9197 | 0.9390 | 0.9201 | 0.9197 | 0.8833 | ||
| 1.5 | 0.9910 | 0.9815 | 0.9911 | 0.9768 | 0.9921 | 0.9815 | 0.9941 | 0.9921 | 0.9614 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.1265 | 0.1491 | 0.1269 | 0.1277 | 0.1326 | 0.1490 | 0.1296 | 0.1326 | 0.1193 | |
| 0.8 | 0.5962 | 0.6858 | 0.5978 | 0.6004 | 0.6129 | 0.6855 | 0.6089 | 0.6129 | 0.5829 | ||
| 1.5 | 0.9853 | 0.9880 | 0.9881 | 0.9893 | 0.9812 | 0.9879 | 0.9777 | 0.9812 | 0.9583 | ||
| (1, 2, 3) | 0.3 | 0.0682 | 0.0656 | 0.0684 | 0.0649 | 0.0708 | 0.0655 | 0.0702 | 0.0708 | 0.0644 | |
| 0.8 | 0.1700 | 0.2006 | 0.1701 | 0.1653 | 0.1760 | 0.2004 | 0.1752 | 0.1760 | 0.1615 | ||
| 1.5 | 0.4832 | 0.5536 | 0.4833 | 0.4716 | 0.4998 | 0.5534 | 0.4968 | 0.4998 | 0.4693 | ||
| (1, 4, 7) | 0.3 | 0.0614 | 0.0566 | 0.0616 | 0.0622 | 0.0627 | 0.0565 | 0.0622 | 0.0627 | 0.0563 | |
| 0.8 | 0.1023 | 0.1200 | 0.1026 | 0.1050 | 0.1075 | 0.1200 | 0.1068 | 0.1075 | 0.0958 | ||
| 1.5 | 0.2542 | 0.3022 | 0.2550 | 0.2556 | 0.2655 | 0.3021 | 0.2594 | 0.2655 | 0.2424 | ||
| (30, 30, 30) | (0.1, 0.2, 0.3) | 0.3 | 0.6299 | 0.7004 | 0.6293 | 0.6281 | 0.6327 | 0.7002 | 0.6342 | 0.6327 | 0.6375 |
| 0.8 | 0.9882 | 0.9980 | 0.9872 | 0.9872 | 0.9815 | 0.9979 | 0.9853 | 0.9815 | 0.9970 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.3210 | 0.3791 | 0.3212 | 0.3210 | 0.3243 | 0.3789 | 0.3224 | 0.3243 | 0.3233 | |
| 0.8 | 0.9772 | 0.9924 | 0.9775 | 0.9779 | 0.9740 | 0.9922 | 0.9726 | 0.9740 | 0.9898 | ||
| 1.5 | 0.9831 | 0.9960 | 0.9834 | 0.9834 | 0.9796 | 0.9960 | 0.9787 | 0.9796 | 0.9960 | ||
| (1, 2, 3) | 0.3 | 0.0923 | 0.1072 | 0.0922 | 0.0934 | 0.0938 | 0.1070 | 0.0930 | 0.0938 | 0.0925 | |
| 0.8 | 0.4785 | 0.5411 | 0.4780 | 0.4733 | 0.4823 | 0.5410 | 0.4789 | 0.4823 | 0.4820 | ||
| 1.5 | 0.9566 | 0.9798 | 0.9557 | 0.9545 | 0.9517 | 0.9797 | 0.9566 | 0.9517 | 0.9653 | ||
| (1, 4, 7) | 0.3 | 0.0661 | 0.0715 | 0.0663 | 0.0663 | 0.0672 | 0.0714 | 0.0648 | 0.0672 | 0.0657 | |
| 0.8 | 0.2282 | 0.2809 | 0.2285 | 0.2284 | 0.2318 | 0.2808 | 0.2298 | 0.2318 | 0.2301 | ||
| 1.5 | 0.6982 | 0.7659 | 0.6990 | 0.7012 | 0.7032 | 0.7655 | 0.7009 | 0.7032 | 0.7100 | ||
| (50, 50, 50) | (0.1, 0.2, 0.3) | 0.3 | 0.8573 | 0.9046 | 0.8577 | 0.8546 | 0.8547 | 0.9045 | 0.8604 | 0.8547 | 0.8593 |
| 0.8 | 0.9811 | 0.9980 | 0.9815 | 0.9825 | 0.9759 | 0.9979 | 0.9853 | 0.9759 | 0.9834 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.5427 | 0.6028 | 0.5425 | 0.5443 | 0.5418 | 0.6026 | 0.5400 | 0.5418 | 0.5442 | |
| 0.8 | 0.9832 | 0.9980 | 0.9834 | 0.9880 | 0.9759 | 0.9980 | 0.9740 | 0.9759 | 0.9872 | ||
| 1.5 | 1 | 1 | 1 | 1 | 0.9979 | 1 | 0.9940 | 0.9981 | 1 | ||
| (1, 2, 3) | 0.3 | 0.1421 | 0.1671 | 0.1431 | 0.1432 | 0.1435 | 0.1671 | 0.1474 | 0.1435 | 0.1434 | |
| 0.8 | 0.7210 | 0.7900 | 0.7214 | 0.7203 | 0.7204 | 0.7900 | 0.7262 | 0.7204 | 0.7232 | ||
| 1.5 | 0.9803 | 0.9974 | 0.9807 | 0.9817 | 0.9751 | 0.9974 | 0.9843 | 0.9751 | 0.9826 | ||
| (1, 4, 7) | 0.3 | 0.0912 | 0.1040 | 0.0917 | 0.0907 | 0.0917 | 0.1040 | 0.0918 | 0.0917 | 0.0904 | |
| 0.8 | 0.4020 | 0.4659 | 0.4018 | 0.4066 | 0.4017 | 0.4659 | 0.4021 | 0.4017 | 0.4032 | ||
| 1.5 | 0.9031 | 0.9405 | 0.9032 | 0.9070 | 0.8990 | 0.9405 | 0.8932 | 0.8990 | 0.9077 | ||
| (5, 10, 15) | (0.1, 0.2, 0.3) | 0.3 | 0.2790 | 0.3346 | 0.2791 | 0.2535 | 0.2573 | 0.2974 | 0.2605 | 0.2573 | 0.2371 |
| 0.8 | 0.9598 | 0.9388 | 0.9570 | 0.9217 | 0.9432 | 0.9342 | 0.9484 | 0.9432 | 0.9406 | ||
| 1.5 | 0.9844 | 0.9483 | 0.9815 | 0.9475 | 0.9731 | 0.9500 | 0.9787 | 0.9731 | 0.9759 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.1522 | 0.1955 | 0.1526 | 0.1346 | 0.1494 | 0.1743 | 0.1490 | 0.1494 | 0.1339 | |
| 0.8 | 0.7601 | 0.8070 | 0.7609 | 0.7018 | 0.7535 | 0.7860 | 0.7532 | 0.7535 | 0.7063 | ||
| 1.5 | 0.9910 | 0.9612 | 0.9913 | 0.9340 | 0.9962 | 0.9748 | 0.9992 | 0.9962 | 0.9567 | ||
| (1, 2, 3) | 0.3 | 0.0702 | 0.0835 | 0.0709 | 0.0604 | 0.0650 | 0.0669 | 0.0652 | 0.0650 | 0.0607 | |
| 0.8 | 0.2018 | 0.2644 | 0.2014 | 0.1817 | 0.1900 | 0.2236 | 0.1912 | 0.1900 | 0.1741 | ||
| 1.5 | 0.5940 | 0.6651 | 0.5928 | 0.5571 | 0.5611 | 0.6205 | 0.5649 | 0.5611 | 0.5412 | ||
| (1, 4, 7) | 0.3 | 0.0562 | 0.0690 | 0.0567 | 0.0460 | 0.0568 | 0.0610 | 0.0580 | 0.0568 | 0.0496 | |
| 0.8 | 0.1202 | 0.1546 | 0.1210 | 0.1068 | 0.1190 | 0.1353 | 0.1190 | 0.1190 | 0.1036 | ||
| 1.5 | 0.3310 | 0.3986 | 0.3312 | 0.3016 | 0.3252 | 0.3689 | 0.3250 | 0.3252 | 0.2971 | ||
| (20, 30, 40) | (0.1, 0.2, 0.3) | 0.3 | 0.7142 | 0.7419 | 0.7143 | 0.7108 | 0.7071 | 0.7609 | 0.7093 | 0.7071 | 0.7068 |
| 0.8 | 0.9955 | 0.9509 | 0.9960 | 0.9980 | 0.9872 | 0.9970 | 0.9921 | 0.9872 | 0.9960 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.3951 | 0.4407 | 0.3953 | 0.3944 | 0.3994 | 0.4314 | 0.3953 | 0.3994 | 0.3872 | |
| 0.8 | 0.9892 | 0.9612 | 0.9895 | 0.9936 | 0.9984 | 0.9869 | 0.9899 | 0.9984 | 0.9857 | ||
| 1.5 | 0.9910 | 0.9614 | 0.9911 | 0.9950 | 1 | 0.9872 | 0.9911 | 1 | 0.9872 | ||
| (1, 2, 3) | 0.3 | 0.1062 | 0.1227 | 0.1066 | 0.1022 | 0.1060 | 0.1196 | 0.1058 | 0.1060 | 0.1044 | |
| 0.8 | 0.5572 | 0.5975 | 0.5576 | 0.5541 | 0.5521 | 0.6064 | 0.5540 | 0.5521 | 0.5530 | ||
| 1.5 | 0.9843 | 0.9452 | 0.9847 | 0.9860 | 0.9758 | 0.9892 | 0.9806 | 0.9758 | 0.9845 | ||
| (1, 4, 7) | 0.3 | 0.0743 | 0.0821 | 0.0747 | 0.0732 | 0.0756 | 0.0772 | 0.0747 | 0.0756 | 0.0735 | |
| 0.8 | 0.2872 | 0.3234 | 0.2874 | 0.2876 | 0.2904 | 0.3157 | 0.2908 | 0.2904 | 0.2843 | ||
| 1.5 | 0.8011 | 0.8166 | 0.8020 | 0.8044 | 0.8100 | 0.8240 | 0.8004 | 0.8100 | 0.7957 | ||
| (25, 50, 75) | (0.1, 0.2, 0.3) | 0.3 | 0.9182 | 0.8505 | 0.9184 | 0.9294 | 0.9129 | 0.9361 | 0.9153 | 0.9129 | 0.9227 |
| 0.8 | 0.9744 | 0.8811 | 0.9750 | 0.9872 | 0.9704 | 0.9796 | 0.9750 | 0.9704 | 0.9825 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7) | 0.3 | 0.6392 | 0.6680 | 0.6398 | 0.6375 | 0.6365 | 0.6702 | 0.6384 | 0.6365 | 0.6380 | |
| 0.8 | 0.9682 | 0.9054 | 0.9685 | 0.9704 | 0.9649 | 0.9631 | 0.9750 | 0.9649 | 0.9722 | ||
| 1.5 | 1 | 0.9654 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (1, 2, 3) | 0.3 | 0.1631 | 0.2058 | 0.1636 | 0.1649 | 0.1630 | 0.1932 | 0.1652 | 0.1630 | 0.1627 | |
| 0.8 | 0.8022 | 0.7839 | 0.8030 | 0.8105 | 0.7976 | 0.8452 | 0.7991 | 0.7976 | 0.8058 | ||
| 1.5 | 0.9745 | 0.8811 | 0.9750 | 0.9872 | 0.9704 | 0.9796 | 0.9750 | 0.9704 | 0.9825 | ||
| (1, 4, 7) | 0.3 | 0.0970 | 0.1217 | 0.0972 | 0.0959 | 0.0969 | 0.1094 | 0.0989 | 0.0969 | 0.0961 | |
| 0.8 | 0.4922 | 0.5291 | 0.4924 | 0.4896 | 0.4902 | 0.5205 | 0.4941 | 0.4902 | 0.4900 | ||
| 1.5 | 0.9466 | 0.8923 | 0.9470 | 0.9486 | 0.9433 | 0.9435 | 0.9531 | 0.9433 | 0.9500 |
| \(n_i\) | \(\sigma_i^2\) | \(\Delta_i\) | AG | AF | B2 | JF | MBF | PB | WE | WA |
|---|---|---|---|---|---|---|---|---|---|---|
| (10, 10, 10, 10, 10) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.0713 | 0.0867 | 0.0716 | 0.0779 | 0.0863 | 0.0729 | 0.0779 | 0.0718 |
| 0.8 | 0.2172 | 0.3403 | 0.2173 | 0.2424 | 0.3400 | 0.2350 | 0.2424 | 0.2271 | ||
| 1.5 | 0.6316 | 0.8549 | 0.6315 | 0.6984 | 0.8542 | 0.6897 | 0.6984 | 0.6722 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.0562 | 0.0586 | 0.0565 | 0.0543 | 0.0536 | 0.0540 | 0.0543 | 0.0515 | |
| 0.8 | 0.0917 | 0.1191 | 0.0915 | 0.0848 | 0.1074 | 0.0843 | 0.0848 | 0.0791 | ||
| 1.5 | 0.2110 | 0.3094 | 0.2109 | 0.1840 | 0.2835 | 0.1850 | 0.1840 | 0.1747 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.0522 | 0.0541 | 0.0523 | 0.0560 | 0.0540 | 0.0525 | 0.0560 | 0.0502 | |
| 0.8 | 0.0658 | 0.0788 | 0.0657 | 0.0713 | 0.0785 | 0.0671 | 0.0713 | 0.0647 | ||
| 1.5 | 0.1066 | 0.5250 | 0.1065 | 0.1165 | 0.1479 | 0.1072 | 0.1165 | 0.1046 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0953 | 0.1154 | 0.0951 | 0.0923 | 0.1153 | 0.0964 | 0.0923 | 0.0940 | |
| 0.8 | 0.4644 | 0.5647 | 0.4635 | 0.4510 | 0.5645 | 0.4635 | 0.4510 | 0.4611 | ||
| 1.5 | 0.9652 | 0.9788 | 0.9633 | 0.9312 | 0.9785 | 0.9611 | 0.9312 | 0.9563 | ||
| (30, 30, 30, 30, 30) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.3557 | 0.4704 | 0.3556 | 0.3598 | 0.4700 | 0.3573 | 0.3598 | 0.3601 |
| 0.8 | 0.9824 | 0.9871 | 0.9824 | 0.9666 | 0.9870 | 0.9716 | 0.9666 | 0.9803 | ||
| 1.5 | 0.9834 | 0.9872 | 0.9844 | 0.9676 | 0.9871 | 0.9731 | 0.9676 | 0.9815 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.1442 | 0.1996 | 0.1443 | 0.1461 | 0.1994 | 0.1445 | 0.1461 | 0.1456 | |
| 0.8 | 0.7418 | 0.8700 | 0.7420 | 0.7478 | 0.8700 | 0.7439 | 0.7478 | 0.7586 | ||
| 1.5 | 0.9843 | 0.9931 | 0.9848 | 0.9708 | 0.9930 | 0.9700 | 0.9708 | 0.9906 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.0773 | 0.0898 | 0.0779 | 0.0791 | 0.0893 | 0.0784 | 0.0791 | 0.0768 | |
| 0.8 | 0.2581 | 0.3510 | 0.2590 | 0.2618 | 0.3508 | 0.2606 | 0.2618 | 0.2614 | ||
| 1.5 | 0.7472 | 0.8595 | 0.7477 | 0.7509 | 0.8593 | 0.7537 | 0.7509 | 0.7579 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0621 | 0.0666 | 0.0622 | 0.0621 | 0.0664 | 0.0624 | 0.0621 | 0.0614 | |
| 0.8 | 0.1158 | 0.1557 | 0.1160 | 0.1167 | 0.1552 | 0.1164 | 0.1167 | 0.1159 | ||
| 1.5 | 0.3113 | 0.4370 | 0.3112 | 0.3182 | 0.4366 | 0.3153 | 0.3182 | 0.3193 | ||
| (50, 50, 50, 50, 50) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.5882 | 0.6978 | 0.5883 | 0.5996 | 0.6975 | 0.5964 | 0.5996 | 0.5881 |
| 0.8 | 0.9955 | 0.9825 | 0.9960 | 1 | 0.9823 | 0.9990 | 1 | 0.9901 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.2161 | 0.3051 | 0.2163 | 0.2215 | 0.3050 | 0.2178 | 0.2215 | 0.2152 | |
| 0.8 | 0.9432 | 0.9636 | 0.9433 | 0.9472 | 0.9633 | 0.9440 | 0.9472 | 0.9321 | ||
| 1.5 | 0.9890 | 0.9787 | 0.9892 | 0.9901 | 0.9785 | 0.9872 | 0.9901 | 0.9750 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.0912 | 0.1121 | 0.0919 | 0.0929 | 0.1120 | 0.0921 | 0.0929 | 0.0909 | |
| 0.8 | 0.4361 | 0.5477 | 0.4363 | 0.4465 | 0.5474 | 0.4435 | 0.4465 | 0.4374 | ||
| 1.5 | 0.9502 | 0.9665 | 0.9508 | 0.9574 | 0.9663 | 0.9558 | 0.9574 | 0.9473 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0621 | 0.0701 | 0.0628 | 0.0626 | 0.0700 | 0.0621 | 0.0626 | 0.0610 | |
| 0.8 | 0.1617 | 0.2248 | 0.1619 | 0.1657 | 0.2245 | 0.1624 | 0.1657 | 0.1602 | ||
| 1.5 | 0.5205 | 0.6456 | 0.5206 | 0.5292 | 0.6455 | 0.5302 | 0.5292 | 0.5197 | ||
| (4, 6, 10, 14, 16) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.1680 | 0.2295 | 0.1682 | 0.1448 | 0.1922 | 0.1309 | 0.1448 | 0.1313 |
| 0.8 | 0.8345 | 0.9315 | 0.8335 | 0.7925 | 0.8584 | 0.7690 | 0.7925 | 0.7730 | ||
| 1.5 | 0.9833 | 0.9844 | 0.9814 | 0.9795 | 0.9198 | 0.9739 | 0.9795 | 0.9758 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.0848 | 0.1110 | 0.0847 | 0.0790 | 0.0988 | 0.0745 | 0.0790 | 0.0736 | |
| 0.8 | 0.3790 | 0.5455 | 0.3790 | 0.3366 | 0.5029 | 0.3283 | 0.3366 | 0.3282 | ||
| 1.5 | 0.9103 | 0.9756 | 0.9072 | 0.8771 | 0.9474 | 0.8920 | 0.8771 | 0.8947 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.0633 | 0.0625 | 0.0632 | 0.0605 | 0.0501 | 0.0583 | 0.0605 | 0.0570 | |
| 0.8 | 0.1252 | 0.1750 | 0.1253 | 0.1103 | 0.1469 | 0.1081 | 0.1103 | 0.1069 | ||
| 1.5 | 0.3671 | 0.5264 | 0.3674 | 0.3126 | 0.4614 | 0.3090 | 0.3126 | 0.3093 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0523 | 0.0537 | 0.0525 | 0.0544 | 0.0471 | 0.0479 | 0.0544 | 0.0474 | |
| 0.8 | 0.0727 | 0.0902 | 0.0735 | 0.0707 | 0.0791 | 0.0635 | 0.0707 | 0.0630 | ||
| 1.5 | 0.1466 | 0.2189 | 0.1475 | 0.1359 | 0.1930 | 0.1195 | 0.1359 | 0.1197 | ||
| (12,18, 30, 42, 48) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.5134 | 0.6239 | 0.5129 | 0.4915 | 0.6184 | 0.4904 | 0.4915 | 0.4905 |
| 0.8 | 0.9969 | 0.9466 | 0.9959 | 0.9824 | 0.9863 | 0.9910 | 0.9824 | 0.9959 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.1999 | 0.2816 | 0.2000 | 0.1936 | 0.2697 | 0.1906 | 0.1936 | 0.1904 | |
| 0.8 | 0.9238 | 0.9291 | 0.9239 | 0.9091 | 0.9556 | 0.9046 | 0.9091 | 0.9183 | ||
| 1.5 | 0.9977 | 0.9543 | 0.9980 | 0.9853 | 0.9882 | 0.9815 | 0.9853 | 0.9980 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.0812 | 0.1162 | 0.0814 | 0.0817 | 0.1028 | 0.0804 | 0.0817 | 0.0772 | |
| 0.8 | 0.3714 | 0.4900 | 0.3713 | 0.3630 | 0.4614 | 0.3567 | 0.3630 | 0.3459 | ||
| 1.5 | 0.9016 | 0.9321 | 0.9017 | 0.9085 | 0.9309 | 0.8963 | 0.9085 | 0.8825 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0611 | 0.0718 | 0.0618 | 0.0615 | 0.0670 | 0.0605 | 0.0615 | 0.0607 | |
| 0.8 | 0.1502 | 0.2101 | 0.1506 | 0.1445 | 0.2032 | 0.1460 | 0.1445 | 0.1443 | ||
| 1.5 | 0.4561 | 0.5788 | 0.4561 | 0.4425 | 0.5779 | 0.4465 | 0.4425 | 0.4472 | ||
| (20, 30, 50, 70, 80) | (0.1, 0.2, 0.3, 0.4, 0.5) | 0.3 | 0.7441 | 0.8321 | 0.7448 | 0.7426 | 0.8292 | 0.7403 | 0.7426 | 0.7293 |
| 0.8 | 0.9692 | 0.9509 | 0.9695 | 0.9759 | 0.9722 | 0.9731 | 0.9759 | 0.9614 | ||
| 1.5 | 0.9891 | 0.9729 | 0.9895 | 0.9959 | 0.9822 | 0.9921 | 0.9899 | 0.9934 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5) | 0.3 | 0.7590 | 0.8306 | 0.7593 | 0.7602 | 0.8469 | 0.7569 | 0.7602 | 0.7499 | |
| 0.8 | 0.9782 | 0.9444 | 0.9785 | 0.9844 | 0.9874 | 0.9828 | 0.9844 | 0.9747 | ||
| 1.5 | 0.9821 | 0.9449 | 0.9825 | 0.9882 | 0.9882 | 0.9872 | 0.9882 | 0.9787 | ||
| (1, 2, 3, 4, 5) | 0.3 | 0.1040 | 0.1441 | 0.1047 | 0.1027 | 0.1374 | 0.1018 | 0.1027 | 0.1008 | |
| 0.8 | 0.5771 | 0.6735 | 0.5773 | 0.5709 | 0.7017 | 0.5684 | 0.5709 | 0.5719 | ||
| 1.5 | 0.9472 | 0.9101 | 0.9475 | 0.9473 | 0.9861 | 0.9499 | 0.9473 | 0.9550 | ||
| (1, 4, 7, 11, 15) | 0.3 | 0.0651 | 0.0805 | 0.0657 | 0.0654 | 0.0765 | 0.0652 | 0.0654 | 0.0643 | |
| 0.8 | 0.2322 | 0.3155 | 0.2326 | 0.2275 | 0.3055 | 0.2278 | 0.2275 | 0.2250 | ||
| 1.5 | 0.7033 | 0.7689 | 0.7038 | 0.6962 | 0.7887 | 0.6987 | 0.6962 | 0.6995 |
| \(n_i\) | \(\sigma_i^2\) | \(\Delta_i\) | AG | AF | B2 | MBF | PB | WA |
|---|---|---|---|---|---|---|---|---|
| (10, 10, 10, 10, 10, 10, 10) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.0916 | 0.1176 | 0.0921 | 0.1175 | 0.0954 | 0.0960 |
| 0.8 | 0.3745 | 0.6158 | 0.3750 | 0.6154 | 0.4288 | 0.4279 | ||
| 1.5 | 0.8713 | 0.9725 | 0.8715 | 0.9723 | 0.9247 | 0.9068 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.0668 | 0.0655 | 0.0666 | 0,0651 | 0.0660 | 0.0658 | |
| 0.8 | 0.1501 | 0.2255 | 0.1502 | 0.2253 | 0.1578 | 0.1603 | ||
| 1.5 | 0.4021 | 0.6756 | 0.4023 | 0.6753 | 0.4519 | 0.4519 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0565 | 0.0520 | 0.0568 | 0.0520 | 0.0597 | 0.0588 | |
| 0.8 | 0.0819 | 0.0944 | 0.0820 | 0.0942 | 0.0831 | 0.0834 | ||
| 1.5 | 0.1612 | 0.2385 | 0.1609 | 0.2381 | 0.1693 | 0.1695 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0560 | 0.0485 | 0.0561 | 0.0483 | 0.0577 | 0.0570 | |
| 0.8 | 0.0632 | 0.0608 | 0.0635 | 0.0604 | 0.0625 | 0.0633 | ||
| 1.5 | 0.0855 | 0.1044 | 0.0858 | 0.1042 | 0.0860 | 0.0880 | ||
| (30, 30, 30, 30, 30, 30, 30) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.2204 | 0.3303 | 0.2209 | 0.3301 | 0.2301 | 0.2298 |
| 0.8 | 0.9598 | 0.9703 | 0.9596 | 0.9701 | 0.9616 | 0.9640 | ||
| 1.5 | 0.9957 | 0.9778 | 0.9960 | 0.9768 | 0.9911 | 0.9941 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.0978 | 0.1334 | 0.0980 | 0.1324 | 0.0977 | 0.0985 | |
| 0.8 | 0.4679 | 0.6533 | 0.4680 | 0.6530 | 0.4867 | 0.4904 | ||
| 1.5 | 0.9702 | 0.9822 | 0.9704 | 0.9820 | 0.9630 | 0.9757 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0651 | 0.0712 | 0.0657 | 0.0711 | 0.0670 | 0.0636 | |
| 0.8 | 0.1652 | 0.2417 | 0.1657 | 0.2414 | 0.1725 | 0.1706 | ||
| 1.5 | 0.5243 | 0.6901 | 0.5247 | 0.6900 | 0.5453 | 0.5461 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0552 | 0.0575 | 0.0555 | 0.0572 | 0.0550 | 0.0537 | |
| 0.8 | 0.0852 | 0.1060 | 0.0857 | 0.1050 | 0.0853 | 0.0860 | ||
| 1.5 | 0.1810 | 0.2765 | 0.1813 | 0.2755 | 0.1841 | 0.1839 | ||
| (50, 50, 50, 50, 50, 50, 50) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.3722 | 0.5118 | 0.3725 | 0.5111 | 0.3782 | 0.3787 |
| 0.8 | 0.9969 | 0.9853 | 0.9978 | 0.9850 | 0.9988 | 0.9948 | ||
| 1.5 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.1362 | 0.1907 | 0.1365 | 0.1907 | 0.1387 | 0.1380 | |
| 0.8 | 0.7502 | 0.8843 | 0.7507 | 0.8840 | 0.7686 | 0.7657 | ||
| 1.5 | 0.9943 | 0.9863 | 0.9948 | 0.9852 | 0.9988 | 0.9968 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0762 | 0.0914 | 0.0764 | 0.0910 | 0.0787 | 0.0758 | |
| 0.8 | 0.2660 | 0.3817 | 0.2661 | 0.3812 | 0.2739 | 0.2697 | ||
| 1.5 | 0.8052 | 0.9063 | 0.8058 | 0.9051 | 0.8202 | 0.8145 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0600 | 0.0629 | 0.0601 | 0.0612 | 0.0605 | 0.0594 | |
| 0.8 | 0.1093 | 0.1499 | 0.1099 | 0.1479 | 0.1123 | 0.1099 | ||
| 1.5 | 0.2902 | 0.4275 | 0.2906 | 0.4271 | 0.2981 | 0.2971 | ||
| (4, 6, 8, 10, 12, 14, 16) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.1201 | 0.1540 | 0.1208 | 0.1419 | 0.1025 | 0.1067 |
| 0.8 | 0.6065 | 0.7846 | 0.6068 | 0.7525 | 0.5561 | 0.5596 | ||
| 1.5 | 0.9622 | 0.9456 | 0.9617 | 0.9273 | 0.9852 | 0.9544 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.0718 | 0.0814 | 0.0720 | 0.0750 | 0.0629 | 0.0704 | |
| 0.8 | 0.2169 | 0.3329 | 0.2173 | 0.3132 | 0.1798 | 0.1944 | ||
| 1.5 | 0.6726 | 0.8591 | 0.6734 | 0.8274 | 0.6240 | 0.6532 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0622 | 0.0549 | 0.0621 | 0.0509 | 0.0587 | 0.0609 | |
| 0.8 | 0.1035 | 0.1204 | 0.1041 | 0.1114 | 0.0856 | 0.0900 | ||
| 1.5 | 0.2357 | 0.3452 | 0.2356 | 0.3250 | 0.1935 | 0.2023 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0561 | 0.0497 | 0.0564 | 0.0481 | 0.0518 | 0.0563 | |
| 0.8 | 0.0653 | 0.0714 | 0.0654 | 0.0677 | 0.0607 | 0.0641 | ||
| 1.5 | 0.1078 | 0.1385 | 0.1080 | 0.1305 | 0.0939 | 0.0947 | ||
| (12, 18, 24, 30, 36, 42, 48) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.3332 | 0.4900 | 0.3336 | 0.4627 | 0.3226 | 0.3149 |
| 0.8 | 0.9737 | 0.9882 | 0.9740 | 0.9750 | 0.9798 | 0.9740 | ||
| 1.5 | 0.9950 | 1 | 0.9940 | 0.9920 | 0.9906 | 0.9912 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.1277 | 0.1966 | 0.1284 | 0.1814 | 0.1228 | 0.1205 | |
| 0.8 | 0.7067 | 0.8465 | 0.7070 | 0.8059 | 0.6955 | 0.6979 | ||
| 1.5 | 0.9990 | 0.9970 | 0.9995 | 0.9649 | 0.9950 | 0.9988 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0712 | 0.0870 | 0.0718 | 0.0798 | 0.0675 | 0.0678 | |
| 0.8 | 0.2458 | 0.3612 | 0.2452 | 0.3438 | 0.2307 | 0.2322 | ||
| 1.5 | 0.7381 | 0.8667 | 0.7380 | 0.8494 | 0.7108 | 0.7255 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0544 | 0.0636 | 0.0541 | 0.0593 | 0.0557 | 0.0530 | |
| 0.8 | 0.0998 | 0.1537 | 0.0993 | 0.1388 | 0.0956 | 0.0938 | ||
| 1.5 | 0.2679 | 0.4137 | 0.2675 | 0.3839 | 0.2616 | 0.2539 | ||
| (20, 30, 40, 50, 60, 70, 80) | (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) | 0.3 | 0.5722 | 0.7108 | 0.5720 | 0.7148 | 0.5596 | 0.5577 |
| 0.8 | 0.9990 | 0.9614 | 0.9987 | 0.9911 | 0.9960 | 0.9980 | ||
| 1.5 | 1 | 0.9614 | 1 | 1 | 1 | 1 | ||
| (0.1, 0.4, 0.7, 1.1, 1.5, 1.9, 2.3) | 0.3 | 0.1939 | 0.2736 | 0.1935 | 0.2647 | 0.1885 | 0.1875 | |
| 0.8 | 0.9270 | 0.9103 | 0.9262 | 0.9315 | 0.9254 | 0.9286 | ||
| 1.5 | 0.9921 | 0.9261 | 0.9919 | 0.9509 | 0.9901 | 0.9941 | ||
| (1, 2, 3, 4, 5, 6, 7) | 0.3 | 0.0835 | 0.1185 | 0.0832 | 0.1079 | 0.0834 | 0.0799 | |
| 0.8 | 0.4142 | 0.5701 | 0.4140 | 0.5418 | 0.4040 | 0.4014 | ||
| 1.5 | 0.9426 | 0.9747 | 0.9423 | 0.9664 | 0.9433 | 0.9380 | ||
| (1, 4, 7, 11, 15, 19, 23) | 0.3 | 0.0551 | 0.0709 | 0.0542 | 0.0658 | 0.0563 | 0.0551 | |
| 0.8 | 0.1388 | 0.2142 | 0.1369 | 0.2012 | 0.1352 | 0.1343 | ||
| 1.5 | 0.4446 | 0.6286 | 0.4442 | 0.6072 | 0.4363 | 0.4414 |
5 Discussion
In this paper, an extensive Monte-Carlo simulation study is conducted to investigate the performance of the tests for equality of normal distributed and independent groups’ means under unequal variances under several scenarios. It is rather rare to encounter normally distributed data and (Bono et al. 2017) showed that the data obtained from health, educational, and social sciences research are often not normally distributed. (Blanca et al. 2013) discussed the negative effect of non-normality on the power and Type I error probability of the parametric tests. It is reality that the normality assumption is crucial for the considered tests in this study. Here, it is focused on the performance of the considered tests under normality to fill the gap mentioned the introduction part. Firstly, the ability of the test to control the Type I error probability is examined and the boxplots in Figs.3, 4 and 5 are used to summarize the results. The tests which can control the Type I error probability are obtained as robust tests with respect to the (Bradley 1978)’s limits. Then, the penalized power of the robust tests is calculated. The reason for using this method was to consider two possible types of error.
According to the results of the Monte-Carlo simulation study, the AF, AG, B2, MBF, PB, WA test control the Type I error probability for \(k=3, 5, 7\) in the interval \([0.0495, 0.0505]\). The GF can control only for \(k=3\) and the WE test can control only for \(k=3, 5\). Besides the controlling of the Type I error probability of these tests, the penalized power properties are also investigated under similar scenarios. The results are indicated that the AF and MBF tests are superior than others in the higher heteroscedasticity levels. Also, it is concluded that the penalized power of the other tests is quite close to the intended level.
As a result of this study, the robust tests are obtained and can be used in most of the situations except for a higher level of heteroscedasticity and small sample sizes. Using the results of the simulation study, researchers can use appropriate tests for their studies.
6 Summary and Future works
The doex package contains the several tests for testing equality of normally distributed groups’ means under unequal variances. Most of these tests are not available in any R package. Thus, we fill this gap by implementing the package in the statistical software literature. The fact that the package contains tests such as the GF, PB, and FA with complex calculation steps provides a significant benefit to multidisciplinary researchers. Furthermore, the performance of the considered tests is investigated under normal distributions in detail in an extensive Monte-Carlo simulation study. Considering the number of methods discussed, this article is the most comprehensive performance investigation study in the literature. Recommendations were made to the researchers by using the interesting outputs from the simulation study.
It is always optimistic idea to encountered normal distribution in real life. The performance of the considered tests can be also investigated under the various distributions or to focus the tests are proposed for non-normal distributions. Thus, it is planned to expand the package by adding methods used to test the equality of the log-normal (Tian and Wu 2007) and inverse-Gaussian (Tian 2006; Ma and Tian 2009) distributed and independent groups’ means in further studies.
7 Acknowledgement
This study is supported by the Eskisehir Technical University Scientific Research Projects Commission under grant No. 20DRP047.
CRAN packages used
doex, asbio, coin, lawstat, onewaytests, welchADF, WRS2, car, rstatix, inferr
CRAN Task Views implied by cited packages
ClinicalTrials, Econometrics, Finance, MixedModels, Robust, Survival, TeachingStatistics
Note
This article is converted from a Legacy LaTeX article using the texor package. The pdf version is the official version. To report a problem with the html, refer to CONTRIBUTE on the R Journal homepage.