1 Introduction
One-way independent groups designs are fundamental and widely used designs in observational and experimental research. One-way fixed effect analysis of variance (ANOVA) is the most common test used in these designs. One-way ANOVA is a parametric method based on comparing the means of independent groups. The assumptions of normality in each independent group and homogeneity of variance between groups must be satisfied to perform one-way ANOVA. However, it is often not possible to find datasets that meet both of these assumptions in practice. Therefore, there are a great number of alternative tests to one-way ANOVA for analyzing one-way independent groups designs. Among these tests, Welch’s heteroscedastic F test (Welch 1951), Welch’s heteroscedastic F test with trimmed means and Winsorized variances (Welch 1951), Alexander-Govern test (Alexander and Govern), James second order test (James 1951), Brown-Forsythe test (Brown and Forsythe 1974b,a) and Kruskal-Wallis test (Kruskal and Wallis 1952) are provided in onewaytests package version 1.5 (Dag et al. 2017). The onewaytests package is compact and comprehensive since it provides descriptive statistics, assessment of the assumptions, and summarizes results using various graphics in addition to performing statistical analysis. This helps researchers evaluate their data from multiple angles.
In this paper, we present the upgraded version of onewaytests package, onewaytests 3. This version includes Alvandi’s F test (Sadooghi-Alvandi et al. 2012), Alvandi’s generalized p-value test (Sadooghi-Alvandi et al. 2012), approximate F test (Asiribo and Gurland 1990), adjusted Welch’s heteroscedastic F test (Welch 1951), B square test (Özdemir and Kurt 2006), Box F test (Box 1954), Cochran F test (Cochran 1937), generalized test equivalent to parametric bootstrap test (Weerahandi and Krishnamoorthy 2019), generalized test equivalent to fiducial test (Weerahandi and Krishnamoorthy 2019), Johansen F test (Johansen 1980), modified Brown-Forsythe test (Mehrotra 1997), permutation F test (Berry et al. 2002), Scott-Smith F test (Scott and Smith 1971), Welch-Aspin test (Aspin 1948), and Weerahandi’s generalized F test (Weerahandi 1995) in addition to the tests in version 1.5. Along with the newly added 15 tests, there are a total of 22 tests for one-way independent groups designs in this extended version. Additionally, two independent samples tests including Student’s t-test, Welch’s t-test, and Mann-Whitney U test are added to the package.
The graphical approaches in the package are restructured from scratch to enhance the functionality and customization options of the graphical approaches. We add new features and enhancements providing even more flexibility in visualizing data while maintaining the core functionality. With these updates and improvements, the new version of the package offers a more comprehensive and versatile set of tools for data visualization, empowering users to create visually compelling and informative graphics effortlessly.
There is a user-friendly web application based on onewaytests package. This web-tool is updated with the addition of 15 one-way independent groups tests, along with graphical improvements. Also, web-tool is made compatible with mobile devices. The web-tool is freely available at http://www.softmed.hacettepe.edu.tr/onewaytests.
The paper is organized as follows. First, one-way tests in independent groups designs with theoretical backgrounds are presented. Second, an extensive application of onewaytests 3 package is demonstrated by using a real dataset. Third, the upgraded web application is introduced. At last, general conclusions and further research are provided.
2 One-way tests in independent groups designs
In this section, the statistical tests that are used to test the equality of several populations in the one-way independent groups designs are explained. Among these tests, Alvandi’s F test (Sadooghi-Alvandi et al. 2012), Alexander-Govern test (Alexander and Govern), Alvandi’s generalized p-value test (Sadooghi-Alvandi et al. 2012), ANOVA, approximate F test (Asiribo and Gurland 1990), adjusted Welch’s heteroscedastic F test (Welch 1951), B square test (Özdemir and Kurt 2006), Brown-Forsythe test (Brown and Forsythe 1974b,a), Box F test (Lix and Keselman 1998), Cochran F test (Cochran 1937), Weerahandi’s generalized F test (Weerahandi 1995), James second order test (James 1951, 1954), Johansen F test (Johansen 1980), Modified Brown-Forsythe test (Mehrotra 1997), Permutation F test (Berry et al. 2002), Scott Smith F test (Scott and Smith 1971), Welch-Aspin test (Aspin 1948), Welch’s heteroscedastic F test (Welch 1951), generalized test equivalent to parametric bootstrap test (Weerahandi and Krishnamoorthy 2019), and generalized test equivalent to fiducial test (Weerahandi and Krishnamoorthy 2019) test the null hypothesis \(H_0: \mu_1=\mu_2= \ldots =\mu_k\) versus alternative \(H_1:\) at least one \(\mu_j\) (\(j=1,2,\ldots, k\)) is different. All of these tests are parametric, with the exception of Permutation F test. Permutation F test evaluates mean differences without depending on the theoretical F distribution. Rather, it relies on an empirical F distribution produced using permutations. Therefore, this test does not require the assumption of normality (Winkler et al. 2014). Welch’s heteroscedastic F test with trimmed means and Winsorized variances (Welch 1951) tests the equality of population trimmed means, \(H_0: \mu_{t1}=\mu_{t2}= \ldots =\mu_{tk}\) versus alternative \(H_1:\) at least one \(\mu_{tj}\) is different, where \(\mu_{tj}\) represents the trimmed mean of the jth population (\(j=1,2,\ldots, k\)). Kruskal-Wallis test (Kruskal and Wallis 1952) tests the null hypothesis \(H_0: \theta_1=\theta_2= \ldots =\theta_k\) versus alternative \(H_1:\) at least one \(\theta_j\) (\(j=1,2,\ldots, k\)) is different. For Kruskal-Wallis test, \(\theta_j\) represents sum of the ranks of the jth population. One-way independent tests that are used in this study are shown in Table 1.
| Existing Methods in the (Dag et al. 2018) | |||
|---|---|---|---|
| Name | Function | Method* | Test Type\(\dagger\) |
| One-way ANOVA | aov.test | aov | \(Param\) |
| Welch’s heteroscedastic F test | welch.test | welch | \(Param\) |
| Welch’s heteroscedastic F test with trimmed and Winsorized Variances | welch.test | welch_tw | \(Param\) |
| Brown-Forsythe test | bf.test | bf | \(Param\) |
| Alexander-Govern test | ag.test | ag | \(Param\) |
| James Second Order test | james.test | james | \(Param\) |
| Kruskal-Wallis test | kw.test | kw | \(NonParam\) |
| Methods Formulated in the Extension | |||
| Name | Function | Method* | Test Type\(\dagger\) |
| Alvandi’s F test | af.test | af | \(Param\) |
| Alvandi’s generalized p-value test | agp.test | agp | \(Param\) |
| Approximate F test | ap.test | ap | \(Param\) |
| Adjusted Welch’s heteroscedastic F test | aw.test | aw | \(Param\) |
| B square test | b2.test | b2 | \(Param\) |
| Box F test | box.test | box | \(Param\) |
| Cochran F test | cochran.test | cochran | \(Param\) |
| Generalized test equivalent to Param bootstrap test | gp.test | gtb | \(Param\) |
| Generalized test equivalent to fiducial test | gp.test | gtf | \(Param\) |
| Johansen F Test | johansen.test | johansen | \(Param\) |
| Modified Brown-Forsythe test | mbf.test | mbf | \(Param\) |
| Permutation F test (PF) | pf.test | pf | \(NonParam\) |
| Scott-Smith test | ss.test | ss | \(Param\) |
| Welch-Aspin test | wa.test | wa | \(Param\) |
| Weerahandi’s generalized F test | wgf.test | wgf | \(Param\) |
Alvandi’s F test
Sadooghi-Alvandi et al. (2012) presented a practical test that can be used when the group variances are heterogeneous. The test statistic for Alvandi’s F test is \[F_{AF}=\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2)(\bar{X}_{j}-\bar{X}_{w})^2, \label{eq:texor_dummy_1} \tag{1}\] where \[\label{eq:Xw} \bar{X}_{w}=\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2)\bar{X_j}/\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2). \tag{2}\] The weights for each group are \(w_j=({n_j/S_j^2})/{\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2)}\). The null hypothesis is rejected when \(F_{AF}\geqslant(k-1) F_{k-1, r}\) where r is obtained with \(r={2m_T}/({m_T-(k-1)})\) and \(m_T={\sum\limits_{j=1}^k (1-w_j)(n_{j}-1)/(n_j-3)}\).
Alexander-Govern test
Alexander and Govern proposed the following test statistic as an alternative to ANOVA when group variances are not homogeneous:
\[\label{AG} \chi_{AG}^2 = \sum_{j=1}^{k} z_j^2. \tag{3}\] In Equation (3), \[z_j = c+\frac{(c^3+3c)}{b}-\frac{(4c^7+33c^5+240c^3+855c)}{(10b^2+8bc^4+1000b)},\] where \(c=[\alpha \times ln (1+t_j^2/v_j)] ^{1/2}\), \(b=48\alpha^2\), \(\alpha =v_j-0.5\) and \(v_j = (n_j - 1)\). The t statistic for each group is
\[\label{t} t_j=\frac{\bar{X}_{j}-X^+}{S_j^{'}}. \tag{4}\] The variance-weighted mean is \(X^+=\sum_{j=1}^{k}w_j\bar{X}_{j}\), the standard error for the jth group is \(S_j^{'}=\left[{\sum_{i=1}^{n_j}(X_{ij}-\bar{X}_{j})^2}/{n_j (n_j-1)}\right]^{1/2}\) and weights for each group are \[\label{eq:wj} {w}_{j}={(1/S_j^{'2})/}{\sum_{j=1}^{k}(1/S_j^{'2})}. \tag{5}\] The null hypothesis is rejected when \(\chi_{AG}^2 \geqslant \chi_{k-1}^2\).
Alvandi’s generalized p-value test
Sadooghi-Alvandi et al. (2012) presented a generalized p-value test. Alvandi’s generalized p-value can be computed by simulation according to the following two steps (Cavus et al. 2021). The first step is computing \(T_w\) value (observed value of Cochran’s statistic) \[T_w=\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2)(\bar{X}_{j}-\bar{X}_{w})^2, \label{eq:texor_dummy_2} \tag{6}\] where \(\bar{X}_{w}\) is calculated as Equation (2). Second step starts with generating independent variables \(x_j\) from N(0,1) and \(U_j\) from \(\chi^2_{n_j-1}\) (j=1,2,...,k). Then T value is calculated as follows: \[T=\displaystyle\sum\limits_{j=1}^n \frac{n_j-1}{U_j}[X_j-q_j\tilde{X}]^2. \label{eq:texor_dummy_3} \tag{7}\] \(\tilde{X}\) in the equation can be computed as \(\displaystyle\sum\limits_{j=1}^k q_jx_j\) where \[q_j=\sqrt[]{{(n_j/S_j^2)}/{\displaystyle\sum\limits_{j=1}^k (n_j/S_j^2)}}. \label{eq:texor_dummy_4} \tag{8}\] The second step repeats a large number of times (\({M}\)). Let \({M(t)}\) be the number of times that \(T \geqslant {T_w}\), then Alvandi’s generalized p-value is obtained as \({M(t)}/{M}\).
ANOVA
The one-way fixed effects analysis of variance (ANOVA) is a powerful test if the assumptions of normality and variance homogeneity are met. The test statistic for ANOVA is \[\label{anova} F=\frac{\sum\limits_{j=1}^k(n_j)(\bar{X}_{j}-\bar{X}_{..})^2/(k-1)}{\sum\limits_{i=1}^n\sum\limits_{j=1}^k(X_{ij}-\bar{X}_{j})^2/(n-k)}. \tag{9}\] In Equation (9), \(n\) is the total number of observations and \(\bar{X}_{..}\) is the overall mean, where \(\bar{X}_{..}=\displaystyle\sum_{j}^{k} n_j(\bar{X}_{j})/n\). The null hypothesis is rejected when \(F \geqslant F_{k-1, \ n-k}\).
Approximate F test
(Asiribo and Gurland 1990) proposed the following test statistic that has approximately F distribution with \(f_{1}\) and \(f_{2}\) degrees of freedom: \[U_{0}=\frac{{n}{-k}}{k-1}\frac{{\displaystyle\sum\limits_{j=1}^k {n_{j}{(\bar{X}_{j}-\bar{X}_{..})^2}}}{}}{{\displaystyle\sum\limits_{j=1}^k f_jS_j^2}}, \label{eq:texor_dummy_5} \tag{10}\]
\[f_1=\frac{(\displaystyle\sum\limits_{j=1}^k 1-\frac{n_j}{n} )\sigma_j^2)^2}{\frac{1}{n^2}(\displaystyle\sum\limits_{j=1}^k n_j\sigma_j^2)^2+\displaystyle\sum\limits_{j=1}^k (1-\frac{2n_j}{n})\sigma_j^4} , \ f_2=\frac{(\displaystyle\sum\limits_{j=1}^k f_j\sigma_j^2)^2}{\displaystyle\sum\limits_{j=1}^k f_j\sigma_j^4}.\] The null hypothesis is rejected when \(U_{0}>\hat{b}F_{f_{1},\ f_{2}}\) where \(b=\frac{{n}{-k}}{n(k-1)}({\displaystyle\sum\limits_{j=1}^k (n-n_j)\sigma_j^2})/({\displaystyle\sum\limits_{j=1}^k f_j\sigma_j^2}).\)
Adjusted Welch’s heteroscedastic F test
Welch (1951) proposed a heteroscedastic alternative to ANOVA that is robust to the violation of variance homogeneity assumption. The test statistic for adjusted Welch’s heteroscedastic F test is calculated as follows (Cavus et al. 2021): \[\label{eq:welch_1} F_w=\frac{\displaystyle\sum_{j=1}^kw_{j}(\bar{X}_{j}-\sum_{j=1}^kh_{j}\bar{X}_{j})^2}{(k-1)+2(k-2)(1/(k+1))\displaystyle\sum_{j=1}^k(1/n_{j}-1)(1-h_{j})^2}. \tag{11}\] In Equation (11), \(w_{j}=n_{j}/((n_{j}-1)/(n_{j}-3)s_j^2)\), and \(h_{j}=w_{j}/\displaystyle\sum_{j=1}^kw_{j}\). The null hypothesis is rejected when \(F_w\) > \(F_{{k-1}, 1/ \nu}\) where \(\nu\) is obtained with \(v=\frac{(k^2-1)/3}{\displaystyle\sum_{j=1}^k(1-h_{j})^2/(n_{j}-1)}\).
B square test
(Özdemir and Kurt 2006) developed a new and simple approximation procedure which intends to create an easy and applicable alternative to ANOVA. The test statistic is \[B^2=\displaystyle\sum\limits_{j=1}^k[c \lbrace log (1+\frac{(\frac{\bar{X}_j{-X^+}}{S_{\bar{X}_j}})^2}{v_j})\rbrace^\frac 1 2]^2 , \label{eq:texor_dummy_6} \tag{12}\] where \(c=({4v_j^2+({5(2z_c^2+3)})/{24}})/({4v_j^2+v_j+({4z_c^2+9})/{12}})v_j^\frac1 2\), \(X^+=\displaystyle\sum\limits_{j=1}^k w_j\bar{X}_j\), the weights are \(w_j={({1}/{S_{\bar{X}_j}^2}})/({\displaystyle\sum\limits_{j=1}^k ({1}/{S_{\bar{X}_j}^2})})\) and \(v_j=n_j-1\). The null hypothesis is rejected when \(B^2>\chi_{k-1}^2\).
Brown-Forsythe test
Brown and Forsythe (1974b,a) proposed the following test statistic as a modification of ANOVA: \[F_{BF}=\frac{\displaystyle\sum_{j=1}^{k} n_{j}(\bar{X}_{j}-\bar{X}_{..})^2}{\displaystyle\sum_{j=1}^{k}(1-n_j/n)S_j^2}. \label{eq:texor_dummy_7} \tag{13}\] The null hypothesis is rejected when \(F_{BF}>F_{k-1, f}\). The degrees of freedom for denominator is calculated as \(\left(\displaystyle\sum_{j=1}^{k}c_j^2/(n_j-1)\right)^{-1}\) where \(c_j=({(1-n_j/n)S_j^2})/{\left[ \displaystyle\sum_{j=1}^{k} (1-n_j/n)S_j^2\right ]}\).
Box F test
Box F test statistic was presented by (Box 1954) as follows:
\[F_{BOX}=\frac{\displaystyle\sum\limits_{j=1}^kn_j(\bar{X}_{j}-\bar{X}_{})^2}{\displaystyle\sum\limits_{i=1}^k (1-\frac{n_j}{n})S_j^2}. \label{eq:texor_dummy_8} \tag{14}\]
The null hypothesis is rejected when \(F_{BOX}>F_{v_1, v_2}\) where \(v_1={[\displaystyle\sum\limits_{j=1}^k (n-n_{j})S_j^2]^2}/({\displaystyle\sum\limits_{j=1}^k n_{j}S_j^2 )^2+n\displaystyle\sum\limits_{j=1}^k (n-2n_{j})S_j^4})\) and \(v_2={[\displaystyle\sum\limits_{j=1}^k (n_{j}-1)S_j^2]^2}/{\displaystyle\sum\limits_{j=1}^k (n_{j}-1)S_j^4}\).
Cochran F test
Cochran (1937) proposed the following test statistic: \[\label{eq:Cochran} T_{Cochran}={\displaystyle\sum\limits_{j=1}^k ({n_j}/{S_j^2})}(\bar{X}_j-\bar{X}_w)^2, \tag{15}\] where \(\bar{X}_{w}\) is calculated as Equation (2). The null hypothesis is rejected when \(T_{Cochran}>\chi_{k-1}^2\).
Generalized test equivalent to parametric bootstrap test
(Weerahandi and Krishnamoorthy 2019) showed that the parametric bootstrap test presented by (Krishnamoorthy et al. 2007) is a regular generalized test. The test is carried out as follows. The null hypothesis is rejected when the generalized p-value given by the following formula is less than the desired nominal level, say 0.05:
\[\label{Pval} p=Pr \left( \sum_{j=1}^k \frac{r_j {Z_j}^2}{U_j} - \frac { \displaystyle\sum_{j=1}^k \frac{ n_j r_j } {s_j^2 U_j} \left( \frac{s_j Z_j}{\sqrt{n_j}} \right) } { \sum_{j=1}^k \frac{ n_j r _j } {s_j^2 U_j} } \geq \sum_{j=1}^k \frac{n_j \left(\bar{x}_j - \bar{x}_s \right)^2 }{s_j^2} \right) , \tag{16}\] where \(Z_i \sim N(0,1)\), \(U_i \sim \chi_{r_i}^2\), \(r_j = n_j -1\), \(\bar{x}_j\) and \(s_j^2\) are the observed sample mean and the observed sample variance from the \(j^{th}\) population, and \(\bar{x}_s\) is the overall weighted sample mean.
Generalized test equivalent to fiducial test
(Weerahandi and Krishnamoorthy 2019) also showed that the fiducial test developed by (Li et al. 2011) is also a regular generalized test. The null hypothesis is rejected when the generalized p-value given by the formula below is less than the desired nominal level:
\[\label{Pval2} p=Pr \left(\sum_{j=1}^k T_j^2 - \frac{ (\sum_{j=1}^k T_j \sqrt[]{n_j} /s_j )^2} {\sum_{j=1}^k n_j/s_j^2} \geq \sum_{j=1}^k \frac{n_j \left(\bar{x}_j - \bar{x}_s \right)^2 }{s_j^2} \right) , \tag{17}\] where \(T_j \sim t_{r_j}\).
James second order test
James (1951) presented an alternative test to ANOVA. This test statistic is \[J=\sum_{j}t_j^2, \label{eq:texor_dummy_9} \tag{18}\] where \(t_j\) is given in Equation (4). The test statistic, J, is compared to a critical value, \(h(\alpha)\), where \[\begin{split} h(\alpha)& =r+\frac{1}{2}(3\chi_4+\chi_2) T + \frac{1}{16}(3\chi_4+\chi_2)^2 \left(1-\frac{k-3}{r}\right)T^2\\ &+\frac{1}{2}(3\chi_4+\chi_2) (8R_{23}-10R_{22}+4R_{21}-6R_{12}^2+8R_{12}R_{11}-4R_{11}^2) \\ &+(2R_{23}-4R_{22}+2R_{21}-2R_{12}^2+4R_{12}R_{11}-2R_{11}^2)(\chi_2-1)\\ &+\frac{1}{4}(-R_{12}^2+4R_{12}R_{11}-2R_{12}R_{10}-4R_{11}^2+4R_{11}R_{10}-R_{10}^2)(3\chi_4-2\chi_2-1)\\ &+(R_{23}-3R_{22}+3R_{21}-R_{20})(5\chi_6+2\chi_4+\chi_2)\\ &+\frac{3}{16}(R_{12}^2-4R_{23}+6R_{22}-4R_{21}+R_{20})(35\chi_8+15\chi_6+9\chi_4+5\chi_2)\\ &+\frac{1}{16}(-2R_{22}+4R_{21}-R_{20}+2R_{12}R_{10}-4R_{11}R_{10}+R_{10}^2)(9\chi_8-3\chi_6-5\chi_4-\chi_2)\\ &+\frac{1}{4}(-R_{22}+R_{11}^2)(27\chi_8+3\chi_6+\chi_4+\chi_2)+\frac{1}{4}(R_{23}-R_{12}R_{11})(45\chi_8+9\chi_6+7\chi_4+3\chi_2). \end{split}\] For any integers s and t, \(R_{st}=\sum_{j}(n_j-1)^{-s}w_j^t\), \(\chi_{2s}=r^s/\left[ \right (k-1)(k+1)\ldots(k+2s-3)]\) where r is the \((1-\alpha)\) centile of a \(\chi^2\) distribution with \(k-1\) degrees of freedom, \(T=\sum_j(1-w_j)^2/(n_j-1)\) and \(w_j\) is defined in Equation (5). The null hypothesis is rejected when \(J \geqslant h(\alpha)\).
Johansen F test
(Johansen 1980) proposed the following test statistics: \[T_{JF}=\frac{\sum_{j=1}^k(w_j(\bar{x}_j-\sum_{j=1}^k(h_j\bar{x}_j))^{2})}{c}, \label{eq:texor_dummy_10} \tag{19}\] where \(w_j=n_j/\sigma_j\), \(h_j=w_j/\sum\limits_{j=1}^k(w_j)\), \(c=(k-1)+2A-6A/(k+1)\) and \(A=\displaystyle\sum\limits_{j=1}^k({(1-h_j)^{2}}/{(n_j-1})\). The null hypothesis is rejected when \(T_{JF} \geqslant F_{v_{1},v_{2}}\) where \(v_1=k-1\) and \(v_2=k-1(k+1)/3A\).
Kruskal-Wallis test
Kruskal and Wallis (1952) presented the nonparametric alternative to ANOVA. The test statistics is
\[\label{KW} \chi_{KW}^2=\frac{1}{S^2}\left(\sum_{j=1}^k \frac{R_j^2}{n_j}-\frac{n(n+1)^2}{4}\right), \tag{20}\]
where \(r_{ij}\) denote the rank of \(X_{ij}\) when \(n=n_1+ \ldots +n_k\) observations are ranked from smallest to largest. \(R_j=\sum_{i=1}^{n_j} r_{ij}\) is the sum of ranks assigned to the observations in the jth group and \(\bar{R}_j=R_j/n_j\) is the average rank for these observations and
\[S^2=\frac{1}{n-1}\left(\sum_{j=1}^k \sum_{i=1}^{n_j} r_{ij}^2 - \frac{n(n+1)^2}{4}\right).\] The null hypothesis is rejected when \(\chi_{KW}^2\geq \chi_{k-1}^{2}\).
Modified Brown-Forsythe test
(Mehrotra 1997) proposed a modification of Brown-Forsythe test to overcome the problem of higher than acceptable rate of false positives. The test statistics is \[F_{MBF}=\frac{\displaystyle\sum\limits_{j=1}^kn_{j}(\bar{X}_{j}-\bar{X}_{..})^2}{\displaystyle\sum\limits_{j=1}^k (1-\frac{n_j}{n})s_j^2}. \label{eq:texor_dummy_11} \tag{21}\] The null hypothesis is rejected when \(F_{MBF}>F_{f_1,f_2}\). \[\label{mbf_df} f_1=\frac{(\displaystyle\sum\limits_{j=1}^k \sigma_j^2-{\displaystyle\sum\limits_{j=1}^k n_j\sigma_j^2}/{n})^2}{\displaystyle\sum\limits_{j=1}^k \sigma_j^4+({\displaystyle\sum\limits_{j=1}^kn_j\sigma_j^2}/{n})^2-2{\displaystyle\sum\limits_{j=1}^kn_j\sigma_j^4}/{n}}, \ f_2=\frac{[\displaystyle\sum\limits_{j=1}^k (1-{n_j}/{n})\sigma_j^2]^2}{{\displaystyle\sum\limits_{j=1}^k {(1-{n_j}/{n})^2\sigma_i^4}/{(n_j-1)}}}. \tag{22}\] The degrees of freedom of the test statistic can be calculated by using the Equation (22).
Permutation F test
(Berry et al. 2002) presented a permutational test as an alternative to ANOVA. The test statistics is
\[F_{PF}=\frac{{T-n\bar{\bar{x}}^2}/{(k-1)}}{{V-T}/{(n-k)}}, \label{eq:texor_dummy_12} \tag{23}\] where \(T=\displaystyle\sum\limits_{j=1}^kn_j\bar{x}_j^2\), \(V=\displaystyle\sum\limits_{j=1}^k \displaystyle\sum\limits_{i=1}^{n_j} x_{ij}^2\) and \(\bar{\bar{x}}=\frac 1 n \displaystyle\sum\limits_{j=1}^k n_jx_j\). The null hypothesis is rejected when \(F_{PF}>F_{k-1, n-k}\).
Scott Smith F test
(Scott and Smith 1971) proposed the following test statistic where group variances are not homogeneous:
\[F_{SS}=\displaystyle\sum\limits_{j=1}^k\frac{n_j(\bar{X_j}-\bar{X})^2}{S_j^{*2}}, \label{eq:texor_dummy_13} \tag{24}\] where \(S_j^{*2}=(n_j-1)/(n_j-3)S_j^{2}\). The null hypothesis is rejected when \(F_{SS}>\chi_{k}\).
Welch-Aspin test
Aspin (1948) presented a modification of Welch test. The test statistic is \[F_{WA}=\frac{\displaystyle\sum\limits_{j=1}^k(\bar{X}_{j}-{X^+})^2/S_j^2}{(k-1)[1+\frac{2k-2}{k^2-1}\Lambda}, \label{eq:texor_dummy_14} \tag{25}\] where \(\Lambda=\Sigma[(1-w_j)^2/v_j]\), \(w_j={1/S_j^2}/{\displaystyle\sum\limits_{j=1}^k1/S_j^2}\) and \(X^+\) is the overall mean. The null hypothesis is rejected when \(F_{WA}> F_{v_1,v_2}\) where \(v_1=k-1\) and \(v_2=(k^2-1)/(3\Lambda)\).
Welch’s heteroscedastic F test
Welch (1951) proposed a robust test that can be used when homogeneity of variance is not met. The test statistic is
\[F_w=\frac{\sum_{j}w_j(\bar{X}_{.j}-X_{..}^{'})^2/(k-1)}{\left[1+\frac{2}{3}((k-2)\nu) \right ]}, \label{welch} \tag{26}\]
where \(w_j=n_j/S_j^2\), \(S_j^2=\sum_{i}(X_{ij}-\bar{X}_{.j})^2/(n_j-1)\), \(X_{..}^{'} ={\sum_{j}w_j\bar{X}_{.j}}/{\sum_{j}w_j},\) and \[\nu =\frac{3\sum_{j}\left[\left(1-\frac{w_j}{\sum_{j}w_j}\right)^2/(n_{j}-1)\right]}{k^2-1}.\]
The test statistics follow an F distribution with degrees of freedom \(k-1\) and \(1/ \nu\).
Welch’s heteroscedastic F test with trimmed means and Winsorized variances
(Welch 1951) presented a robust procedure for independent groups design by replacing the usual means and variances with trimmed means and Winsorized variances. Let \(X_{(1)j}\leq X_{(2)j}\leq \ldots \leq X_{(n_j)j}\) be the ordered observations in the jth group and \(g_j= \|\epsilon n_j \|\), \(\epsilon\) is the proportion to be trimmed in each tail of the distribution. After trimming, the effective sample size for the jth group becomes \(h_j=n_j-2g_j\). Then jth sample trimmed mean is \[\bar{X}_{tj}=\frac{1}{h_j}\sum_{i=g_j+1}^{n_j-g_j}X_{(i)j},\]
and jth sample Winsorized mean is
\[\bar{X}_{wj}=\frac{1}{n_j}\sum_{i=1}^{n_j}Y_{ij},\] where \[Y_{ij} = \begin{cases} X_{(g_{j}+1)j} & if X_{ij} \leq X_{(g_j+1)j} ,\\ X_{ij} & if X_{(g_j+1)j}< X_{ij}< X_{(n_j-g_j)j},\\ X_{(n_j-g_j)j} & if X_{ij}\geq X_{(n_j-g_j)j}. \end{cases}\] The sample Winsorized variance is \[s_{wj}^2=\frac{1}{(n_j-1)}\sum_{i=1}^{n_j}(Y_{ij}-\bar{X}_{wj})^2,\] where \(q_j={(n_j-1)s_{wj}^2}/{h_j(h_j-1)}\), \(w_j=\frac{1}{q_j}\), \(U=\sum_{j}w_j\), \(\tilde{X}=\frac{1}{U}\sum_{j}w_j\bar{X}_{tj}\). The test statistic is \[\label{Welch2} F_{WT}=\frac{A}{B+1}, \tag{27}\] where \(A=\frac{1}{k-1}\sum_{j}w_j(\bar{X}_{tj}-\tilde{X})^2,\) \(B=\frac{2(k-2)}{k^2-1}\sum_{j}\frac{(1-w_j/U)^2}{h_j-1}\). The null hypothesis is rejected when \(F_{WT}>F_{k-1,\nu'}\) where \(\nu' =\left(\frac{3}{k^2-1}\sum_j \frac{(1-w_j/U)^2}{h_j-1} \right)^{-1}.\)
Weerahandi’s generalized F test
(Weerahandi 1995) provided an extension of the classical F-test for the case of unequal variances. The generalized p value of that test is obtained by using the following formula involving an expected value of the form \(EG_{k'} ()\), where \(G_{k'}\) is the cdf of the chi-squared distribution with \(k'\) degrees of freedom, and \(k'=k-1\).
\[p = 1 - EG_{k'} (s_B \left(\frac{n_1 s_1^2}{Y_1}, \frac{n_2 s_2^2}{Y_2}, ..., \frac{n_k s_k^2}{Y_k} , \right) \label{eq:texor_dummy_15} \tag{28}\] where \(Y_j \sim \chi_{r_j}^2\) and \[s_B (w_1,...,w_k) = \sum_{j=1}^{k} w_j \left( \bar{x}_j -\bar{x}_B\right)^2\] is the weighted between group sum of squares, \(w_j\) is the \(j^{th}\) weight, and \(\bar{x}_B\), is the overall weighted sample mean based on the same weights.
3 Implementation of R package
The onewaytests package is a comprehensive collection of functions specifically designed for analyzing data from one-way independent groups designs. This versatile package provides a wide range of tools to facilitate statistical analysis and exploration of such designs. The package includes 22 one-way tests for independent groups designs. In addition to the one-way tests, the package offers functions for generating descriptive statistics, conducting goodness-of-fit tests to check the fundamental assumptions, performing pairwise comparisons, and employing graphical approaches to enhance data interpretation.
In this section, we work on penguins data set, originally published in the work done by (Gorman et al. 2014), available in the palmerpenguins package (Horst et al. 2020). For implementation of the package, we utilize body mass as a response variable and species as a grouping variable.
After successfully installing and loading the onewaytests package, researchers have access to a wide range of functions specifically designed for analyzing one-way independent groups designs. These functions are incredibly useful when researchers compare k populations with respect to continuous outcomes. The one-way tests provided by this package not only calculate the test statistics and p-values, but they also provide the pairwise comparisons for valuable insights by evaluating the hypothesis of the statistical process. With the onewaytests package, researchers can easily conduct thorough and in-depth analyses of their one-way independent groups designs, gaining valuable insights and making well-informed conclusions.
In this part, we use the functions, originally developed in our previously published work (Dag et al. 2018) to describe data and assess the main assumptions, variance homogeneity and normality.
The describe() functions return the output involving sample size,
mean, standard deviation, median, minimum value, maximum value, 25th
percentile, 75th percentile, skewness, kurtosis and number of missing
values (NA) for one-way layout.
R> library(onewaytests)
R> library(palmerpenguins)
R> describe(body_mass_g ~ species, data = penguins)
n Mean Std.Dev Median Min Max 25th 75th Skewness Kurtosis NA
Adelie 151 3700.662 458.5661 3700 2850 4775 3350.0 4000 0.28249381 2.405611 0
Chinstrap 68 3733.088 384.3351 3700 2700 4800 3487.5 3950 0.24194125 3.463681 0
Gentoo 123 5076.016 504.1162 5000 3950 6300 4700.0 5500 0.06878276 2.257871 0Researchers perform variance homogeneity tests with the homog.test()
function. It offers three variance homogeneity tests; Levene, Bartlett,
and Fligner-Killeen tests.
R> homog.test(body_mass_g ~ species, data = penguins, method = "Levene")
Levene's Homogeneity Test (alpha = 0.05)
-----------------------------------------------
dependent var : body_mass_g
grouping var : species
F = 5.335495
num df = 2
denum df = 339
p-value = 0.005230535
Result : Variances are not homogeneous.
----------------------------------------------- Levene’s homogeneity test results reveal that the variances between penguin species are not homogeneous (F = 5.335495, p-value = 0.005230535).
Researchers can assess normality using the nor.test() function. This
function offers six normality tests: Shapiro-Wilk, Shapiro-Francia,
Lilliefors (also known as Kolmogorov-Smirnov test), Anderson-Darling,
Cramer-von Mises, and Pearson Chi-square tests.
R> nor.test(body_mass_g ~ species, data = penguins, method = "SW")
Shapiro-Wilk Normality Test (alpha = 0.05)
--------------------------------------------------
data : body_mass_g and species
Level Statistic p.value Normality
1 Adelie 0.9807079 0.03239702 Reject
2 Chinstrap 0.9844938 0.56050824 Not reject
3 Gentoo 0.9859276 0.23361649 Not reject
--------------------------------------------------Shapiro-Wilk normality test results state that body masses of chinstrap and gentoo species are normally distributed since the associated p-values (0.56050824 and 0.23361649, respectively) are greater than 0.05. On the other hand, there is enough evidence to reject the normality of body mass for adelie group (p-value = 0.03239702).
In the onewaytests package, the first argument is specified as a formula where the left-hand side represents sample values and the right-hand side represents groups. The formula must have one variable on each side, with the left side being numeric and the right side being a factor. Otherwise, an error message is returned by each function.
One-way tests in independent groups designs: onewaytests(...)
In our previously published paper (Dag et al. 2018), we introduced
this package that included seven one-way tests. In this latest version
of the package, we have expanded its capabilities by adding 15 new
functions specifically designed for one-way independent groups designs.
These additional functions provide users with a wider range of options
for conducting their analyses. The complete list of all the one-way
tests, their corresponding functions, and the method name of the
onewaytests() function can be found in Table
1.
In this section, we introduce a generic function called onewaytests()
that encompasses a total of 22 one-way tests. By utilizing this
function, researchers can conveniently apply any of the available
one-way tests. To specify the desired method, researchers simply need to
provide the method argument within the onewaytests() function. The
list of all methods is provided under Method in Table
1.
All one-way tests can be utilized with separate functions, offering
flexibility and versatility. To facilitate ease of use and
accessibility, we have meticulously gathered all methods for
researchers’ convenience. The onewaytests() function provides four
different types of output. We present each of four kinds to introduce
the outputs. As an example, we have incorporated Alvandi’s F test,
Scott-Smith test, Weerahandi’s generalized F test, and James second
order test, which can also be applied using af.test(), ss.test(),
wgf.test(), james.test() functions, respectively, to demonstrate the
practical usage of the onewaytests() function. The test results can be
displayed by setting the verbose argument to TRUE. The results are
returned as an object of class htest.
R> out <- onewaytests(body_mass_g ~ species, data = penguins, method = "af")
Alvandi's F Test
data: body_mass_g and species
F = 318.69, num df = 2.000, denom df = 97.832, p-value < 2.2e-16For more comprehensive results, the summary() function can be used, as
demonstrated in the example below.
R> summary(out)
Alvandi's F Test (alpha = 0.05)
-------------------------------------------------------
dependent var : body_mass_g
grouping var : species
F = 318.6896
num df = 2
denom df = 97.83241
p-value < 2.2e-16
Result : Difference is statistically significant.
-------------------------------------------------------
* statistic is distributed as F distribution.Here, Alvandi’s test statistic is distributed as F with the degrees of
freedom for numerator (num df) and denominator (denom df). Also,
p-value is the significance value of the test statistic. Since the
p-value is lower than 0.05, there is enough evidence to conclude that
the difference between the penguin species in terms of body mass is
statistically significant (F = 318.6896, p-value < 2.2^-16).
R> out <- onewaytests(body_mass_g ~ species, data = penguins, method = "ss",
verbose = FALSE)
R> summary(out)
Scott-Smith Test (alpha = 0.05)
-------------------------------------------------------------
dependent var : body_mass_g
grouping var : species
X-squared = 639.8695
df = 3
p-value < 2.2e-16
Result : Difference is statistically significant.
-------------------------------------------------------------
* statistic is distributed as chi-squared distribution. In the output, Scott-Smith test statistic is distributed as \(\chi^2\)
with the degrees of freedom (df). Also, p-value is the significance
value of the test statistic. Since the p-value is lower than 0.05, one
can conclude that the difference between the species is statistically
significant (\(\chi_{AG}^2\) = 639.8695, p < 2.2^-16).
R> out <- onewaytests(body_mass_g ~ species, data = penguins, method = "wgf",
verbose = FALSE)
R> summary(out)
Weerahandi's Generalized F Test (alpha = 0.05)
----------------------------------------------------------
dependent var : body_mass_g
grouping var : species
p-value < 2.2e-16
Result : Difference is statistically significant.
----------------------------------------------------------
* p-value is obtained using Monte Carlo simulation. Here, p-value is the significance value of the test statistic. Since
the p-value is lower than 0.05, there is sufficient evidence to conclude
that the difference between the penguin species in terms of body mass is
statistically significant.
R> out <- onewaytests(body_mass_g ~ species, data = penguins, method = "james",
verbose = FALSE)
R> summary(out)
James Second Order Test (alpha = 0.05)
-----------------------------------------------------------------------------------
dependent var : body_mass_g
grouping var : species
Jtest = 637.3792
CriticalValue = 6.108713
Result : Difference is statistically significant.
-----------------------------------------------------------------------------------
* test statistic is sum of the squared standardized differences and compared to a
critical value. In the output, Jtest is the James second order test statistic (J),
criticalValue is the critical value (\(h(\alpha)\)) associated to the
significance level (\(\alpha\)). The null hypothesis is rejected when J
exceeds \(h(\alpha)\). We conclude that there is a statistically
significant difference between the penguin species since
\(\textit{J} = 637.3792\) is larger than \(h(\alpha) = 6.108713\).
The onewaytests
package provides the capability to perform pairwise comparisons when a
statistically significant difference is observed. It offers several
options for controlling the type I error rate, including bonferroni,
holm (Holm 1979), hochberg (Hochberg 1988), hommel (Hommel 1988),
BH (Benjamini and Hochberg 1995), BY (Benjamini and Yekutieli 2001), and none in paircomp()
function. The default is set to "bonferroni". The development of
paircomp() function is extensively placed in the article work done by
(Dag et al. 2018). The paircomp() function is capable of
recognizing the output of the onewaytests() function. In this part, we
include the pairwise comparison following to Alvandi’s F test to
protect the compactness of the paper.
R> out <- onewaytests(body_mass_g ~ species, data = penguins, method = "af",
verbose = FALSE)
R> paircomp(out, adjust.method = "bonferroni")
Bonferroni Correction (alpha = 0.05)
-----------------------------------------------------
Level (a) Level (b) p.value No difference
1 Adelie Chinstrap 1.000000e+00 Not reject
2 Adelie Gentoo 2.765295e-48 Reject
3 Chinstrap Gentoo 1.639844e-34 Reject
----------------------------------------------------- According to the result obtained by adjusting with Bonferroni method, Gentoo group has statistically greater body mass than Adelie and Chinstrap groups.
Graphical approaches based on ggplot2 package
In this part, we rewrite the gplot function from scratch. The gplot
function offers different graphic types, box-and-whisker plot, violin
plot, box-and-whisker plot with violin lines and error bars. These
options can be selected in the type argument with "boxplot",
"violin", "boxplot-violin", and "errorbar", respectively. There
exists width argument involving three numeric values. These numeric
values specify the widths for the boxes of box-and-whisker plots
(defaults to 0.3), violin plots (defaults to 1.0), and the little lines
at the tops and bottoms of the error bars (defaults to 0.2),
respectively. Also, there exists the dots argument to draw
observations on the plots. The binwidth argument offers to change the
size of dots. Users can change the colors of the plots with
color_manual argument. Moreover, researchers can set color palette
using wes_palette function available in
wesanderson
package (FantasticFox1 palette is used as a default). Additionally, the
theme of plot can be changed with theme argument. The list of all
theme is available with ggtheme function available in
ggplot2 package
(Wickham 2016). We use the classic dark-on-light ggplot2 theme as
a default theme by setting theme_bw() to theme argument. Researchers
can use standard deviation or standard error with option argument
while drawing error bars. Also, researchers can add mean bar to error
bars with add argument. Additionally, the labels of x and y axes and
title can be changed with xlab, ylab, and title arguments,
respectively.
In this part, we draw some graphic types to illustrate the usage of
gplot function. These graphics involve box-and-whisker plot with
violin line and dots (Figure 1a), box-and-whisker plot with violin line
(Figure 1b), box-and-whisker plot (Figure
1c),
violin plot (Figure 1d), mean \(\pm\) standard deviation graph
(Figure 1e), and mean \(\pm\) standard deviation graph
with mean bar (Figure 1f). These graphics can be obtained via
following codes:
# Box-and-whisker plot with violin lines and dots
R> gplot(body_mass_g ~ species, data = penguins, type = "boxplot-violin",
width = c(0.4, 0.95, NA), binwidth = 50, ylab = "body mass (grams)")
# Box-and-whisker plot with violin lines
R> gplot(body_mass_g ~ species, data = penguins, type = "boxplot-violin",
width = c(0.4, 0.95, NA), dots = FALSE, ylab = "body mass (grams)")
# Box-and-whisker plot
R> gplot(body_mass_g ~ species, data = penguins, type = "boxplot",
width = c(0.4, NA, NA), dots = FALSE, ylab = "body mass (grams)")
# Violin plot
R> gplot(body_mass_g ~ species, data = penguins, type = "violin",
width = c(NA, 0.95, NA), dots = FALSE, ylab = "body mass (grams)")
# Error bar (mean +- standard deviation)
R> gplot(body_mass_g ~ species, data = penguins, type = "errorbar",
width = c(NA, NA, 0.3), binwidth = 50, ylab = "body mass (grams)",
option = "sd")
# Error bar (mean +- standard deviation) with mean bar
R> gplot(body_mass_g ~ species, data = penguins, type = "errorbar",
width = c(NA, NA, 0.3), binwidth = 50, ylab = "body mass (grams)",
option = "sd", bar = TRUE)
|
|
| a. Box-and-whisker plot with violin lines and dots | b. Box-and-whisker plot with violin lines |
|
|
| c. Box-and-whisker plot | d. Violin plot |
|
|
| e. Mean ± standard deviation graph | f. Mean ± standard deviation graph with bar |
4 Web-based tool
There is an easy-to-use web application available for users, which is built upon the onewaytests package, providing a convenient and accessible way to perform statistical analysis. This web-tool has recently undergone significant updates and enhancements. One of the major improvements is the addition of 15 one-way independent groups tests, allowing users to conduct a wide range of statistical comparisons. Furthermore, the web-tool now features enhanced graphics, providing users with clear and visually appealing representations of their data. Moreover, we have made the web-tool compatible with a mobile phone application due to the growing popularity of mobile devices. This means that users can conveniently access and utilize the web-tool on their smartphones or tablets, making statistical analysis more accessible and convenient than ever before. Researchers can freely access the web application by visiting the following link, http://www.softmed.hacettepe.edu.tr/onewaytests.
We utilize the shiny package (Chang et al. 2017) to create the web-interface of the onewaytests package. This tool is specifically designed to cater to the needs of new R users who may not have extensive programming experience, as well as applied researchers who require a straightforward interface for conducting their analyses. Our web tool aims to simplify the process of performing one-way tests and empower users to efficiently explore and interpret their data by harnessing the power of the shiny package. There is an example well-known data set called iris data collected by (Anderson 1935) placed in the web-tool, also available in R, to help users learn the usage of the tool.
|
|
| a. Data upload | b. Describe data |
|
|
| c. One-way tests | d. Graphics |
Researchers can obtain descriptive statistics using the "Describe data" tab (Figure 2b). In this tab, the assumptions of variance homogeneity and normality can be assessed. This tab provides Levene’s test, Bartlett’s test, and Fligner-Killeen test to check for variance homogeneity. It also offers various normality tests (Shapiro-Wilk, Cramer-von Mises, Lilliefors (Kolmogorov-Smirnov), Shapiro-Francia, Anderson-Darling, Pearson Chi-Square tests) and plots (Q-Q plot and Histogram with normal curve) to assess the normality of data in each subgroup.
Users can perform 22 one-way tests (Scott-Smith test, Box F test, Johansen F test, Generalized tests equivalent to Parametric Bootstrap and Fiducial tests, Alvandi’s F test, Alvandi’s generalized p-value, approximate F test, B square test, Cochran test, Weerahandi’s generalized F test, modified Brown-Forsythe test, adjusted Welch’s heteroscedastic F test, Welch-Aspin test, Permutation F test, one-way analysis of variance, Welch’s heteroscedastic F test, Welch’s heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test, Kruskal-Wallis test) using the "One-way tests" tab (Figure 2c). This tab also allows users to make pairwise comparisons (Bonferroni, Holm, Hochberg, Hommel, Benjamini-Hochberg, Benjamini-Yekutieli, no corrections) to investigate the group(s) leading to the differences.
Additionally, researchers can summarize their research findings using well-designed ggplot2 graphics (box-and-whisker plot, violin plot, and error bars) in the "Graphics" tab (Figure 2d). The web tool is freely available at http://www.softmed.hacettepe.edu.tr/onewaytests.
5 Conclusion
One-way tests in independent groups designs are highly popular and extensively utilized statistical techniques in a multitude of fields. These fields encompass medical sciences, engineering, social sciences, economics, psychology, and biology. These techniques allow the researchers for the identification of potential differences among groups, which can further enhance the scientific knowledge and contribute to the advancement of various disciplines.
In this paper, we present the latest version of onewaytests, which provides a variety of tools for one-way independent groups designs. We make significant enhancements to the package by introducing 15 new one-way tests (Scott-Smith test, Box F test, Johansen F test, Generalized tests equivalent to Parametric Bootstrap and Fiducial tests, Alvandi’s F test, Alvandi’s generalized p-value, approximate F test, B square test, Cochran test, Weerahandi’s generalized F test, modified Brown-Forsythe test, adjusted Welch’s heteroscedastic F test, Welch-Aspin test, Permutation F test) in addition to the existing seven tests (one-way analysis of variance, Welch’s heteroscedastic F test, Welch’s heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test, Kruskal-Wallis test) in the work done by (Dag et al. 2018). Users now have a total of 22 tests available for comprehensive assessments.
In order to enhance the functionality and customization options of the graphical approaches, we undertake a complete overhaul of their function. Through this process, we not only add new features and enhancements, but also ensure that the core functionality of the function remains intact. This comprehensive revision allows us to fully optimize the graphical approaches, resulting in a more versatile tool for the users.
In this study, we improve the web tool of onewaytests package, ensuring its compatibility with a wide range of mobile devices. This significant update allows users to effortlessly access and utilize the web tool on their smartphones or tablets. This mobile compatibility opens up a world of possibilities, enabling researchers and statisticians to seamlessly integrate statistical analysis into their daily routines, regardless of their location or device.
The onewaytests package also provides additional capabilities to enhance your data analysis which involve generating descriptive statistics, examining the homogeneity of variances and normality of data in each group using a variety of tests and plots. Also, it has capability to perform two sample tests. Furthermore, the package offers functionality for conducting pairwise comparison methods after the statistically significant difference among groups is obtained. These features allow users to explore their data from multiple angles in a compact manner and draw more accurate conclusions.
Currently, the onewaytests package provides a comprehensive set of tools for conducting analysis on one-way independent groups designs. These tools include conducting one-way tests, performing pairwise comparisons, obtaining descriptive statistics, conducting two-sample tests, utilizing graphical approaches, and assessing variance homogeneity and normality through the use of tests and plots. Additionally, it is worth noting that the package and its associated web-tool will be regularly updated to ensure that users have access to the latest features and improvements.
6 Supplementary materials
Supplementary materials are available in addition to this article. It can be downloaded at RJ-2026-014.zip
7 CRAN packages used
onewaytests, palmerpenguins, wesanderson, ggplot2, shiny
8 CRAN Task Views implied by cited packages
ChemPhys, NetworkAnalysis, Phylogenetics, Spatial, TeachingStatistics, WebTechnologies
9 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.