RobustBF: An R Package for Robust Solution to the Behrens-Fisher Problem

Abstract:

Welch’s two-sample t-test based on least squares (LS) estimators is generally used to test the equality of two normal means when the variances are not equal. However, this test loses its power when the underlying distribution is not normal. In this paper, two different tests are proposed to test the equality of two long-tailed symmetric (LTS) means under heterogeneous variances. Adaptive modified maximum likelihood (AMML) estimators are used in developing the proposed tests since they are highly efficient under LTS distribution. An R package called RobustBF is given to show the implementation of these tests. Simulated Type I error rates and powers of the proposed tests are also given and compared with Welch’s t-test based on LS estimators via an extensive Monte Carlo simulation study.

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Published

Dec. 15, 2021

Received

Jul 13, 2021

Citation

Güven, et al., 2021

Volume

Pages

13/2

713 - 733


1 Introduction

Testing the equality of two population means is one of the most encountered problems in applied sciences. Student’s t-test, which is uniformly most powerful unbiased, is commonly used under normality and homogeneity of variances assumptions. The well-known Behrens-Fisher (BF) problem arises when the assumption of homogeneity of variances is not met. This problem can be defined as testing the null hypothesis (1)H0:μ1=μ2 when Yi1,Yi2,...,Yini (i=1,2) are independent random samples from N(μi,σi2) distribution. endorsed Behrens’ solution to the BF problem by using the fiducial theory. Many researchers studied this problem. For example, proposed a test statistic and provided its degrees of freedom approximately. It should be noted that degrees of freedom provided by can also be obtained by using the Satterthwaite approximation; see . This is why the mentioned degrees of freedom is also known as Welch-Satterthwaite degrees of freedom in the literature. calculated the Type I error rates of the Welch’s two-sample t-test and Aspin-Welch test for different sets of degrees of freedom and nominal significance levels and concluded that Welch’s t-test could be used in practice with little loss of accuracy. considered the test suggested by for the BF problem and compared its Type I error rates with those of . They concluded that this test is a very practical solution to the BF problem besides being stable in regard to size and having adequate power. calculated the Wald score and likelihood ratio statistics and showed that the test based on Wald statistics has the same asymptotic properties as the Welch’s t-test. presented a review of basic concepts and applications concerning the BF problem under fiducial, Bayesian and frequentist approaches. developed a test based on the Jackknife estimator of the common population variance and compared the powers of the proposed test with those of Welch’s t-test and test. According to the results of their study, the proposed test is more powerful than the Cochran-Cox test for all cases, while it is more preferable to Welch’s t-test for some cases. developed a computational approach test (CAT) for the BF problem and compared it with Welch’s t-test, Cochran-Cox text, Generalized p-value test, and Singh–Saxena–Srivastava test under the normal and t-model. They found that Welch’s t-test, Cochran-Cox text, and CAT are robust under the heavier tailed t-models besides having similar size and power.

When the literature is examined, it can be seen that Welch’s t-test has a very good performance as compared to other tests in the case of heteroscedasticity and unequal sample sizes in the context of normality. The power of Welch’s t-test decreases very rapidly when the underlying distribution is long-tailed symmetric (LTS) since the least squares (LS) estimators are not robust to the violation of normality. It is known that non-normal distributions are more common in real-life problems. proposed a two-sample trimmed t-test and compared its performance of it with Welch’s t-test for both normal and long-tailed samples. proposed Welch-type statistics based on modified maximum likelihood (MML) estimators and showed that the proposed test is more powerful than ’s trimmed t-test. In addition, investigated the analogous test based on the robust bisquare estimators BS82 and showed that this test statistic gives misleading Type I errors.

In this study, a robust version of Welch’s t-test for the BF problem is proposed when the underlying distribution is LTS. A second test using the fiducial model, which is a special case of a functional model given by , is also proposed; see for more information about the fiducial approach. The reason for including a robust version of fiducial-based test into this study is to see its performance in the context of BF problem and to make comprehensive comparisons with its rivals (i.e., robust version of Welch’s t-test and the traditional Welch’s t-test). Both of the proposed tests are based on adaptive modified maximum likelihood (AMML) estimators, see and . To the best of our knowledge, this is the first study using AMML estimators for testing the equality of two LTS means under heterogeneous variances. These estimators are efficient and easy to compute for LTS samples, see .

The R packages stats by and asht by include Welch’s t-test based on LS estimators and BF test under normality, respectively. WRS2 by contains Yuen’s test based on the trimmed sample means. Different from the earlier studies, we provide an R package RobustBF computing the values of the proposed test statistics and/or the corresponding p-values.

The rest of this study is organized as follows. Firstly, AMML estimators are given. Secondly, the robust Welch test and robust test based on the fiducial approach are developed. Thirdly, an extensive Monte Carlo simulation study is conducted to compare the performances of the proposed tests with the traditional Welch’s t-test based on LS estimators. The proposed tests are applied to a real data set via RobustBF package. This paper is finalized some concluding remarks.

2 AMML Estimators

Assume that Yi1,Yi2,...,Yini (i=1,2) be independent random samples from LTS(p,μi,σi) distribution (2)f(y)=1kβ(1/2,p1/2)σ(1+(yμ)2kσ2)p,<y<;<μ<;σ>0;p2, where μ is the location parameter, σ is the scale parameter, p is the shape parameter, and k=2p3 . It should be noted that E(y)=μ, V(y)=σ2, and t=(ν/k)(y/σ) has Student’s t distribution with ν=2p1 degrees of freedom.

The log-likelihood (lnL) function is given by (3)lnL=Nln(kβ(1/2,p1/2))i=12niln(σi)pi=12j=1niln(1+(yijμi)2kσi2), where N=n1+n2. Then, the likelihood equations are obtained as follows (4)lnLμi=2pkσij=1nig(zij)=0

(5)lnLσi=niσi+2pkσij=1nizijg(zij)=0 where (6)g(zij)=zij1+(1/k)zij2andzij=yijμiσi.

By solving the above likelihood equations, (4) and (5), simultaneously, the maximum likelihood (ML) estimators of the parameters μi and σi are obtained. However, these equations involve nonlinear functions of the parameters, and so ML estimators cannot be obtained explicitly. Hence, numerical methods can be used to solve these equations. Numerical methods may cause convergence problems like non-convergence of iterations, convergence to wrong roots, or multiple roots . MML methodology proposed by overcomes these mentioned problems by providing explicit solutions to likelihood equations. In MML methodology, firstly, the standardized statistics are ordered in ascending way, i.e., zi(1)zi(2)...zi(ni). Then, likelihood equations in (4) and (5) are rewritten in terms of zi(j) and g(zi(j)) (i=1,2;j=1,2,...,ni) as shown in (7) and (8) since summation is invariant to ordering, i.e., j=1nizi(j)=j=1nizij. (7)lnLμi=2pkσij=1nig(zi(j))=0

(8)lnLσi=niσi+2pkσij=1nizi(j)g(zi(j))=0. Here, zi(j)=yi(j)μiσi and g(zi(j))=zi(j)1+(1/k)zi(j)2. The nonlinear function g(zi(j)) is linearized utilizing the first two terms of the Taylor series expansion around the expected values of the ordered statistics E(zi(j))=ti(j) as follows (9)g(zi(j))αij+βijzi(j), where (10)αij=(2/k)ti(j)3(1+(1/k)ti(j)2)2andβij=1(1/k)ti(j)2(1+(1/k)ti(j)2)2. Since ti(j) values cannot be obtained exactly, approximate values of ti(j) which do not affect the efficiencies of the resulting estimators are used, (11)ti(j)f(z)dz=jni+1,i=1,2;j=1,2,...,ni. Secondly, modified likelihood equations are obtained by inserting the approximation (9) into Eqs. (7) and (8) (12)lnLμi=2pkσij=1ni(αij+βijzi(j))=0

(13)lnLσi=niσi+2pkσij=1nizi(j)(αij+βijzi(j))=0. Finally, MML estimators of μi and σi are found by solving Eqs. (12) and (13). They are given as follows (14)μ^i=j=1niβijyi(j)miandσ^i=Bi+Bi2+4niCi2ni(ni1), where (15)Bi=2pkj=1niαij(yi(j)μ^i),Ci=2pkj=1niβij(yi(j)μ^i)2andmi=j=1niβij; see . The asymptotic properties of the MML estimators μ^i and σ^i can be demonstrated with the help of the following theorems.

Theorem 1. μ^i is the minimum variance bound (MVB) estimator and is asymptotically normally distributed with mean μi and variance σi2/Mi (Mi=2pmi/k).

Theorem 2. (ni1)σ^i2/σi2 is distributed as chi-square (more accurately a multiple of chi-square) with (ni1) degrees of freedom.

For proofs of theorems, see, e.g. .

MML estimators have the same asymptotic properties as the ML estimators and are as efficient as ML estimators, even for small samples. They are easy to compute and robust to the outliers.

It should be noted that the shape parameter p is assumed to be known in the MML methodology. However, in some real-life applications, it may be possible to assume that the data comes from a certain type of distribution, namely LTS distribution, but there is no opportunity to specify the value of the shape parameter. Hence, proposed AMML methodology, which is a new version of MML methodology, see and . This methodology relaxes the assumption of the known shape parameter. AMML estimators are computed in two iterations. In the first iteration, initial tij values are calculated from the sample data, as shown below (16)tij=(yijT0i)/S0ii=1,2;j=1,...,ni. Here, T0i and S0i are the initial estimates of μi and σi and they are calculated as (17)T0i=med{yij}andS0i=1.483med{yijT0i}i=1,2;j=1,...,ni, respectively. Using the tij values in (16), αij and βij coefficients are calculated as follows (18)αij=(1/k)tij1+(1/k)tij2andβij=11+(1/k)tij2. Then, the AMML estimates of the parameters μi and σi are obtained using Eq. (14) and αij and βij values given in Eq. (18). To distinguish these estimates from the MML estimates, they are represented by μ^i(AMML) and σ^i(AMML) in the rest of the paper. In the second iteration, tij values are revised as follows (19)tij=(yijμ^i(AMML))/σ^i(AMML)i=1,2;j=1,...,ni and recalculate the αij and βij values using the equalities in (18) for these tij values. Then final AMML estimates of μi and σi are obtained.

It should be noted that in AMML methodology, yij observations are used rather than the ordered yi(j) observations since tij values are calculated from the sample observations. In addition, the shape parameter p is taken to be 16.5 in the calculations of αij and βij coefficients since this value makes AMML estimators efficient for normal and near normal distributions. It also makes them robust to mild outliers. The reason why we use AMML methodology in the proposed tests is that it provides the same asymptotic properties as MML methodology and, as mentioned before, relaxes the assumption of known shape parameter p.

3 Proposed Test Statistics

In this section, we propose two different tests for testing the equality of two LTS means.

Robust Welch (RW) Test

In this subsection, we briefly introduce Welch’s t-test proposed by under normal theory and then give the robust version of it. Welch’s t-test based on LS estimators is defined as W={(x¯1x¯2)(μ1μ2)}/{(s12/n1)+(s22/n2)}. It is known that W is approximately distributed as Student’s t with degrees of freedom (20)f=1{c2/(n11)+(1c2)/(n21)}, where c=(s12/n1)/{(s12/n1)+(s22/n2)}. Here, x¯i and si2 (i=1,2) are the sample means and sample variances, respectively. The value of W test can be obtained using t.test function available in R.

In this study, we propose the following test statistics based on AMML estimators as a robust alternative to Welch’s t-test (21)RW=(μ^1(AMML)μ^2(AMML))(μ1μ2)(σ^1(AMML)2/M1)+(σ^2(AMML)2/M2). As we shall see at the end of this section, the null distribution of RW is approximately distributed as Student’s t based upon Theorems 1 and 2 . The approximate degrees of freedom for this test is obtained using the approximation as follows.

Let (22)c1=σ12(n11)M1,c2=σ22(n21)M2

(23)Q1=(n11)σ^1(AMML)2σ12andQ2=(n21)σ^2(AMML)2σ22 where Q1 and Q2 are independent chi-square random variables with degrees of freedom (n11) and (n21), respectively (see Theorem 2). If the linear combination of Q1 and Q2 is written as (24)Q=c1Q1+c2Q2=σ^1(AMML)2M1+σ^2(AMML)2M2, then νQ/E(Q) has an approximate χ2 distribution with the following degrees of freedom ν=[c1Q1+c2Q2]2([c1Q1]2/ν1)+([c2Q2]2/ν2)=((σ^1(AMML)2/M1)+(σ^2(AMML)2/M2))2(σ^1(AMML)2/M1)2/(n11)+(σ^2(AMML)2/M2)2/(n21). Here, ν1=n11,ν2=n21andE(Q)=σ12M1+σ22M2. RW in (21) can be rewritten as follows RW=((μ^1(AMML)μ^2(AMML))(μ1μ2))/(σ12/M1)+(σ22/M2)(σ^1(AMML)2/M1)+(σ^2(AMML)2/M2)/(σ12/M1)+(σ22/M2). Since this expression is equivalent to ZQ/E(Q), it is obvious that RW is approximately distributed as Student’s t with ν degrees of freedom. Here, ZN(0,1) (see Theorem 1) and Q/E(Q)χν2/ν.

To verify the null distribution of the RW, the probabilities (25)p1=Pr(|RW|t1α/2,ν) are simulated from 10,000 Monte Carlo runs for various combinations of the sample sizes n1 and n2. The results are demonstrated in Table 1. Here, ν is the degrees of freedom for RW.

Robust Fiducial (RF) Based Test

In this section, fiducial-based test is proposed using the concept of fiducial inference and pivotal model; see and . Let denote the RW test based on the observed values as (26)RW=(μ^1(AMML)μ^2(AMML))(μ1μ2)σ^1(AMML)2M1+σ^2(AMML)2M2. First, the fiducial distribution of RW is derived using pivotal quantities and fiducial distribution of the parameters of interest. Then, the corresponding p-value is obtained. Here, (μ^i(AMML),σ^i(AMML)2) are the observed values of (μ^i(AMML),σ^i(AMML)2) (i=1,2). Let Zi=μ^i(AMML)μiσi/Mi and Qi=(ni1)σ^i(AMML)2σi2 are mutually independent pivotal quantities. They have asymptotically N(0,1) and χ(ni1)2 distributions, respectively (see Theorems 1 and 2). Using pivotal quantities Zi and Qi, data generating equations are obtained as given below (27)μ^i(AMML)=μi+(σi/Mi)Zi and (28)σ^i(AMML)2=σi2Qi/(ni1). Given (μ^i(AMML),σ^i(AMML)2), Eqs. (27) and (28) are expressed as follows (29)μ^i(AMML)=μi+(σi/Mi)zi and (30)σ^i(AMML)2=σi2qi/(ni1). Here, (zi,qi) are the observed values of (Zi,Qi). Eqs. (29) and (30) have the unique solutions as given below μi=μ^i(AMML)ziqi/(ni1)σ^i(AMML)Mi and σi2=(ni1)σ^i(AMML)2qi. Since ZiQi/(ni1) is distributed as a ti variable with (ni1) degrees of freedom, the fiducial distribution of μi is the same as that of Tμi=μ^i(AMML)tiσ^i(AMML)Mi for given (μ^(AMML),σ^(AMML)2). Therefore, the fiducial distribution of RW in (26) is derived by utilizing the fiducial distribution of μi as follows (31)TRF=((t1σ^1(AMML))/M1)((t2σ^2(AMML))/M2)(σ^1(AMML)2)/M1+(σ^2(AMML)2)/M2, where t1t(n11) and t2t(n21). Since (32)RW0=(μ^1(AMML)μ^2(AMML))(σ^1(AMML)2)/M1+(σ^2(AMML)2)/M2 is the observed value of TRF under H0:μ1=μ2, the corresponding p-value is given by (33)p=Pr(TRFRW0). An algorithm for calculating the fiducial p-value in Eq.(33) via Monte Carlo simulation study is given as follows

Algorithm 1

Step 1

For the given data, compute μ^i(AMML), σ^i(AMML) (i=1,2) and then RW0 utilizing Eq. (32).

Step 2

Generate tit(ni1),(i=1,2).

Step 3

Compute TRF2 utilizing Eq. (31).

Step 4

Let Fl=1 if TRF2>RW02, else Fl=0

Step 5

Repeat the steps 2-4 K times.

Step 6

Compute the simulated p-value using p=1Kj=1KFj.

It should be noted that the squares of TRF and RW0 in Steps 3 and 4 are taken since the alternative hypothesis is two-sided, i.e., H1:μ1μ20; see .

4 Monte Carlo Simulation

In this section, Type I error rates and powers of the proposed tests (RW and RF) are compared with those of the W test under the specified nominal level α=0.05. The plan of the simulation study is outlined as follows:

We use the following population distributions while generating samples.

Population 1 Population 2
(a) Cauchy(0,1) Cauchy(0,1)
(b) 5×Cauchy(0,1) Cauchy(0,1)
(c) Normal(0,32)/Uniform(0,1) Normal(0,1)/Uniform(0,1)
(d) 0.8Normal(0,42)+0.2Normal(0,42)Uniform[0,1] 0.8Normal(0,1)+0.2Normal(0,1)Uniform[0,1]
(e) 3t2 t2
(f) 2t5 t5
(g) Logistic(0,3) Logistic(0,1)
(h) Laplace(0,1) Laplace(0,6)

Here, ta: Student’s t distribution with a degrees of freedom.

10,000 different samples are considered for each of size ni(i=1,2). Sample sizes are taken as (n1,n2)=(6,6),(6,10),(10,10),(10,15),(10,30), (20,20),(20,30),(20,50),(30,50) and (50,50) while comparing the Type I error rates and powers of the tests. Simulations are conducted in R software.

To compute the Type I error rates of the RW, RF, and W tests, firstly, samples are generated under the null hypothesis H0:μ1=μ2 for given (n1,n2). Then AMML and LS estimates of the parameters are calculated. The probability in Eq. (25) gives the Type I rates of the RW test. It should be noted that this probability shows that how close the distribution of the RW test is to Student’s t with degrees of freedom ν. RF is carried out using Algorithm 1 with K=5,000. The fiducial p-value for the RF is computed in the final step of the mentioned algorithm. This procedure is repeated for each of the 10,000 samples. The proportion of the 10,000 p-values that are less than the nominal level α=0.05 gives Type I error rates of the RF.

To compute the power of the tests, similar steps are followed, but a constant d is added to the observations in the first population. Any test can be considered powerful if it achieves maximum power and adheres to the prescribed significance level.

5 Results

The results of the Monte Carlo simulation study are given in Tables 1-9. The Type I error rates and power of the tests are given in Table 1 and Tables 2-9, respectively.

Numerical results of Table 1 can be summarized for Models (a)-(h) as follows

The numerical results of Tables 2-9 can be summarized as follows. It should be noted that the first line of Tables 2-9, that is, d=0.00 presents simulated Type I error rates of the tests.

Overall, the RW test can be recommended for testing the equality of two LTS means under the assumption of heterogeneous variances since it has the best performance with respect to size and power. Although the performance of the RF test is not as good as the RW test, it has better performance than the traditional Welch’s t-test.

6 Using RobustBF package

In the RobustBF package, we show the implementation of the proposed tests (RW and RF), based on AMML estimators, and W test, based on LS estimators, using the data representing the values of 10(y2.0) (y is the pollution level (measurement of lead) in water samples from two lakes). It has been shown that long-tailed symmetric distribution provides a plausible model for the mentioned data; see and also reference therein.

To run RobustBF package, we first install the package and then load it by typing:

> install.packages("RobustBF")
> library(RobustBF)

respectively. Next the pollution level data are inputted for each lakes (Lake 1 and Lake 2) in terms of the vectors as shown below

y1 <- c(-1.48, 1.25, -0.51, 0.46, 0.60, -4.27, 0.63, -0.14, -0.38, 1.28,
         0.93, 0.51, 1.11, -0.17, -0.79, -1.02, -0.91, 0.10, 0.41, 1.11)
y2 <- c(1.32, 1.81, -0.54, 2.68, 2.27, 2.70, 0.78, -4.62, 1.88, 0.86,
        2.86, 0.47, -0.42, 0.16, 0.69, 0.78, 1.72, 1.57, 2.14, 1.62)

The value of the RW test, its degrees of freedom with the corresponding p-value, AMML estimates of the location parameters (μ^1(AMML), μ^2(AMML)), and AMML estimates of the scale parameters (σ^1(AMML), σ^2(AMML)) are given by using the function

> RW(y1,y2)

The p-value and AMML estimates of the location and scale parameters are given for the RF test by using the function

> RF(y1,y2,iter=5000)

It should be noted that the p-value for the RF test is obtained using a computational approach, and it is based on the replication number in Algorithm 1, denoted as iter in the RF function. When the above-mentioned functions in the RobustBF package are performed, the following results are obtained

> RW(y1,y2)
    
       Robust Welch's Two Sample t-Test
    
data:  y1 and y2
RW = -3.1602, df = 36.892, p-value = 0.0031
alternative hypothesis: true difference between in means is not equal to 0
sample estimates:
  mean of y1  mean of y2   sd of y1   sd of y2 
      0.0626      1.2391     1.0861     1.2876
> RF(y1,y2,iter=5000)

      Robust Fiducial Based Test
    
data:  y1 and y2
p-value = 0.0032
alternative hypothesis: true difference in means is not equal to 0
sample estimates:
mean of y1 mean of y2   sd of y1   sd of y2 
    0.0626     1.2391     1.0861     1.2876
    
    

We also use t.test function in R to test the null hypothesis H0:μ1=μ2 and obtain its p-value as 0.0243. It can be seen from these results, RW, RF, and W tests reject the null hypothesis at α=0.05 significance level since the p-values corresponding to these tests are all less than 0.05. However, p-values for RW and RF tests are much smaller than the ones obtained for W. Results of the RW and RF tests are more reliable since the AMML estimates of the σ1 and σ2 (σ^1(AMML)=1.0861, σ^2(AMML)=1.2876) are less than the corresponding LS estimates (σ^1(LS)=1.2819, σ^2(LS)=1.6542). It should be noted that RW and RF tests reject the null hypothesis while W fails to reject it at the significance level α=0.01. These results are in agreement with the simulation results in the context of long-tailed symmetric distributions.

7 Conclusion

Reviewing the literature shows that comparing two means is a commonly encountered problem, especially in applied sciences when the usual normality and homogeneity of variances assumptions are violated. For this reason, in this study, we present RobustBF package and propose RW and RF tests to test the equality of two LTS means when the variances are unknown and arbitrary. The first test included in the package is a robust version of Welch’s t-test, and the other one is a robust fiducial-based test. The proposed tests are based on AMML estimators. Also, we use t.test function available in R to compare the proposed tests with Welch’s t-test in terms of Type I error rates and powers. Examining the results of the simulation study reveals that Type I error rates of the RW test are closer to the nominal level in general. Therefore, the RW test verifies the obtained null distribution for long-tailed symmetric samples. This test is followed by RF. RF does not require the knowledge of sampling distribution of the test statistics. W test appears to be conservative except for the t5, Logistic and Laplace distributions. RW shows the best power performance among the others besides being robust for the contamination model for the scenarios considered in this study. Therefore, the proposed RW test can be recommended for testing the equality of two LTS means under heterogeneity of variances. W test performs poorly in almost all cases. According to our knowledge, the proposed tests presented in the RobustBF package are not available in any other R tool.

Table 1: Simulated Type I error rates of the RW, RF and W tests for Models (a)-(h).
Model (a) Model (b)
n1 n2 RW RF W RW RF W
6 6 0.023 0.014 0.015 0.031 0.025 0.019
6 10 0.026 0.016 0.017 0.033 0.027 0.020
10 10 0.025 0.020 0.018 0.031 0.027 0.020
10 15 0.023 0.016 0.017 0.029 0.027 0.020
10 30 0.030 0.025 0.020 0.029 0.028 0.020
20 20 0.024 0.022 0.020 0.028 0.026 0.021
20 30 0.028 0.025 0.024 0.027 0.026 0.022
20 50 0.026 0.024 0.021 0.028 0.027 0.020
30 50 0.027 0.025 0.021 0.026 0.026 0.022
50 50 0.030 0.028 0.020 0.027 0.026 0.020
Model (c) Model (d)
n1 n2 RW RF W RW RF W
6 6 0.035 0.024 0.022 0.047 0.036 0.030
6 10 0.037 0.030 0.020 0.053 0.047 0.037
10 10 0.030 0.025 0.020 0.046 0.042 0.033
10 15 0.031 0.026 0.018 0.046 0.043 0.033
10 30 0.031 0.029 0.022 0.046 0.044 0.030
20 20 0.030 0.027 0.019 0.042 0.040 0.027
20 30 0.030 0.028 0.021 0.046 0.044 0.030
20 50 0.030 0.030 0.022 0.045 0.044 0.030
30 50 0.031 0.029 0.020 0.042 0.041 0.026
50 50 0.030 0.028 0.019 0.044 0.044 0.025
Model (e) Model (f)
n1 n2 RW RF W RW RF W
6 6 0.040 0.030 0.034 0.050 0.042 0.042
6 10 0.050 0.045 0.035 0.054 0.042 0.046
10 10 0.044 0.038 0.038 0.049 0.043 0.044
10 15 0.045 0.042 0.033 0.054 0.048 0.049
10 30 0.042 0.040 0.037 0.055 0.052 0.051
20 20 0.044 0.042 0.028 0.054 0.049 0.047
20 30 0.040 0.037 0.038 0.052 0.049 0.048
20 50 0.041 0.041 0.026 0.051 0.049 0.046
30 50 0.043 0.042 0.036 0.053 0.053 0.049
50 50 0.044 0.043 0.028 0.054 0.052 0.049
Model (g) Model (h)
n1 n2 RW RF W RW RF W
6 6 0.053 0.041 0.045 0.048 0.032 0.044
6 10 0.056 0.052 0.051 0.044 0.034 0.043
10 10 0.055 0.048 0.047 0.044 0.036 0.042
10 15 0.055 0.052 0.048 0.045 0.039 0.044
10 30 0.052 0.050 0.044 0.045 0.039 0.045
20 20 0.054 0.053 0.049 0.044 0.041 0.044
20 30 0.054 0.053 0.048 0.047 0.044 0.047
20 50 0.055 0.055 0.049 0.050 0.046 0.049
30 50 0.054 0.054 0.048 0.046 0.045 0.046
50 50 0.054 0.053 0.049 0.054 0.051 0.052
Table 2: Simulated powers of the RW, RF and W tests for Model (a).
d RW RF W d RW RF W
0.00 0.023 0.014 0.015 0.00 0.024 0.022 0.020
1.60 0.19 0.15 0.11 0.60 0.10 0.09 0.04
n=(6,6) 3.20 0.51 0.46 0.30 n=(20,20) 1.20 0.35 0.33 0.09
4.80 0.74 0.70 0.46 1.80 0.65 0.64 0.17
6.40 0.84 0.82 0.56 2.40 0.84 0.83 0.25
8.00 0.91 0.89 0.64 3.00 0.94 0.94 0.33
d RW RF W d RW RF W
0.00 0.026 0.016 0.017 0.00 0.028 0.025 0.024
1.50 0.21 0.17 0.10 0.50 0.09 0.08 0.03
n=(6,10) 3.00 0.57 0.53 0.29 n=(20,30) 1.00 0.31 0.29 0.07
4.50 0.78 0.76 0.44 1.50 0.59 0.58 0.13
6.00 0.89 0.88 0.57 2.00 0.80 0.79 0.20
7.50 0.94 0.93 0.62 2.50 0.92 0.91 0.27
d RW RF W d RW RF W
0.00 0.025 0.020 0.018 0.00 0.026 0.024 0.021
1.00 0.13 0.11 0.06 0.46 0.10 0.09 0.04
n=(10,10) 2.00 0.43 0.39 0.18 n=(20,50) 0.92 0.32 0.30 0.06
3.00 0.70 0.67 0.32 1.38 0.60 0.59 0.12
4.00 0.85 0.83 0.42 1.84 0.81 0.81 0.18
5.00 0.92 0.91 0.51 2.30 0.92 0.92 0.25
d RW RF W d RW RF W
0.00 0.023 0.016 0.017 0.00 0.027 0.025 0.021
0.80 0.11 0.09 0.05 0.40 0.09 0.09 0.03
n=(10,15) 1.60 0.36 0.33 0.14 n=(30,50) 0.80 0.30 0.29 0.06
2.40 0.64 0.61 0.24 1.20 0.60 0.59 0.10
3.20 0.80 0.79 0.35 1.60 0.83 0.83 0.16
4.00 0.90 0.89 0.43 2.00 0.94 0.94 0.22
d RW RF W d RW RF W
0.00 0.030 0.025 0.020 0.00 0.030 0.028 0.020
0.70 0.12 0.10 0.05 0.32 0.08 0.08 0.030
n=(10,30) 1.40 0.37 0.35 0.11 n=(50,50) 0.64 0.26 0.26 0.05
2.10 0.65 0.63 0.21 0.96 0.55 0.54 0.08
2.80 0.80 0.79 0.30 1.28 0.79 0.78 0.11
3.50 0.90 0.89 0.39 1.60 0.93 0.92 0.16
Table 3: Simulated powers of the RW, RF and W tests for Model (b).
d RW RF W d RW RF W
0.00 0.031 0.025 0.019 0.00 0.028 0.026 0.021
5.40 0.22 0.19 0.13 2.00 0.11 0.11 0.04
n=(6,6) 10.80 0.52 0.50 0.34 n=(20,20) 4.00 0.34 0.34 0.11
16.20 0.72 0.71 0.49 6.00 0.61 0.60 0.19
21.60 0.83 0.83 0.60 8.00 0.80 0.79 0.30
27.00 0.90 0.90 0.68 10.00 0.90 0.90 0.38
d RW RF W d RW RF W
0.00 0.033 0.027 0.020 0.00 0.027 0.026 0.022
5.30 0.22 0.21 0.13 2.00 0.10 0.10 0.05
n=(6,10) 10.60 0.53 0.52 0.35 n=(20,30) 4.00 0.33 0.33 0.10
15.90 0.72 0.71 0.50 6.00 0.62 0.62 0.20
21.20 0.82 0.82 0.59 8.00 0.81 0.80 0.29
26.50 0.90 0.90 0.68 10.00 0.90 0.90 0.37
d RW RF W d RW RF W
0.00 0.031 0.027 0.020 0.00 0.028 0.027 0.20
3.60 0.17 0.16 0.09 2.00 0.11 0.11 0.05
n=(10,10) 7.20 0.47 0.46 0.23 n=(20,50) 4.00 0.35 0.34 0.11
10.80 0.71 0.71 0.38 6.00 0.62 0.62 0.20
14.40 0.84 0.84 0.49 8.00 0.80 0.80 0.29
18.00 0.92 0.92 0.58 10.00 0.91 0.91 0.37
d RW RF W d RW RF W
0.00 0.029 0.027 0.020 0.00 0.026 0.026 0.022
3.60 0.17 0.16 0.09 1.60 0.10 0.10 0.04
n=(10,15) 7.20 0.49 0.48 0.23 n=(30,50) 3.20 0.32 0.32 0.08
10.80 0.72 0.72 0.39 4.80 0.62 0.62 0.15
14.40 0.85 0.85 0.50 6.40 0.82 0.82 0.23
18.00 0.92 0.91 0.57 8.00 0.92 0.92 0.30
d RW RF W d RW RF W
0.00 0.029 0.028 0.020 0.00 0.027 0.026 0.020
3.40 0.16 0.16 0.09 1.12 0.08 0.08 0.03
n=(10,30) 6.80 0.45 0.45 0.22 n=(50,50) 2.24 0.26 0.26 0.05
10.20 0.69 0.69 0.36 3.36 0.53 0.53 0.09
13.60 0.83 0.83 0.47 4.48 0.76 0.76 0.13
17.00 0.90 0.90 0.55 5.60 0.90 0.90 0.18
Table 4: Simulated powers of the RW, RF and W tests for Model (c).
d RW RF W d RW RF W
0.00 0.035 0.024 0.022 0.00 0.030 0.027 0.019
5.00 0.23 0.20 0.14 1.70 0.10 0.10 0.04
n=(6,6) 10.00 0.55 0.52 0.37 n=(20,20) 3.40 0.33 0.32 0.11
15.00 0.76 0.75 0.53 5.10 0.60 0.59 0.19
20.00 0.87 0.86 0.62 6.80 0.80 0.80 0.29
25.00 0.92 0.92 0.70 8.50 0.92 0.91 0.38
d RW RF W d RW RF W
0.00 0.037 0.030 0.020 0.00 0.030 0.028 0.021
4.80 0.23 0.21 0.14 1.70 0.10 0.10 0.04
n=(6,10) 9.60 0.56 0.54 0.36 n=(20,30) 3.40 0.33 0.33 0.11
14.40 0.75 0.74 0.52 5.10 0.62 0.61 0.20
19.20 0.86 0.86 0.62 6.80 0.81 0.81 0.30
24.00 0.92 0.92 0.69 8.50 0.92 0.92 0.37
d RW RF W d RW RF W
0.00 0.030 0.025 0.020 0.00 0.030 0.030 0.022
3.00 0.15 0.14 0.08 1.70 0.11 0.11 0.05
n=(10,10) 6.00 0.44 0.42 0.22 n=(20,50) 3.40 0.35 0.35 0.11
9.00 0.71 0.69 0.37 5.10 0.63 0.63 0.20
12.00 0.84 0.84 0.48 6.80 0.82 0.82 0.30
15.00 0.92 0.92 0.56 8.50 0.92 0.92 0.38
d RW RF W d RW RF W
0.00 0.031 0.026 0.018 0.00 0.031 0.029 0.20
2.60 0.13 0.12 0.06 1.30 0.09 0.09 0.03
n=(10,15) 5.20 0.38 0.37 0.18 n=(30,50) 2.60 0.31 0.30 0.07
7.80 0.64 0.63 0.32 3.90 0.58 0.58 0.13
10.40 0.79 0.79 0.44 5.20 0.79 0.79 0.20
13.00 0.90 0.90 0.54 6.50 0.92 0.92 0.29
d RW RF W d RW RF W
0.00 0.031 0.029 0.022 0.00 0.030 0.028 0.019
2.60 0.14 0.13 0.07 0.96 0.08 0.08 0.03
n=(10,30) 5.20 0.39 0.39 0.18 n=(50,50) 1.92 0.25 0.25 0.05
7.80 0.64 0.64 0.32 2.88 0.54 0.53 0.09
10.40 0.80 0.80 0.44 3.84 0.78 0.77 0.13
13.00 0.90 0.90 0.54 4.80 0.91 0.91 0.20
Table 5: Simulated powers of the RW, RF and W tests for Model (d).
d RW RF W d RW RF W
0.00 0.047 0.036 0.030 0.00 0.42 0.040 0.027
2.00 0.15 0.13 0.11 0.80 0.11 0.10 0.07
n=(6,6) 4.00 0.42 0.40 0.34 n=(20,20) 1.60 0.30 0.29 0.17
6.00 0.69 0.67 0.57 2.40 0.57 0.56 0.32
8.00 0.84 0.83 0.71 3.20 0.78 0.78 0.46
10.00 0.91 0.91 0.78 4.00 0.91 0.91 0.57
d RW RF W d RW RF W
0.00 0.053 0.047 0.037 0.00 0.046 0.044 0.030
2.00 0.16 0.15 0.12 0.80 0.13 0.13 0.09
n=(6,10) 4.00 0.43 0.42 0.34 n=(20,30) 1.60 0.37 0.36 0.25
6.00 0.70 0.69 0.59 2.40 0.65 0.65 0.47
8.00 0.84 0.84 0.72 3.20 0.82 0.82 0.62
10.00 0.91 0.91 0.79 4.00 0.91 0.91 0.71
d RW RF W d RW RF W
0.00 0.046 0.042 0.033 0.00 0.045 0.044 0.030
1.30 0.12 0.12 0.09 0.80 0.11 0.11 0.06
n=(10,10) 2.60 0.36 0.34 0.26 n=(20,50) 1.60 0.31 0.31 0.17
3.90 0.63 0.62 0.46 2.40 0.57 0.56 0.32
5.20 0.81 0.81 0.60 3.20 0.79 0.79 0.46
6.50 0.92 0.92 0.71 4.00 0.92 0.92 0.58
d RW RF W d RW RF W
0.00 0.046 0.043 0.033 0.00 0.042 0.041 0.026
1.30 0.13 0.12 0.09 0.64 0.11 0.11 0.05
n=(10,15) 2.60 0.37 0.36 0.26 n=(30,50) 1.28 0.30 0.30 0.14
3.90 0.65 0.64 0.47 1.92 0.56 0.56 0.27
5.20 0.83 0.82 0.62 2.56 0.79 0.79 0.39
6.50 0.92 0.91 0.71 3.20 0.92 0.92 0.51
d RW RF W d RW RF W
0.00 0.046 0.044 0.030 0.00 0.044 0.044 0.025
1.30 0.13 0.13 0.09 0.48 0.10 0.10 0.05
n=(10,30) 2.60 0.37 0.36 0.25 n=(50,50) 0.96 0.27 0.27 0.10
3.90 0.65 0.65 0.47 1.44 0.53 0.53 0.19
5.20 0.82 0.82 0.62 1.92 0.78 0.77 0.30
6.50 0.91 0.91 0.71 2.40 0.92 0.92 0.39
Table 6: Simulated powers of the RW, RF and W tests for Model (e).
d RW RF W d RW RF W
0.00 0.040 0.030 0.034 0.00 0.044 0.042 0.028
2.00 0.16 0.13 0.13 0.80 0.11 0.11 0.07
n=(6,6) 4.00 0.44 0.40 0.38 n=(20,20) 1.60 0.30 0.29 0.17
6.00 0.70 0.67 0.62 2.40 0.55 0.55 0.31
8.00 0.84 0.83 0.76 3.20 0.78 0.78 0.46
10.00 0.92 0.91 0.85 4.00 0.91 0.91 0.57
d RW RF W d RW RF W
0.00 0.050 0.045 0.035 0.00 0.040 0.037 0.038
1.90 0.15 0.14 0.11 0.80 0.10 0.10 0.08
n=(6,10) 3.80 0.41 0.40 0.33 n=(20,30) 1.60 0.31 0.30 0.23
5.70 0.67 0.66 0.55 2.40 0.59 0.58 0.42
7.60 0.83 0.83 0.71 3.20 0.81 0.81 0.60
9.50 0.90 0.90 0.77 4.00 0.92 0.92 0.74
d RW RF W d RW RF W
0.00 0.044 0.038 0.038 0.00 0.041 0.041 0.026
1.26 0.12 0.10 0.10 0.80 0.11 0.11 0.06
n=(10,10) 2.52 0.35 0.33 0.29 n=(20,50) 1.60 0.30 0.30 0.17
3.78 0.62 0.60 0.50 2.40 0.58 0.58 0.32
5.04 0.80 0.80 0.67 3.20 0.79 0.79 0.46
6.30 0.91 0.91 0.78 4.00 0.91 0.91 0.57
d RW RF W d RW RF W
0.00 0.045 0.042 0.033 0.00 0.043 0.042 0.041
1.24 0.12 0.12 0.09 0.64 0.11 0.11 0.08
n=(10,15) 2.48 0.33 0.33 0.23 n=(30,50) 1.28 0.31 0.31 0.21
3.72 0.60 0.60 0.44 1.92 0.59 0.58 0.38
4.96 0.80 0.80 0.59 2.56 0.81 0.81 0.56
6.20 0.90 0.90 0.68 3.20 0.93 0.93 0.70
d RW RF W d RW RF W
0.00 0.042 0.040 0.037 0.00 0.044 0.043 0.028
1.24 0.13 0.13 0.11 0.48 0.10 0.10 0.05
n=(10,30) 2.48 0.36 0.36 0.29 n=(50,50) 0.96 0.27 0.27 0.10
3.72 0.63 0.63 0.52 1.44 0.55 0.55 0.20
4.96 0.81 0.81 0.69 1.92 0.78 0.78 0.30
6.20 0.91 0.91 0.80 2.40 0.92 0.92 0.41
Table 7: Simulated powers of the RW, RF and W tests for Model (f).
d RW RF W d RW RF W
0.00 0.050 0.042 0.042 0.00 0.054 0.049 0.047
0.90 0.13 0.10 0.11 0.40 0.11 0.11 0.10
n=(6,6) 1.80 0.33 0.29 0.30 n=(20,20) 0.80 0.27 0.26 0.24
2.70 0.60 0.54 0.56 1.20 0.53 0.51 0.47
3.60 0.80 0.77 0.77 1.60 0.76 0.75 0.69
4.50 0.92 0.90 0.89 2.00 0.90 0.90 0.85
d RW RF W d RW RF W
0.00 0.054 0.042 0.046 0.00 0.052 0.049 0.048
0.90 0.13 0.11 0.12 0.40 0.11 0.11 0.09
n=(6,10) 1.80 0.37 0.34 0.34 n=(20,30) 0.80 0.29 0.29 0.25
2.70 0.63 0.59 0.59 1.20 0.55 0.54 0.48
3.60 0.83 0.81 0.80 1.60 0.77 0.77 0.71
4.50 0.92 0.92 0.90 2.00 0.92 0.92 0.87
d RW RF W d RW RF W
0.00 0.049 0.043 0.044 0.00 0.051 0.049 0.046
0.64 0.11 0.10 0.10 0.40 0.12 0.11 0.10
n=(10,10) 1.28 0.32 0.29 0.29 n=(20,50) 0.80 0.30 0.30 0.27
1.92 0.59 0.55 0.54 1.20 0.57 0.57 0.51
2.56 0.80 0.78 0.76 1.60 0.79 0.78 0.72
3.20 0.93 0.92 0.90 2.00 0.92 0.92 0.88
d RW RF W d RW RF W
0.00 0.054 0.048 0.049 0.00 0.053 0.053 0.049
0.60 0.12 0.11 0.10 0.32 0.11 0.11 0.10
n=(10,15) 1.20 0.31 0.29 0.28 n=(30,50) 0.64 0.29 0.28 0.25
1.80 0.56 0.54 0.52 0.96 0.55 0.55 0.49
2.40 0.78 0.77 0.73 1.28 0.78 0.78 0.71
3.00 0.92 0.91 0.88 1.60 0.92 0.92 0.87
d RW RF W d RW RF W
0.00 0.055 0.052 0.051 0.00 0.054 0.052 0.049
0.60 0.12 0.11 0.11 0.26 0.11 0.11 0.10
n=(10,30) 1.20 0.32 0.31 0.29 n=(50,50) 0.52 0.30 0.29 0.25
1.80 0.58 0.57 0.54 0.78 0.55 0.54 0.47
2.40 0.80 0.79 0.75 1.04 0.79 0.79 0.71
3.00 0.93 0.92 0.89 1.30 0.93 0.93 0.88
Table 8: Simulated powers of the RW, RF and W tests for Model (g).
d RW RF W d RW RF W
0.00 0.053 0.041 0.045 0.00 0.054 0.053 0.049
1.90 0.13 0.11 0.12 0.86 0.11 0.11 0.10
n=(6,6) 3.80 0.33 0.29 0.30 n=(20,20) 1.72 0.30 0.29 0.26
5.70 0.60 0.56 0.57 2.58 0.55 0.54 0.50
7.60 0.81 0.78 0.78 3.44 0.78 0.78 0.74
9.50 0.92 0.91 0.90 4.30 0.92 0.91 0.89
d RW RF W d RW RF W
0.00 0.056 0.052 0.051 0.00 0.054 0.053 0.048
1.90 0.13 0.12 0.11 0.86 0.12 0.11 0.10
n=(6,10) 3.80 0.35 0.33 0.32 n=(20,30) 1.72 0.30 0.30 0.27
5.70 0.61 0.59 0.58 2.58 0.56 0.56 0.51
7.60 0.82 0.81 0.80 3.44 0.80 0.79 0.76
9.50 0.93 0.92 0.91 4.30 0.92 0.92 0.90
d RW RF W d RW RF W
0.00 0.055 0.048 0.047 0.00 0.055 0.055 0.049
1.24 0.11 0.10 0.10 0.82 0.11 0.11 0.10
n=(10,10) 2.48 0.28 0.27 0.25 n=(20,50) 1.64 0.29 0.29 0.26
3.72 0.53 0.51 0.49 2.46 0.53 0.53 0.49
4.96 0.75 0.74 0.72 3.28 0.76 0.75 0.71
6.20 0.90 0.89 0.87 4.10 0.90 0.90 0.87
d RW RF W d RW RF W
0.00 0.055 0.052 0.048 0.00 0.054 0.054 0.048
1.24 0.11 0.11 0.10 0.64 0.11 0.11 0.10
n=(10,15) 2.48 0.29 0.28 0.27 n=(30,50) 1.28 0.27 0.27 0.24
3.72 0.54 0.53 0.50 1.92 0.50 0.50 0.46
4.96 0.76 0.75 0.72 2.56 0.74 0.74 0.70
6.20 0.90 0.90 0.88 3.20 0.89 0.89 0.86
d RW RF W d RW RF W
0.00 0.052 0.050 0.044 0.00 0.054 0.053 0.049
1.24 0.12 0.11 0.10 0.50 0.11 0.11 0.09
n=(10,30) 2.48 0.30 0.29 0.27 n=(50,50) 1.00 0.26 0.26 0.23
3.72 0.55 0.54 0.51 1.50 0.50 0.49 0.45
4.96 0.76 0.76 0.73 2.00 0.73 0.73 0.68
6.20 0.90 0.90 0.88 2.50 0.90 0.89 0.86
Table 9: Simulated powers of the RW, RF and W tests for Model (h).
d RW RF W d RW RF W
0.00 0.048 0.032 0.044 0.00 0.044 0.041 0.044
1.16 0.13 0.10 0.12 0.54 0.11 0.10 0.10
n=(6,6) 2.32 0.35 0.30 0.31 n=(20,20) 1.08 0.30 0.29 0.26
3.48 0.61 0.56 0.56 1.62 0.57 0.56 0.49
4.64 0.80 0.77 0.76 2.16 0.79 0.79 0.71
5.80 0.91 0.90 0.88 2.70 0.93 0.92 0.86
d RW RF W d RW RF W
0.00 0.044 0.034 0.043 0.00 0.047 0.044 0.047
0.84 0.11 0.09 0.10 0.44 0.11 0.10 0.09
n=(6,10) 1.68 0.30 0.26 0.27 n=(20,30) 0.88 0.28 0.27 0.24
2.52 0.57 0.52 0.51 1.32 0.56 0.54 0.47
3.36 0.78 0.74 0.72 1.76 0.79 0.78 0.69
4.20 0.91 0.89 0.69 2.20 0.92 0.92 0.86
d RW RF W d RW RF W
0.00 0.044 0.036 0.042 0.00 0.050 0.046 0.049
0.80 0.12 0.10 0.11 0.34 0.10 0.09 0.09
n=(10,10) 1.60 0.32 0.30 0.28 n=(20,50) 0.68 0.27 0.26 0.23
2.40 0.57 0.55 0.50 1.02 0.50 0.49 0.42
3.20 0.79 0.77 0.73 1.36 0.74 0.73 0.64
4.00 0.91 0.90 0.86 1.70 0.90 0.89 0.82
d RW RF W d RW RF W
0.00 0.045 0.039 0.044 0.00 0.046 0.045 0.046
0.64 0.11 0.10 0.10 0.34 0.11 0.10 0.09
n=(10,15) 1.28 0.29 0.27 0.25 n=(30,50) 0.68 0.29 0.28 0.24
1.92 0.55 0.52 0.48 1.02 0.55 0.54 0.45
2.56 0.78 0.76 0.71 1.36 0.79 0.78 0.69
3.20 0.91 0.90 0.85 1.70 0.92 0.92 0.85
d RW RF W d RW RF W
0.00 0.045 0.039 0.045 0.00 0.054 0.051 0.052
0.48 0.11 0.09 0.10 0.30 0.10 0.09 0.08
n=(10,30) 0.96 0.28 0.26 0.24 n=(50,50) 0.60 0.25 0.25 0.21
1.44 0.54 0.51 0.46 0.90 0.48 0.48 0.40
1.92 0.77 0.75 0.68 1.20 0.73 0.72 0.62
2.40 0.91 0.90 0.85 1.50 0.90 0.89 0.80

CRAN packages used

RobustBF, asht, WRS2

CRAN Task Views implied by cited packages

Robust

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.

Footnotes

    References

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    For attribution, please cite this work as

    Güven, et al., "RobustBF: An R Package for Robust Solution to the Behrens-Fisher Problem", The R Journal, 2021

    BibTeX citation

    @article{RJ-2021-107,
      author = {Güven, Gamze and Acıtaş, Şükrü and Şamkar, Hatice and Şenoğlu, Birdal},
      title = {RobustBF: An R Package for Robust Solution to the Behrens-Fisher Problem},
      journal = {The R Journal},
      year = {2021},
      note = {https://rjournal.github.io/},
      volume = {13},
      issue = {2},
      issn = {2073-4859},
      pages = {713-733}
    }