Welch’s two-sample
Testing the equality of two population means is one of the most
encountered problems in applied sciences. Student’s
When the literature is examined, it can be seen that Welch’s
In this study, a robust version of Welch’s
The R packages stats by R Core Team (1970) and
asht by Fay (2020) include
Welch’s
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
Assume that
The log-likelihood
By solving the above likelihood equations, (4) and (5),
simultaneously, the maximum likelihood (ML) estimators of the parameters
Theorem 1.
Theorem 2.
For proofs of theorems, see, e.g. Şenoğlu and Tiku (2001; Güven et al. 2019).
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
It should be noted that in AMML methodology,
In this section, we propose two different tests for testing the equality of two LTS means.
In this subsection, we briefly introduce Welch’s 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
Let
To verify the null distribution of the
In this section, fiducial-based test is proposed using the concept of
fiducial inference and pivotal model; see Fisher (1933, 1935) and Dawid and Stone (1982). Let denote the
Algorithm 1
For the given data, compute
Generate
Compute
Let
Repeat the steps 2-4
Compute the simulated
It should be noted that the squares of
In this section, Type I error rates and powers of the proposed tests
(
We use the following population distributions while generating samples.
Population 1 | Population 2 | ||
(a) | |||
(b) | |||
(c) | |||
(d) | |||
(e) | |||
(f) | |||
(g) | |||
(h) |
Here,
10,000 different samples are considered for each of size
To compute the Type I error rates of the
To compute the power of the tests, similar steps are followed, but a
constant
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,
Overall, the
In the RobustBF package, we show the implementation of the proposed
tests (
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(y1,y2)
The
> RF(y1,y2,iter=5000)
It should be noted that the 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
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
t.test
function available in R to compare the
proposed tests with Welch’s
Model (a) | Model (b) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
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) | ||||||||||
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) | ||||||||||
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) | ||||||||||
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 |
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 | ||
3.20 | 0.51 | 0.46 | 0.30 | 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 | ||
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 | ||
3.00 | 0.57 | 0.53 | 0.29 | 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 | ||
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 | ||
2.00 | 0.43 | 0.39 | 0.18 | 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 | ||
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 | ||
1.60 | 0.36 | 0.33 | 0.14 | 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 | ||
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 | ||
1.40 | 0.37 | 0.35 | 0.11 | 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 |
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 | ||
10.80 | 0.52 | 0.50 | 0.34 | 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 | ||
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 | ||
10.60 | 0.53 | 0.52 | 0.35 | 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 | ||
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 | ||
7.20 | 0.47 | 0.46 | 0.23 | 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 | ||
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 | ||
7.20 | 0.49 | 0.48 | 0.23 | 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 | ||
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 | ||
6.80 | 0.45 | 0.45 | 0.22 | 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 |
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 | ||
10.00 | 0.55 | 0.52 | 0.37 | 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 | ||
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 | ||
9.60 | 0.56 | 0.54 | 0.36 | 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 | ||
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 | ||
6.00 | 0.44 | 0.42 | 0.22 | 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 | ||
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 | ||
5.20 | 0.38 | 0.37 | 0.18 | 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 | ||
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 | ||
5.20 | 0.39 | 0.39 | 0.18 | 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 |
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 | ||
4.00 | 0.42 | 0.40 | 0.34 | 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 | ||
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 | ||
4.00 | 0.43 | 0.42 | 0.34 | 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 | ||
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 | ||
2.60 | 0.36 | 0.34 | 0.26 | 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 | ||
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 | ||
2.60 | 0.37 | 0.36 | 0.26 | 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 | ||
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 | ||
2.60 | 0.37 | 0.36 | 0.25 | 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 |
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 | ||
4.00 | 0.44 | 0.40 | 0.38 | 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 | ||
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 | ||
3.80 | 0.41 | 0.40 | 0.33 | 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 | ||
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 | ||
2.52 | 0.35 | 0.33 | 0.29 | 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 | ||
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 | ||
2.48 | 0.33 | 0.33 | 0.23 | 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 | ||
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 | ||
2.48 | 0.36 | 0.36 | 0.29 | 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 |
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 | ||
1.80 | 0.33 | 0.29 | 0.30 | 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 | ||
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 | ||
1.80 | 0.37 | 0.34 | 0.34 | 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 | ||
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 | ||
1.28 | 0.32 | 0.29 | 0.29 | 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 | ||
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 | ||
1.20 | 0.31 | 0.29 | 0.28 | 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 | ||
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 | ||
1.20 | 0.32 | 0.31 | 0.29 | 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 |
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 | ||
3.80 | 0.33 | 0.29 | 0.30 | 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 | ||
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 | ||
3.80 | 0.35 | 0.33 | 0.32 | 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 | ||
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 | ||
2.48 | 0.28 | 0.27 | 0.25 | 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 | ||
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 | ||
2.48 | 0.29 | 0.28 | 0.27 | 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 | ||
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 | ||
2.48 | 0.30 | 0.29 | 0.27 | 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 |
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 | ||
2.32 | 0.35 | 0.30 | 0.31 | 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 | ||
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 | ||
1.68 | 0.30 | 0.26 | 0.27 | 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 | ||
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 | ||
1.60 | 0.32 | 0.30 | 0.28 | 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 | ||
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 | ||
1.28 | 0.29 | 0.27 | 0.25 | 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 | ||
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 | ||
0.96 | 0.28 | 0.26 | 0.24 | 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 |
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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} }