Supplement to Accompany
“The Sensitivity of Three Methods to Non-normality and Unequal Variances
in Interval Estimation of Effect Sizes”
By L.-T. Chen and C.-Y. J. Peng
January, 2014
Table of Contents
Table A Population Characteristics for CR-3 Design with Coefficient (1, -0.5, -0.5) .................................. 2
Table B Population Characteristics for CR-4 Design with Coefficient (0.5, 0.5, -0.5, -0.5) ........................ 4
Example 1 ..................................................................................................................................................... 6
Example 2 ..................................................................................................................................................... 7
Additional Analyses: Eta-Squared Analyses of the Factors Manipulated .................................................... 8
1
Table A Population Characteristics for CR-3 Design with Coefficient (1, -0.5, -0.5)
Population ES (𝛿Bonett =
Variances (skewness, kurtosis)
1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (0, 0): 1 (0, 0): 1 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 1 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 1 (1, 1.5)
1 (0, 0): 1 (0, 0): 1 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 1 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2) : 1 (0, 0): 1 (0, 0)
1 (1.75, 3.75): 1 (0, 0): 1 (0, 0)
1 (1, 1.5): 1 (0, 0): 1 (0, 0)
1 (0.25, ̶ 0.75): 1 (0, 0): 1 (0, 0)
1 (0, ̶ 1.2; uniform): 1 (0, 0): 1 (0, 0)
1 (0, 0): 1 (0, 0): 2.25 (0, 0)
1 (0, 0): 1 (0, 0): 2.25 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 2.25 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 2.25 (1, 1.5)
1 (0, 0): 1 (0, 0): 2.25 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 2.25 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2) : 1 (0, 0): 2.25 (0, 0)
1 (1.75, 3.75): 1 (0, 0): 2.25 (0, 0)
1 (1, 1.5): 1 (0, 0): 2.25 (0, 0)
1 (0.25, ̶ 0.75): 1 (0, 0): 2.25 (0, 0)
1 (0, ̶ 1.2; uniform): 1 (0, 0): 2.25 (0, 0)
∑𝑘𝑗=1 𝑐j μj
∑𝑘 σ2
√ 𝑗=1 𝑗
𝑘
=
ψ
)
∑𝑘 σ2
√ 𝑗=1 𝑗
𝑘
0
𝜇1 = 𝜇2 = 𝜇3
=0
0.2
𝜇1 = 1, 𝜇2
= 𝜇3 = 0.8
0.5
𝜇1 = 1, 𝜇2 = 𝜇3
= 0.5
0.8
𝜇1 = 1, 𝜇2 = 𝜇3
= 0.2
𝜇1 = 0
𝜇2 = 0
𝜇3 = 0
𝜇1 = 1
𝜇2 = 0.762
𝜇3 = 0.762
𝜇1 = 1
𝜇2 = 0.405
𝜇3 = 0.405
𝜇1 = 1
𝜇2 = 0.048
𝜇3 = 0.048
2
1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (0, 0): 1 (0, 0): 4 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 4 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 4 (1, 1.5)
1 (0, 0): 1 (0, 0): 4 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 4 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2) : 1 (0, 0): 4 (0, 0)
1 (1.75, 3.75): 1 (0, 0): 4 (0, 0)
1 (1, 1.5): 1 (0, 0): 4 (0, 0)
1 (0.25, ̶ 0.75): 1 (0, 0): 4 (0, 0)
1 (0, ̶ 1.2; uniform): 1 (0, 0): 4 (0, 0)
1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (0, 0): 1 (0, 0): 8 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 8 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 8 (1, 1.5)
1 (0, 0): 1 (0, 0): 8 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 8 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2) : 1 (0, 0): 8 (0, 0)
1 (1.75, 3.75): 1 (0, 0): 8 (0, 0)
1 (1, 1.5): 1 (0, 0): 8 (0, 0)
1 (0.25, ̶ 0.75): 1 (0, 0): 8 (0, 0)
1 (0, ̶ 1.2; uniform): 1 (0, 0): 8 (0, 0)
𝜇1 = 0
𝜇2 = 0
𝜇3 = 0
𝜇1 = 1
𝜇2 = 0.717
𝜇3 = 0.717
𝜇1 = 1
𝜇2 = 0.293
𝜇3 = 0.293
𝜇1 = 1
𝜇2 = −0.131
𝜇3 = −0.131
𝜇1 = 0
𝜇2 = 0
𝜇3 = 0
𝜇1 = 1
𝜇2 = 0.635
𝜇3 = 0.635
𝜇1 = 1
𝜇2 = 0.087
𝜇3 = 0.087
𝜇1 = 1
𝜇2 = −0.461
𝜇3 = −0.461
3
Table B Population Characteristics for CR-4 Design with Coefficient (0.5, 0.5, -0.5, -0.5)
Population ES (𝛿Bonett =
Variances (skewness, kurtosis)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (1, 1.5)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 1 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (1.75, 3.75): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (1, 1.5): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (0.25, ̶ 0.75): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (0, ̶ 1.2; uniform): 1 (0, 0): 1 (0, 0): 1 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (2,6; chi-squared with df
=2)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (1, 1.5)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 2.25 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2): 1 (0, 0): 1 (0, 0): 2.25 (0,
0)
1 (1.75, 3.75) : 1 (0, 0): 1 (0, 0): 2.25 (0, 0)
1 (1, 1.5) : 1 (0, 0): 1 (0, 0): 2.25 (0, 0)
1 (0.25, ̶ 0.75) : 1 (0, 0): 1 (0, 0): 2.25 (0, 0)
1 (0, ̶ 1.2; uniform) : 1 (0, 0): 1 (0, 0): 2.25 (0, 0)
0
𝜇1 = 𝜇2
= 𝜇3 = 𝜇4 = 0
𝜇1
𝜇2
𝜇3
𝜇4
=0
=0
=0
=0
4
∑𝑘𝑗=1 𝑐j μj
∑𝑘 σ2
√ 𝑗=1 𝑗
𝑘
=
ψ
)
∑𝑘 σ2
√ 𝑗=1 𝑗
𝑘
0.2
𝜇1 = 𝜇2 = 1,
𝜇3 = 𝜇4 = 0.8
0.5
𝜇1 = 𝜇2 = 1,
𝜇3 = 𝜇 4
= 0.5
0.8
𝜇1 = 𝜇2 = 1,
𝜇3 = 𝜇4 = 0.2
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.771
𝜇4 = 0.771
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.427
𝜇4 = 0.427
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.083
𝜇4 = 0.083
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (1, 1.5)
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 4 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2): 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (1.75, 3.75) : 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (1, 1.5) : 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (0.25, ̶ 0.75) : 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (0, ̶ 1.2; uniform) : 1 (0, 0): 1 (0, 0): 4 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (2,6; chi-squared with df =2)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (1.75, 3.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (1, 1.5)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (0.25, ̶ 0.75)
1 (0, 0): 1 (0, 0): 1 (0, 0): 8 (0, ̶ 1.2; uniform)
1 (2,6; chi-squared with df =2): 1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (1.75, 3.75) : 1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (1, 1.5) : 1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (0.25, ̶ 0.75) : 1 (0, 0): 1 (0, 0): 8 (0, 0)
1 (0, ̶ 1.2; uniform) : 1 (0, 0): 1 (0, 0): 8 (0, 0)
𝜇1
𝜇2
𝜇3
𝜇4
=0
=0
=0
=0
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.735
𝜇4 = 0.735
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.339
𝜇4 = 0.339
𝜇1 = 1
𝜇2 = 1
𝜇3 = −0.058
𝜇4 = −0.058
𝜇1
𝜇2
𝜇3
𝜇4
=0
=0
=0
=0
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.668
𝜇4 = 0.668
𝜇1 = 1
𝜇2 = 1
𝜇3 = 0.171
𝜇4 = 0.171
𝜇1 = 1
𝜇2 = 1
𝜇3 = −0.327
𝜇4 = −0.327
5
Example 1
Condition examined: k =3, the first group sampled from a uniform distribution with variance
ratio 2.25 and ES = 0.2
The first group sampled from a uniform distribution with variance ratio 2.25 and ES = 0.2, the
scores of the first group were generated from the CALL RANUNI function and the scores had .5
subtracted from them, followed by the square root of 1/12 divided from them, and then had 1
added to them. The scores of the second group were generated from the function RAND
(‘NORMAL’, .762, 1). The scores from the third group were generated from the function RAND
(‘NORMAL’, .762’, 1.5).
6
Example 2
Condition examined: k =4, the last group sampled from a chi-square distribution with df = 2 with
variance ratio 8 and ES = 0.2
The scores of the first two groups were generated from the function RAND('NORMAL',1,1).
The scores of the third group were generated from the function RAND('NORMAL',.668,1). The
scores of the fourth group were generated from RAND (‘CHISQUARE’, 2) and the scores had 2
subtracted from them, followed by 2 divided from them, the square root of 8 multiplied by them,
and then had .668 added to them.
7
Additional Analyses: Eta-Squared Analyses of the Factors Manipulated
To examine the relative contributions of factors to the variance of coverage probability
and of interval width, we computed eta-squared (  2 ) for six main factors, and their interactions.
Results are shown in Tables 9 (for k = 3) and 10 (for k = 4). We decided to report eta-squared,
instead of F-tests of effects or its p-value, because there is only one observation per cell,
resulting in zero degrees of freedom for error term. If an eta-squared was ≥ 8%, its corresponding
effect was considered substantial in explaining the variability in coverage probabilities or
interval widths. The 8% cutoff was chosen because (1) it falls between a medium effect (.0588)
and a large effect (.1379), according to Cohen (1969), and (2) it helped to identify patterns of
substantial effects in this simulation study. The substantial eta-squared values are bolded in
Tables 9 and 10.
The eta squares were obtained from imposing a 3  6  4  4  3 factorial linear model on
either the estimated coverage probability or the estimated interval width, as the dependent
variable. The independent variables were the CI methods (= 3), population distributions (= 6),
population variance ratios (= 4), population ESs (= 4), and sample size patterns (= 3). Because
either the first group or the last was sampled from a non-normal distribution, two separate
ANOVAs were performed to correspond to these two pairings. The pairing of the first group
with a non-normal distribution will be referred to as First Non-normal in subsequent paragraphs.
The pairing of the last group with a non-normal distribution will be referred to as Last Nonnormal in subsequent paragraphs. These two configurations will be compared to aid our
understanding of the confounding of population distributions and variance ratios with weights
(for k = 3), or without weights (for k = 4). When k = 3, the weights assigned to the three groups
were (1, −0.5, −0.5). Hence, the confounding of non-normal distributions paired with the first or
8
the last group, and variance ratios was further confounded with the weight, because large
variances were always assigned to the last group in the unequal variance conditions. When k = 4,
the weights assigned to the four groups were (0.5, 0.5, −0.5, −0.5), all equal in absolute values.
Hence, the confounding of non-normal distributions paired with the first or the last group, and
variance ratios was not confounded with the weight, even when large variances were always
assigned to the last group in the unequal variance conditions. Results are presented in the next
four sections.
Eta-squared analysis of coverage probabilities for k = 3 (Table 9)
First Non-normal conditions. Using 8% as a criterion, we identified five effects that
were substantial under the First Non-normal conditions for coverage probabilities: method,
sample size pattern, method by sample size pattern, variance ratio by sample size pattern, and
method by variance ratio by sample size pattern. Together they explained 77.61% of the total
variance. Among these five effects, method by sample size pattern was the most influential effect,
accounting for nearly one third of the total variance in coverage probabilities. Figure 1 further
illustrates that Bonett’s and the BCa methods performed similarly to each other, yet the
noncentral method performed poorly for the sample size patterns of (10 10 10) and (10 10 30).
The other four effects each explained about 10% of the total variance. These effects were further
elaborated on in Figures A to D.
Last Non-normal conditions. Using the same criterion, we identified five effects that
were substantial under the Last Non-normal conditions for coverage probabilities: distribution,
ES, sample size pattern, method by sample size pattern, and variance ratio by sample size pattern.
Together they explained 55.56% of the total variance. Among these five effects, method by
sample size pattern was once again the most influential effect, accounting for 15.44% of the total
9
variance in coverage probabilities. Figure 2 further illustrates that Bonett’s and the BCa methods
performed similarly to each other, yet the noncentral method performed poorly for the sample
size patterns of (10 10 30) and (30 10 10). The other four effects each explained about 10% of
the total variance. These effects were further elaborated on in Figures E to H.
Eta-squared analysis of interval widths for k =3 (Table 9)
Results for both the First Non-normal and the Last Non-normal conditions were similar.
Using 8% as a criterion, we identified four effects that were substantial: sample size pattern,
method by sample size pattern, variance ratio by sample size pattern, and variance ratio
(primarily for the First Non-normal conditions). Together they explained approximately 80% of
the total variability in interval widths. Among these four effects, the sample size pattern
accounted for the largest eta-squared, about 54%. The narrowest to the widest intervals were
given by the sample size patterns of (30 10 10), (10 10 30), and (10 10 10), respectively (see
Figures 3 and 4). Because the sample size pattern (30 10 10) was always paired the largest
sample size (30) with the largest weight (1), it therefore yielded the narrowest interval, regardless
of the methods, distributions, variance ratios, population ESs, or the pairing of the first or the last
group with a non-normal distribution. The interaction of variance ratio by sample size pattern
(Figures I and J) modified the main effect of sample size pattern and the main effect of variance
ratio under the First Non-Normal conditions only (Figure K). Likewise, the interaction of method
by sample size pattern (Figures L and M) modified the main effect of sample size pattern.
Eta-squared analysis of coverage probabilities for k = 4 (Table 10)
As stated before, the weights assigned to the four groups were (0.5, 0.5, −0.5, −0.5) for k
= 4. Weights were not confounded with distributions and variance ratios, even when large
variances were always assigned to the last group in the unequal variance conditions.
10
First Non-normal conditions. Using 8% as a criterion, we identified five effects that
were substantial under the First Non-normal conditions for coverage probabilities: ES, sample
size pattern, method by ES, method by sample size pattern, and method by variance ratio by
sample size pattern. Together they explained 78.34% of the total variance. Among these five
effects, method by sample size pattern was the most influential effect, accounting for a little
more than a quarter of the total variance in coverage probabilities. Figure 5 further illustrates that
Bonett’s and the BCa methods performed similarly to each other, yet the noncentral method
performed poorly for the sample size patterns of (10 10 10 30) and (30 10 10 10). In other words,
the unequal sample sizes affected the performance of the noncentral method, but not Bonett’s
method or the BCa method. The other four effects explained 8.22% to 17.43% of the total
variance. These effects were further elaborated on in Figures N to Q.
Last Non-normal conditions. Using the same criterion, we identified five effects that
were substantial under the Last Non-normal conditions for coverage probabilities: distribution,
variance ratio, ES, sample size pattern, and method by sample size pattern. Together they
explained 55.09% of the total variance. Among these five effects, the distribution was the most
influential effect, accounting for 13.13% of the total variance in coverage probabilities. Figure 6
further illustrates the impact of distributions on coverage probabilities. The coverage
probabilities obtained from normal distributions were similar to those obtained from uniform and
the slightly non-normal distribution with skewness = 0.25 and kurtosis =  0.75 , than the other
three distributions. The other four effects each explained about 10% of the total variance. These
effects were further elaborated on in Figures R to U.
Eta-squared analysis of interval widths for k = 4 (Table 10)
11
Results for both the First Non-normal and the Last Non-normal conditions were similar.
Using 8% as a criterion, we identified three effects that were substantial: sample size pattern,
method by sample size pattern, and variance ratio by sample size pattern. Together they
explained 87.86% of the total variability in interval widths under the First Non-normal
conditions and 74.39% of the total variability under the Last Non-normal conditions. Among
these three effects, the sample size pattern accounted for the largest eta-squared, more than 50%.
The narrowest to the widest intervals were given by the sample size patterns of (10 10 10 30),
(30 10 10 10), (10 10 10 10), respectively (Figures 7 and 8). Because the width of a CI is
inversely related to SE which is determined from the total sample size. The two unequal sample
size patterns therefore resulted in narrower intervals than the equal sample size pattern,
regardless of the methods, distributions, variance ratios, population ESs, or the pairing of the first
or the last group with a non-normal distribution. The interactions of method or variance ratio by
sample size pattern modified the main effect of sample size pattern (Figures 9 to 12). Regardless
of the pairing of the first or the last group with a non-normal distribution, the CIs obtained from
the noncentral method were similar and narrower for the two unequal sample size patterns than
those obtained under equal sample sizes (Figures 9 and 10). The narrowest to the widest CIs of
Bonett’s and the BCa’s were obtained from (10 10 10 30), (30 10 10 10), and (10 10 10 10)
patterns respectively. Furthermore, the intervals widened as the variance ratio increased for the
sample size patterns of (10 10 10 10) and (30 10 10 10), whereas they narrowed as the variance
ratio increased for the sample size pattern of (10 10 10 30) (see Figures 11 and 12). This
phenomenon was observed under both the First Non-normal and the Last Non-normal conditions.
12
Table 9
Eta-squared Analysis of Coverage Probabilities and Interval Widths for k =3: Normal
Distributions and the First or the Last Group Sampled from a Non-normal Distribution
Eta-squared in Percentage
Coverage Probability
Interval Width
First
Last
First
Last
NonNonNonNonnormal
normal
normal
normal
Simulation Main and Interaction Effects
method
2.6292
4.5135
2.0721
12.8979
distribution
0.0836
0.5608
1.5079
11.7111
variance ratio
0.7823
4.1192
9.5077
7.0713
ES
3.0590
1.1842
3.7592
8.3606
sample size pattern
9.5146
11.5478
54.2837
53.5709
method*distribution
1.3540
1.3024
0.4664
1.6978
method*variance ratio
1.0899
0.5670
4.9144
4.2434
distribution*variance ratio
0.0869
4.0755
0.1018
0.2478
method*ES
5.0448
1.4207
0.3620
0.5277
distribution*ES
0.8258
5.2493
0.0919
0.4361
variance ratio*ES
1.6482
2.7950
0.1446
0.2471
method*sample size pattern
33.9598
15.4382
9.6285
8.9634
distribution*sample size pattern
0.4279
3.0740
0.1446
0.0535
variance ratio*sample size pattern
10.1565
8.5072
9.1724
9.4594
ES*sample size pattern
1.6214
2.7770
0.1373
0.2063
method*distribution*variance ratio
0.2206
0.3484
0.0937
0.3170
method*distribution*ES
0.0903
0.7804
0.1581
0.6418
method*variance ratio*ES
0.5550
0.2558
0.0659
0.1398
distribution*variance ratio*ES
0.3611
1.6495
0.0223
0.1107
method*distribution*sample size pattern
0.2073
0.0810
0.0963
0.0368
method*variance ratio*sample size pattern
4.0937
4.2321
11.0826
6.7060
distribution*variance ratio*sample size pattern
0.2407
1.3217
0.0267
0.0251
method*ES*sample size pattern
1.9424
0.9450
0.0242
0.0147
distribution*ES*sample size pattern
0.2808
1.2484
0.0139
0.0395
variance ratio*ES*sample size pattern
0.4781
0.7881
0.0231
0.0482
Note. Percentage of main or interaction effects greater than 8 is bolded. Since no four-way or five-way
interaction is greater than 1%, they are not presented in this table.
13
Table 10
Eta-squared Analysis of Coverage Probabilities and Interval Widths for k =4: Normal
Distributions and the First or the Last Group Sampled from a Non-normal Distribution
Eta-squared in Percentage
Coverage Probability
Interval Width
First
Last
First
Last
NonNonNonNonnormal
normal
normal
normal
Simulation Main and Interaction Effects
method
0.6806
0.9492
1.9842
5.3726
distribution
0.1342
0.0459
1.1261
13.1343
variance ratio
3.5490
0.2139
0.1272
8.4433
ES
2.6004
5.4840
8.2166
10.3457
sample size pattern
17.4257
12.2309
64.3490
53.7211
method*distribution
0.0827
1.0433
0.0898
1.7593
method*variance ratio
1.6374
1.0380
0.4449
1.3670
distribution*variance ratio
0.1830
4.9492
0.0119
0.3171
method*ES
4.0626
0.3022
0.7896
11.2437
distribution*ES
0.1854
5.9127
0.0224
0.7394
variance ratio*ES
0.8641
2.4371
0.1207
0.3438
method*sample size pattern
27.9892
10.9392
11.2921
9.5761
distribution*sample size pattern
0.1811
1.7964
0.0321
0.0377
variance ratio*sample size pattern
7.8313
6.6847
12.2142
11.0974
ES*sample size pattern
1.1263
0.9454
0.1479
0.2337
method*distribution*variance ratio
0.0719
0.3339
0.0241
0.5290
method*distribution*ES
0.0669
0.8766
0.0452
1.1646
method*variance ratio*ES
0.2277
0.2207
0.0590
0.2555
distribution*variance ratio*ES
0.3738
1.9016
0.0225
0.1695
method*distribution*sample size pattern
0.0773
0.1111
0.0407
0.0785
method*variance ratio*sample size pattern
5.4825
5.0270
13.4683
7.1443
distribution*variance ratio*sample size pattern
0.2676
1.1382
0.0316
0.0168
method*ES*sample size pattern
1.4167
0.3488
0.0218
0.0338
distribution*ES*sample size pattern
0.1707
0.7510
0.0273
0.0393
variance ratio*ES*sample size pattern
0.4672
0.4575
0.0395
0.0560
Note. Percentage of main or interaction effects greater than 8 is bolded. Since no four-way or five-way
interaction is greater than 1%, they are not presented in this table.
14
Figure 1. The interaction of method by sample size pattern on coverage probability under First
Non-normal for k =3
15
Figure 2. The interaction of method by sample size pattern on coverage probability under Last
Non-normal for k =3
16
Figure 3. The main effect of sample size pattern on interval width under First Non-normal for k
=3
17
Figure 4. The main effect of sample size pattern on interval width under Last Non-normal for k
=3
18
Figure 5. The interaction of method by sample size pattern on coverage probability under First
Non-normal for k =4
19
Figure 6. The main effect of population distribution on coverage probability under Last Nonnormal for k =4
20
Figure 7. The main effect of sample size pattern on interval width under First Non-normal for k
=4
21
Figure 8. The main effect of sample size pattern on interval width under Last Non-normal for k
=4
22
Figure 9. The interaction of method by sample size pattern on interval width under First Nonnormal for k =4
The eta-squared for the interaction of method by sample size pattern under the First Non-normal
conditions was 11.29% demonstrating the impact of sample size patterns on interval widths to be
different across three methods. Figure 9 shows that the interval widths obtained from the
noncentral method were similar under the sample size patterns of (10 10 10 30) and (30 10 10
10); the CIs obtained from these two sample size patterns were narrower than the CIs obtained
from equal sample sizes. The interval widths obtained from Bonett’s method and the BCa
method from the narrowest to the widest was in the order of (30 10 10 10), (10 10 10 30), and
(10 10 10 10).
23
Figure 10. The interaction of method by sample size pattern on interval width under Last Nonnormal for k =4
The eta-squared for the interaction of method by sample size pattern under the Last Non-normal
conditions was 9.58% demonstrating the impact of sample size patterns on interval widths to be
different across three methods. Figure 10 shows that the interval widths obtained from the
noncentral method were similar under the sample size patterns of (10 10 10 30) and (30 10 10
10); the CIs obtained from these two sample size patterns were narrower than the CIs obtained
from the equal sample sizes. The CIs obtained from Bonett’s method and the BCa method from
the narrowest to the widest was in the order of (30 10 10 10), (10 10 10 30), and (10 10 10 10).
24
Figure 11. The interaction of variance ratio by sample size pattern on interval width under First
Non-normal for k =4
The eta-squared for the interaction of variance ratio by sample size pattern under the First Nonnormal conditions was 12.21% demonstrating the impact of variance ratios on interval widths to
be different across three sample size patters. Figure 11 shows that the CIs obtained from the
sample size patterns of (10 10 10 10) and (30 10 10 10) became wider as the variance ratio
increased, whereas the CIs obtained from the sample size pattern of (10 10 10 30) became
narrower as the variance ratio increased.
25
Figure 12. The interaction of variance ratio by sample size pattern on interval width under Last
Non-normal for k =4
The eta-squared for the interaction of variance ratio by sample size pattern under the Last Nonnormal conditions was 11.10% demonstrating the impact of variance ratios on interval widths to
be different across three sample size patters. Figure 12 shows that the CIs obtained from the
sample size patterns of (10 10 10 10) and (30 10 10 10) became wider as the variance ratio
increased; whereas the CIs obtained from the sample size pattern of (10 10 10 30) became
narrower as the variance ratio increased.
26
Figure A. The method main effect on coverage probability under First Non-normal for k =3
The eta-squared for the main effect of method under the First Non-normal conditions was
12.90% demonstrating the impact of methods on coverage probabilities. Figure A shows that
Bonett’s and the BCa methods performed similarly to each other and they performed better than
the noncentral method.
27
Figure B. The main effect of sample size pattern on coverage probability under First Non-normal
for k =3
The eta-squared for the main effect of sample size pattern under the First Non-normal conditions
was 9.51% demonstrating the impact of sample size patterns on coverage probabilities. Figure B
shows when the sample size pattern was (30 10 10), the mean of the coverage probabilities was
closer to the nominal CI, than when the sample size patterns were (10 10 10) or (10 10 30). Yet,
when the sample size pattern was (30 10 10), the range of the coverage probabilities was the
largest.
28
Figure C. The interaction of variance ratio by sample size pattern on coverage probability under
First Non-normal for k =3
The eta-squared for the interaction of variance ratio by sample size pattern under the First Nonnormal conditions was 10.16% demonstrating the impact of variance ratio on coverage
probabilities to be different across three sample size patterns. Figure C shows when the sample
size patterns were (10 10 10) and (10 10 30), the mean coverage probabilities were higher when
the variance ratio was 4 or 8, than when the variance ratio was 1 or 2.25. When the sample size
pattern was (30 10 10), the coverage probabilities decreased as the variance ratio increased.
29
Figure D. The interaction of method by variance ratio by sample size pattern on coverage
probability under First Non-normal for k =3
30
The eta-squared for the interaction of method by variance ratio by sample size pattern under the
First Non-normal conditions was 11.08% demonstrating the interaction of variance ratio by
sample size pattern on coverage probability to be different across three methods. Figure D shows
for the noncentral method and the sample size patterns of (10 10 10) and (10 10 30), the
coverage probabilities increased as the variance ratio increased. When the sample size pattern
was (30 10 10), the coverage probabilities decreased as the variance ratio increased for the
noncental method. For Bonett’s and the BCa methods, their coverage probabilities were
relatively consistent across different variance ratios and sample size patterns, compared to the
corresponding coverage probabilities produced by the noncentral method.
31
Figure E. The main distribution effect on coverage probability under Last Non-normal for k =3
The eta-squared for the main effect of distributions under the Last Non-normal conditions was
11.71% demonstrating the impact of distributions on coverage probabilities. Figure E shows that
the coverage probabilities obtained from normal distributions were similar to those obtained
from uniform and slightly non-normal distribution (skewness = .25 and kurtosis = −.75) than the
other three distributions.
32
Figure F. The main effect of ES on coverage probability under Last Non-normal for k =3
The eta-squared for the main effect of ES under the Last Non-normal conditions was 8.36%
demonstrating the impact of ES on coverage probability. Figure F shows that the coverage
probabilities obtained from ES = 0 were more similar to those obtained from ES = 0.2 or 0.5 than
ES = 0.8.
33
Figure G. The main effect of sample size pattern on coverage probability under Last Non-normal
for k =3
The eta-squared for the main effect of sample size pattern under the Last Non-normal conditions
was 11.55% demonstrating the impact of sample size patterns on coverage probabilities. Figure
G shows that the coverage probabilities obtained from the sample size pattern of (10 10 10) was
more similar to the sample size pattern of (10 10 30) than to (30 10 10).
34
Figure H. The interaction of variance ratio by sample size pattern on coverage probability under
Last Non-normal for k =3
The eta-squared for the interaction of variance ratio by sample size pattern under the Last Nonnormal conditions was 8.51% demonstrating the impact of variance ratios on coverage
probabilities to be different across three sample size patterns. Figure H shows that when the
sample size pattern was (10 10 30), the coverage probabilities increased as the variance ratio
increased. When the sample size pattern was (30 10 10), the coverage probabilities decreased as
the variance ratio increased. When the sample size pattern was (10 10 10), the coverage
probabilities obtained from equal variances were more similar to those obtained from variance
ratios of 2.25 or 4, than from the variance ratio of 8.
35
Figure I. The interaction of variance ratio by sample size pattern on interval width under First
Non-normal for k =3
The eta-squared for the interaction of variance ratio by sample size pattern under the First Nonnormal conditions was 9.17% demonstrating the impact of variance ratios on interval widths to
be different across three sample size patterns. Figure I shows that the interval widths became
narrower as the variance ratio increased under sample size patterns of (10 10 10) and (10 10 30).
The interval widths did not change much across different variance ratios under the sample size
pattern of (30 10 10).
36
Figure J. The interaction of variance ratio by sample size pattern on interval width under Last
Non-normal for k =3
The eta-squared for the interaction of variance ratio by sample size pattern under the Last Nonnormal conditions was 9.46% demonstrating the impact of variance ratios on interval widths to
be different across three sample size patterns. Figure J shows that the interval widths became
narrower as the variance ratio increased under sample size patterns of (10 10 10) and (10 10 30).
The interval widths did not change much across different variance ratios under the sample size
pattern of (30 10 10).
37
Figure K. The main effect of variance ratio on interval width under First Non-normal for k =3
The eta-squared for the main effect of variance ratio under the First Non-normal conditions was
9.51% demonstrating the impact of variance ratios on interval widths. Figure K shows that the
interval widths became narrower as the variance ratio increased.
38
Figure L. The interaction of method by sample size pattern on interval width under First Nonnormal for k =3
The eta-squared for the interaction of method by sample size pattern under the First Non-normal
conditions was 9.63% demonstrating the impact of sample size patterns on interval widths to be
different across three methods. Figure L shows that the interval widths obtained from the
noncentral method from the narrowest to the widest was in the order of (30 10 10), (10 10 30),
and (10 10 10). The interval widths obtained from Bonett’s method and the BCa method were
similar under the sample size patterns of (10 10 30) and (30 10 10); the interval widths obtained
from these two sample size patterns were narrower than the interval widths obtained from the
equal sample size pattern.
39
Figure M. The interaction of method by sample size pattern on interval width under Last Nonnormal for k =3
The eta-squared for the interaction of method by sample size pattern under the Last Non-normal
conditions was 8.96% demonstrating the impact of sample size patterns on interval widths to be
different across three methods. Figure M shows that the interval widths obtained from the
noncentral method from the narrowest to the widest was in the order of (30 10 10), (10 10 30),
and (10 10 10). The interval widths obtained from Bonett’s method and the BCa method were
similar under the sample size patterns of (10 10 30) and (30 10 10) and the interval widths from
these two sample size patterns were narrower than those obtained from the equal sample size
pattern.
40
Figure N. The main effect of ES on coverage probability under First Non-normal for k =4
The eta-squared for the main effect of ES under the First Non-normal conditions was 8.22%
demonstrating the impact of ES on coverage probabilities. Figure N shows that the coverage
probabilities were closer to the nominal CI when ES = 0 or 0.8 than when ES = 0.2 or 0.5.
41
Figure O. The main effect of sample size pattern on coverage probability under First Non-normal
for k =4
The eta-squared for the main effect of sample size pattern under the First Non-normal conditions
was 17.43% demonstrating the impact of sample size patterns on coverage probabilities. Figure
O shows that when the sample size pattern was (30 10 10 10), the mean of the coverage
probabilities was closer to the nominal CI, than when the sample size patterns were (10 10 10 10)
or (10 10 10 30). Yet, when the sample size pattern was (30 10 10 10), the range of the coverage
probabilities was the largest.
42
Figure P. The interaction of method by ES on coverage probability under First Non-normal for k
=4
The eta-squared for the interaction of method by ES under the First Non-normal conditions was
11.24% demonstrating the impact of ES on coverage probabilities to be different across three
methods. Figure P shows that coverage probabilities produced by the noncentral method were
closer to the nominal CI when ES = 0 or 0.8 than when ES = 0.2 or 0.5. The coverage
probabilities produced by Bonett’s method and the BCa method were consistently close to the
nominal CI regardless of ESs.
43
Figure Q. The interaction of method by variance ratio by sample size pattern on coverage
probability under First Non-normal for k =4
44
The eta-squared for the interaction of method by variance ratio by sample size pattern under the
First Non-normal conditions was 13.47% demonstrating the interaction of variance ratio by
sample size pattern on coverage probability to be different across three methods. Figure Q shows
that for the noncentral method and the sample size pattern of (10 10 10 30), the coverage
probabilities increased as the variance ratio increased. When the sample size patterns were (10,
10, 10, 10) and (30, 10, 10, 10), the coverage probabilities decreased as the variance ratio
increased for the noncentral method. For Bonett’s and the BCa method, their coverage
probabilities were relatively consistent across different variance ratios and sample size patterns,
compared to the corresponding coverage probabilities produced by the noncentral method.
45
Figure R. The interaction of method by sample size pattern on coverage probability under Last
Non-normal for k =4
The eta-squared for the interaction of method by sample size pattern under the Last Non-normal
conditions was 10.94% demonstrating the impact of sample size patterns on coverage
probabilities to be different across three methods. Figure R further illustrates that Bonett’s and
the BCa methods performed similarly to each other, yet the noncentral method performed poorly
for the sample size patterns of (10 10 10 30) and (30 10 10 10).
46
Figure S. The main effect of variance ratio on coverage probability under Last Non-normal for k
=4
The eta-squared for the main effect of variance ratio under the Last Non-normal conditions was
8.44% demonstrating the impact of variance ratio on coverage probability. Figure S shows that
the coverage probabilities obtained from ES = 0 were more similar to those obtained from ES =
0.2 or 0.5 than ES = 0.8.
47
Figure T. The main effect of ES on coverage probability under Last Non-normal for k =4
The eta-squared for the main effect of ES under the Last Non-normal conditions was 10.35%
demonstrating the impact of ES on coverage probabilities. Figure T shows that the coverage
probabilities obtained from ES =0 were similar to those obtained from ES = 0.2 or 0.5 than ES =
0.8.
48
Figure U. The main effect of sample size pattern on coverage probability under Last Non-normal
for k = 4
The eta-squared for the main effect of sample size pattern under the Last Non-normal conditions
was 12.23% demonstrating the impact of sample size pattern on coverage probabilities. Figure U
shows that the mean coverage probabilities from the closest to the farthest from the nominal CI
were in the order of sample size patterns of (10 10 10 10), (10 10 10 30), and (30 10 10 10).
49