Page 88 - Contributed Paper Session (CPS) - Volume 8

CPS2202 Oladugba Abimibola Victoria et al.
∑ 2
= =1 (6)
−1
Where
2
= √ − √( − 1) − (7)
( −1)

2
1
= 2 + (8)

ℎ
= the unbiased estimate of variance for the subgroup; = sample size
2

of the subgroup; MSE = pooled within group error variance
ℎ
Hartley Test or Fmax Test (HF): The test statistic is just a simple ratio between
the largest subgroup variance and the smallest:
2

= (9)
2

A table of values created by Hartley evaluates the test statistic with degrees of
freedom of k and − 1 (if all the groups have the same size). When the

groups have slightly different sample sizes, the harmonic mean may serve as
the adjusted sample size n Zhang (1998).

Harmonic mean = (10)
∑ −1

Cochran’s G Test (CG): The test statistics is:
2

= (11)

2
∑( − ̅ )
= − (12)
2 = maximum variance; k = number of groups; N = number of all sample
sizes

3. Result
The summary of the results from the analysis for one-way and two-way
ANOVA are shown in Tables 1, 2, 3 and 4 for the seven HOV test considered.
Table 1: Type I error rates summary for One-way ANOVA: Minimum and maximum empirical
Type I error rates and number of times the Type I error rates was rejected when it is true
Equal sample Normal Distribution Non-normal Distribution
size
Test Minimum Maximum Number Minimum Maximum Number
empirical empirical of times empirical empirical of times
Type I Type I error Type I Type I error Type I error Type I
error error was error was
rejected rejected
Bartlett 0.075 0.969 0 0.010 0.870 4
Levene 0.057 0.783 0 0.040 0.680 3
Brown-Forsythe 0.237 0.960 0 0.140 0.980 0
O’Brien 0.140 0.807 0 0.036 0.780 1
Z-variance 0.203 0.776 0 0.024 0.591 4
Hartley F-max 0.384 0.789 0 0.011 0.441 2
Cochran’s G 0.731 0.750 0 0.009 0.485 3

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