6.7.7. ANOVA
All parameters input and estimated in this section are based on a single factor ANOVA. However, different types of ANOVA can also be accommodated simply by entering the relevant statistics in place of existing parameters. For instance, it is possible to apply the sample size and power of the test algorithms introduced below to a multi-way ANOVA problem, by treating each of the factors separately.
6.7.7.1. Sample Size for ANOVA
Estimation of the sample size is based on the following relationship:
where:
·
is the power of the
test statistic the p-value of which can be determined by consulting the OC
Curves published by Pearson and Hartley (1951), pp. 112-130.
· n is the sample size (i.e. number of cases in each group)
· k is the number of groups
· δ is the minimum detectable difference
· s2 is the error mean square
However, since itself depends
on n, an iterational algorithm should be employed and this is an extremely
laborious process. Here we completely automate this process by reproducing the
OC Curves with an iterational algorithm based on the noncentral F-distribution. First defining the noncentrality parameter as:
and inserting this into the above equation we obtain:
Then n is augmented until the following equality is obtained within the given tolerance limits:
where:
·
is the numerator
degrees of freedom,
·
is the denominator
degrees of freedom,
·
is the critical value
from F-distribution for Type I error level of α,
·
is the critical value
from noncentral F-distribution for power of the test corresponding to
Type II error level of β.
The user is expected to enter:
· Number of Groups
· Minimum Detectable Difference
· Error Mean Square
· Power of the Test
· Confidence Level
and the program will output the estimated sample size.
Example
Example 10.6 on p. 211 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → ANOVA and the Sample Size option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: ANOVA
Sample Size
|
Number of Groups = |
4.0000 |
|
Minimum Detectable Difference = |
3.5000 |
|
Error Mean Square = |
9.3833 |
|
Power of the Test = |
0.8000 |
|
Confidence Level = |
0.9500 |
|
Phi = |
1.7000 |
|
Estimated Sample Size = |
17.7100 |
6.7.7.2. Maximum Number of Groups for ANOVA
Estimation of the maximum number of groups is similar to that of the sample size described in the previous section. An iterational algorithm is employed based on the noncentral F-distribution.
The user is expected to enter:
· Total number of cases (i.e. n x k)
· Minimum Detectable Difference
· Error Mean Square
· Power of the Test
· Confidence Level
and the program will output the estimated group size.
Example
Example 10.8 on p. 213 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → ANOVA and the Maximum Number of Groups option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: ANOVA
Maximum Number of Groups
|
Total Number of Cases (nk) = |
50.0000 |
|
Minimum Detectable Difference = |
4.5000 |
|
Error Mean Square = |
9.3833 |
|
Power of the Test = |
0.8000 |
|
Confidence Level = |
0.9500 |
|
Phi = |
1.6860 |
|
Number of Groups = |
4.3567 |
6.7.7.3. Minimum Detectable Difference for ANOVA
Rearranging the equation given in section 6.7.7.1. Sample Size for ANOVA we obtain:
where:
is the noncentrality parameter. Then the minimum detectable difference is estimated employing an iterational algorithm similar to the one in section 6.7.7.1. Sample Size for ANOVA.
The user is expected to enter:
· Sample Size per Group (n)
· Number of Groups (k)
· Error Mean Square
· Power of the Test
· Confidence Level
and the program will output the estimated and the minimum detectable
difference computed according to the above equation.
Example
Example 10.7 on p. 212 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → ANOVA and the Minimum Detectable Difference option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: ANOVA
Minimum Detectable Difference
|
Sample Size = |
10.0000 |
|
Number of Groups = |
4.0000 |
|
Error Mean Square = |
9.3833 |
|
Power of the Test = |
0.9000 |
|
Confidence Level = |
0.9500 |
|
Phi = |
1.9883 |
|
Minimum Detectable Difference = |
5.4476 |
6.7.7.4. Power of the Test for ANOVA
Power of the test can be computed directly from the equation given in section 6.7.7.1. Sample Size for ANOVA where there is no need for an iterational solution.
The user is expected to enter:
· Sample Size per Group (n)
· Number of Groups (k)
· Error Mean Square
· Minimum Detectable Difference
· Confidence Level
and the program will output the estimated and its corresponding p-value.
Example
Example 10.5 on p. 209 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → ANOVA and the Power of the Test option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: ANOVA
Power of the Test
|
Sample Size = |
10.0000 |
|
Number of Groups = |
4.0000 |
|
Error Mean Square = |
7.5888 |
|
Minimum Detectable Difference = |
4.0000 |
|
Confidence Level = |
0.9500 |
|
Phi = |
1.6234 |
|
Power of the Test = |
0.7349 |