           Research Design in Occupational Education Copyright 1997. James P. Key. Oklahoma State University Except for those materials which are supplied by different departments of the University (ex. IRB, Thesis Handbook) and references used by permission. MODULE S6 - ANALYSIS OF VARIANCE

The purpose of the Analysis of Variance (ANOVA) Technique is to test for significant differences among two or more groups.

In single classification ANOVA, you are trying to find out if there is any relationship between a dependent variable (such as student achievement) and several classifications of one independent variable (such as different instructional materials).

In multiple classification ANOVA, you are trying to find out the relationship between one dependent variable (such as student achievement) and classifications of two or more independent variables (such as several methods of instruction and different instructional materials).

Therefore, the factor determining whether to use single or multiple classification ANOVA is the number of independent variables.

Since the variance (or its square root, the standard deviation) is really an average distance of the raw scores in a distribution of numbers from the mean of that distribution, this functional relationship between the variance and the mean can be used to determine mean differences by analyzing variances.

In essence, the ANOVA method is to calculate the variances of each subgroup being compared. The average variance of these subgroups is then compared to the variance of the total group (created by artificially combining the subgroups). If the average variance of the subgroups is about the same as the variance of the total group, then no significant difference exists among the means of the subgroups. However, if the average variance of the subgroups is smaller than the variance of the total group, then the means of the subgroups are significantly different.

The first step in computing ANOVA is to calculate the sums of squares of the deviations of the observations from their mean (hereafter referred to as sums of squares or SS) for each of the separate groups being compared and add them together to form the within groups SS. Next, compute the SS for the total group made by combining the subgroups. Subtract the within group sum of squares from the total group sum of squares to derive the among group sum of squares. Divide the among and within sums of squares by their degrees of freedom to obtain their mean squares (variances), then divide the among mean square by the within mean square to obtain the calculated F value. Finally, determine if the calculated F value is sufficiently large to reject the null hypothesis. If the calculated F is >= table value at the chosen level of significance, the null hypothesis is rejected; if the calculated value is < table value, then null hypothesis is accepted.

The among group mean square or variance and the within group mean square or variance determine the size of F.

The hypothesis being tested is: There are no significant differences among the means of achievement of the groups being taught by the three different methods.

Example

 Lecture Discussion Demonstration x1 x2 x3 1 3 5 2 4 6 3 5 7 4 6 8 5 7 9 15 25 35 Step 1: Within Group Variation = for each group added together.

 x1  x2  x3  1 -2 4 3 -2 4 5 -2 4 2 -1 1 4 -1 1 6 -1 1 3 0 0 5 0 0 7 0 0 4 1 1 6 1 1 8 1 1 5 2 4 7 2 4 9 2 4 15 0 10 25 0 10 35 0 10 SS Within = 30

Step 2 - Total = x1  x2  x3  1 -4 16 3 -2 4 5 0 0 2 -3 9 4 -1 1 6 1 1 3 -2 4 5 0 0 7 2 4 4 -1 1 6 1 1 8 3 9 5 0 0 7 2 4 9 4 16 30 10 30 SS Total = 70

Step 3: Among Group Variation =     3 -2 4 20 5 0 0 0 7 2 4 20 15 0 8 40

SS Among = 40

The degrees of freedom for the different sums of squares

Among group df equals the number of groups minus one (k - l)

Within groups df equals the number of groups times the number within each group minus one k(n - l)

Total group df equals the total number of subjects minus 1 (kn - l) and can be used as a cross check since among df plus within df must equal total df.

 Source of Variation Sums of Deviations Squared df Mean Square (Variance) F Among (Between) 40 2 20 8 Within (Error) 30 12 2.5 Total 70 14

Significant at the .01 level of confidence

F.05 with 2 and 12 df = 3.88 8 > 3.88
F.01 with 2 and 12 df = 6.93 8 > 6.93

Therefore, reject null hypothesis.

Assumptions

1. Representative Sample (Random)

2. Normal Distribution for the Populations

3. Interval Measures

4. Homoscedasticity

5. Independent Observations

1. State the purpose of the ANOVA Technique.

2. Name the factor determining whether ANOVA single or multiple classification be used.

3. State the relationship between the variance and the mean which allows us to determine differences between means by analyzing variances.

4. State which group variances are compared to see if there are differences between means.

5. List the steps necessary to compute F.

6. Name the two variances or mean squares that determine the size of F.

7. List the assumptions underlying the analysis of variance test.

8. Assume you are testing for differences in the number of chin ups junior high boys can do after varying weeks of practice.

 1 Weeks Practice 2 Weeks Practice 3 Weeks Practice x1 x2 x3 2 4 6 3 5 7 4 6 8 5 7 9 6 8 10 State the null hypothesis and test at the .01 level of significance. Calculate the sums of squares, degrees of freedom, and F value.           