This problem will introduce the learner into a technique called Analysis of Variance. For this course we will only conduct a simple One-Way ANOVA and touch briefly on the important elements of this technique. The One-Way ANOVA is an extension of the independent –t test that can only look at two independent sample means. We can use the One-Way ANOVA to look at three or more independent sample means. Use the following data to conduct a One-Way ANOVA:
Scores Group
1 1
2 1
3 1
2 2
3 2
4 2
4 3
5 3
6 3
Notice the group (grouping) variable, which is the independent variable or factor is made up of three different groups. The scores are the dependent variable.
Use the instructions for conduction an ANOVA on page 438 of the text for SPSS or Excel.
a) What is the F-score; Are the results significant, and if so, at what level (P-value)?
b) If the results are significant to the following: Click analyze, then click Compare Means, and then select one-way ANOVA like you did previously. Now click Post Hoc. In this area check Tukey. If there is a significant result, we really do not know where it is. Is it between group 1 and 2, 1 and 3, or 2 and 3? Post hoc tests let us isolate where the level of significance was. So if the results come back significant, conduct the post hoc test as I mentioned above and explain where the results were significant.
c) What do the results obtained from the test mean?
To conduct a One-Way ANOVA and answer the given questions, you can follow these steps:
Step 1: Organize the data
First, organize the data into two columns: one for the scores and another for the group numbers. This will make it easier to input the data into SPSS or Excel.
Step 2: Perform the One-Way ANOVA
a) In SPSS:
- Open SPSS and go to Analyze > Compare Means > One-Way ANOVA.
- Select the dependent variable (scores) and move it to the Dependent List.
- Select the independent variable (group) and move it to the Factor.
- Click OK to perform the analysis.
- Look for the F-score in the ANOVA table.
b) In Excel:
- Enter the scores for each group in separate columns (e.g., A1:A3 for group 1, B1:B3 for group 2, and C1:C3 for group 3).
- Select a blank cell where you want to display the F-score.
- Use the following formula: =F.TEST(A1:A3, B1:B3, C1:C3)
- Press Enter to calculate the F-score.
Step 3: Interpret the results
a) The F-score is a measure of the variation between group means compared to the variation within groups. In this case, the F-score will tell you if there is a significant difference between the means of the three groups.
- To determine if the results are significant, compare the calculated F-score to the critical F-value. You can do this by looking up the critical F-value based on the degrees of freedom and significance level (e.g., using an F-table or an online calculator). If the calculated F-score is greater than the critical F-value, then the results are significant.
- To determine the p-value, you can use the F-distribution table or an online calculator. The p-value is the probability of observing a result as extreme as the calculated F-score, assuming the null hypothesis (no difference between means) is true. If the p-value is less than your chosen significance level (e.g., 0.05), then the results are considered significant.
b) If the results are significant, you can proceed with post hoc tests (e.g., Tukey's test) to identify which groups differ significantly from each other.
- In SPSS, follow the instructions provided by clicking on Analyze > Compare Means > One-Way ANOVA and then selecting Tukey for the post hoc test.
- In Excel, you can use the Data Analysis Toolpak to perform post hoc tests by going to Data > Data Analysis > Anova: Two-Factor Without Replication. Then, select the range of your data and choose Tukey's test from the options.
c) The results obtained from the One-Way ANOVA and any post hoc tests provide information about whether there are significant differences between the means of the three groups.
- If the overall ANOVA test is significant, it indicates that there is at least one significant difference between the means of the groups.
- The post hoc tests (e.g., Tukey's test) help identify which specific group means differ significantly from each other. For example, it will show if there is a significant difference between group 1 and 2, group 1 and 3, or group 2 and 3.
Overall, the One-Way ANOVA allows you to determine if there are significant differences between the means of three or more independent groups, while post hoc tests help identify which specific group means differ significantly.
To conduct a One-Way ANOVA on the given data, follow these steps:
Step 1: Set up the data
- Organize the data with the scores in one column and group identifiers in the other column.
- Scores: 1, 2, 3, 2, 3, 4, 4, 5, 6
- Group: 1, 1, 1, 2, 2, 2, 3, 3, 3
Step 2: Calculate the sum of squares (SS) and degrees of freedom (df)
- Calculate the sum of squares within groups (SSW), sum of squares between groups (SSB), and total sum of squares (SST).
- Calculate the degrees of freedom for each SS.
- SSW df = N - k (number of observations - number of groups)
- SSB df = k - 1 (number of groups - 1)
- SST df = N - 1 (number of observations - 1)
Step 3: Calculate the mean squares (MS)
- Divide each sum of squares by its respective degrees of freedom to calculate the mean squares.
- MSW = SSW / dfW (Mean Square Within Groups)
- MSB = SSB / dfB (Mean Square Between Groups)
Step 4: Calculate the F-score
- Calculate the F-score by dividing the MSB by the MSW.
- F = MSB / MSW
Step 5: Determine the significance level (P-value)
- Look up the F-score in the F-distribution table or use statistical software such as SPSS or Excel to find the corresponding P-value.
- Compare the P-value to a predetermined significance level (commonly α = 0.05) to determine if the results are significant.
a) The F-score and significance level:
- Conduct the calculations based on the steps provided above.
- F = (SSB / dfB) / (SSW / dfW)
- Substitute the values obtained from the data into the formula.
- Calculate the degrees of freedom using N = 9 observations and k = 3 groups.
- Determine the P-value using the F-distribution table or statistical software.
b) Post hoc test (Tukey test):
- If the results of the One-Way ANOVA are significant, conduct a post hoc test, such as the Tukey test, to determine where the significant differences lie.
- In the software, go to Analyze > Compare Means > One-Way ANOVA and select the Tukey post hoc test.
- This test will compare all group pairs and identify significantly different groups.
c) Interpretation of results:
- The results obtained from the One-Way ANOVA with significant F-score and P-value indicate that there is a statistically significant difference between at least two of the group means.
- The post hoc test, such as the Tukey test, will identify which specific group pairs have significant differences.
- The results provide evidence that the dependent variable (scores) is influenced by the independent variable (group).
- Further analysis and interpretation of the specific significant differences will depend on the findings of the post hoc test.