One-Way ANOVA - Variance Analysis


Where Independent Samples T-test was used to compare a sample across two groups, there might be situations when a dependent variable might be categorized on more than two variables and then the sample is to be compared across three groups. for instance Comparing Work Stress in Junior, Middle and Senior level employees, comparing Job Satisfaction in College, Graduate and Postgraduate Education level employees or making comparison of Commitment to Change in professions as Doctors, Engineers, Teachers, Bankers and Marketers.

It is important to understand that we have one variable that is categorized/divided into various groups/samples and those samples are then compared with each other. This is the ultimate objective of One Way ANOVA (Analysis of Variance).Example of research question: Is there a difference in optimism scores for young, middle-aged and old participants?

What you need: Two variables:

  • One categorical independent variable with three or more distinct categories. This can also be a continuous variable that has been recoded to give three equal groups (e.g. age group: participants divided into three age categories, 29 and younger, between 30 and 44, 45 or above)
  • One continuous dependent variable (e.g. optimism scores).

A few example scenarios/hypothesis in which we would use One-Way ANOVA are identified for the understanding of the readers

  1. The average sale of the new brand of gasoline is same in all the 3 metro cities.
  2. There are differences in Work Morale across 4 occupations.
  3. Is there a change in confidence scores over the 3 time periods?

It is important to note that in each of the above hypothesis, there is one continuous variable (Average Sale, Work Morale and Confidence Scores) that is compared across different groups (3 Metro Cities, 4 Occupations and 3 Time Periods). Now to run the One Way ANOVA, follow the following steps

Click on the Analysis Tab, Select Compare Means > One-Way ANOVA

Select the variable, that you would want to compare across different groups, In this case we would select stress with Intrinsic_Factors from the variable window and put it in the dependent list and would compare the variable across different occupations. The suggessted hypothesis for this test is that "The are differece in Stress with intrinsic factors across the 4 occupations". Adding the Variable the dialog box should look like

Now Click on Option and Select Homogeneity of variance test and press continue

Two Tables are shown in the output window, Here each of the tables are explained, the first table is the Test of Homogeneity of Variances, this table shows, if the Variances in the Data across the groups are similar or not, to explain it further, in this test we are checking Stress with intrinsic factors across the 4 occupations, Now the test would check if the Variances in the Data for Intrinsic Stress are same for each of the occupation groups i-e Banker, Teacher, Marketer and Engineer, if the value of Sig is greater than 0.05 we would say that Equal Variances are assumed, otherwise Equal Variances not Assumed. In this case we would say that variance in the data for stress are similar in the 4 occupations

The Next table of ANOVA (Also shown below) shows that if differeces exist in the Stress with Intrinsic Job Factors across the four occupations or not, Sig value of 0.50 shows that there are differences across the four occupations, if the value would have been greater than 0.50, we would have inferred that there exist no differences in Stress with Intrinsic Factors across occupations. meaning all occupants feel similar kind of stress pertinent to the intrinsic job factors.

Reporting ANOVA Table

The ANOVA summary table suggests, the Stress relating to Intrinsic Job Factors across the four occupations under study differed significantly (F3,138 = 2.665, p = .050)
Since now we know that differences do exist, we need to evaluate that between which occupations does the differences exist, and for this we would conduct a Post Hoc Analysis. for this purpose, Select One-Way ANOVA from the Menu and after selecting the Continuous variable add grouping variable, press Post Hoc button, you will see the following dialog box

There are two groups, Equal Variances Assumed and Equal Variances Not Assumed, In this case the Test of Homogeneity of Variances revealed Equal Variance Assumed, so we select a test from Equal Variances Assumed, in this case we select LSD, you can select any, mostly LSD, Bonferroni, TUkey or Tukey's-b are used. After selection of LSD press continue, then press OK, apart from the other table new table of Multiple Comparisons is also displayed, that would make comparisons for Intrinsic Factors between each of the occupations.

The above table shows the differences prevalent between two occupations, The Table Shows that No Differences exist in Stress with Intrinsic Factors between Banker and Teacher since the Sig. value is greater tha .05, however there are significant differences how Stresst with intrinsic factors affects Banker and Teacher, Since the Significance  (Sig.) value is less than .05.

Reporting Multiple Comparisons Table

Post-hoc analysis [LSD] were conducted to explore differences pertinent to Stresst with Intrinsic Job Factors among the four occupations groups. There was a significant difference between Banker and Marketing Job [mean difference = .47636, p < .01]. however no differences were recorded between anyother occupations. Finally we would reject the null hypothesis and accept the alternate hypothesis.