One-Way Between-Subjects Analysis of Variance (Chapter 12)
1. The DV is quantitative in nature and measured on a level that at least approximates interval characteristics.
2. The IV is between-subjects in nature (it can be either qualitative or quantitative – generally, it’s qualitative).
3. The IV has three or more levels.
Inference of a Relationship Using One-Way ANOVA
- Remember with the independent-groups t-test we were testing whether the difference between two sample means was greater than would be expected from sampling error alone.
- With the one-way ANOVA, we’re doing the same thing – but we have to account for the differences among a set of three or more sample means.
- When using the one-way ANOVA, how do we account for differences among the set of sample means?
- We focus on the amount of variability (or variance) among the sample means.
- What we’re looking at is a test of how much of the variability among sample means is due to sampling error and how much is due to the IV.
- We compute this with a ratio:
F ratio = sampling error + effect of IV
sampling error
EXAMPLE: Does caffeine influence anagram performance?
Question 1: Is there a relationship between the IV and DV?
- Again, this question is an application of the hypothesis-testing process.
Step 1: State the null and alternative hypotheses:
H1 : 3 population means are not equal
Step 2: State the expected results, assuming H0 is true (i.e., determine critical values):
- We’ll be using the F test now – we need to use critical values from an F distribution
Step 3: Compute the relevant values for your test statistic:
- The F test is a ratio of variances:
F = Variance Between the Group Means = sampling error + effect of IV
Variance Within Each Group sampling error
- We potentially have two different things causing variability among scores:
- Sampling error (which includes the influence of disturbance variables)
- The IV (different dosages of caffeine)
- We can split up the total variability in anagram scores in a way that allows us to evaluate these "sources of variance"
- We use SS to analyze variability and to break it apart because they can be meaningfully added and subtracted, unlike variance and standard deviation.
- SStotal = SSbetween + SSwithin
- SStotal is the sum of squared deviations of all scores regardless of which group it came from, from the "Grand mean" across all scores – typical SS formula
- SSbetween includes [variability due to sampling error + variability due to caffeine]
- the sum of squared deviations of each Group mean from the Grand mean
- Here we’re computing the amount of variability among the group means – how far do the group means deviate from the Grand mean?
- SSwithin includes [variability due to sampling error]
- SSwithin is an index of sampling error alone. For this, we take the sum of squared deviations of each score from its Group mean, instead of the Grand Mean.
- We construct a "Summary Table" showing the steps in computing the F test:
Source SS df MS F
Between Groups
Within Groups
Total
- Our F ratio is going to be a ratio of Between/Within-group variances (remember, MS is another term for variance estimate)
COMPUTATIONAL FORMULAS:
- There are three main values we need to calculate to get each of our SS values:
I. 
II. 
III. 
- These are the intermediate values. Once we have calculated these values, we can calculate our SS values. (CANNOT HAVE NEGATIVE VALUES!!)
- SSbetween = III – II
- SSwithin = I – III
- SStotal = I – II
- Now we can start filling in our summary table:
Source SS df MS F
Between Groups
Within Groups
Total
- SStotal = SSbetween + SSwithin, but it’s good to compute all 3 for a math check.
Step 4: Compute the test statistic:
**The test statistic is our F-ratio
- The F ratio will be a ratio of MSbetween/MSwithin, so we need to convert our SS values to MS (variance) values – by dividing by df.
- First, we determine the df for each source:
- dfbetween = k – 1
- dfwithin = N – k
- dftotal = N – 1
also dftotal = dfbetween + dfwithin,
- Now we compute our variance estimates (MSbetween & MSwithin) by dividing SS/df.
**Remember that Mean Square is another name for a variance estimate. But in the ANOVA we have broken apart the total variance, so we now have two MSs.
Step 5: Compare results to critical value
- How do we get our critical value?
- Use Appendix F (p. 600).
- There are only positive critical values for F, because it’s a ratio of variances, and variances cannot be negative.
So, Step 5: Do we reject or not reject the null hypothesis that all group means are equal?
