1. Intro to Statistics for the Behavioral Sciences PSYC 1900
Lecture 12: One-Way Analysis of Variance

2. Analysis of Variance (ANOVA)
Probably the most used statistical technique in psychology
Like t-tests, it is used to compare different sample means
But, there are no restrictions on the number of sample means
And, the effects of more than one independent variable may be analyzed

3. One-way (ANOVA)
Statistical technique used to test for differences in the means of several groups
Groups are defined using only one independent variable
Usually groups are on a nominal scale
Sometimes, they can be on an ordinal scale

4. General Approach by Example
Let’s say, for example, that we are interested in determining if different emotional states make one more receptive to political advertisements.
Three groups:
Control, Anger, & Happy
Participants view an add while in one of these states and report their attitudes toward the candidate

5. Here is the data for each group:
0=Control
1=Angry
2=Happy
The null hypothesis would be that the means of all three populations are equal.
By population, we only mean a population of scores obtained under certain conditions.
The alternative hypothesis is that at least one of the population means differs from at least one of the others.
Many alternatives are possible, but any one invalidates the null.

6. Assumptions
If we make a certain set of assumptions, ANOVA can be used to test the probability of the null hypothesis.
Normality
We assume that scores are relatively normally distributed.
Primarily refers to sampling distribution and ANOVA is robust to moderate violations
Independence of Observations
Knowing one score tells us nothing about other observations.

7. Assumptions Homogeneity of Variance
Each population of scores has the same variance.
The ‘e’ subscript refers to error.
It is variance unrelated to any group differences, just within group variability
Treatments (e.g., emotions) would be expected to make a linear shift in populations means.
Simply adds a constant to each score

8. Logic of One-way ANOVA
Each sample variance is an estimate of the variance of the population from which it was drawn.
Because we assume that all populations have the same variance, each can also serve as an estimate of the common population variance.
We can pool the three estimates to obtain a more accurate estimate of the common error variance.
This value is termed MSwithin or MSerror.
It does not depend on the validity of the null as it is calculated using each sample separately.

9. Logic of One-way ANOVA Now, let’s assume that the null is true.
If the null is true, the 3 samples can be thought of as drawn from the same parent population.
We use the variance of the 3 means to estimate the population variance.
Variance of sample means is square of standard error and is less than population variance by a factor of n.
This value is the MSgroups

10. Logic of One-way ANOVA
We now have two estimates of the population variance.
MSwithin is independent of the null
MSgroups is only an estimate of the population variance if null is true.
If null is false, MSgroups also estimates variance between means
If the null is true, these two estimates should be relatively equal.
To the extent that they differ, the likelihood is reduced that the null is true.
Here, MSwithin = 1.40, MSgroups = 5.43

11. ANOVA and the F Statistic
Once we have the two variance estimates, we use the F distribution to compare the ratio of one to the other.
If the null were true, F should approximately equal 1.
If the null is false, F should be larger than one.
The F distribution allows us to calculate the probability of a given F value if the null is true.

12. Calculation Formulae for ANOVA
In calculating ANOVA’s, we will make use of Sums of Squares (SS).
SS are sums of squared deviations around some specified point (usually a mean)
Squared deviations are central components in calculating variances, and SS are easy to work with.

13. Calculation Formulae for ANOVA
Notation:
Individual scores for person i in group j:
Means for group j:
Grand mean (mean of all scores):

14. Calculation Formulae for ANOVA
SStotal is the sum of squared deviations of all observations from the grand mean.
SSgroup is the sum of squared deviations of the group means from the grand mean.
SSerror is the sum of squared deviations within each group.

15. Calculation Formulae for ANOVA
Now we convert SS into Mean Squares (MS).
We divide SS by associated df’s.
dftotal=N-1
dfgroup= #groups-1
dferror=(#groups)(n-1)
F is the ratio of these two estimates of population variance.
Critical values of F are found using F(dfgroup, dferror),