ANOVA

ANOVA is used to compare two independent groups.

ANOVA vs. multiple t-tests

ANOVA is more powerful (less chance of a type II error)
The actual alpha is known and controlled; if you used lots of t-tests for more than 2 groups, alpha keeps growing.
A one-way ANOVA is the same as an independent groups t-test.

General Linear Model

This says that the probability that your sample mean is the same as the population sample mean is a linear combination of three factors:

mu, the population mean (estimated by the overall sample mean xbar dot)
the between-group variance, which should hopefully be large
(estimated by group mean - overall mean)
an error term, or the within group variance, which should hopefully be small
(estimated by individual mean - group mean)
if j is the number of groups and there are i people in each group, then xij = xbar dot + alpha j + eij where
- xbar dot is the overall mean for the entire experiment (which approaches mu for large N)
- alpha j is the difference between the overall mean and the mean of the jth group, and
- eij is the difference between the mean of the jth group and the ith person in that group.

Note that for all linear models, if we want our groups to be orthogonal, (which is an ANOVA assumption), then we can only do calculations for j-1 groups. This is very important.

Components of ANOVA

Numerator
- SSB = sum nj * (xbarj - xbardot)**2
  In plain English this means for each group, take the square of the difference between the group mean and the overall mean, and then multiply by nj, the sample size of the group. Then add them up.
  If all n's are the same, then it is just n * the squares of group - overall differences.
- dfb = j-1
  For orthogonal groups, the between-group df is always 1 less than the number of groups.
- MSB = SSB/df
  This is called the mean square difference between groups. It is the numerator for the ANOVA statistic.
Denominator
- SSW = sum over i,j (xij = xbarj)**2
  In plain English this means for every single individual in every single group, take the difference between their score and the group mean score, square it, then add them all up. Don't multiply by n.
- dfw = N-j
  This means that for j groups and an overall sample size of N, take the total sample and subtract the number of groups.
- MSW = SSW/dfw
  This is called the mean square difference within groups. It is the denominator of the ANOVA statistic.
F is called the ANOVA statistic. It is the numerator/denominator, i.e., MSB/MSW.

Interpreting ANOVA results

Calculate SSB/SSW. Compare it with the critical value of the F distribution in the table. To do this, you need three numbers:

dfw for the denominator (down the left column) = N-J
dfb for the numerator (across the top) = J-1
alpha (pick the right one for the right page)

If the calculated F statistic exceeds the critical F statistic in the table, reject the null hypothesis. This give a type I error of .05, and a p value of whatever SPSS calculates.

Effect of Sample Size

If you increase the sample sizes n, you will increase the obtained F statistic because n appears in the numerator, not the denominator.

Now this means that you can get significance for some small factor just because you have a big N. You really ought to look at the effect size, which is (xbar experimental - xbar control)/s control
which means the difference between the two means in standard deviation units.

Back to Statistics course

Lorraine Sherry
lsherry@carbon.cudenver.edu
Updated December 7, 1996