One-Way Anova

What is an ANOVA

• More than 2 sample means being compared

• Probability that the means are from the same population

Why not use t-tests?

We want to limit type 1 errors; when we say there is an effect when there is none. 

• Doing repeated t-tests exposes us to type 1 errors over andover again

• More comparisons means more type 1 errors

Hypothesis

• Null is all pop. equal

• Alternative is not all pop. equal

Assumptions of ANOVA

1. Observations are normally distributed within each population

2. Population variances are equal (homogeneity of variance)

   • Still use an estimate of population variance

   • Pooled variance across each group; we assume that variances pulled from pop of equal variance.

3. Observations are independent

   • Each person is in their own condition and only provides one observation/Between subjects one-way.

Partitioning variance

Dividing up the variance to see how miuch of the variance is due to the manipulation; this is what the F statistic represents.

F statistic

MS group over MS error

ANOVA solving steps

1. Source table

2. Degrees of freedom

3. Total SS

4. Group SS

5. Error SS

6. Mean SS column

7. F statistics

Source Table

Left column:

• Group

• Error

• Total

Top row:

• Sum of Squares (SS)

• Degrees of freedom (df)

• Mean SS

• F

Degrees of freedom

Group: k-1

Error: (n-1)+(n-1)+(n-1)…

Total: N-1

Total SS

• Compute the grand mean

• Take each row of observations and do (X-x̂ )2

• Add up the sum of all the (X-x̂ )2 columns

If you dont make a data table what can you do

• MS = df/ss

  • So if you have a df or an ss

Group SS

SSgroup = Sum of n