Question

What distribution is used for ANOVA? What are limitations of ANOVA?

How do you calculate various portions of the ANOVA process (ex. SST, SSB, SSW)? What do they measure?

What is MSB? MSW? What is the F ratio equal to? What is a computational shortcut for anova (consider how you can calculate SSW)?

How do you calculate degrees of freedom in anova?

Answer 1

1) For ANOVA F distribution is used because the statistic is the ratio of the square of the normally distributed random variables.

The limitations of ANOVA are:- a) All samples should be independent.

b) The equal variances between the groups should be satisfied otherwise we cannot perform ANOVA.

2) Let the data table for anova be

Treatments		Observations
1		y11	y12	.	.	y1n
2		y21	y22			y2n
3		.				.
.		.				.
a		ya1	ya2			yan

Let a = number of treatments

n = number of observations in each treatment.

to calculate Sum of squares

let yi. = sum of observations in each treatment.

y.. = sum of all observations.

SST measures the total variance of the observations.

SSB measures variation between the treatment groups.

SSW measures variance with in the treatment groups.

3) MSB = mean square of variation between the groups = SSB/dfB

dfB = degrees of freedom for groups

MSW = mean square of variation with in the groups = SSW/dfW

dfW = degrees of freedom within groups

F ratio = MSB/MSW

computational shortcut for anova (consider how you can calculate SSW)?

SSW = SST - SSB

4) Degrees of freedom

dfB = number of groups -1 = a-1

dftotal = total number of observations - 1 = a*n -1

dfW = degrees of freedom within groups = dftotal - dfB = a*n - 1 - (a-1) = a(n-1)