Question

Remember that there is a common notation for the number of levels in a factor and the total number of scores in the entire study. Using these symbols, what are the formulas for each of the following:
dftotal= syntax error

dfwithin=
(also known as the denominator degrees of freedom or the error term degrees of freedom)

dfbetween=
(also know as the numerator degrees of freedom or the treatment degrees of freedom)

Answer 1

In an experiment, the factor (also called an independent variable) is an explanatory variable manipulated by the experimenter. Each factor has two or more levels (i.e., different values of the factor). Combinations of factor levels are called treatments.

In an experiment,

• Factor means "the variability due to the factor of interest."​ Sometimes Factor is labelled as "Treatment" or "Between" to make it clear that it explains the variation between the groups.
• Error means "the variability within the groups" or "unexplained random error." Sometimes it is labeled as "Within" to make it clear that it explains the variation within the groups.
• Total means "the total variation in the data from the grand mean" (that is, ignoring the factor of interest).

Now the following are the notations used in experiment:

represents the total number of valid observations.

represents the total number of groups (levels of the independent variable).

TOTAL DEGREES OF FREEDOM:

Imagine a set of three numbers, whose mean is 3. There are lots of sets of three numbers with a mean of 3, but for any set you can freely pick the first two numbers, any number at all, but the third (last) number is out of your hands as soon as you picked the first two.

Say our first two numbers are the same as in the previous set, 1 and 6, giving us a set of two freely picked numbers, and one number that we still need to choose, x: [1, 6, x]. For this set to have a mean of 3, thus x has to be 2, because (1 + 6 + 2) / 3 is the only way to get to 3. So, the first two values were free for you to choose, the last value is set accordingly to get to a given mean. This set is said to have two degrees of freedom, corresponding with the number of values that you were free to choose (that is, that were allowed to vary freely).

The fact that in an ANOVA, we don’t have just one set of numbers, but there is a system (design) to the numbers. In the simplest form we test the mean of one set of numbers against the mean of another set of numbers (one-way ANOVA). But the resoning for degrees of freedom is same.

The general rule then for any set is that if equals the number of values in the set, the degrees of freedom equals – 1. Thus as ​ represents the total number of scores in the entire study, thus total degrees of freedom is ​.

TREATMENT OR BETWEEN DEGREES OF FREEDOM:

This is all about means and not about single observations. The value depends on the exact design of your test. Basically, the value represents the number of cell means that are free to vary to get to a given grand mean. The grand mean is just the mean across all groups and conditions of your entire sample. We will call the number of cells (or cell means) .

Let us consider a one-way ANOVA. Suppose we have two groups that we want to compare, so we have two cells. If we know the mean of one of the cells and the grand mean, the other cell must have a specific value such that (cell mean 1 + cell mean 2) / 2 = grand mean. Thus for a two-group design, degrees of freedom is 1.

Now moving on to three groups. We now have three cells, so we have three means and a grand mean. And now there are 2 cell means which are free to vary to get to the given grand mean.

Thus treatment degrees of freedom is number of groups or levels of independent variable minus 1. That is, .

ERROR OR WITHIN DEGREES OF FREEDOM:

This is about how the single observations in the cells relate to the cell means. Basically the within degrees of freedom is the total number of observations in all cells minus the degrees of freedom lost because the cell means are set (that is, minus the number of cell means or groups/conditions: ). Thus .

Say if we have 150 observations across four groups. That means we will have