One of the questions an instrutor dreads most from a mathematically unsophisticated audience is, "What exactly is degrees of freedom?" It's not that there's no answer. The mathematical answer is a single phrase, "The rank of a quadratic form." The problem is translating that to an audience whose knowledge of mathematics does not extend beyond high school mathematics. It is one thing to say that degrees of freedom is an index and to describe how to calculate it for certain situations, but none of these pieces of information tells what degrees of freedom means.
As an alternative to "the rank of a quadratic form", I've always enjoyed Jack Good's 1973 article in the American Statistician "What are Degrees of Freedom?" 27, 227-228, in which he equates degrees of freedom to the difference in dimensionalities of parameter spaces. However, this is a partial answer. It explains what degrees of freedom is for many chi-square tests and the numerator degrees of freedom for F tests, but it doesn't do as well with t tests or the denominator degrees of freedom for F tests.
At the moment, I'm inclined to define degrees of freedom as a way of keeping score. A data set contains a number of observations, say, n. They constitute n individual pieces of information. These pieces of information can be used either to estimate parameters or variability. In general, each item being estimated costs one degree of freedom. The remaining degrees of freedom are used to estimate variability. All we have to do is count properly.
A single sample: There are n observations. There's one parameter (the mean) that needs to be estimated. That leaves n-1 degrees of freedom for estimating variability.
Two samples: There are n1+n2 observations. There are two means to be estimated. That leaves n1+n2-2 degrees of freedom for estimating variability.
One-way ANOVA with g groups: There are n1+..+ng observations. There are g means to be estimated. That leaves n1+..+ng-g degrees of freedom for estimating variability. This accounts for the denominator degrees of freedom for the F statistic.
The primary null hypothesis being tested by one-way ANOVA is that the g population means are equal. The null hypothesis is that there is a single mean. The alternative hypothesis is that there are g individual means. Therefore, there are g-1--that is g (H1) minus 1 (H0)--degrees of freedom for testing the null hypothesis. This accounts for the numerator degrees of freedom for the F ratio.
There is another way of viewing the numerator degrees of freedom for the F ratio. The null hypothesis says there is no variability in the g population means. There are g sample means. Therefore, there are g-1 degrees of freedom for assessing variability among the g means.
Multiple regression with p predictors: There are n observations with p+1 parameters to be estimated--one regression coeffient for each of the predictors plus the intercept. This leaves n-p-1 degrees of freedom for error, which accounts for the error degrees of freedom in the ANOVA table.
The null hypothesis tested in the ANOVA table is that all of coefficients of the predictors are 0. The null hypothesis is that there are no coefficients to be estimated. The alternative hypothesis is that there are p coefficients to be estimated. herefore, there are p-0 or p degrees of freedom for testing the null hypothesis. This accounts for the Regression degrees of freedom in the ANOVA table.
There is another way of viewing the Regression degrees of freedom. The null hypothesis says the expected response is the same for all values of the predictors. Therefore there is one parameter to estimate--the common response. The alternative hypothesis specifies a model with p+1 parameters--p regression coefficients plus an intercept. Therefore, there are p--that is p+1 (H1) minus 1 (H0)--regression degrees of freedom for testing the null hypothesis.
Okay, so where's the quadratic form? Let's look at the variance of
a single sample. If y is an n by 1 vector of observations,