Degrees of freedom (df)

The number of independent units of information in a sample used in the estimation of a parameter or calculation of a statistic. In the simplest example of a 2x2 table, if the marginal totals are fixed, only one of the four cell frequencies is free to vary and the others will be dependent on this value not to alter the marginal totals. Thus, the df is only 1. Similarly, it can easily be worked out that in a contingency table with r rows and c columns, the df = (r-1)(c-1). In parametric tests, the idea is slightly different that the n bits of data have n degrees of freedom before we do any statistical calculations. As soon as we estimate a parameter such as the mean, we use up one of the df which was initially present. This is why in most formulas, the df is (n-1). In the case of a two-sample t-test with and observations, to do the test we
calculate both means. Thus, the In linear regression, when the linear equation is calculated, two parameters are estimated (the intercept and the slope). The df used up is then 2: Non-parametric tests do not estimate parameters from the sample data, therefore, df do not occur in them.
In simple linear regression, the df is partitioned similar to the total sum of squares (TSS). The df for TSS is N-k. Although there are n deviations, one df is lost due to the constraint they are subject to: they must sum to zero. TSS equals to RSS + ESS. In one-way ANOVA, the df for RSS is N-2 because two parameters are estimated in obtaining the fitted line. ESS has only one df associated with it. This is because the n deviations between the fitted values and the overall mean are calculated using the same
estimated regression line which is associated with two df (see above). One of them is lost because the of the constraint that the deviations must sum to zero. Thus, there is only one df associated with ESS. Just like TSS = RSS + ESS, their df have the same
relationship: N-1 = (N-2) + 1.