![]() So, we say there are only two (or \(n-1\)) degrees of freedom. ![]() So even though two of the data points can be any value, the third is restricted because the mean has to be 5. If we know the mean is 5, then if we know any two of the three numbers, we can figure out the third. If we know the mean is 5, do we really have three data points that could be any value? Not really. What does it mean to “limit the freedom of data points”? Let’s say we have a distribution with three scores and the mean is 5. Whenever you estimate a parameter, you spend (or lose) a degree of freedom because a parameter will limit the freedom of those data points. The more data (represented by \(n\)) you have, the more degrees of freedom you have. We find it’s helpful to think about degrees of freedom as a budget. Technically, the degrees of freedom of an estimate is the number of independent pieces of information that went into calculating the estimate. For our Fingers data set, which has a sample of 157 students, degrees of freedom for the empty model is 156, which is \(n-1\). The next column, labeled df, shows us degrees of freedom. We already have discussed the column labeled SS. We will start by looking at the last row of the supernova table for the Height2Group model. Just a note: df stands for degrees of freedom, MS stands for Mean Square, and F, well, that stands for the F ratio. Let’s now look at the next three columns in the table: df, MS, and F. We have already discussed how we analyze the SS column and break it into parts. Analysis of variance is when we break apart, or partition, variation. One definition of “analyze” is to break up into parts. We want to take a moment to appreciate why the ANOVA tables (printed by the function supernova()) are called ANalysis Of VAriance tables. Let’s go back to our analysis of variance table (reprinted below). The F ratio provides a solution to this problem, giving us an indicator of the amount of error reduced by a model that adjusts for the number of parameters it takes to realize the reduction in error. But it does not take into account the cost of that reduction. Yes, it tells us whether we are reducing error. PRE alone, therefore, is not a sufficient guide in our quest to reduce error.
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