In linear regression, the degrees of freedom of the residuals is:. You have 3 regressors bp, type, age and an intercept term. That's my way of looking at it. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why are the Degrees of Freedom for multiple regression n - k - 1? For linear regression, why is it n - 2? Asked 4 years, 6 months ago.
Active 8 months ago. Viewed 99k times. Improve this question. Jwan Jwan 1 1 gold badge 4 4 silver badges 12 12 bronze badges. Get this straightened out and then we can consider the explanation. Use the self study tag. Add a comment. First, forget about statistics. You couldn't care less what a degree of freedom is. You believe that variety is the spice of life. Unfortunately, you have constraints. You have only 7 hats. Yet you want to wear a different hat every day of the week. On the first day, you can wear any of the 7 hats.
On the second day, you can choose from the 6 remaining hats, on day 3 you can choose from 5 hats, and so on. But after you choose your hat for day 6, you have no choice for the hat that you wear on Day 7. You must wear the one remaining hat. Degrees of freedom are often broadly defined as the number of "observations" pieces of information in the data that are free to vary when estimating statistical parameters. You have a data set with 10 values.
Each value is completely free to vary. But suppose you want to test the population mean with a sample of 10 values, using a 1-sample t test. You now have a constraint—the estimation of the mean. What is that constraint, exactly? By definition of the mean, the following relationship must hold: The sum of all values in the data must equal n x mean, where n is the number of values in the data set. So if a data set has 10 values, the sum of the 10 values must equal the mean x If the mean of the 10 values is 3.
With that constraint, the first value in the data set is free to vary. The second value is also free to vary, because whatever value you choose, it still allows for the possibility that the sum of all the values is But to have all 10 values sum to 35, and have a mean of 3.
It must be a specific number:. You end up with n - 1 degrees of freedom, where n is the sample size. Another way to say this is that the number of degrees of freedom equals the number of "observations" minus the number of required relations among the observations e. For a 1-sample t-test, one degree of freedom is spent estimating the mean, and the remaining n - 1 degrees of freedom estimate variability.
Notice that for small sample sizes n , which correspond with smaller degrees of freedom n - 1 for the 1-sample t test , the t-distribution has fatter tails. This is because the t distribution was specially designed to provide more conservative test results when analyzing small samples such as in the brewing industry.
As the sample size n increases, the number of degrees of freedom increases, and the t-distribution approaches a normal distribution. Let's look at another context. A chi-square test of independence is used to determine whether two categorical variables are dependent.
For this test, the degrees of freedom are the number of cells in the two-way table of the categorical variables that can vary, given the constraints of the row and column marginal totals. So each "observation" in this case is a frequency in a cell. Consider the simplest example: a 2 x 2 table, with two categories and two levels for each category:. It doesn't matter what values you use for the row and column marginal totals. Once those values are set, there's only one cell value that can vary here, shown with the question mark—but it could be any one of the four cells.
Once you enter a number for one cell, the numbers for all the other cells are predetermined by the row and column totals. They're not free to vary. So the chi-square test for independence has only 1 degree of freedom for a 2 x 2 table. Similarly, a 3 x 2 table has 2 degrees of freedom, because only two of the cells can vary for a given set of marginal totals. For a table with r rows and c columns, the number of cells that can vary is r -1 c The degrees of freedom then define the chi-square distribution used to evaluate independence for the test.
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