Question

​For linear regression calculated for a sample of n = 20 pairs of X and Y values, what is the value for degrees of freedom for the predicted portion of the Y-score variance, MSregression?

Answer 1

The degrees of freedom for the model on this case is given by
`df_(model)=df_(regression)=k=1` where k =1 represent the number of variables.

The degrees of freedom for the error on this case is given by
`df_(error)=N-k-1=20-1-1=18`. Since we know we can find N.

And the total degrees of freedom would be
`df=N-1=20 -1 =19`

Explanation:

Previous concepts

Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"

When we conduct a multiple regression we want to know about the relationship between several independent or predictor variables and a dependent or criterion variable.

Solution to the problem

If we assume that we have
`k=1` independent variables and we have
`j=1,\dots,j` individuals, we can define the following formulas of variation:

`SS_(total)=\sum_(j=1)^n (y_j-\bar y)^2`

`SS_(regression)=SS_(model)=\sum_(j=1)^n (\hat y_(j)-\bar y)^2`

`SS_(error)=\sum_(j=1)^n (y_(j)-\hat y_j)^2`

And we have this property

`SST=SS_(regression)+SS_(error)`

The degrees of freedom for the model on this case is given by
`df_(model)=df_(regression)=k=1` where k =1 represent the number of variables.

The degrees of freedom for the error on this case is given by
`df_(error)=N-k-1=20-1-1=18`. Since we know we can find N.

And the total degrees of freedom would be
`df=N-1=20 -1 =19`