Calculate ANOR

November 5, 2008 by  
Filed under ANOR, How To Calculate

Getting from Sum of Squares (SS) to mean squares (ms)

The F test is a ratio of variance (understood/not understood). To find the variance, we begin by partitioning the Sum of Squares (SS) of the regression into explained and unexplained components. Explained variance is simply the SSy multiplied by r2 (the coefficient of determination). The result is the SSregression (the understood portion of the regression).

The unexplained, not yet understood portion of the regression is found by multiplying the SSy by 1-r2 (the coefficient of nondetermination). The result is the SSerror (the non-understood portion of the regression).

To get from Sum of Squares to variance, we divided each SS by its respective degrees of freedom. The resulting variance terms are called mean squares (a reminder that variance is the average of the squared deviations from a distributions mean).

Degrees of Freedom

The degrees of freedom (df) for Regression is k-1 (columns minus one). Since a simple linear regression has only 2 columns, the df for an Analysis of Regression always equals 1. The df for Error is N-k (number of people minus the number of columns). And Totalerror = N-1.

If it seems like it’s getting hard to keep track of all this, there is good news. An Analysis of Regression using a summary table that organizes all of the important information. Simply fill in the blanks of the table and the hard part is done.

EXAMPLE

In order to calculate an Analysis of Regression for this data,

X Y
11          1
4          2
8          8
2        12
7        11
16          2

We fill in the blanks for the Analysis of Regression’s summary table:

SS df ms

Regression _____ ____ ____

Error         _____ ____ ____

Total         _____ ____ ____

Let’s start with the degrees of freedom. Since this is a simple linear regression, we know that dfregression = 1. Two columns minus 1 = 1. We know that dferror is equal to N-k (the number of people minus the number of columns); so 6-2 = 4. The total degrees of freedom is equal to N-1; 6-1 = 5.

We know that SStotal equals SSy. In this example, the SSy = 122. We partition this into SSregression and SSerror by multiplying the SStotal by r2and 1-r2, respectively. So 122 is partitioned into 41.14 (explained dispersion) and 80.86 (unexplained dispersion).

With this in mind, let’s update the summary table with what we know:

SS df ms

Regression     41.14     1     ____

Error              80.86     4     ____

Total            122.00     5     ____

Variance (which in a F-test is given the special designation of mean squares) is calculated by dividing the SS term by its respective degrees of freedom. Updating the summary table gives us:

SS df ms

Regression     41.14     1     41.14

Error              80.86     4     20.21

Total            122.00     5     25.20

 

Testing F

The F statistic is the mean squares of Regression divided by the mean squares of Error. Use the mean squares from the summary table:

SS df ms

Regression     41.14     1     41.14

Error              80.86     4     20.21

Total            122.00     5     25.20

So F = 41.14 / 20.21. When divided through, you get: F = 2.04We test the significance of this F by comparing it to the critical value in the F Table. We enter the table by going across to the dfregression (1) and down the dferror (in this case it’s 4). So the critical value = 7.71. In order to be significant, the F we calculated would have to be larger than 7.71. Since it isn’t, the pattern we see is likely to be due to chance.

 

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