Coefficient of Determination ANOVA table

What's the general way to determine the Coefficient of Determination from the ANOVA table

SSR/SST

Sum of Squares Regression divided by the Sum of Squares Total

If you have a negative correlation, is the Coefficient of Determination positive?

Yes.  For example, say your data has a strong negative correlation of -0.8.  Therefore, the coefficient of determination will be -0.8 squared, for a value of 0.64.

Adjusted R Squared

Can the Adjusted R Squared be a negative value?

No.  Typically, the adjusted R Square cannot be a negative value and will range from 0 to 1.

Coefficient of Determination Interpretation

In a short answer, the Coefficient of Determination Interpretation is the "goodness of fit" of a regression.  It's an overall measure of the accuracy of a regression.  The Coefficient of Determination Interpretation is the accuracy of the predictor of the independent variable on the dependent variable value.





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Coefficient of Determination Definition


The Coefficient of Determination, also known as R Squared, is interpreted as the goodness of fit of a regression. The higher the coefficient of determination, the better the variance that the dependent variable is explained by the independent variable.  The coefficient of determination is the overall measure of the usefullness of a regression. For example, you are looking at an ANOVA table, and you see that your R2 is given at 0.95. This means that the variation in the regression is 95% explained by the independent variable. That is a good regression.

Now, looking at at different ANOVA table, you see that you have a Coefficient of Determination, or R2, of 0.50. That means that the variation in the regression is 50% explained by the independent variable. This is not a good regression.

The Coefficient of Determination can be calculated as the Regression sum of squares, RSS, divided by the total sum of squares, SST

Coefficient of Determination = RSS
                                                  SST

Some things to consider about the Coefficient of Determination, aka R2. The regression must be examined for Multicollinearity. Multicollinearity, correlated independent variables, can have the effect of causing a higher R Squared (Coefficient of Determination).

That means that the Coefficient of determination will increase, as you add more independent variables, even if those independent variable do not assist in explaining the variation of the dependent variable. This brings us to the topic of Adjusted R Squared, or the Adjusted Coefficient of Determination, that fixes this problem. The adjusted R Squared can be negative, but must always be less than or equal to the Coefficient of Determination.

The adjusted R Square is not always better thatn the R Square. Only if the new variables added explain more of the variation. Also, the adjusted R square, is better when looking at samples.

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Coefficient of Determination Forumla

Coefficient of Determination Formula


Let's Consider an example for the Coefficient of Determation, when you are given the Total Variation and the Unexplained Variation.

Second - using the formula, [ Total Variation = Unexplained Variation + Explained Variation]

Thus, if you have the Total Variation and Unexplained Variation, you can solve for the Explained Variance, the Coefficient of Determination.

For example, the Total Variation is 100 and the Unexplained Variation is 20.
Therefore 100= 20 + Explained Variation, which solves to equal 80. This is the Coefficient of Determination (or known as R Squared).

Second Example. Total Variation 100, Unexplained Variation 10. Therefore, the Explained Variation is equal to 90.

A good rule of thumb to remember this is pretty simple. Total Variation is made up of Unexplained and Explained Variation.

Another Shortcut to calculate Coefficient of Determination

R Squared = (explained variation)/(total variation) = 1 - (unexplained var/total var)

You can quickly get this information from the ANOVA table. In some situations, they give you some of the information and request that you are able to determine the rest of the information.

RSS = .3
SST = 1

R2 = (.3/1) for 30%

The RSS is the, Regression Sum of Squares, which is the explained variation in the ANOVA table.
The SST is the, Sum of Squares total, which is the total variation in the ANOVA table.

Calculate Coefficient of Determination

Correlation is .80
0.80 squared is 0.64
The Coefficient of Determination is 0.64


1stExample
Given: Correlation 0.75


What is the Coefficient of Determination?
a. 0.5895
b. 0.8574
c. 0.5625
d. 0.9857
e. 0.6514


Answer (C). 0.75 x 0.75 = 0.5625. So the variation in the independent variable, explains 56.25% of the variation in the dependent variable.


2nd Practice Question


Given Correlation 0.97


What is the Coefficient of Determination?
a. 0.8254
b. 0.7854
c. 0.8541
d. 0.9874
e. 0.9409


Answer (e) 0.9409. 0.97 x .97 = 0.9409.  So, the independent variable explains 94.09% of teh dependent variable.  Notice, this is a very high Coefficient of Determination value, intrepreted that this is a pretty good regression.


3rd Practice Question


Given Correlation 0.25


a. 0.07854
b. 0.234
c. 0.0625
d. 0.0671
d. 0.984


Answer (c).  The Coefficient of Determination is 0.0625.  To get the answer, you square 0.25.
So, in summary, the easiest way to calculate the Coefficient of Determination is just to square the Coefficient of Correlation. That's it, it's that simple!

Problems with the Coefficient of Determination

What are some problems or drawbacks to the Coefficient of Determination?

First, let's consider that the Coefficient of Determination will increase as more independent variables are added. It does not matter if those independent variables help to explain the variation of the dependent variable, the R Square (Coefficient of Determination) will increase as more independent variables are added. This brings us to the concept of Adjusted R Squared. The adjusted R Squared takes into account only the independent variables that assist in explaining the variation of the dependent variable.

The adjusted R Squared is different than the Coefficient of Determination, because the adjusted R Squared will only increase if the independent variables are helpful in an explanatory nature. The adjusted R Squared may be negative and must be lower than the original R Square (original Coefficient of Determination)

What are some things that the Coefficient of Determination does not measure?

  1. If there is Collinearity in the independent variables
  2. If the correct regression was selected
  3. If the independent variable was the absolute cause of the dependent variable change
  4. If there is an omitted variable bias

The Coefficient of Determination, R2, chart

When there is only one independent variable, the calculation of the Coefficient of Determination is the square of the Correlation between the Independent Variable and the Dependent Variable. Please see the table below.

Correlation Coefficient                            Coefficient of Determination
1                                                                1
0.95                                                           0.90250
0.90                                                           0.81
0.85                                                           0.72250
0.80                                                           0.64
0.75                                                           0.5625
0.70                                                           0.49
0.65                                                           0.4225
0.60                                                           0.36
0.55                                                           0.3025
0.50                                                           0.25
0.45                                                           0.2025
0.40                                                           0.16
0.35                                                           0.1225
0.30                                                           0.09
0.25                                                           0.0625
0.20                                                           0.04
0.15                                                           0.0225
0.10                                                           0.01
0.05                                                           0.0025
0                                                                0

** It is importance to notice the speed of decrease in each site. For the correlation coefficient, the decrease is at a constant decrease of (-.05). For the Coefficient of Determination, the decrease picks up at a faster rate.

For example, at a 0.90 correlation, the independent variable only explains 81% of the dependent variable.

For example, at a 0.80 correlation, the independent variable only explains 64% of the dependent variable.

For example, at a .60 correlatoin, the independent variable only explains 36% of the dependent variable.
So, for a -0.10 decrease in the correlation coefficient from 0.9 to 0.8, the decrease in the Coefficient of Determination (RSquared) decreased by 17%. Also, note that at a value of 0, the model does not correctly explain the variation of the dependent variable at all.

Also, note the Range for the Coefficient of Determination will range from 1 to 0.

What is the Coefficient of Non-Determination?

The Coefficient of Non-Determination is equal to, [ 1 - The Coefficient of Determination]. For example, say the Coefficient of Determination is 80%. The Coefficient of Non-Determination would be 1 - (.8) = 0.2. So, 20% of the regression would be unexplained variation.

Adjusted R Squared

What is the Adjusted R Squared?

The adjusted R squared is different than the Coefficient of Determination, because the Coefficient of Determination will increase are more independent variables are added to the regression.  This may occur whether or not the independent variables add to the explanatory power of that regression.

The Adjusted R Squared will only increase if the independend variables that are added to the regression help the overall explanatory power of the regression.

Coefficient of Determination related to Correlation

Can the Coefficient of Determination be affected by false correlation values?

In short, the answer is yes.  The Coefficient of Determination, as we've seen in the previous examples, is highly dependent on the overall correlation when you are using just two variables.  If there were problems in the corrrelation, that lead to a spurrious correlation or an unnecessarily high correlation, than the Coefficient of Determination would be too high.  It would need to be adjusted downwards to account for the improper correlation figure.


Coefficient of Determination Calculator

Where can I find a Coefficient of Determination Calculator? Take the Correlation and Multiply it by itself

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What is the Coefficient of Multiple Determination?

The Coefficient of Multiple Determination is the variation of the criterion variable that is explained by the covariations of the predictor variable.



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Coefficient of Determination Value

If you have a Coefficient of Determination with a value of 1;

What does that mean?

If you R Squared (Coefficient of Determination) is 1, that means that you have a perfect explanatory regression.  The variation of the independent variable explains all of the variation in the dependent variable at this value.