This page includes Step-by-Step instructions to Calculate the Coefficient of Partial Determination by hand.

Two Predictor Variables

We first consider a first-order multiple regression model with two predictor variables:

                                   

SSE(X1) measures the variation in Y when X1 is included in the model. 

SSE(X1, X2) measures the variation in Y when both X1 and X2 are included in the model. 

Hence, the relative marginal reduction in the variation in Y associated with X1  and X2 is already in the model is :

 

This measure is the coefficient of partial determination between Y and , X2  given that X1  is in the model. We denote this measure by :

Thus,  measures the proportionate reduction in the variation in Y remaining after X1 is included in the model that is gained by also including X2 in the model. In the following parts, we will measure the proportionate reduction in the variation in Y (Water Consumption) remaining after X1 (Temperature) is included in the model that is gained by also including  X2   (Time mowing the grass) in the model.

The coefficient of partial determination between Y and X1, given that X2  is in the model, is defined correspondingly:

                                   

General Case

The generalization of coefficients of partial determination to three or more X variables in the model is immediate. For instance:

         

 

              

 

 

C1 C2

C3

 C4 C5 C6

C7

 C8 C9
X1 (Temperature)

X2 (Time mowing the grass)

Y

(Water Consumption)

(Regression X1)

(Regression X1 )

(Regression X1 )

(Regression )

(Regression )

(Regression )
99

1.6

48

46.70

1.30

1.6900

47.835

0.165

0.02722
85

1.75

27

26.40

0.60

0.3600

28.570

-1.570

2.46490
97

1.75

48

43.80

4.20

17.6400

46.690

1.310

1.71610
75

1.85

16

11.90

4.10

16.8100

14.720

1.280

1.63840
92

1.15

32

36.55

-4.55

20.7025

31.640

0.360

0.12960
85

1.5

25

26.40

-1.40

1.9600

25.445

-0.445

0.19802
83

1.25

20

23.50

-3.50

12.2500

19.300

0.700

0.49000

Totals

 SSE(X1)

71.413

6.6642

Procedure  

  1. Either print the table above or make your own copy on a sheet of paper.
  2. Enter the X1’s into column C1 as in the table above.
  3. Enter the X2’s into column C2 as in the table above.
  4. Enter the Y’s into column C3 as in the table above.
  5. Regressing Y on X1, you will get the estimated regression function , where is the expected value of Y for this one predictor variable model. If you don’t know how to do a simple linear regression, please click here.
  6. Enter the expected value of the Y’s into column C4 as in the table above. Expected values of Y are obtained by substituting the appropriate X1 values (in column C1) into the estimated regression function as showing above.
  7. For each row in column C5, take each individual Y value and subtract .
  8. For each row in column C6, square the answer for that row of C5.
  9. Add the values in C6 and put that answer in the Total cell under the column.

This total value is the sum of squared error,

  1. Regressing Y on X1 and X2, you will get the estimated regression function , where is the expected value of Y for this two predictor variables model. If you don’t know how to do a multiple linear regression, please click here.
  2. Enter the expected value of the Y’s into column C7 as in the table above. Expected values of Y are obtained by substituting the appropriate X1 values (in column C1) and X2 values (in column C2) into the estimated regression function as showing above.
  3. For each row in column C8, take each individual Y value and subtract .
  4. For each row in column C9, square the answer for that row of C8.
  5. Add the values in C9 and put that answer in the Total cell under the column.
  6. This total value is the sum of squared error,
  7. Then, calculate the Coefficient of Partial Determination:

So the Coefficient of Partial Correlation is:    

                

   

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