7.2.2. Polynomial Regression
With this option, polynomials can be fitted on bivariate data. There are no restrictions on the degree of polynomials, but you are reminded that high degree polynomials will take longer to compute and number overflow problems may occur.
Several dependent variables may be selected by clicking on [Dependent] and an independent variable by clicking on [Variable] from the Variables Available list. If more than one dependent variable is selected, the analysis will be repeated as many times as the number of dependent variables, each time only changing the dependent variable. The same dialogue will also allow you to edit the confidence level and select a regression with or without a constant term. Next, another dialogue will prompt for the degree of the polynomial to be fitted.
All output options available for the Linear Regression procedure will be available.
Fitted values for the Polynomial Regression are extremely sensitive to slight changes in coefficients. Therefore, use of the truncated coefficient values from the formatted output display is not recommended in reconstructing a fitted polynomial equation. To run predictions, you are advised to use one of the three methods explained at the beginning of section 7.2.1. Linear Regression. These functions use the full 16-digit precision of the estimated coefficients. The estimated coefficients will also be saved in full precision automatically in the file POLYCOEF.TXT, in the order of constant term (if any), X^1, X^2, ..., X^r.
Example
Table 4.4.1 on p. 295 from Elliot, M. A., J. S. Reisch, N. P. Campbell (1989). The following results are given on p. 297.
Open REGRESS, select Statistics 1 → Regression Analysis → Polynomial Regression and select X (C17) as [Variable] and Y (C18) as [Dependent]. The following set of outputs has been obtained by using these variables with only changing the degree of polynomial. Here we will only print the estimated regression coefficients:
Polynomial Regression
Regression results
Valid Number of Cases: 19, 0 Omitted
Dependent variable: Y
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
37.3890 |
0.43640 |
85.6754 |
0.0000 |
|
X |
3.12686 |
0.15099 |
20.7087 |
0.0000 |
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
40.3017 |
1.13349 |
35.5554 |
0.0000 |
|
X |
0.66658 |
0.91352 |
0.72968 |
0.4761 |
|
X^2 |
0.45397 |
0.16688 |
2.72031 |
0.0151 |
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
32.7673 |
3.11320 |
10.5253 |
0.0000 |
|
X |
10.4109 |
3.90298 |
2.66743 |
0.0176 |
|
X^2 |
-3.38682 |
1.51356 |
-2.23765 |
0.0408 |
|
X^3 |
0.47011 |
0.18442 |
2.54915 |
0.0222 |
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
6.92654 |
7.28551 |
0.95073 |
0.3579 |
|
X |
55.8348 |
12.4946 |
4.46873 |
0.0005 |
|
X^2 |
-31.4866 |
7.60544 |
-4.14001 |
0.0010 |
|
X^3 |
7.76246 |
1.95731 |
3.96587 |
0.0014 |
|
X^4 |
-0.67507 |
0.18076 |
-3.73460 |
0.0022 |
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
36.2391 |
22.7989 |
1.58951 |
0.1360 |
|
X |
-9.16153 |
49.5645 |
-0.18484 |
0.8562 |
|
X^2 |
23.3871 |
41.2381 |
0.56712 |
0.5803 |
|
X^3 |
-14.3460 |
16.4561 |
-0.87178 |
0.3991 |
|
X^4 |
3.59360 |
3.16090 |
1.13689 |
0.2761 |
|
X^5 |
-0.31740 |
0.23467 |
-1.35255 |
0.1993 |
|
|
Coefficient |
standard error |
t-statistic |
Probability |
|
Constant |
157.882 |
73.6834 |
2.14271 |
0.0533 |
|
X |
-330.976 |
192.285 |
-1.72128 |
0.1109 |
|
X^2 |
364.043 |
201.286 |
1.80858 |
0.0956 |
|
X^3 |
-199.361 |
108.401 |
-1.83911 |
0.0908 |
|
X^4 |
58.1131 |
31.7588 |
1.82983 |
0.0922 |
|
X^5 |
-8.60699 |
4.81303 |
-1.78827 |
0.0990 |
|
X^6 |
0.50964 |
0.29560 |
1.72410 |
0.1103 |