Example 9. Polynomial Regression: Fish Length Data

Fingerlings of a particular species of fish were put into four tanks that were maintained at respective temperatures of 25, 27, 29 and 31 degrees Celsius. After the first 2 weeks, and thereafter, every week for 21 weeks, one fish was randomly selected from each tank and its length was measured. The following data set consists of observations on the ages of the fish and the lengths of the fish at each tank (temperature). The variables are shown below.
X: Age of fish (in days)
Y1: Length of fish at temperature of 25 degrees C
Y2: Length of fish at temperature of 27 degrees C
Y3: Length of fish at temperature of 29 degrees C
Y4: Length of fish at temperature of 31 degrees C

Source: Freund, R.J., and Littell, R.C. (1991). SAS System for Regression. SAS Institute Inc.

Table 9: Fish Length Data

  X     Y1     Y2     Y3     Y4
 14    620    625    590    590
 21    910    820    910    910
 28   1315   1215   1305   1205
 35   1635   1515   1730   1605
 42   2120   2110   2140   1915
 49   2300   2320   2725   2035
 56   2600   2805   2890   2140
 63   2925   2940   3685   2520
 70   3110   3255   3920   2710
 77   3315   3620   4325   2870
 84   3535   4015   4410   3020
 91   3710   4235   4485   3025
 98   3935   4315   4515   3030
105   4145   4435   4480   3025
112   4465   4495   4520   3040
119   4510   4475   4545   3177
126   4530   4535   4525   3180
133   4545   4520   4560   3180
140   4570   4600   4565   3257
147   4605   4600   4626   3166
154   4600   4600   4566   3214

Questions:

1. Fit a regression between Y3 and X.
   
   
2. Is a simple linear regression adequate?
   
   
3. Is there merit in using a polynomial regression model; if so, of what degree?

Keywords: Polynomial regression, sequential SS, polynomial plots