When higher degree polynomial models are being used

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when higher-degree polynomial models are being used, since minor extrapolations can have serious errors. Nevertheless, polynomial response models have proven to be extremely useful for summarizing relationships. The polynomial model is built sequentially, starting either with a first- degree polynomial and adding progressively higher-order terms as needed. The lowest-degree polynomial that accomplishes the degree of approximation needed or warranted by the data is adopted. It is common practice to retain in the model all lower-degree terms, regardless of their significance, that are contained in (or are subsets of) any significant term. Example. Data are from a growth experiment with blue-green algae Spirulina platensis for the treatment where CO 2 is bubbled through the culture. The measure of algae density is the dependent variable. Consider a cubic polynomial model with the data for the first replicate. The (ordinary) least squares fit of the model is given by Y i = . 00948 + . 53074 X i + 0 . 00595 X 2 i . 00119 X 3 i + ǫ i , ( . 16761) ( . 09343) ( . 01422) ( . 00062) PAGE 45
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2.3 Qualitative Independent Variables c circlecopyrt HYON-JUNG KIM, 2017 Day rep1 rep2 Day rep1 rep2 1 .530 .184 8 4.059 3.892 2 1.183 .664 9 4.349 4.367 3 1.603 1.553 10 4.699 4.551 4 1.994 1.910 11 4.983 4.656 5 2.708 2.585 12 5.100 4.754 6 3.006 3.009 13 5.288 4.842 7 3.867 3.403 14 5.374 4.969 where the standard errors of the estimates are given in parentheses. Assuming that a cubic model is adequate, we can test the hypotheses a) that a quadratic polynomial model is adequate. b) that a linear trend model is adequate. SSE(full)= 0 . 013658, SSE(Red)= 1 . 458 . c) With two replicates per each day, we can test the adequacy of a quadratic polynomial model. We fit the quadratic model with whole data (including rep1 and rep2) to obtain SSE(reduced) = . 7984 and pure-error sum of squares = . 6344 . Note that the natural polynomials with terms X i , X 2 i , and X 3 i , etc. give (nearly) linearly dependent columns of the X matrix leading to multicollinearity problems. For the cubic polynomial model (as in the above example), a set of orthogonal polynomials can be defined: O 0 i = 1 , O 1 i = 2 X i 15 , O 2 i = . 5 X 2 i 7 . 5 X i + 20 , O 3 i = 5 3 X 3 i 37 . 5 X 2 i + 698 . 5 3 X i 340 . where O 1 i , O 2 i , and O 3 i are linear combinations of the natural polynomials X i , X 2 i , and X 3 i . For the data above, Y i = 3 . 48164 + . 19198 O 1 i . 04179 O 2 i . 00072 O 3 i + ǫ i , ( . 03123) ( . 00387) ( . 00433) ( . 00037) PAGE 46
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2.4 Data Transformations c circlecopyrt HYON-JUNG KIM, 2017 The orthogonal polynomials can be obtained using the Gram Schmidt orthogonalization procedure or with a function ‘poly’ in R. 2.4 Data Transformations There are many situations in which transformations of the dependent or independent vari- ables are helpful in least squares regression. Transformations of the dependent variable are often used as remedies for nonnormality and for heterogeneous variances of the errors. Note that data transformation definitely requires a ‘trial and error’ approach. In building the model, we try a transformation and then check to see if the transformation eliminated the problems with the model. If it doesn’t help, we try another transformation until we have an adequate model.
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