The Value of Orthogonal Polynomials in the Analysis of Change-Over Trials with Diary Cows

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Abstract
The standard analysis for Latin and Youden squares may need modification for change-over trials with dairy cows. It may be necessary to introduce parameters for residual effects of diets. Also the error terms for successive periods cannot usually be regarded as independently distributed. It is useful to represent the records for successive periods by a series of orthogonal polynomials. The linear polynomial has the highest variance, and the quadratic may be more variable than higher-order polynomials. The variability of the logarithm of milk yield, in four-period and five-period trials with the Colworth herd, has been analysed. Correlations between polynomials were not large enough to affect routine analyses seriously. Separate analyses were made for trials in which cows were fed for production and for those in which equalized feeding was used; and the ratios of variances of the polynomials did not vary significantly from trial to trial within either feeding method. Information on the treatment parameters can be obtained from each degree of polynomial. The standard least-squares analysis (here called Method 1) gives equal weight to each set of information. Patterson [1951] suggested a weighted analysis (Method 2) and Cox [1958] suggested discarding the information given by linear, and perhaps by quadratic, polynomials (Method 3). The precision of the estimates of treatment means, given by these methods, is compared. It depends on the ratios of variances of polynomials, and for the Colworth herd Method 2 is to be preferred. The weights should be based where possible on ratios obtained from the pooled results of past experiments this may cause some bias in the estimated variance of treatment parameters, but it will be small. If some observations are missing, the analysis by Method 2 is complex. A suggestion for reducing the labour of analysis is made.[long dash]Author.