A Note on the Statistical Aspects of Pregnancy Diagnosis

A NOTE ON THE STATISTICAL ASPECTS OF PREGNANCY DIAGNOSIS Bv D. E.

WALTERS

A.R.C. Statistics Group, Department of Applied Biology, Cambridge INTRODUCTION

Pregnancy diagnosis using the hormone levels at various stages in the reproductive cycle would appear to be ideal for the application of classical discriminant analysis. By this technique, a linear function of the hormone levels of an individual would be calculated and then used to classify the individual as either pregnant or non-pregnant. The function used would have been evaluated previously from a random sample of both pregnant and non-pregnant individuals, and would be that function which maximized the separation of the two groups. The evaluation and utilization of a discriminant function from experimental data is most satisfactory and effective when the variates (hormone levels) may be assumed to be normally distributed, with the same variance/covariance structure for the two groups of individuals. The error rates of classification may then be estimated from normal theory. The failure of these assumptions for hormone levels, together with other factors discussed below precluded the successful application of classical discriminant analysis for pregnancy diagnosis, and prompted an examination of alternative methods. THE PROBLEM OF DISCRIMINATION

Classification problems on two populations, such as that posed by the desire to diagnose pregnancy at an early stage, are generally solved by examining a sample of individuals whose status is known, so that a criterion may be established which maximizes the discrimination between the two samples, and hopefully between the corresponding populations. Homogeneity of the variance/covariance matrix results in a linear function of the variates being optimum for this purpose, as illustrated in Fig. 1. Suppose that each individual is represented by two variates (x 1 and x2 ), which could perhaps be the hormone levels at days 2 I and 24 of the reproductive cycle. The two groups are represented by crosses ( X ) and dots (•) in Fig. I. The DISCRIMINANT is that function rp---,-- ax 1

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which best separates the two groups. A new individual is then classified as being in Group I or Group 2 depending on whether its representation on

BRITISH VETERINARY JOURNAL, 132, 5

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Fig. I falls below or above the discriminant line. The full statistical theory of this technique may be found in any text on multivariate analysis, such as Kendall ( I975). A large number of variates may thus be combined in a linear function to give a single value (

STATISTICAL ASPECTS

from the body of the data, which individual is then classified from a criterion derived from the main body. Although rather tedious to perform, and requiring a mathematical derivation of the criterion, this modified jacknife procedure of Lachenbruch & Mickey has the attraction of being scrupulously fair in the estimation of error rates. APPLICATIONS TO PREGNANCY DIAGNOSIS

The distribution of progesterone levels in a sample of both pregnant (P) and non-pregnant (NP) animals is shown in Fig. 2, for days 2I, 24, 28 and 42 of the reproductive cycle. 80

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There are obvious distributional differences between the groups. In particular the variances were found to be inhomogeneous and the distributions were far from being normal. In an attempt to improve matters a logarithmic transformation was carried out on the observations resulting in the distribution shown in Fig. 3 of the paper by Heap et al., page 458 of this issue. It will be noted that apart from the values corresponding to the maximum concentration of So ngjml, the distribution is quite regular for the pregnant animals, having the appearance of normality. The non-pregnant animals however continued to give quite irregular results, possibly due to the sample containing pregnant animals which subsequently lost the foetus and so were wrongly designated as being non-pregnant.

BRITISH VETERINARY JOURNAL, 132,---5

Despite these irregularities, a discriminant analysis was carried out on the transformed data, and revealed another important aspect of multivariate classification. It was found that the success rates, both using normal theory and the modified jacknife procedure improved very little when the observations at 24, 28 and 42 days were included in the analysis, in addition to the 2 I day value. Thus the correlation between the variates was such that a substantial portion of the information in the whole data was contained in the first variate, the hormone level at day 2 I. Since early diagnosis is very desirable this repre.sents a useful property and the small gain in discrimination by including later observations would hardly justify the cost and inconvenience. This emphasis on early diagnosis, and the distributional difficulties previously mentioned certainly detract from the applicability of classical discriminant analysis, so that an empirical criterion based on the progesterone level on the twenty-first day of the cycle may well prove just as useful. One such method, the results for which are reported elsewhere in this issue (Heap, Holdsworth, Gadsby, Laing & Walters) consists of fixing the discriminant point at one standard deviation below the mean of the pregnant animals. Using this criterion, or indeed any criterion based on a statistic computed from the data, a realistic estimate of the error rate may be calculated. The statistical ramifications of the presence of pregnant animals in the ostensibly non-pregnant group have already been mentioned. Their presence, however, also gives a slightly different interpretation to the results of the classification exercise. A discrimination procedure would not, in such circumstances, classify the individuals as pregnant or non-pregnant, but rather as individuals who will, or will not give birth. CONCLUSION

It has been found that the use of standard multivariate statistical techniques in pregnancy diagnosis have proved less than ideal, due mainly to distributional difficulties and the desire to diagnose after only one observation has been taken. There is considerable scope therefore for deriving a simple criterion of diagnosis, but great care needs to be taken in estimating success rates from experimental data. The use of the modified jacknife procedure of Lachenbruch & Mickey (Ig68) should safeguard against the computation and quotation of over optimistic success rates. REFERENCES

LACHENBRUCH, P. A. & MICKEY, M. R. (1g68). Technometrics, 10, K ENDALL, M .G. ( 1975). lvlultivariate analysis. London: Griffin.

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