27.) exp {-j(r-- i)2/y2,u 2},

Transcription

1 Q. Jl exp. Physiol. (197) 55, STATISTICAL ANALYSIS OF GRANULE SIZE IN THE GRANULAR CELLS OF THE MAGNUM OF THE HEN OVIDUCT: By PETER A. K. COVEY-CRuiMP. From the Department of Statistics, University of Glasgow. (Received for publication 3rd February 197) Statistical analysis of granule size in the granular cells of the fowl oviduct supports the hypothesis that these granules can be allotted to six different size groups. An investigation of the ultrastructure of the hen oviduct by Wyburn et at. (197) showed tubular gland cells in the magnum which contained electron dense granules. Because the sections taken through the magnum did not necessarily pass through the centres of the granules, the observed diameters of intersection could be much smaller than the actual diameters of the granules. We have therefore considered the following statistical problem: Spherical particles (granules) are imbedded in a medium which is cut by a plane, and as a result, the circular sections of these particles intersected by the plane are observed. The sizes of the particles possibly form a non-homogeneous population, and it is desired to infer from the observations something about the distribution of the diameters of the particles. Initially attention was concentrated on two hypotheses regarding this distribution, whose probability density function (p.d.f.) we denote by f(r). Hypothesis (a). The diameters of the particles come from a homogeneous population approximately described by a normal distribution with unknown mean,u and unknown variance a2: that is, f(r) has the form f(r) = V(2 ) exp {-j(r- )2/f2}. Hypothesis (b). There is a mixture of different kinds of particle, particles of any one kind being very similar in size, though not all exactly equal. (It seems somewhat unrealistic to postulate that particles of one kind are all of exactly the same size.) In order to translate this hypothesis into a parametric form for f(r), with a manageable number of parameters, it was assumed that the diameters of each different kind of particle have a common small coefficient of variation, so that the statistical variation is proportional to the mean for each kind of particle, an assumption which seems fairly realistic in this context. Then in terms of f(r), the hypothesis (b) becomes: For some integer m >2, f(r) has the form f(r) = m 1I 27.) exp {-j(r-- i)2/y2,u 2}, i=1 YUi-,/(2 233

2 234 Covey-Crump where a1+(o2+ M = 1. The ois, ui's and y are all unknown: (xi is the proportion of the ith kind of particle in the population; Pi is the mean diameter for the ith kind of particle; y is the common coefficient of variation for different kinds of particle. Thus for any fixed value of m, f(r) contains 2m unknown parameters. The aim is to determine which of these two hypotheses better fits the observed data (sizes of circular sections) and, if (b) is significantly better than (a), what value m has (how many essentially different sizes of particle there are). EBtimation and Significance Tests The information about the distribution of particle diameters given by observations of circular sections of them is indirect, or 'blurred by noise', and before we can use such observations for inference, we must know the relationship between the distribution of particle diameters and that of their sections. This has been derived by Kendall and Moran (1963, p. 86) who show that if the centres of the particles have a Poisson distribution (that is, are at random in the medium), the p.d.f. +(x) of the diameters of their circles of intersection in the random plane is given in terms of the p.d.f. of their diameters f(r) thus: i(x)= 2 dr where H = r f(r) dr. They further show that if M and m are the mean and harmonic mean of the diameters of intersection, then the mean and variance of the particle diameters are given by,=-jim and u2= 2Mm- 2. Now if we assume a particular parametric form for f(r) - as we do in setting up hypotheses (a) and (b) - this result enables us to calculate the corresponding parametric form for i(x), and then we may use standard methods for estimating the unknown parameters from observations on x. The estimation method adopted was that of maximum-likelihood. Estimation of parameters in a mixture of distributions is notoriously difficult from a computational point of view. Here we have the additional difficulty of indirect information. Two ALGOL programs were developed to calculate and plot the p.d.f. q(x) and its integral, the cumulative distribution function (c.d.f.), for the section diameters obtained when particular values are assigned to the parameters in f(r). These were compared with the empirical plots of the data, and the parameters were adjusted to give a reasonably good fit by eye. These values of the parameters were then taken as initial estimates for the maximum likelihood calculations. A search procedure was employed and the values of the parameters which gave the largest value of the likelihood were obtained. We consider first the hypothesis (a) that the particle sizes are described by a single normal distribution. Since the two unknown parameters in f(r) are just the mean,u and the variance a2 we can calculate initial estimates for them from the mean and harmonic mean of the data using the formulae above; this gave

3 Ultrastructure of the Hen Oviduct 235 -L= 1P92 and ao2 = 36. The likelihood of the data was now calculated for a range of values about (,uo, a2) and the maximum value of thelikelihood occurred with the maximum likelihood estimates (m.l.e.) i = 1.85 and 82 = '49 which gave a log-likelihood of -33*76. The search procedure revealed a high negative correlation between the two parameters,u, a2. The theoretical c.d.f., calculated using these values, is shown by the broken line in Fig. 1, and the p.d.f. is shown by the broken line in Fig. 2. The c.d.f. 1. z D.* zll I.- LL -6 - X -42 I-- 3 i,'i i DIAMETER OF SECTION (pm) FIGa. 1. Empirical cumulative distribution function with the calculated cumulativre distribution functions for hypothesis (a) (broken line) and hypothesis (6) (solid line) superimposed. did not differ significantly from the empirical c.d.f. using the Kolmogorov- Smirnov test. However we note that the residuals display a pattern which suggests very strongly that hypothesis (a) is not true. This of course is a reflection of the fact that the Kolmogorov-Smirnov test may have low power for the alternatives which we are considering. We now consider hypothesis (b) that there is a mixture of several different particle sizes present with a much smaller allowed variation. The exrperimenters placed high prior probability on there being between five and seven different sizes of particle corresponding to the different proteins known to be present. We initially tried a mixture of five different particle sizes. Various values of the ten parameters were tried and plots of the p.d.f. and c.d.f. were produced. The c.d.f. was reasonably informative about the mixing proportions and the p.d.f. plots gave estimators of the locations of the different sizes, but this visual procedure yielded very little information on the coefficient of variation. Then these initial estimates were improved by calculating the likelihood of the data for a range of values of the unknown parameters. In particular the coefficient of variation was found to be reasonably independent of the other

4 236 Covey-Crump parameters, and when estimating the means, it proved most satisfactory to estimate them in decreasing order of magnitude. The maximum value of the log-likelihood obtained was -315x76. However when the calculated c.d.f. was compared with the data a pattern in the residuals was still apparent and so a mixture of six different sizes was tried. The maximum value of the log-likelihood in this case was which is a significantly better fit and the model accounts for the pattern in the residuals observed above. The calculated c.d.f. (see the solid line in Fig. 1) U) z 3 2 U) 1 FIG DIAMETER OF SECTIONS (t,m) Histogram of the data with the probability density functions for hypothesis (a) (broken line) and hypothesis (b) (solid line) superimposed. agrees with the observations very well in the sense that it would be very difficult to decide where to add another component to the mixture. The m.l.e.'s of the parameters gave six basic sizes of particle with a coefficient of variation of only 3.3 per cent. The mixture is composed of 7 per cent at diameter -61,um, 9 per cent at 1 19,um, 19 per cent at 1 63,um, 15 per cent at 1*94,um, 37 per cent at 2-49,um and 13 per cent at 3-15,um. The p.d.f. for the section diameters is shown by the solid line in Fig. 2. The increase in log-likelihood from hypothesis (a) to hypothesis (b) is On large sample theory twice the increase is distributed as a chi-square deviate on 1 degrees of freedom; the calculated difference is significant at -1 per cent. Thus the hypothesis that there are six basic sizes of particle is much more acceptable than the hypothesis of just one single size. Moreover the excellence of the fit so obtained, and the lack of pattern in the residuals for this fit does provide rather convincing evidence of its general acceptability. An Alternative Hypothesis for the Mixture We have also considered a different model for the mixture where the statistical variation is the same for each of the different kinds of particle. Hypothesis (c). The diameters of the particles come from the population described by the p.d.f. f(r) where for some integer m >2, f(r) has the form f(r) = where 1±+C Cm -= 1. m I1 (2ir) exp { (r-iu)2//u2},

5 Ultrastructure of the Hen Oviduct 237 For any fixed value of m, f(r) has 2m unknown parameters as before; in this case, a2 is the unknown common variance. The m.l.e.'s were calculated for hypothesis (c) in an exactly similar manner. It sufficed to use five components to obtain a reasonable fit. The m.l.e. of the variance was *13 and the mixture consists of 8 per cent at diameter -66,m, 9 per cent at 1-16,u.m, 3 per cent at 1-8,um, 4 per cent at 2-47,um and 13 per cent at 3-13,um. The maximized log-likelihood is and thus this hypothesis is significantly better than hypothesis (a) at -2 per cent. The calculated p.d.f. is shown in Fig. 3. cn z > 3- Ld m 2 LL ir 1 LUI :13 D 1 ~~2 3 4 DIAMETER OF SECTIONS Ipm) FIG. 3. Histogram of the data with the probability density function for hypothesis (c) imposed. super- In testing between hypotheses (b) and (c) we note that they do not belong to the same parametric family and we must use Cox's result on separate hypotheses [Cox, 1962], to which the reader is referred for fuller details. The principal difficulty in applying his method is the calculation of the variance of the test statistic, and we have used a simplification which will tend to overestimate the variance, and thus the test of separation between the two hypotheses will be less powerful. Applying a two-sided test since there is no prior preference for either hypothesis, the test shows that hypothesis (b) is significantly better than hypothesis (c) at.1 per cent, or possibly much more significant. Thus hypothesis (b) is much preferable to hypothesis (c). This result can be interpreted as showing that the experimental error can be considered as variation within the particle types rather than measurement error, and supports the hypothesis that these particles are a mixture of (probably) six different sizes of granules. REFERENCES Cox, D. R. (1962). 'Further results on Tests of Separate Families of Hypotheses.' J. Roy. Stat. Soc. B, 24, 46. KENDALL, M. G. and MoRAN, P. A. P. (1963). Geometrical Probability. London: Griffin. WYBuFRN, G. M., JOENSTON, H. S., DRAPER, M. H. and DAvmsoN, MAIDA F. (197). 'The fine structure of the Infundibulum and Magnum of the oviduct of Gallus domesticus.' Q. JZ exp. Phy8iol. 55,