Ronald A. Fisher - An Attempted Reconstruction lyrics

3. AN ATTEMPTED RECONSTRUCTION. A framework for dating the experiments is afforded by the statement (p. 318) : "This experiment was practically confined to a small plant group, and is now, after eight years, concluded in all essentials." Mendel's paper was presented on the 8th of February, 1865 ; if he first grew his experimental peas in 1857 he could then be reporting on eight seasons' work. His monastery had sent him for two years to the University at Vienna, where he had studied mathematics, physics, and biology. He returned and took up teaching duties in the Technical High School in 1853 ; he may then have undertaken work in the monastery garden for three years before starting his investigation of peas. On this basis parts of the experiment can be definitely dated (p. 320): "In all thirty-four more or less distinct varieties of peas were obtained from several seedsmen and subjected to a two-years' trial . . . For fertilization twenty-two of these were selected and cultivated during the whole period of the experiments." It was evidently in the second trial year (1858) that the first cross-pollinations were made, namely, crosses for the two seed-characters wrinkled and green, and the two plant characters white flowers and dwarf, Of these the two first are said (p. 331) to have shown segregation for six years, which must be 1859-64, the two named plant characters for five (1860-64), while the three other plant characters used by Mendel, constricted pods, yellow pods, and terminal flowers, for which only four segregating generations are mentioned, may have been first crossed a year later (1859). In 1858 the recessiveness of the two seed characters must have appeared in the ripe seeds from the flowers cross-pollinated, for these would be round (or yellow) irrespective of the shape (or colour) of the self-fertilized seeds borne by the same plants. From the cross round by wrinkled sufficient seed was sown to raise 253 plants in 1859, while from the cross yellow by green 258 plants were raised. It is not improbable that about 250 plants heterozygous for each of the other two factors were also grown in 1859, but we are only told the numbers of plants raised from their seed in 1860, and these do not exceed what could have been bred from forty plants of each kind. In any case, ground for some 600 to 1000 crossbred plants must have been needed in 1859, and it may be noted that in this year the number of self-fertilized lines was reduced from 38 to 22, releasing probably the ground occupied by sixteen rows. The area of the experiments may well have been the same in the three years 1857, 1858, and 1859. The heterozygous plants grown in 1859 from white-flowered parents, and those from dwarf parents, must have established the recessiveness of these characters, and so confirmed the fact of dominance in reciprocal crosses observed with the seed-characters in the previous year. In 1859 too, when the pods were ripe, seeds on plants heterozygous for wrinkled and green showed segregation in 3 : 1 ratios. For wrinkled seeds 253 plants gave 7324 seeds, an average of 29 to a plant. 5474 were round and 1850 wrinkled. The deviation from the expected 3 : 1 is less than its standard error of random sampling. For green seeds 258 plants gave 8023 seeds, an average of 31 to a plant. 6022 were yellow and 2001 green. The agreement with expectation is here even closer. Mendel does not test the significance of the deviation, but states the ratios as 2.96 : 1 and 3.01 : 1, without giving any probable error. The yield per plant seems low. Possibly only four or five pods on each plant were left to ripen, the remainder being consumed green ; it is possible again that little room was allowed for each plant. The discovery, or demonstration, whichever it may have been, of the 3 : 1 ratio was evidently the critical point in Mendel's researches. The importance of the work was demonstrated, if not to Mendel himself, at least to his a**ociates, and, in the following years, the area of the experimental site must have been greatly enlarged. Perhaps for the same reason, in this year also three new crosses were initiated, using the factors for constricted pods, yellow pods, and terminal flowers. That Mendel was satisfied with the two approximate ratios so far obtained would be intelligible if, either previously or immediately upon reviewing the 1859 results, he had convinced himself as to their explanation, and framed the entire Mendelian theory of genetic factors and gametic segregation. His confidence and lack of scepticism shows itself in three distinct ways. (a) He has numerous opportunities in subsequent years of testing on a large scale whether or not the ratios really remained constant from year to year. If he made any such verification he does not record the data. (b) The test of significance of deviations from expectation in a binomial series had been familiar to mathematicians at least since the middle of the eighteenth century. Mendel's mathematical studies in Vienna may have given little attention to the theory of probability; but we know that he was engaged in other researches of a statistical character, in meteorology, and in connection with sun-spots, so that it is scarcely conceivable, had the matter caused him any anxiety, that he knew of no book or friend that would enable him to examine objectively whether or not the observed deviations from expectation conformed with the laws of chance. He goes so far as to give "by way of illustration" the cla**ification of the seeds from "the first ten individuals" of each of these two series (p. 327). In both cases the variations are no larger than the deviations to be expected, but Mendel does not say so. The average numbers of seeds from these two samples are above those for the whole series, being 44 against 29 in the first case and 48 against 31 in the second. Indeed, only three of the twenty plants give less than the average number for its experiment. Possibly some poor-yielding plants were rejected when the list was made up, in which case Mendel's statement, though it may be entirely honest, cannot be entirely literal. Possibly, again, the first ten plants had happened in each case to have been grown in more favourable conditions than the majority of the rest Mendel also gives examples of extreme deviations in both directions from each series. These extreme cases, again, cannot be judged more extreme than would be expected among samples of about 250 plants, but Mendel gives no grounds for this opinion, and, indeed, does not express it. (c) The third point on which Mendel seems more incurious than we could imagine him being, were he not already satisfied, is in not comparing the outcome of reciprocal crosses. He alludes to the point at issue in a footnote to his concluding remarks (p. 355) : "In Pisum it is placed beyond doubt that for the formation of the new embryo a perfect union of the elements of both reproductive cells must take place. How could we otherwise explain that among the offspring of the hybrids both original types reappear in equal numbersand with all their peculiarities ? If the influence of the egg-cell upon the pollen-cell were only external, if it fulfilled the role of a nurse only, then the result of each artificial fertilization could be no other than that the developed hybrid should exactly resemble the pollen parent, or atany rate do so very closely. This the experiments have in nowise confirmed. An evident proof of the complete union of the contents of the two gametes is afforded by the experience gained on all sidesthat it is immaterial, as regards the form of the hybrid, which of the original species is the seed parent and which the pollen parent." If, in 1859, any doubt as to the equivalence of the contributions of the two parents had entered Mendel's mind, he would surely have made a separate enumeration of the seeds borne by the two types of heterozygous plants derived from reciprocal pollinations. Their equivalence as regards dominance had been indicated in the previous year. Their equivalence in genie content Mendel seems early to have felt very sure of. In 1930, as a result of a study of the development of Darwin's ideas, I pointed out that the modern genetical system, apart from such special features as dominance and linkage, could have been inferred by any abstract thinker in the middle of the nineteenth century if he were led to postulate that inheritance was particulate, that the germinal material was structural, and that the contributions of the two parents were equivalent. I had at that time no suspicion that Mendel had arrived at his discovery in this way. From an examination of Mendel's work it now appears not improbable that he did so and that his ready a**umption of the equivalence of the gametes was a potent factor in leading him to his theory. In this way his experimental programme becomes intelligible as a carefully planned demonstration of his conclusions. In 1860 the obstacles to the extension of his experimental programme had been overcome. In this year the two experiments with seed characters were completed by demonstrating that the 3 : 1 ratios observed in the previous year were genetically 1:2:1 ratios. In addition to an unknown number of wrinkled seeds, which came true for this character, 565 plants were raised from round seeds, of which 193 yielded round seeds only, while 372 behaved like their parents. Although at least a couple of pods from each of these 372 plants must have been allowed to ripen, the seed numbers are not reported and, perhaps, were not counted. In the second experiment some green seeds were sown, which duly gave green seeds only, while of 519 plants raised from yellow seeds 166 yielded yellow only and 353 were heterozygous. Again, no seed counts are reported from the 353 heterozygous plants. The ratios in both cases show deviations from the expected 2 : 1 ratio of less than their standard errors. This pair of experiments occupied the space of something more than 1084 plants. They were continued with smaller numbers for the next four years, but no further counts are given. For the two plant characters white flowers and dwarf, which in this year (1860) first showed segregation, provision was made on a larger scale. Of 929 plants 224 bore white flowers, while of 1064 plants 277 were dwarfed. In both cases the deviation is less than the standard error of random sampling. In addition to making provision for over 3000 plants from the crosses made in 1858 Mendel must in this year have raised perhaps 250 heterozygous plants from each of the three crosses started in 1859. His cultures were therefore probably increased this year by about 3000 plants. In 1861 provision was made for 1000 plants each for completing the experiments with the first two plant characters, these being families of 10 plants each from a hundred of the 1860 crop, chosen as showing the dominant characters, coloured flowers, and tall stems respectively. The families from 36 pla nts had only coloured flowers, while those from 64 contained one or more white -flowered plants. The proportionate numbers among the 640 plants of these families was apparently not counted. Again, the families from 28 plants were exclusively tall, while 72 showed segregation of dwarfs. We are not told what was the frequency of dwarfs among these 720 plants. In neither case does the ratio depart significantly from the 2 : 1 ratio expected, although in the second case the deviation does exceed the standard deviation of random sampling. In this year also the three crosses of plant characters started in 1859 required provision for nearly 1000 plants each. Of 1181 plants counted 299 had constricted pods, of 580 plants 152 had yellow pods, and of 858 plants 207 had terminal inflorescences. The deviation is below the standard in every case. Apart from progenies grown from recessive plants, these experiments account in all for 4619 plants. The total was thus probably greater than in the previous year, but the increase was not great. So far as this, the first series of experiments, is concerned, there only remained in 1862 to provide for 3000 plants to establish the 2 : 1 ratios among the progenies of plants segregating for constricted pods, yellow pods, and terminal flowers. Out of a hundred parents tested there were respectively 29, 40, and 33 h*mozygous. Of these the first and third conform well with expectation. In the second case the observed frequencies, 40 h*mozygous to 60 heterozygous, shows a relatively large, but not a significant, deviation. It is remarkable as the only case in the record in which Mendel was moved to verify a ratio by repeating the trial. A second series of a hundred progenies, presumably grown in 1863, gave 65 : 35, as near to expectation as could be desired. Although in 1861 only 580 plants had been available to display the 3 : 1 ratio for yellow pods, and in these two trials respectively 600 and 650 more must have appeared, they do not seem to have been counted, and are not reported in the paper. In connection with these tests of h*mozygosity by examining ten offspring formed by self -fertilization, it is disconcerting to find that the proportion of plants miscla**ified by this test is not inappreciable. If each offspring has an independent probability, .75, of displaying the dominant character, the probability that all ten will do so is (.75) 10, or .0563. Consequently, between 5 and 6 per cent. of the heterozygous parents will be cla**ified as h*mozygotes, and the expected ratio of segregating to non-segregating families is not 2 : 1 but 1.8874 : 1.1126. or approximately 377.5 : 222.5 out of 600. Now among the 600 plants tested by Mendel 201 were cla**ified as h*mozygous and 399 as heterozygous. Although these numbers agree extremely closely with his expectation of 200 : 400, yet, when allowance is made for the limited size of the test progenies, the deviation is one to be taken seriously. It seems extremely improbable that Mendel made any such allowance, or that the numbers he records as segregating are "corrected" values, rounded off to the nearest integer, obtained by dividing the numbers observed to segregate by .9437. We might suppose that sampling errors in this case caused a deviation in the right direction, and of almost exactly the right magnitude, to compensate for the error in theory. A deviation as fortunate as Mendel's is to be expected once in twenty-nine trials. Unfortunately the same thing occurs again with the trifactorial data. These seven experiments of the first series require, as we have seen, a total of four or five thousand plants in the years 1860 and 1861. Apart from the continuation of heterozygous series they account for only 3000 in 1862 and for 1000 in 1863. The pollinations for his second series of experiments were, therefore, probably carried out in 1861. The large trifactorial experiment could not indeed have been finished had it started later, and, as the factor for white flowers first showed segregation in 1860, it is difficult to place it earlier. The bifactorial experiment took a year less, and might have been started in 1860, since the ripened seeds of 1859 had established the 3 : 1 ratios of the two factors. I shall suppose that both were initiated in 1861, and that the same is true of the important but smaller experiments devoted to determining the gametic ratios. To 1862, then, are ascribed the fifteen doubly heterozygous plants of the bifactorial experiment, of which the 556 seeds displayed the first 9:3:3:1 ratio reported. All these were sown in 1863, even the thirty- two wrinkled-green seeds, which suggests that in this year space was abundant. (It was, indeed, in this same year that we have supposed Mendel to depart from his usual practice, and repeat the determination of a frequency ratio, at the expense of growing 1000 additional plants. Even with these additions the summary (Table VI) shows 1863 as less crowded than most of the other years.) The plants from these seeds, cla**ified by the seeds they bore, exhibited independent segregation of the two factors. Mendel's cla**ification of the 529 plants which came to maturity is shown in Table I. Table I. Cla**ification of Plants grown in the Bifactorial Experiment The numbers are close to expectation at all points, but they are not very large. In relation to possible linkage, for example, they may be regarded as excluding, at the 5 per cent. level of significance, recombination fractions less than 44.9 per cent., which is not very strong negative evidence ; yet on this point also Mendel evidently felt that further data would be superfluous, for he certainly could have obtained many more for the counting. The 138 plants, for example, recorded in the table above as being doubly heterozygous, doubtless bore over 4000 seeds segregating in the 9 : 3 : 3 : 1 ratio, and, even if the bulk of the crop were needed when green, at least ten seeds from each plant must have been allowed to ripen in order to cla**ify the plant on which they grew. The trifactorial experiment required 24 hybrid plants grown in 1862, which gave 639 offspring in 1863. In order to distinguish heterozygotes from h*mozygotes among the plants with coloured flowers progenies from at least 473 of these must have been grown. If, as in other cases, Mendel used a progeny of ten plants for such discrimination the experiment must have needed 4730 plants in 1864. Of this experiment Mendel says (p. 335) : "Among all the experiments this demanded the most time andtrouble", and the extent of the third filial generation explains this remark. It was evidently on the completion of this extensive work that Mendel felt that his researches were ripe for publication. It may have constituted the whole of his experimental work with peas in the last year before his paper was read. Even so, probably this year saw more experimental plants than were grown in any previous year. Since the factor for coloured flowers used in this experiment obscures the cotyledon-colour of unopened seeds, not all of the vast number of seeds borne by these three generations was easily available to supplement the bifactorial and trifactorial data reported, yet even what was easily available must have been much more extensive than any data which Mendel published. Mendel's trifactorial cla**ification of the 639 plants of the second generation is shown in Table II, which follows Mendel's notation, in which a stands for wrinkled seeds, b for green seeds, and c for white flowers. Table II. Cla**ification of Plants grown in the Trifactorial Experiment In order to discriminate CC from Cc plants progenies from these, which are seen to number 463 together, must have been grown on in 1864. In addition to abundant new unifactorial data the additional bifactorial data supplied by the experiments is seen to be large. 175 of the plants were heterozygous for both of the two seed characters, and, if 30 seeds from each had been cla**ified, these would have given 5250 seeds, nearly ten times as many as the 556 reported from the bifactorial experiment. The cla**ification of these plants as double heterozygotes must indeed have required that about half this number of seeds from each plant were examined. In the following year also nine- sixteenths of the progeny of 127 F2 plants, or about 815 F3 plants, must have borne seeds segregating in the 9:3:3:1 ratio, so that a further 24,000 seeds could have been so cla**ified in 1864. Evidently, however, Mendel felt that the complete cla**ification of 529 plants in the bifactorial experiment was sufficient; he does not even add, for the simultaneous segregation of Aa and Bb, the 639 plants completely cla**ified in the trifactorial experiment, which suffice to raise the recombination fraction significantly higher than 46.56 per cent. (from 44.9 per cent.). Table III. Comparison of Numbers reported with Uncorrected and Corrected Expectations In the case of the 600 plants tested for h*mozygosity in the first group of experiments Mendel states his practice to have been to sow ten seeds from each self-fertilized plant. In the case of the 473 plants with coloured flowers from the trifactorial cross he does not restate his procedure. It was presumably the same as before. As before, however, it leads to the difficulty that between 5 and 6 per cent of heterozygous plants so tested would give only coloured progeny, so that the expected ratio of those showing segregation to those not showing it is really lower than 2:1, while Mendel's reported observations agree with the unconnected theory. The comparisons are shown in Table III. A total deviation of the magnitude observed, and in the right direction, is only to be expected once in 444 trials ; there is therefore here a serious discrepancy. If we could believe that Mendel changed his previous practice, and in 1863 went to the great labour of back-crossing the 473 doubtful plants, the data could be explained, for in such progenies miscla**ification would be only about one-fiftieth part as frequent as in progenies by self-fertilization. Equally, if we could suppose that larger progenies, say fifteen plants, were grown on this occasion, the greater part of the discrepancy would be removed. However, even using families of 10 plants the number required is more than Mendel had a**igned to any previous experiment, and there is no reason for thinking that he ever grew so many as 7000 experimental plants in one year, apart from his routine tests. Such explanations, moreover, could not explain the discrepancy observed in the first group of experiments, in which the procedure is specified, without the occurrence of a coincidence of considerable improbability. An explanation of a different type is that the selection of plants for testing favoured the heterozygotes. In the first series of experiments the selection might have been made in the garden, or, if the whole crop was harvested, on the dry plants. In either case the larger plants might have been unconsciously preferred. It is also not impossible that, in some crosses at least, the heterozygotes may have been on the average larger than sister h*mozygotes. The difficulties to accepting such an explanation as complete are three. (i) In the tri-factorial experiment there was no selection, for all plants grown must have been tested. The results here do not, however, differ in the postulated direction from those of the first series. On the contrary, they show an even larger discrepancy. (ii) It is improbable that the supposed compensating selection of heterozygotes should have been equally effective in the case of five different factors. (iii) The total compensation for all five factors (21.5 plants) must be supposed to be greater than would be needed (16.8 plants) if families of 11 had been grown, and less than would be needed (30.0) if 9 only had been grown, though nearly exactly right for the actual number 10 of F3 plants in each progeny (22.5). The possibility that the data for the trifactorial experiment do not represent objective counts, but are the product of some process of sophistication, is not incapable of being tested. Fictitious data can seldom survive a careful scrutiny, and, since most men underestimate the frequency of large deviations arising by chance, such data may be expected generally to agree more closely with expectation than genuine data would, The twenty-seven cla**es in the trifactorial experiment supply twenty-six degrees of freedom for the calculation of x2. The value obtained is 15.3224, decidedly less than its average value for genuine data, 26, though this value by itself might occur once in twenty genuine trials. This total may be subdivided in various ways; one relevant subdivision is to separate the nine degrees of freedom created by the dis crimination of h*mozygous and heterozygous plants with coloured flowers from the remaining seventeen degrees of freedom based on discriminations made presumably in the previous year. To the total the 9 supply 6.3850, leaving only 8.9374 for the remaining 17. If anything, therefore, the subnormality in the deviations from expectation is more pronounced among the seventeen degrees of freedom than among the nine. If there has been sophistication there is no reason to think that it was confined to the final cla**ification made in 1864. To 1863 belong probably the bifactorial experiment and the five comparisons, each of four equal expected frequencies, supplied by the experiments on gametic ratios. The bifactorial experiment, having nine cla**es, supplies eight degrees of freedom for comparison, and gives a x2 of only 2.8110—almost as low as the 95 per cent. point. The fifteen degrees of freedom of gametic ratios supply only 3.6730, which is beyond the 99 per cent. point. In the same year also should be included the verified 2 : 1 ratio for yellow pods, giving 0.125 for one degree of freedom. Putting together the comparisons available for 1863 we have :— Table IV. Measure of Deviation Expected and Observed in 1863 The discrepancy is strongly significant, and so low a value could scarcely occur by chance once in 2000 trials. There can be no doubt that the data from the later years of the experiment have been biased strongly in the direction of agreement with expectation. One natural cause of bias of this kind is the tendency to give the theory the benefit of doubt when objects such as seeds, which may be deformed or discoloured by a variety of causes, are being cla**ified. Such an explanation, however, gives no a**istance in the case of the tests of gametic ratios and of other tests based on the cla**ification of whole plants. For completeness it may be as well to give in a single table the x2 values for all the experiments recorded. Table V. Deviations Expected and Observed in all Experiments The bias seems to pervade the whole of the data, apart, possibly, from the illustrations of plant variation. Even the 14 degrees of freedom available before 1863 give only 7.1872, a value which would be exceeded about 12 times in 13 trials. Table VI. Approximate Numbers of Plants grown in different Years What I have inferred respecting the extent of Mendel's cultures is summarized by years in Table VI. I have arbitrarily allowed sixty plants for each of the stock lines and fifty for each segregating line which was continued with smaller numbers after the completion of the main experiments. I have included also in 1862 and 1863 the two small experiments devoted to the demonstration of gametic ratios. Some of the totals for years may be correct to the nearest hundred, but I do not expect all to be so. I feel justified in concluding only that the experiment was greatly enlarged after the first three years and that, with only ten plants to a family, the year 1864 was probably the fullest of all.