'14 



NATURE 



[October 17, 1907 



largely on the sinuosities of the regression curve, and not 

 on a uniform decrease of parallax with magnitude. Miss 

 Gibson's values for the seventy-two Newcomb stars 

 arc: — Correlation coelVicient, +o-i±o-i; Correlation ratio, 

 oS±oi. Kor the 173 stars given in the Yale memoir, Dr. 

 Lee has shown that Correlation ratio = 0-28 ±006. 



Now let us sec how Mr. Hinks faces such results. In 

 the report of the British Association discussion, published 

 with his approval in the Royal Statistical Society's 

 Journal, he refers, in the jirst place, to the theoretical 

 question, namely, he speaks of the theoretical possibility 

 tiiat the relation between luminosity and distance is con- 

 trolled by a logarithmic curve, and makes the suggestion 

 that this curve (although referred to several times in Miss 

 Gibson's memoir) had been overlooked by us, and that 

 accordingly she ought not to have used the correlation 

 coefficient, which might screen under such a value as 

 0.3 the really perfect correlation which would flow from 

 the logarithmic relation. What was the meaning of 

 Mr. Hinks's appeal to the logarithmic curve if he 

 did not at the British Association suppose Miss Gibson's 

 value of the relationship of magnitude and paraIla.K to be 

 an underestimation? If that appeal was made to show 

 that she ought to have used the correlation ratio, then he 

 had clearly not studied her paper before criticising it. The 

 charge made at the British Association is indirectly re- 

 peated in the last words of his letter in Nature, where he 

 talks about the propriety of calculating the correlation 

 ratio, as if it had not actually been given on p. 452 of 

 the memoir. The divergence between the correlation 

 coefficient and the correlation ratio shows the trained 

 statistician that the sinuosities of the parallax-magnitude 

 curve are not solely humps due to random sampling. 



Well, let us come to our one point of agreement at 

 present : " Astronomers do not believe that magnitude is 

 closely related to parallax." I am glad Mr. Hinks accepts 

 this view, and I will refrain from quoting the work of 

 some great astronomers to show that it has not always 

 been their opinion. Mr. Hinks states in his letter that I 

 asked for details of the reason for the switchback 

 character of Miss Gibson's diagram of parallax and 

 magnitude. If he will read my letter carefully, he will 

 find I asked for no such thing. I asked for details of 

 his reasoning at the British Association that the dips and 

 humps were produced by selection of proper motions, 

 which is an entirely different point. 



He said at Leicester : — " The second peak belongs to 

 stars of the fourth and fifth magnitude ; they were not 

 representative of the average star of that magnitude, but 

 had been chosen because of their exceftionally large proper 

 motion." I very pertinently asked him why this selection 

 had not also been applied to stars of the second to third and 

 to stars of the sixth magnitudes, and further to demonstrate 

 that if it had been thus applied it could possibly have pro- 

 duced the desired effect. To produce an effect on the 

 correlation of A and B by selecting C, a third character, 

 A and B must both be fairly highly correlated with C, 

 and, further, to produce humps we must show that the 

 selection was concentrated at certain points of the range. 

 Mr. Hinks, to give a logical reply, must therefore show :• — 

 (i) that parallax and proper motion are highly corre- 

 lated; {2) that proper motion and magnitude arc highly 

 correlated ; (3) that the selection of astronomers has been 

 discontinuous along the magnitude range. Instead of 

 proving (3), Mr. Hinks has pointed out that the " humps 

 in the curve are due to the presence of individual stars of 

 low or high parallax in the special groups, or rather that 

 some of them are. Quite so ; any statistician knows that 

 with a population of seventy-two the averag-es of eleven 

 subclasses will be largely influenced by individuals ; but 

 the statistician calls this a result of random sampling, and 

 does not suggest discontinuous selection by a third 

 variable with a rclativelv low correlation to at least one 

 of the two characters. Did Miss Gibson, however, lay any 

 special stress on these humps? On the contrary, she 

 says : — " It is possible that a curve of a somewhat com- 

 plex character — a quartic curve, for instance — might fit the 

 observations." But she concludes: — "Examining Fig. i, 

 we see that on the present data nothing better than a 

 horizontal straight line at the mean parallax, or a zero 

 correlation coefficient is likely to be found to fit the 



NO. 1 98 1, VOL. 76] 



observations." I think this will show that we laid no 

 special weight on the humps. On the other hand, the 

 high value of the correlation ratio compared with the value 

 of the correlation coefficient does suffice to suggest that 

 astronomers should be cautious about assuming even a 

 moderately low, but continuously descending relationship 

 between luminosity and distance. The rise in parallax of 

 the faintest stars in Miss Gibson's diagram is again 

 manifest in the Yale results, and is probably not due to 

 mere random sampling. The thirty-six stars of magnitude 

 eight to nine in the Yale data have a parallax nearly three 

 times as great as those of magnitude six to seven, which 

 number thirty-one. Will Mr. Hinks assert here again 

 that the former group have been selected by proper motion 

 and the latter have not? 



Let me further remind him that the correlation between 

 magnitude and proper motion has not even been mentioned 

 by him in his argument, and yet this vital relationship, 

 for whicii I have further determinations, is lower than 0'2. 

 Accordingly, it would need a very high relationship 

 between parallax and proper motion to reduce by a proper 

 motion selection the magnitude-parallax relationship to a 

 small value. 



(b) The second point in Miss Gibson's memoir was 

 involved in the statement that the correlation between 

 parallax and proper motion was " quite significant and 

 important, but not half-way up the scale of correlation " 

 (p. 449). Mr. Hinks says that I am misled by Miss 

 Gibson's results, and asserts that his scatter diagram 

 shows " considerable correlation." Mr. Hinks is an 

 astronomer, and therefore knov/s the value of exact 

 numerical work. Miss Gibson obtained in her memoir the 

 value 0-4 for the proper motion components. Does he 

 really suppose that his scatter diagram can demonstrate 

 that the value of the whole proper motion correlation is 

 greater or less than this?' My estimate, however, was 

 based, unfortunately for Mr. Hinks, not on Miss Gibson's 

 results, but on determinations of the correlations of proper 

 motion and parallax, which involve not only component, 

 but total proper motion. Actually the Newcomb stars 

 exaggerate the result, and the relation between the two 

 characters is sensibly under Miss Gibson's value. Thus 

 for the Yale stars, when we deal with the whole oroper 

 motion, it is only 0-36. Mr. Hinks says that Miss Gibson 

 " has naturally obtained a smaller result than if she had 

 used the whole proper motion in seconds of the arc on a 

 great circle as is usually done." If Mr. Hinks had 

 studied the subject of correlation a little more fully, he 

 would have known that one correlation is related to the 

 others, and if he had actually worked out the theoretical 

 relationship between them he would not have attributed 

 the lowness of our estimate of the parallax and proper 

 motion correlation to dealing with component instead of 

 total proper motions. The multiplying factor is, I think. 

 ^2/{i + r), where r is the mutual correlation of the com- 

 ponents = o-3. 



Mr. Hinks would support his view of high relationship 

 between parallax and proper motion by selecting seventeen 

 stars by magnitude, and leaving the reader to form a 

 mental impression of their correlation. He asserted that 

 our results on seventy-two stars were vitiated by select- 

 ing- by proper motion, yet he does not hesitate him- 

 self to select, not seventy-two, but seventeen stars bv 

 magnitude. And what is the result even of this selection? 

 Why, that 40 per cent, of the value he reaches for correla- 

 tion depends upon the fact that he has not sufficiently 

 counteracted the overwhelming influence due to his in- 

 clusion of a Centauri in the seventeen ! If he leaves this 

 star out (or reduces its influence by introducingr another 

 seventeen stars) the value of the correlation differs from 

 Miss Gibson's value by less than the probable error of the 

 difference. , , . • tc 



I must now state the conclusion to which I feel myselt 

 driven, namely, that astronomers in the near future will not 

 suppose a very close, but a " quite significant and import- 

 ant " relationship between proper motion and parallax. 

 The relationship is more intense than that of parallax and 

 magnitude, but, as shown bv the Yale data, it is probablv 

 less than the value originally fixed by Miss Gibson, and I 

 1 For the total proper mot:on the Newcomb stars give o-jSio-os. 



