Miscellanea 
191 
on the assumption that w' = 4 in fourfold tables, and consequently should not be used when, as is 
almost always the case, the marginal totals are obtained from the data" {loc. cit. p. 91). 
I hold those tables are quite correctly calculated for n' = A, and those who attempt to modify 
them by assuming ?i' = 2 will be dealing with an entirely different problem. Namely, they will 
be considering not the improbability of the given sample as one of all possible samples of the 
given size, which it really is, but one of the indefinitely smaller number of samples that have 
fixed marginal totals. We do not find the probable error of r for a tetrachoric table* on the 
assumption that the marginal totals are fixed. We find it on the assumption that the marginal 
totals also vary from sample to sami)le, and when we have found it, then we substitute in the 
result the values of not only the marginal totals, but the cell-contents, «, 6, c, d of the sample 
itself for those of the unknown population. With we go through an exactly similar process of 
reasoning. If by this pi'ocedure we in some mysterious manner tied our degrees of freedom down 
to the values of the cell-contents used in our formula and adopted from our sample there could 
be no probable error for r, for the values of «, c, and d are all required and used. I trust my 
critic will pardon me for comparing him with Don Quixote tilting at the windmill ; he must either 
destroy himself, or the whole theory of probable errors, for they are invariably based on using 
sample values for those of the sampled population unknown to us. For example here is an 
argument for Don Quixote of the simplest nature : In the sth category of a population N the 
frequency is 71^, a sample shows in a total M. The standard deviation of this frequency is 
But we don't know the population sampled and accordingly obtain an approximate value of the 
above standard deviation by writing for ^ and taking for the standard deviation of 
tils - . In doing this it is not a question even of using a marginal total, we have used 
a cell frequency found from our sample. We have therefore according to our critic reduced our 
possibilities of freedom by selecting out of all possible samples those with 7n, in the sth cell — this 
is exactly parallel to our reducing our freedom by "fixing" marginal proportiou.s or moment- 
coefficients. But if nig be fixed, it is ridiculous to talk of a variation of the Wg frequency. There- 
fore either ^1^=0 or mg = M, or the usual theory and practice of probable errors are wholly at 
fault. I think this will illustrate what I mean by Don Quixote and the windmill. 
II. 
Is Tuberculosis to be regarded from the Aetiological Standpoint as an acute disease 
of Childhood ? By Dr Kr. F. Andvord (Christiania). Tubercle, Vol. ill. No. 3, 
December, 1921. 
This paper is, we must confess, unconvincing. The author holds that in a community that 
has long been subject to tuberculosis the time of infection should be fixed in the infantile years 
for the great majority of cases and consequently we should protect children for the first three or 
four years from infection. 
As evidence of his views he takes a graph of what he calls a "population frame" which is 
really the well-known " number living in a stationary population " (l^,) and represents within this 
graph the numbers dying from tuberculosis and the numbers who have suffered from it at each 
age. We are doubtful if his graphs for deaths are correctly drawn. They are made to rise 
suddenly for about a year and then fall till age 7 but we suspect that they should fall from birth 
till age 7. We cannot justify his chart (No. VIII) which gives the whole population and the 
* Phil. Trans. Vol. 195 A, p. 14. 
