412 Multiple Cases of Disease in the Same House 
by the weighting of each contribution to the contingency with the square of the 
number of cases occurring in houses of the given category. A little thought 
seems to indicate that this weighting is a priori probable, and it is possible that 
a direct proof of it may be discoverable. At any rate when discovered, it seems 
so reasonable, that I am inclined to think that (xx) may be safely applied even 
beyond the range (i.e. m and n large and m large compared to n) for which 
it has been demonstrated. 
In actually applying the method to statistical data of disease occurrence, we 
must confine our attention to those frequencies which give at least a whole house. 
Thus we have practically for the case of enteric 
Pl = 3398, p 2 = 56, p 3 = 1, 
and the observed frequencies were 
Pl = 3350, p, = 78, p s - 2, 
(3398-3350y Q2 (78- 56) 2 y (2-l)» 
* - 1 3398 + 2 56 +3 "^~ 
= -678 + 34-571 + 9 = 44-25. 
I think we must say that this is the for four groups, although by the nature 
of the expression for ^ 2 , the houses with no cases do not enter*. Consulting 
Pal in Elderton's Tables (Biometrika, Vol. I. p. 158) we conclude that such an 
improbability would not occur on the average once in a million random samplings, 
and therefore with a high degree of probability cases of enteric do tend to repeat 
in the same houses. If we had judged simply on the group p 2 we should have 
(p. 2 — p.,)/cr.2 = 22/7 roughly = 3, and such an excess would occur about once in 
a 1000 trials. This again would be extremely improbable, but not so improbable 
as the result based on an examination of the entire frequency. The importance of 
using the latter is thus indicated. At the same time the caution given by Troup 
and Maynard as to the influence in houses of age and sex must have full weight, 
and the problem of the number of inhabitants to the house requires further 
consideration, which I postpone for the present. 
I hope shortly to publish Dr Law Webb's data for " cancer houses," which 
were sent to me in 1910, discussed by this method: see below , Miscellanea , p. 430. 
* The upper limit to the number of groups may be considered disputable. Experimentally all 
ii balls might be cast in one compartment, or there are n+1 frequency classes conceivable. This 
is impossible in the case of multiple cases of disease, for n persons have not lived in any one house 
during the period under observation. To fix the upper limit the average number of persons to the house 
and the average duration of occupation would need to be considered. The fact, however, that x 2 for 
tho?e groups which show theoretically at least a unit is found experimentally to give reasonable values 
of P, justifies empirically its use to test vital statistics. 
