Miscellanea 
179 
IV. The Slimination of Spurious Correlation due to position in Time 
or Space. 
By "STUDENT." 
In the Journal of the Royal Statistical Society for 1905*, p. 696, appeared a paper by 
R. H. Hooker giving a method of determining the correlation of variations from the "in- 
stantaneous mean" by correlating corresponding differences between successive values. This 
method was invented to deal with the many statistics which give the successive annual values 
of vital or commercial variables ; these values are generally subject to large secular variations, 
sometimes periodic, sometimes uniform, sometimes accelerated, which would lead to altogether 
misleading values were the correlation to be taken between the figures as they stand. 
Since Mr Hooker published his paper, the method has been in constant use among those who 
have to deal statistically with economic or social problems, and helps to show whether, for 
example, there really is a close connection between the female cancer deatli rate and the quantity 
of imported apples consumed per head ! 
Prof. Pearson, however, has pointed out to me that the method is only valid when the 
connection between the variables and time is linear, and the following note is an effort to extend 
Mr Hooker's method so as to make it applicable in a rather more general way. 
If a'l, .V2, ^3, etc., //i , </2, etc., be corresponding values of the variables .v and y, then if 
Xi, x-i-i .^'3, etc., y.j, etc. are randomly distributed in time and space, it is easy to show that 
the correlation between the corresponding nXS\ dittei'ences is the same as that between x and y. 
Let „Z>^. be the ni\\ difference. 
For ,/)j; = .r, -a'2, . \B^ = x^ -'ix^^x-i-^-x.^. 
Summing for all values and dividing by /V and remembering that since .r, and .Co are mutually 
random S (.I'l, .^■2)=0, we gett 
'^-,Z), = 2<r,-. 
Again, i = tii -y.,, . ■ . j Z)^. , Z>^ = y^ - x., - x^ y., + x-, y<, . 
Summing for all values and dividing by and remembering that .r, and y., and x., and are 
mutually random 
Proceeding successively /■ r, =r j. j^, = ... = /■_,.„ (1). 
Now suppose X,, x-2, x-i, etc. are not random in space or time; the problems arising from 
correlation due to successive positions in space are exactly similai' to those due to successi\'e 
occurrence in time, but as they are to some extent complicated by the second dimension, it is 
perhaps simpler to consider correlation due to time. 
Suppose then .r, = A', + ht^ + et^^ + dt^" + etc., x., = X-, ht-, + cl.r + Jt.? + etc. 
where X^, X^, etc. are independent of time and t-^, ^o, h '^^^'^ successive values of time, so that 
ti^-t,i-i=T, and suppose »/i= Yi + h't^+c'ti- + ctc. as before. 
* The metbod had been used by Miss Cave in Proc. Roy. So<-. Vol. lxxiv. pp. 407 et scq. that is iu 
1904, but being used incidentally in the course of a paper it attracted less attention than Hooker's 
paper which was devoted to describing the method. The papers were no doubt quite independent. 
+ The assumption made is that n is sufficiently large to justify the relations 
(x)/(ri - 1) = S." (.r)/((i - 1) = S," (.r)jn and ,^,"-1 (.r2)/(» - 1) = .SV' (.vi)l{n - 1) = Sj" {.r-i)/«, 
being taken to hold. 
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