Elementary Biometrical Inferences 



531 



the untreated and treated do not have the 

 same parameter. Upon examining the 

 data, one will conclude that the sex ratio 

 is lower following salt treatment than when 

 such a treatment is omitted. (One cannot 

 determine from these delta whether salt 

 raises the number of females or lowers the 

 number of males. One finds only a differ- 

 ence in sex ratio as a function of the 

 presence or absence of salt, the actual mech- 

 anism of the effect remaining unknown.) 



c. Contingency table approach to the chi- 

 square test. Assume that X A = 3 and 

 N A = 6 in sample A, and X B = 5 and 

 N B = 18 in sample B. Are these statistics 

 different at the 5% level of significance? 

 To determine this, one tests the null 

 hypothesis that both samples have the 

 same parameter (p). However, the value 

 of p is completely unknown. If a con- 

 tingency table is constructed, it will give the 



most likely values of X (and hence N — X), 

 a common p for both samples being under- 

 stood. Having determined these ideally- 

 expected values, one can then proceed as 

 before to calculate chi-square. 



The observed data are arranged as shown 

 in Figure A-4A. The best estimates for 

 the values expected according to the un- 

 known p are shown in B. To obtain the 

 value expected in the shaded box in A, for 

 example, multiply together the totals at the 

 end of its column and row and divide by the 

 number N A + N B . This value (6 X 8/24) 

 is 2. 



Since we are dealing with x 2 » recall that 

 it is usually safe to require that no class 

 have an expected frequency less than 2 and 

 that most expected values be at least 5. 

 Note that the other expected values in B 

 can be obtained in a similar manner; this 

 procedure, however, is unnecessary since 

 all the other values are fixed by the mar- 



Classes 



Success 

 Failure 

 Totals 



Samples Totals 

 A B 





A. Actual Data 



B. Expected Data 



C. Difference 

 (A-B) 



Sum of Values 

 in E =X 2 



D. C with Yates' 

 Correction 



E. 



D Squared 



Expected 

 figure A-4- 2X2 contingency table. 



