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# How do I know which elements contribute to a relationship in a two-way frequency table?

Suppose we wish to compare the age frequency profiles of two groups. The data for the four age bands and two groups, those who either complied or refused (to go in an ambulance to hospital), are given below.

 Group Ages 18-40 Ages 40-65 Ages 65-80 Ages 80+ Refused 51 24 13 18 Complied 27 25 31 26

There is a relationship between age and group (chi-square(3) = 16.18, p<0.001). Where, though, exactly is the relationship?

SPSS produces adjusted residuals which may be interpreted as z-values and referred to a standard normal distribution.

```WEIGHT BY CASE.
CROSSTABS
/TABLES=age by group
/CELLS= ASRESID.```

The adjusted residuals for the age group example are

 Group Ages 18-40 Ages 40-65 Ages 65-80 Ages 80+ Refused -3.6 0.1 2.9 1.2 Complied 3.6 -0.1 -2.9 -1.2

We can use the CDFNORM function (in SPSS compute menu) to see the numbers in the cells deviate from what would be expected assuming there is no relationship between age and refusal. For cells (1,1) and (1,3)

```COMPUTE SIG=2*CDFNORM(-3.6).
COMPUTE SIG2=2*(1-CDFNORM(2.9)).
EXE.```

The adjusted residuals -3.6 and 2.9 have two-sided p-values of 0.0003 and 0.004 respectively. This tells us that the 18-40 age group tended to refuse and the 65-80 group tended to comply.

If you are using version 12 of SPSS or later then you can evaluate the p-values for the adjusted residuals directly using the Output Management System (OMS).