EXAMPLE 5 The cocaine study, conclusion

We have seen that desipramine produced markedly more successes and fewer failures than lithium or a placebo. Comparing observed and expected counts gave the chi-square statistic = 10.5. The last step is to assess significance.

Figure 24.2 The density curves for three members of the chi-square family of distributions. The sampling distributions of chi-square statistics belong to this family.

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TABLE 24.1 To be significant at level a, a chi-square statistic must be larger than the table entry for a

	Significance Level a
df	0.25	0.20	0.15	0.10	0.05	0.01	0.001
1	1.32	1.64	2.07	2.71	3.84	6.63	10.83
2	2.77	3.22	3.79	4.61	5.99	9.21	13.82
3	4.11	4.64	5.32	6.25	7.81	11.34	16.27
4	5.39	5.99	6.74	7.78	9.49	13.28	18.47
5	6.63	7.29	8.12	9.24	11.07	15.09	20.51
6	7.84	8.56	9.45	10.64	12.59	16.81	22.46
7	9.04	9.80	10.75	12.02	14.07	18.48	24.32
8	10.22	11.03	12.03	13.36	15.51	20.09	26.12
9	11.39	12.24	13.29	14.68	16.92	21.67	27.88

The two-way table of three treatments by two outcomes for the cocaine study has three rows and two columns. That is, r = 3 and c = 2. The chi-square statistic, therefore, has degrees of freedom

(r − 1)(c −1) = (3 − 1)(2 −1) = (2)(1) = 2

Look in the df = 2 row of Table 24.1. We see that x²= 10.5 is larger than the critical value 9.21 required for significance at the level but smaller than the critical value 13.82 for . The cocaine study shows a significant relationship () between treatment and success.

The significance test says only that we have strong evidence of some association between treatment and success. We must look at the two-way table to see the nature of the relationship: desipramine works better than the other treatments.