Testing for Independence between Two Variables Using an $r \times c$ Table

Is there an association  between gender and height? Yes, males tend to be taller than females. A more formal way of saying this is `height distribution for males tends to be different from females'. Is there an association between shoe size and height? Yes, `height distribution for men who wear size 12 is different from those who wear size 8'. Is there an association betwen GPA and height? No, `height distribution tends to be the same for 3.0 students as well as 3.5 students.'

Two variables A and B are said to be associated  if the distribution of B tends to change with the level of the A variable. Otherwise, they are said to be independent .

Therefore height is associated with gender and shoe size, but independent of GPA.

Now consider the following $3\times 4$ table. 189 students entering a business school program were followed as part of an attrition (i.e. drop out, transfer) study. The students were cross classified according to 4 categories of high school GPA (2.0-2.5, 2.5-3.0, 3.0-3.5, 3.5-4.0) and 3 categories of attrition outcomes (`did not return for 2nd year', `returned for 2nd but not for 3rd year', `returned for 3rd year'). Is there an association between HS GPA and college attrition?

 

Table 9.1: Retention versus HS GPA

 

	2.0-2.5	2.5-3.0	3.0-3.5	3.5-4.0
No 2nd Year	25	3	4	6
No 3rd Year	14	7	4	6
Return for 3rd Year	41	7	28	44

To analyze whether attrition and GPA are independent, we will analyze whether attrition distribution remains the same regardless of GPA level. Here is how a statistical report would present the analyis, in numbered stages.

1.

Hypotheses

H₀: The two variables are independent versus H₁: The two variables are associated.

2.

Test Statistic 

If the two variables are independent, the observed frequencies should be distributed like these expected frequencies  (calculations later):

	2.0-3.5	2.5-3.0	3.0-3.5	3.5-4.0
No 2nd Year	16.08	3.42	7.24	11.26
No 3rd Year	13.12	2.79	5.90	9.18
Return for 3rd Year	50.80	10.80	22.86	35.56

The chi-square test statistic measures whether the observed  and expected frequencies are close:

$\chi^2$	=	$\frac{(25-16.08)^2}{16.08}$	+ $\frac{(3-3.42)^2}{3.42}$	+ $\frac{(4-7.24)^2}{7.24}$	+ $\frac{(6-11.26)^2}{11.26}$
		+ $\frac{(14-13.12)^2}{13.12}$	+ $\frac{(7-2.79)^2}{2.79}$	+ $\frac{(4-5.90)^2}{5.90}$	+ $\frac{(6-9.18)^2}{9.18}$
		+ $\frac{(41-50.80)^2}{50.80}$	+ $\frac{(7 - 10.80)^2}{10.80}$	+ $\frac{(28-22.86)^2}{22.86}$	+ $\frac{(44-35.56)^2}{35.56}$
	=	23.42

3.

P-value:

The area greater than 23.42 under the chi-square curve  with (3-1)(4-1)=6 degrees of freedom is .0007.      

4.

Conclusion: Since P-value < .05, we reject H₀ and conclude with very strong evidence that GPA and attrition are associated. In particular, low HS GPA's are associated with higher attrition rates.