An (equivalent) alternative to performing a partial F test for the significance of adding a new variable to a model while controlling for variables already in the model is to perform a t test using the appropriate partial correlation coefficient. If the dependent variable is Y, the independent variable of interest is X, and the controlling variables are Z1, Z2, ..., Zp, then the t test for H0: pYX|Z1, Z2, €¦, Zp = 0 versus HA: pYX|Z1, Z2, €¦, Zp ‰  0 is given by the test statistic

which has a t distribution under H0 with n €“ p €“ 2 degrees of freedom. The critical region for this test is therefore given by
|T| ‰¥ tn-p-2, 1-α/2
Two variables X and Y are said to have a spurious correlation if their correlation solely reflects each variable's relationship to a third (antecedent) variable Z (and possibly to other variables). For example, the correlation between the total annual income (from all sources) of members of the U.S. Congress (Y) and the number of persons owning color television sets (X) is quite high. Simultaneously, however, a general upward trend has occurred in buying power (Z1) and in wages of all types (Z2), which would naturally be reflected in increased purchases of color TVs, as well as in increased income of members of Congress. Thus, the high correlation between X and Y probably only reflects the influence of inflation on each of these two variables. Therefore, this correlation is spurious because it misleadingly suggests a relationship between color TV sales and the income of members of Congress.
a. How would you attempt to detect statistically whether a correlation between X and Y like the one described is spurious?
b. In a hypothetical study investigating socioecological determinants of respiratory morbidity for a sample of 25 communities, the following correlation matrix was obtained for four variables.

(1) Determine the partial correlations rYX1|X2, rYX1|X3, and rYX1|X2, X3.
(2) Use the results in part (1) to determine whether the correlation of 0.35 between unemployment level (X1) and respiratory morbidity rate (Y) is spurious. (Use the testing formula given in Problem 2 to make the appropriate tests.)

Respiratory Morbidity Rate (Y) Air Pollution Unemployment Level (X) Average Temperature (x) Level ( Unemployment level (%) Average temperature (%) Air pollution level C) Respiratory morbidity rate () 0.41 0.29 0.35 0.65 0.50 0.51