Explain the difference between the information given by the

Explain the difference between the information given by the tests of statistical significance t and Chi Square, and measures of association Cramer\'s V, Gamma and Pearson\'s correlation coefficient, r.

Can a test of significance show a statistically significant relationship while a measure of assocation shows a weak relationship between the variables?

If yes, present an example illustrating how such a situation might occur.

Solution

Sol)

Cramer\'s V test:  For crosstabs with nominal measures we can generally preferred Cramer\'s V test. It is ranges from 0 to 1, and the closer to 1, the stronger the relationship.

Gamma Test:When dealing with two ordinal measures that are related in a crosstabs (or at least an ordinal independent variable and a dependent variable that can be interpreted as dichotomous), These tests range form -1 to +1,

Pearson correlation: When dealing with interval/ratio measures. This also ranges from +1 to -1.

The most common test for crosstabs is the chi square test

Example:

However, the relationship may be practically not significant . The measure of associattion gives the estimate of the strength of the relationship, and also its direction.

If we have a large sample, it is quite possible that even for weak correlation, the test of significance shows a significant relationship. That means the relationship is statistically significant but practically insignificant.

The following is an example.

This is a data on the use of seatbelts among men and women. The sample size is 2239

. always | Rarely or never | Total

Female 964 97 1061

Male 924 254 1178

Total 1888 351 2239

If we do a test of significance using Chi-squared test we get test statistic t=65.1365 (observed)

It follows chi-square with degree of freedom 1

p-value of the test is 6.661338*10^(-16) =0.000000which is very small.

Hence, the test is significant for almost all cases (generally significance level is 0.05, 0.01, etc.)

But, if we find Cramer’s V:

sqrt(65.1365/2239)= 0.1705631

Thus, the observed association is weak but the test is significant.

This means it cannot be denied that there is a relationship between the variables but the relationship is weak and practically insignificant.