Statistical Independence

[alphaqn69]

I'm trying to give my students a brief introduction to chi-square testing, but there's something that's a little confusing to me regarding the terminology of "statistical independence"

Here it goes:

For data in a 2 x 2 contingency table to be "statistically dependent" the percentage distributions of the dependent variable within each category of the independent variable must be identical. And this is where we would use the term "statistically independent"

If this is in fact the case, how is it that there are numerous examples out there on the internet where a chi square test of independence, concludes that we "fail to reject" the null hypothesis of independence, even though the proportions in the cells are not identical?

Or does the term "statistically independent" have some leeway in its use.

Thanks Guys.
Yours in Education
Theo

[arpix]

Consider following example: your IV is gender and the DV is smoker/nonsmoker. The two variables would be considered "statistically" independent if you wouldn't find a relationship between gender and smoking status, meaning that you would have identical or similar percentage distributions. On the other hand if you found that men are more likely to be smokers than women, then what it means is that whether one smokes or not will depend to a certain degree on whether you're aman or a woman (men = 70%smokers, 30%nonsmokers: women = 30%smokers, 70%nonsmokers), the two variables are dependent, you could say there is relationship between the two. Going back to what you wrote... statistically dependent means that the distributions are not identical and not the other way around.
We do reject the nullhypothesis (there is no dependency (independence) between IV and DV) in favor of the alternative hypothesis (there is a dependency or relationship between IV and DV) when we find a statistical result in favor for the alternative hypothesis. The nullhypothesis is not necessarily rejected when proportions of the cell are nonidentical. Two factors contribute to the rejection of it, one being the sample size, the other is the relative discrepancy among the cells. The larger the discrepancy the more likely you will reject the nullhypothis, increase the sample size and with the same discrepancy you will be even more likely to reject it.
Hope that helps.