Chi-square Basics

The Chi-square distribution Positively skewed but becomes symmetrical with increasing degrees of freedom Mean = k where k = degrees of freedom Variance = 2k Assuming a normally distributed dataset and sampling a single z2 value at a time 2(1) = z2 If more than one… 2(N) =

Why used? Chi-square analysis is primarily used to deal with categorical (frequency) data We measure the “goodness of fit” between our observed outcome and the expected outcome for some variable With two variables, we test in particular whether they are independent of one another using the same basic approach.

One-dimensional Suppose we want to know how people in a particular area will vote in general and go around asking them. How will we go about seeing what’s really going on? Republican Democrat Other 20 30 10

Hypothesis: Dems should win district Solution: chi-square analysis to determine if our outcome is different from what would be expected if there was no preference

Plug in to formula 20 30 10 Observed Expected Republican Democrat Other Observed 20 30 10 Expected

Reject H0 The district will probably vote democratic However…

Conclusion Note that all we really can conclude is that our data is different from the expected outcome given a situation Although it would appear that the district will vote democratic, really we can only conclude they were not responding by chance Regardless of the position of the frequencies we’d have come up with the same result In other words, it is a non-directional test regardless of the prediction

More complex What do stats kids do with their free time? 30 40 20 10 TV Nap Worry Stare at Ceiling Males 30 40 20 10 Females

Example for males TV: (100*50)/200 = 25 Is there a relationship between gender and what the stats kids do with their free time? Expected = (Ri*Cj)/N Example for males TV: (100*50)/200 = 25 TV Nap Worry Stare at Ceiling Total Males 30 40 20 10 100 Females 50 70 60 200

df = (R-1)(C-1) R = number of rows C = number of columns 30 (25) TV Nap Worry Stare at Ceiling Total Males (E) 30 (25) 40 (35) 20 (30) 10 (10) 100 Females (E) 20 (25) 30 (35) 40 (30) 50 70 60 20 200

Interpretation Reject H0, there is some relationship between gender and how stats students spend their free time

Other Important point about the non-directional nature of the test, the chi-square test by itself cannot speak to specific hypotheses about the way the results would come out Not useful for ordinal data because of this

Assumptions Normality Inclusion of non-occurences Independence Rule of thumb is that we need at least 5 for our expected frequencies value Inclusion of non-occurences Must include all responses, not just those positive ones Independence Not that the variables are independent or related (that’s what the test can be used for), but rather as with our t-tests, the observations (data points) don’t have any bearing on one another. To help with the last two, make sure that your N equals the total number of people who responded

Measures of Association Contingency coefficient Phi Cramer’s Phi Odds Ratios Kappa These were discussed in 5700