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## The chi-square statistic

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**The chi-square statistic**Chapter 16**The tests we’ve used so far…**• All involve one quantitative variable • One sample t-test: compare mean of sample on a quantitative variable to some meaningful value • Independent samples t-test: compare means of two groups on a quantitative variable • Paired samples t-test: compare pairs of scores on a quantitative variable**What if you don’t have a quantitative variable?**• For instance, if you want to know if more males or females volunteer to participate in psychology studies • Or if you want to know if the percentage of males who prefer blue to pink is different from the percentage of females who prefer blue to pink • No quantitative variable • can’t estimate parameter in the population • need to use a non-parametric statistic**Two different situations**• One qualitative variable, and want to compare which group people are most likely to belong to • Two qualitative variables, and want to know whether the percentages in each category for one group of people differ from the percentages in each category for another group of people**The first situation**• Are people more likely to belong to one group than another? • Null hypothesis = nothing’s going on – people should be equally likely to belong to each group or category • Question: How well does this model fit the data • Chi-square test for goodness of fit**How to test this**• Like all statistical tests, compare the data that you have with what you would have if the null hypothesis was true • Data you have = observed frequencies for each group • What you expect to see if the null were true = expected frequencies for each group • Larger the difference between these, the bigger the chi square statistic, and the smaller the p value**Quantifying this**• For each category: • 1. count the number of cases observed in that category • 2. subtract the number of cases you would expect to see in that category if the null hypothesis were true • 3. square this difference • 4. divide by the number of cases you would expect to see in that category if the null hypothesis were true • 5. add up across categories • If the null hypothesis is true, this should be close to 0**How big is big enough?**• Compare calculated chi square value to critical chi square value • If calculated value is larger, reject the null, since p is smaller than alpha • To get critical chi square, need df • df = number of categories minus 1**Telling the world**• Similar format to t • c2 (df, n = x) = calculated chi square value, p information**Another situation for the goodness of fit test**• If, based on prior research and theory, you have reason to expect that the distribution of people across categories will follow a particular pattern, you can use these values as the expected values • E.g., if there are 60% females in the population, it wouldn’t make sense to compare the distribution of genders volunteering for a study to a 50-50 breakdown**The second type of chi square**• You have different types of participants (e.g., males/females, people with different majors) • You want to know whether which type of participant they are is independent of some other variable • E.g., is gender independent of color preference? • chi square test for independence**Null hypothesis**• Nothing’s going on • the two variables are independent of each other • if, overall, 25% of people like pink, the people who like pink should be equally likely to be male and female • the breakdown of categories for the entire sample should be the same as the breakdown of categories for sub-groups of the sample**Calculating the test for independence**• Exactly the same formula as the test for goodness of fit • For each category: • 1. observed frequency minus expected frequency, squared • 2. divided by expected frequency • 3. add up across categories**How big is big enough?**• Need critical chi square value • df = df one variable * df other variable • df = (number categories – 1) * (number categories – 1)**Effect size still matters**• A chi square could be significant, or not significant, due to sample size • need to know how much of something is going on • phi: square root of (chi squared divided by number of participants)