Nonparametric Tests. Chi-Square Test for Independence

Transcription

1 DDBA 8438: Nonparametric Statistics: The Chi-Square Test Video Podcast Transcript JENNIFER ANN MORROW: Welcome to "Nonparametric Statistics: The Chi-Square Test." My name is Dr. Jennifer Ann Morrow. In today's demonstration, I will review with you nonparametric tests, specifically the chi-square test for independence. I will give you two sample research questions that can be addressed using the chisquare test for independence. I will define what observed and expected frequencies are. I will go over the basic formula for a chisquare test for independence. And I will go over the assumptions of this analysis. And lastly, I will give you an example both using the formula and SPSS. Okay, let's get started. Nonparametric Tests JENNIFER ANN MORROW: Nonparametric tests are statistical tests to use when the tests have not met the assumptions of a parametric test, such as a Pearson correlation. These type of tests are wellsuited to nominal and ordinal data. They are also known as distribution-free tests. They can be used for distributions that are not normally distributed. And lastly, nonparametric tests are not as sensitive as parametric tests. In other words, they're not as powerful, so if you have a choice, choose parametric tests. Chi-Square Test for Independence JENNIFER ANN MORROW: The chi-square test for independence is used when you have two independent nominal variables. It tests to see if the values of one variable are related to or dependent on the values of the second variable. This test calculates the difference between observed and expected frequencies. Sample Research Questions JENNIFER ANN MORROW: Some examples of research questions that can be addressed using chi-square tests for independence are as follows. Question one: Does intention to persist depend on learning community status, where one variable is intention to persist,

2 yes and no, and the other variable is learning community status, yes or no? Question two: Is there a relationship between Greek status and marijuana use? One variable is Greek status, yes or no, and the other variable is marijuana use-- low, moderate, and high. Observed and Expected Frequencies JENNIFER ANN MORROW: Let me tell you about the difference between observed and expected frequencies. Observed frequencies are the frequencies in your sample distribution. These are the frequencies that your participants are giving you. Your expected frequencies are those that are based on theory or prior research or speculation, and sometimes your expected frequencies are going to be equal in each category. Here's an example of a chi-square table. This is also known as a crosstabs table. By looking at this table, you can decipher that there are two variables. The first variable is group, and you have two levels, experimental and control, and those are your rows. The second variable is graduation, and you have two levels, graduated and failed to graduate, and those are represented in the columns. In each cell of the table is the observed frequency. This is the number of participants that fall in each of these categories. For example, there are 73 participants in the experimental group that graduated and 43 of the participants in the control group graduated and so forth. Basic Formula JENNIFER ANN MORROW: The basic formula for a chi-square test for independence is as follows. Your chi-square value is equal to the sum of the observed frequencies minus the expected frequencies squared divided by the expected frequencies. And your degrees of freedom for the chi-square test of independence is C minus 1 times R minus 1, where C is the number of columns and R is the number of rows. Assumptions JENNIFER ANN MORROW: There are two main assumptions that must be met in order to be justified in using a chi-square test for independence. The first is that observations must be independent.

3 No two scores should be related to each other. Lastly, your expected frequencies in each cell must be greater than 5. If you violate either of these assumptions, you cannot use a chi-square test for independence. Examples JENNIFER ANN MORROW: Let's go over an example using the formula. My research question is: Is there a relationship between stress and exam performance? My first variable is exam performance, with two groups: low and high. My second variable is stress, with three groups: low, moderate, and high. My null hypothesis is that there is no relationship between the variables, and my alternative or research hypothesis is that there is a relationship between these variables. Now let's do the example. So here, for my chi-square test for independence, let me draw my crosstabs table... where I have high, moderate, and low, and that is my stress variable. And my exam performance, again, is high and low. And my observed frequencies are as follows: 17 in the high-high cell, 13 in the high-low cell, 32 in the moderate-high cell, 43 in the moderate-low cell, 11 in the low-high cell, and 34 in the low-low cell. Now I'm going to plug in my expected frequencies, and I'll put those in parentheses. I expect there should be 12 participants in the high-high cell, 18 participants in the high-low cell, 30 participants in the moderate-low cell, and 45 participants in the moderate-low cell, 18 participants in the low-high cell, and 27 participants in the low-low cell. The next thing you should do is total the values, your observed frequencies, for each column and row. So if I add up my observed frequencies for this top row, that would be 60. Here, it's 90. Total combined is 150. That's the number of participants for this analysis. This column, the total is 30. This column, the total is 75. And this column, the total is 45. Now I'm going to set my criteria. I'm going to choose an alpha level of 0.05, and I'm going to choose a two-tailed test. My degrees of freedom, again, is equal to C minus 1 times R minus 1, which, in this case, is equal to 2 minus 1 times 2 minus 1. I'm sorry, 3 minus 1 times 2 minus 1, which is equal to 2. So when I look up in the chi-square distribution table, which can be found in the appendix of your textbook, for 2 degrees of freedom, an alpha level of 0.05, and two-tailed, the critical value that I need to surpass is So now let's compute the chi-square test. Again, the formula for a chi-square test for independence is equal to the sum of the observed frequencies minus the sum of the expected frequencies squared divided by the expected frequencies. So for the

4 first cell, I have 17 is my observed frequency minus 12 is my expected frequency, and I square that over the expected frequency, again 12. And I do this for each cell of this crosstab table. So my next one is 32 minus 30 squared over 30 plus 11 minus 18 squared over 18 plus 13 minus 18 squared over 18 plus 43 minus 45 squared over 45 and, lastly, 34 minus 27 squared over 27. And I come up with a chi-square value of 8.22, which, this does surpass my critical value of So how do I interpret this? I would write it: chi-square, my degrees of freedom, which is 2, equals 8.22, comma, P less than 0.05, comma, two-tailed. And what this means is, is that there is... a relationship between stress and exam performance. Now let's do an example using SPSS. Open up your SPSS software and find the data set that you want to use. Click on File, Open, Data. Now search for the data set that you want to use. Once you have found that data set, highlight it, and click Open. And make sure your data view window appears in your screen. To conduct a chi-square test for independence, you need to click on Analyze, Descriptive Statistics, and Crosstabs. Now your crosstabs dialog box will appear on your screen. Now you have to tell SPSS the two variables you want to use for your analysis. For this analysis, I'm going to chose the variables gender and learning community status. So I go to the box here in the left and highlight my variable, gender, and click on the right arrow key to put it in the box for rows. And then I highlight my variable, learning community, and then I click on the right arrow key and highlight that to put it in the box for column. Now, it does not matter which variable goes in rows or which variable goes in columns. You will get the same result. Now I need to click on Statistics, and you should check off, on the top left, chi-square. Then click Okay-- I mean Continue. Now click on Cells. And you should have checked here Observed, Expected, and I always like to ask for the percentages as well, so I click on the Row, Column, and Total. I find those helpful. Then click on Continue. And now click on Okay. And as you can see, SPSS is going to give you a crosstab table, or a chi-square table, for the variables gender and learning community. As you can see here, you have 46 participants that are female that are not in the learning community and 53 participants that are female that are in a learning community, 19 participants that are male that are not in a learning community, and 32 participants that are male that are in a learning community. And SPSS is also going to give you the expected counts for all of those, as well as the percentages within each category. Now let's scroll down a bit further. The next table will have your chi-square statistic test. You look here at the top where it says Pearson chisquare. You have your chi-square value of You have 1 degree

5 of freedom. Again, remember, your degrees of freedom is C minus 1, R minus 1. So your degrees of freedom is 1, and your significance for this statistic is 0.281, so it is not significant. So how do I interpret this? I would have chi-square and 1 degree of freedom equals 1.16 NS, two-tailed. What this is saying is that there is no relationship-- excuse me-- between gender and learning community status. Recap JENNIFER ANN MORROW: All right, let's recap. In this demonstration, we learned about nonparametric tests. Specifically, we learned about the chi-square test for independence. We went over some sample research questions that can be addressed using this statistic. We talked about observed and expected frequencies. We went over the basic formula for this analysis. We talked about the two assumptions. And we went over an example using the formula and SPSS. We have now come to the end of this demonstration. Don't forget to practice what you have learned today. Practice calculating chi-square tests for independence using the formula as well as using SPSS. Thank you, and have a great day.