Chapter 11: Chi-square (χ 2 )

Transcription

1 Chapter 11: Chi-square (χ 2 ) *This chapter corresponds with Chapter 16 in your text ( What to do when you re not normal ). What it is: Chi-square is a nonparametric statistic. This means that it can be used with data that do not meet assumptions required to run other analyses. When to use it: Unlike other analyses, the chi-square is not in the flow chart on pg. 173 of your text, but should be used with categorical outcome variables (such as yes/no or graduated/did not graduate). Chi-square tests do not require: 1) a normal distribution that is often present only in large samples, 2) equal variance between groups, or 3) continuous outcome variables. Questions asked by the one-sample chi-square: Is the number of respondents equally distributed across all levels of an outcome? Examples of research questions that would use chi-square: Are people more or less likely to prefer Coke than Pepsi? Are women versus men equally likely to become medical doctors? *te: You can use another form of the chi-square to ask if the number of respondents in each level of an outcome is equally distributed over the levels of a second variable. For example, if the number of men and women are equally distributed across several professions, such as doctors and nurses. However, we ll focus only on the one-sample chi-square in this class. Using SPSS to Compute Chi-Square (dataset: ch11 example 1.sav) A study examined helping behavior among students at Texas A&M. A confederate (someone working with the researchers) walked across campus and dropped her notebook near a student standing by themselves. Whether or not the student picked up the notebook and returned it to the student was recorded. The research question was: what are students on Texas A&M campus more likely to do: not help a student or help a student by returning a dropped notebook? And now for the famous 8 steps! 1. Statement of the null and research hypothesis: Null H 0 : P helped = P didnothelp (P = percentage of occurrences in each category) The percentage of people who helped is equal to the percentage of people who did not help Research H 1 : P helped P didnothelp The percentage of people who helped is not equal to the percentage of people who did not help

2 2. Set the level of risk at p < Selection of the appropriate test statistic You are testing the proportions of students in each category and must therefore use the chi-square (the chi-square is not in the flow chart in your text) 4. Compute the test statistic value Open the dataset titled ch11 example1.sav. Take a moment to familiarize yourself with the data. te that the data is entered with one participant per row and a single column that includes the categorical variable (1 = helped or 2 = did not). This tells you, for example, that subject 1 did help but subject 2 did not. If you switch to variable view, you should see that the variable called helped has a label indicating that it represents did the student pick up the notebook? You should also see that it has two possible values: 1 = no and 2 = yes. w it s time to run the chi-square. On the toolbar, click on the Analyze drop-down menu, highlight nparametric tests, then click on Chi-Square, as shown:

3 The following pop-up window will appear: Highlight the helped variable and click the arrow to place it in the test variable list box, as shown in the picture below.

4 te that there is an area for expected values where you can choose to assume that the proportion of people in each category should be equal (e.g., the number of people who helped should be equal to the number of people who did not help) or you can set the values to some other distribution (e.g. 80% of people will help, 20% will not). This determination is based on your research question, theory, or previous research. In this example, we want to know whether people are more likely to help or not help. This means we want to test whether our observed frequencies are different from an expected frequency that is equal between the two groups (e.g. people are equally likely to help or not help). Thus, we ll use the default setting and assume that the proportion in each category should be equal. Hit ok and an output file should open with your chi-square test. w navigate to the output window to find your results, which should look like those shown on the next page.

5 Chi-Square Test Frequencies did the student pick up the notebook? Observed N Expected N Residual The observed N tells you that 31 students picked up the notebook (yes) and 14 did not (no) no yes Total 45 Test Statistics The expected N tells you that, if people were equally likely to help versus not help, 22.5 people out of the 45 we tested would have helped and 22.5 people would not have helped. The residual gives you the difference between the expected and observed Ns. This is O-E when we calculate a chi-square by hand. did the student pick up the notebook? Chi-Square a df 1 Asymp. Sig..011 This box contains the information about the chi-square statistic. It shows that the chi-square is 6.42, there was 1 degree of freedom (remember this is r-1, where r = the number of rows), and the p-value is.011 (called Asymp. Sig. in this table). Interpreting the Output The top table gives you a summary of what happened and what was expected to happen in the study. This information is similar to some of the hand calculations for a chi-square and tells you how many people fell into each category. The bottom table gives you the test statistic and associated p-value. These will be used when determining whether or not to reject the null hypothesis. The chi-square is the same one that you would get if you calculated a chi-square by hand for this data. The df row tells you that the degrees of freedom were 1. For a chi-square, r 1 will give you the degrees of freedom, where r is the number of rows in your table (the top table in the output). In this example, we have two rows: no or yes, so our degrees of freedom are 2-1, or df = 1. The Asymp. Sig. gives us the precise p-value associated with the obtained value of chi-square: Determination of the value needed for rejection of the null hypothesis If you were calculating by hand, you would check the critical values table on pg. 342 of your text and see that the critical value at df = 1 and p <.05 is Remember that the

6 critical value gives you the smallest value of chi-square needed to reject the null hypothesis. The obtained value has to be larger than 3.84 to be significant at p <.05. In SPSS, you can look at the p-value and degrees of freedom in the table so you can skip this step. 6. A comparison of the obtained value and the critical value is made SPSS does some of this for you. You should get output that looks like the example above. As shown in the output, the obtained value is 6.42, with 1 degree of freedom, and the significance is p =.011 (this is less than the level of risk set at p <.05). 7 & 8. Decision time! The obtained value of 6.42 is greater than the critical value at 1 degree of freedom with a p-value of.011. This means there is only a 1.1% chance that you would get this pattern of results if the null hypothesis were true and students were equally likely to help versus not help. Therefore the obtained value does exceed the critical value, we reject the null hypothesis and conclude that the proportion of students who helped versus did not help was not equal. Interpretation of the Findings w you report your results. Here s an example of how these results would be reported in a journal article: There was a significant difference in the proportion of students willing to help another student by returning a dropped notebook, χ 2 (1) = 6.42, p <.05. More students returned the notebook (69%) than did not return the notebook (31%). (*te: To get the percentages, just divide the number of students in each category by the total number of students). For someone unfamiliar with stats, you might say: More students helped another student by returning a dropped notebook than did not help.

7 Practice Problem #1 for SPSS (answer in Appendix) A researcher was interested in whether depression, anxiety, or other disorders were most common among college students. Students attending counseling sessions reported the primary symptoms that brought them in. Use SPSS and the data below to conduct the appropriate analysis. Primary Symptoms Depression Depression Other disorder A. What is the null and research hypothesis (in both words and statistical format)? B. What is the level of risk associated with the null hypothesis? C. What is the appropriate test statistic and WHY? D. What is the obtained value for the chi-square and what is its associated p-value? E. What do you conclude about the rates of disorders among college students? Write up your results as you would for a journal article. F. Write up your results as you would for an intelligent person who doesn t know stats.

8 Practice Problem #2 for Hand Calculation (answer in Appendix) In a study of people s tendency to obey authority figures, people were approached on the subway by an official-looking person in uniform and asked to get out of their seat (no explanation was offered). Whether or not each person stood up was recorded by an observer. Hand calculate the analysis using the data below to determine whether people were more likely to obey or not. Left seat? A. What is the null and research hypothesis (in both words and statistical format)? B. What is the level of risk associated with the null hypothesis? C. What is the appropriate test statistic and WHY? D. What is the value needed for rejection of the null hypothesis (i.e., the critical value)? E. What is the obtained value for the chi-square and what is its associated p-value? F. What do you conclude about whether or not people gave up their seats? Write up your results as you would for a journal article. G. Write up your results as you would for an intelligent person who doesn t know stats.

9 Practice Problem #3 for Hand Calculation and SPSS (answer in Appendix) A human resource manager is interested in whether most employees take an early retirement option offered by the company. Hand calculate the analysis using the data below, then use SPSS to check your answer. Took early retirement? A. What is the null and research hypothesis (in both words and statistical format)? B. What is the level of risk associated with the null hypothesis? C. What is the appropriate test statistic and WHY? D. What is the value needed for rejection of the null hypothesis (i.e., the critical value)? E. What is the obtained value for the chi-square and what is its associated p-value? F. What do you conclude about whether or not people take the early retirement option? Write up your results as you would for a journal article. G. Write up your results as you would for an intelligent person who doesn t know stats.