There are two common chi-square tests: the goodness-of-fit test and the test for independence. The goodness-of-fit test is used when there are categorical data on one variable, as we had in the soft drink preference example. Perhaps a researcher is interested in the relationship between two categorical variables. In this case, you might see the chi-square test for independence. Imagine that our researcher has asked cola tasters to indicate their choice of cola and has also categorized them by age. The research hypothesis might be that preference for cola depends on age. The researcher might think that younger people prefer Pepsi, for example, and older people prefer Coke. Or perhaps older people have no preference. The chi-square statistic is the same for this test as for the goodness-of-fit test. The difference is in the hypothesis. The null is that the two variables are independent (i.e., there is no relationship between them). In a research article, you will likely see a table of frequencies (or percentages), a statement as to whether a relationship was found between the variables, and the chi-square statistic and p value.
Conceptual Exercise 2B
For each of the following, decide whether a t test, an F test, or a chi-square test might be appropriate:
1 A new teacher decides to put some of the principles he learned in school to the test. He randomly selects half of his class and consistently praises each student for being on the task for a minimum period of time. With the other half of the class, he periodically gives praise for on-task behavior. He wants to know if periodic praise produces more on-task behavior than consistent praise.
2 Psychiatric walk-in clients are randomly assigned to five therapists for short-term counseling. One therapist specializes in psychoanalytic techniques, one in client-centered techniques, one in behavioral techniques, and one in cognitive techniques. The fifth therapist is eclectic, using techniques from each of the therapies. All clients are rated on various scales designed to measure improvement. The researcher compares mean improvement ratings of the clients for each therapist.
3 A statistics professor wants to know if generally there are more or less equal numbers of psychology, sociology, and business students in her classes. She keeps a tally.
Other Nonparametric Tests.
In addition to chi-square, there are numerous other nonparametric tests that you will see in the literature. We have not tried to present a complete list here; instead, we have included the more common tests.
A nonparametric alternative to a t test for independent groups is the Mann-Whitney U test, which detects differences in central tendency and differences in the entire distributions of rank-ordered data. The Wilcoxon signed-ranks test is an alternative to a t test for dependent groups for rank-ordered data on the same or matched participants.
A nonparametric alternative to the one-way ANOVA is the Kruskal–Wallis H test, used when the data are rank orders of three or more independent groups. When those groups are dependent (i.e., repeated measures), a nonparametric test is the Friedman test.
Pearson’s r Test.
If you earned a lot of money, would you be happy? Is there a relationship between income and happiness? If a researcher were interested in investigating a linear relationship between two continuous variables, he or she would use the Pearson product–moment test to calculate the correlation r. If you are getting a sense of déjà vu, it is probably because we talked about r as a descriptive statistic, but here we are talking about it as an inferential statistic. The important distinction is that the r reported as an inferential statistic will have an associated p value. For example, in a research article, you will read that a positive relationship was found between a measure of need for achievement and years of education and that the relationship was statistically significant. If the relationship was statistically significant, you will also see a p value reported.
Regression.
Regression is related to correlation, but in regression, we are interested in using a predictor variable to predict a criterion variable. Continuing with the example of need for achievement and education, perhaps the researcher was also interested in predicting the need for achievement from education level. If the correlation between the two variables is statistically significant, it is a simple matter of fitting a line through the data and using the equation for the line to predict need for achievement from education level. We say “simple matter” because the calculations are all done by computer, but, certainly, the equation for a straight line is simple:
Y = mX + b
where Y is the criterion variable, X is the predictor variable, m is the slope of the line, and b is the value of Y where the line intercepts the y-axis. Be sure to keep in mind as you read the research that the accuracy of the predicted values will be as good as the correlation is. That is, the closer the correlation is to +1 (or −1), the better the predictions will be.
The statistical procedures we have been discussing all involve an a priori hypothesis about the nature of the population. Hypothesis testing is used a lot in psychology. Some other disciplines tend to prefer post hoc procedures, and you will find confidence interval estimates quite often in the literature you will be reading.
Confidence Intervals
Confidence intervals are used when we are interested in estimating population parameters. We are still making an inference from a sample to a population, and because of that, we are using probability estimates. But instead of reporting a p value indicating the probability that the null is true, we report the probability that our estimate about the population is true. Pollsters describing political candidates often use confidence intervals. For example, you may have read reports that, based on a poll of 1,000 respondents, 83% say they would vote for X if there were an election tomorrow. These statements are typically followed with a statement such as “These results are accurate to within 3 percentage points 19 times out of 20.” What does this mean? It means that, based on a sample of 1,000, the population support for the candidate is probably somewhere between 83% – 3% and 83% + 3%, or somewhere between 80% and 86%. Are the pollsters absolutely sure? No, they say that the estimate should be correct 19 times out of 20, or 95% of the time (19/20 = .95). So a p value of .05 from hypothesis testing becomes a confidence interval of .95, and, similarly, a p value of .01 becomes a confidence interval of .99 (reported as 99 times out of 100). Again, in hypothesis testing, we report a significance test with a p value that indicates the probability that the null is true. In confidence intervals, we report an interval within which we estimate the true population parameter to fall.
Important note to students
If you’re reading this material and starting to get anxious, relax! Our intention here is to discuss these inferential statistics at a conceptual level. As we indicated earlier, when you begin reading the literature, it is unlikely that you will see research using t tests or simple ANOVAs. What you will see are complex statistics that may be completely new to you. Our intention here is to give you enough information to understand what is being described in the literature.
More Complex Statistical Procedures
Multiple Regression.
If predicting someone’s performance using one predictor variable is a good idea, using more than one predictor variable is a better idea. Entire textbooks are devoted to multiple regression analysis techniques, but the basic idea is to use more than one predictor variable, X1, X2, X3, and so on, to predict one criterion variable, Y. As with simple regression, multiple regression requires the fitting of a line through