Lecture 20Current LectureSection 13: Interpreting NHSTs cont.Put Information Together With Bayes Theorem- --P(Null|Sig.) = 1 - p(Alt.|sig.)oOnce you know the probability that the alternative hypothesis is true, it is easy to figure out the probability that the null hypothesis is trueoWe can make statements about both hypotheses because we have done the work of defining what we expect under both hypotheses-ExampleoResearchers are testing whether sitting for long periods of time increases the risk of heart disease later in life. They find a significant correlation between sitting and heart disease with alpha = .05. Given the size of the correlation they expected, they had a power of .65 to detect a significant correlation. If the prior probability of a correlation between these variables was .5, what is the probability that the null and alternative hypotheses are true after observing the significant result? Sec. 14: Chi-Squared Test Psych 240 1st EditionChi-Squared Test-A chi-squared (x2) test is a type of null hypothesis significance test-It is used to try to find evidence that two qualitative variables are related-Specifically used to find evidence that he proportion of scores in each response category on one variable are different based on the value of the other variable-The standard way to display the relationship between two qualitative variables is with a contingency table-A contingency table shows the number of scores at each combination of the levels of multiple variables-Null Hypothesis - the two variables are independent at the population leveloThis means that the proportion of scored at each level of one variable is the same across all levels of the other variable-Alternative Hypothesis - the two variables are related at the population level-Example:oImagine that we sampled 3000 people before flu season and randomly assign hem to either get a flu vaccine, start taking vitamins, or do nothing (control). We track them over the flu season and score whether or not each person go the flu.oIf there is no relationship between the two variables at the population level, then the proportion of people who do and do not get the flu will be the same for thecontrol, vaccine, and vitamin group-oIf there is a relationship between the two variables at the population level, then the proportion of people who do and do not get the flu will be different for the control, vaccine, and vitamin groups-oIs this evidence that the two variables (group and outcome) are related? Psych 240 1st Edition--To answer this question we define the results we would expect to see if the variables were independent, and we measure how much the actual resultsdeviate from this expectation-The expected frequencies are generated by getting as close as we can to the observed frequencies without allowing for any relationship between the variables-We measure how much the observed and expected frequencies deviate from one another with a statistic called chi-squared (x2):Do not have to apply this formula-If we get a significant result from this example, what can we conclude about the relationship between flu outcome (got it or not) and group (do nothing, get a vaccine, or take vitamins)?The results provide evidence that flu outcome and group are related at the population level Psych 240 1st EditionoIf our sample x2 is atypically high relative to the distribution we expect for variables with no relationship, then we get a significant result-A significant result means that our sample provides evidence that the variables are related in the populationoIf our sample x2 is fairly typical for the distribution we expected for variables with no relationship, then we get a non-significant result-A non-significant result means that our sample failed to provide evidence that the variables are related in the population Chi-Square Distributions-If we take lots of random samples from a population with independent variables and compute x2 for each one, we get a x2 distribution--Like the other tests, we set a critical value to ensure we have a low chance (ex. .05) of finding a significant result if the null hypothesis is true-Is this result significant or non-significant? ooSignificant Test for Independence Psych 240 1st Edition-Unless you randomly assign one of the variables, a chi-squared test is like a correlation-Just because the variables are related, doesn’t mean that one is causing the other-Other examples of chi-squared tests:oIs hair color related to whether or not you have freckles?oDoes the prevalence of diabetes vary by gender?oDoes employment status for young adults vary based on whether or not they have a college degree?oDo judges' decisions to grant or deny parole vary from the hour before lunch to the hour afterward? Objectives-Know what a contingency table is and what the values in a contingency table represent. -Be able to recognize research scenarios in which a chi-squared test should be used. -Understand what the null and alternative hypotheses mean for a chi-square test for independence. -Know what it means when a chi-square test produces a significant versus a non-significant result. -Given a research scenario and the outcome of a chi-squared test, be able to recognize appropriate conclusions. oo Psych 240 1st EditionooNHSTs are set up to support the alternative hypothesis, so a significant result usually provides better support for the alternative than a non-significant result provides for the nulloIn the last problem, a significant result meant that the alternative had a 93% chance of being true and the null had a 7% chance. A non-significant result meant that the alternative had a 30% chance of being true and the null had a 70% chance What Affects Conclusions-Alpha does affect the probability that the alternative hypothesis is true if you get a significant result?-The examples in this section assume the "average" psychology study:oMedium effect sizeoNon-directions hypothesisoAlpha=.05oN=40 subjects per group Psych 240 1st Edition--Significant results provide stronger support for the alternative hypothesis if alpha is loweroAlpha has to be set before the data are collected-Decreasing alpha increases the specificity of evidence for the alternative hypothesis: p(Sig.|Null) decreasesoBut it also decreases the sensitivity: p(Sig.|Alt.) decreases-The change in specificity is the most important. That is why significant results give bettersupport for the alternative hypothesis with lower alpha