Cross-Tabs Continued Andrew Martin PS 372 University of Kentucky
Statistical Independence Statistical independence is a property of two variables in which the probability that an observation is in a particular category of one variable and a particular category of the other variable equals the simple or marginal probability of being in those categories. Contrary to other statistical measures discussed in class, statistical independence indicators test for a lack of a relationship between two variables.
Statistical Independence Let us assume two nominal variables, X and Y. The values for these variables are as follows: X: a, b, c, ... Y: r, s, t , ...
Statistical Independence P(X= a ) stands for the probability a randomly selected case has property or value a on variable X. P(Y=r) stands for the probability a randomly selected case has property or value r on variable Y P(X=a, Y=r) stands for the joint probability that a randomly selected observation has both property a and property r simultaneously.
Statistical Independence If X and Y are statistically independent: P(X= a , Y= r ) = [P(X= a )][P(Y= r )] for all a and r .
If gender and turnout are independent: Total obs in column m * Total obs in row v N = mv
Statistical Independence Total obs in column m * Total obs in row v N = mv 210 * 100 300 = 70 70 is the expected frequency. Because the observed and expected frequencies are the same, the variables are independent.
Here, the relationship is not independent (or dependent) because 75 (expected frequency) is less than 100 (observed frequency).
Testing for Independence How do we test for independence for an entire cross-tabulation table? A statistic used to test the statistical significance of a relationship in a cross-tabulation table is a chi-square test (χ 2 ).
Chi-Square Statistic The chi-square statistic essentially compares an observed result—the table produced by the data—with a hypothetical table that would occur if, in the population, the variables were statistically independent.
How is the chi-square statistic calculated? The chi-square test is set up just like a hypothesis test. The observed chi-square value is compared to the critical value for a certain critical region. A statistic is calculated for each cell of the cross-tabulation and is similar to the independence statistic.
How is the chi-square statistic calculated? (Observed frequency – expected frequency) 2
Chi-Square Test <ul><li>The null hypothesis is statistical independence between X and Y. </li></ul><ul><li>H 0 : X, Y Independent </li></ul><ul><li>The alternative hypothesis is X and Y are not independent. </li></ul><ul><li>H A : X, Y Dependent </li></ul>
Chi-Square Test <ul><li>The chi-square is a family of distributions, each of which depends on degrees of freedom. The degrees of freedom equals the number of rows minus one times the number of columns minus one. (r-1)(c-1) </li></ul><ul><li>Level of significance: The probability ( α ) of incorrectly rejecting a true null hypothesis. </li></ul>
Chi-Square Test <ul><li>Critical value: The chi-square test is always a one-tail test. Choose the critical value of chi-square from a tabulation to make the critical region (the region of rejection) equal to α. </li></ul><ul><li>(JRM: Appendix C, pg. 577) </li></ul>
Chi-Square Test <ul><li>The observed chi-2 is the sum of the squared differences between observed and expected frequencies divided by the expected frequency. </li></ul><ul><li>If χ 2 obs ≥ χ 2 crit. , reject null hypothesis. Otherwise, do not reject. </li></ul>
Chi-Square Test <ul><li>Let's assume we want to test the relationship at the .01 level. </li></ul><ul><li>The observed χ 2 is 62.21. </li></ul><ul><li>The degrees of freedom is (5-1)(2-1) = 4. </li></ul><ul><li>The critical χ 2 is 13.28. </li></ul><ul><li>Since 62.21 > 13.28, we can reject the null of an independent relationship. </li></ul><ul><li>Y (attitudes toward gun control) is dependent on X (gender). </li></ul>
Chi-Square Test <ul><li>The χ 2 statistic works for dependent variables that are ordinal or nominal measures, but another statistic is more appropriate for interval- and ratio-level data. </li></ul>
Analysis of Variance <ul><ul><li>For quantitative (or interval- and ratio-level ) data the analysis of variance is appropriate. </li></ul></ul><ul><ul><li>Analysis of variance aka ANOVA. </li></ul></ul><ul><ul><li>The independent variable, however, is generally still nominal or ordinal. </li></ul></ul>
ANOVA tells political scientists ... (1) if there are any differences among the means (2) which specific means differ and by how much (3) whether the observed differences in Y could have arisen by chance or whether they reflect real variation among the categories or groups in X
Two important concepts Effect size —The difference between one mean and the other. Difference of means test —The larger the difference of means, the more likely the difference is not due to chance and is instead due to a relationship between the independent and dependent variables.
Setting up an Example Suppose you want to test the effect of negative political ads on intention of voting in the next election. You set up a control group and a test group. Each group watches a newscast, but the test group watches negative TV ads during the commercial breaks. The control group watches a newscast without a campaign ad. You create a pre- and post-test of both groups to compare the effects of both ads.
Difference of the means Effect = Mean (test group) – Mean (control group)
Difference of the means Although different statistics use different formulas, each means test has two identical properties: (1) The numerator indicates the difference of the means (2) The denominator indicates the standard error of the difference of the means.
Difference of the means A means test will compare the means of two different samples. The larger the N for both samples, the greater confidence that the observed difference in the sample (D) will correctly estimate the population difference ( ∆ ).
Difference of the means In abstract terms: Mean of test group – mean of control group Std. Error (test) + Std. error (control) In concrete terms: Mean (Ads) – Mean (No ads) Std. Error (Ads) + Std. error (No ads)
Hypothesis Test of Means Difference of the means tests the null hypothesis that there is no difference between the means. You can basically test the significance of a means difference by employing a hypothesis test or calculating a confidence interval.
Small-Sample Test of DoM Let's suppose we are looking at a two samples: both measure the level of democracy in a country, but one is a sample of developed countries and another is a sample of developing countries. We want to test whether the level of economic development impacts the level of openness in a democracy. Specifically, we want to test whether the population differences are 0.
Small-Sample Test of DoM Two small samples, so we must use the t distribution and calculate degrees of freedom . When there are two samples, degrees of freedom equals N 1 (first sample) + N 2 (second sample) – 2.
The standard error for the difference of means is .144.
ANOVA ANOVA or analysis of variance allows us to expand on previous methods. This procedure treats the observations in categories of the explanatory or independent variable as independent samples from populations. This makes it possible to test hypotheses such as H0 = μ1 = μ2 =μ3
Variation Three types Total variation —A quantitative measure of the variation in a variable, determined by summing the squared deviation of each observation from the mean.
Variation Explained variation —That portion of the total variation in a dependent variable explained by the variance in the independent variable. Unexplained variation —That portion of the total variation in a dependent variable that is not accounted for by the variation in the independent variable(s).
ANOVA Total variance = Within variance + Between variance Within variance is the unexplained variance Between variance is the explained variance
ANOVA Explained variance refers to the fact that some of the observed differences seems to be due to “membership in” or “having a property of” one category of X. On average, the A's differ fro the B's. Knowing this characteristics can help us tell the value of Y.
ANOVA ANOVA involves quantifying the types of variation and using the numbers to make inferences. The standard measure of variation is the sum of squares, which is a total of squared deviations about the mean.
ANOVA TSS = BSS + WSS TSS = Total sum of squares BSS =Between mean variability WSS = Within group variability Percent explained – (BSS/WSS) X 100
ANOVA The percent of variation explained is called eta-squared (η 2 ). It varies between 0 and 1 like any proportion. 0 means the independent variable explains nothing about the dependent variable. 1 means the independent variable explains all variation in the dependent variable.
ANOVA Often this statistic is explained as follows: X explains 60 percent of the variation in Y and hence is an important explanatory factor. (if η 2 ) = .6