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.

Statistical Independence

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 * 100300 = 70

70 is the expected frequency. Because the observed and expected frequencies are the same, the

variables are independent.

150 * 150300 = 75

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 (χ2).

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

➲ The null hypothesis is statistical

independence between X and Y.

➲ H0: X, Y Independent

➲ The alternative hypothesis is X and Y

are not independent.

➲ HA: X, Y Dependent

Chi-Square Test

➲ 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)

➲ Level of significance: The probability

(α) of incorrectly rejecting a true null

hypothesis.

Chi-Square Test

➲ 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.➲ (JRM: Appendix C, pg. 577)

Chi-Square Test

➲ The observed chi-2 is the sum of the squared differences between observed and expected frequencies divided by the expected frequency.

➲ If χ2obs

≥ χ2crit.,

reject null hypothesis. Otherwise, do not reject.

Chi-Square Test

➲ Let's assume we want to test the

relationship at the .01 level.

➲ The observed χ2 is 62.21.➲ The degrees of freedom is (5-1)(2-1) = 4.➲ The critical χ2 is 13.28.➲ Since 62.21 > 13.28, we can reject the null of

an independent relationship.➲ Y (attitudes toward gun control) is dependent

on X (gender).

Chi-Square Test

➲ The χ2 statistic works for dependent variables that are ordinal or nominal measures, but another statistic is more appropriate for interval- and ratio-level data.

Analysis of Variance

For quantitative (or interval- and ratio-level ) data the analysis of variance is appropriate.

Analysis of variance aka ANOVA.

The independent variable, however, is generally still nominal or ordinal.

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

Percent explained = (between/total) X 100

Ex: Percent explained = (0/total) = 0

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 (η2).

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 η2 ) = .6