Week 12 Chapter 13 – Association between variables measured at the ordinal level & Chapter 14: Association Between Variables Measured at the Interval-Ratio

Week 12

Chapter 13 – Association between variables measured at the ordinal level

& Chapter 14: Association Between Variables

Measured at the Interval-Ratio Level

Chapter 13

Association Between Variables Measured at the Ordinal Level

This Presentation

Two Types of Ordinal Variables Gamma Spearman’s Rho Hypothesis Tests for Gamma and Rho

Two Types of Ordinal Variables

1. Continuous ordinal variables: Have many possible scores Resemble interval-ratio level variables Use Spearman’s Rho: rs

Example: a scale measuring attitudes toward handgun control with scores ranging from 0 to 20

2. Collapsed ordinal variables: Have just a few values or scores Use Gamma: G

Can also use Somer’s d and Kendall’s tau-b (see text website)

Example: social class measured as lower, middle, upper

Gamma

Gamma is used to measure the strength and direction of the relationship between two ordinal level variables that have been arrayed in a bivariate table

Before computing and interpreting Gamma, it will always be useful to find and interpret the column percentages

Gamma

Interpretation: Use the table below as a guide to interpret the strength of

gamma in overall terms

Gamma

In addition to strength, gamma also identifies the direction of the relationship

In a negative relationship, the variables change in different directions Example: As age increases, income decreases (or, as age

decreases, income increases) In a positive relationship, the variables change in the same direction

Example: As education increases, income increases (or, as education decreases, income decreases)

Gamma

Gamma

Gamma

In addition to strength and direction, a hypothesis test of Gamma can also indicate if the two variables share a relationship in the population, or if the two variables are significantly related

Hypothesis Test of Gamma:

Step 1: Make Assumptions and Meet Test Requirements

Random sampling Ordinal level of measurement Normal sampling distribution

Gamma


Step 2: State the Null Hypothesis Ho: γ = 0

No relationship exists between the variables in the population

H1: γ ≠ 0 A relationship exists between the variables in

the population

Gamma


Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling distribution = Z distribution Set alpha (two-tailed) Look up Z(critical) in Appendix A

Gamma


Step 4: Compute the Test Statistic

ds

ds

2ds

NN

NNGwhere

)G1(N

NNG)obtained(Z

Gamma


Step 5: Make a Decision and Interpret the Results Compare Z(obtained) to Z(critical) If Z(obtained) falls in the critical region,

reject Ho

If Z(obtained) does not fall in the critical region, fail to reject Ho

Interpret results

Spearman’s Rho (rs)

Measure of association for ordinal-level variables with a broad range of different scores and few ties between cases on either variable

Computing Spearman’s Rho

1. Rank cases from high to low on each variable

2. Use ranks, not the scores, to calculate Rho





Rho is positive, therefore jogging and self-image share a positive relationship: as jogging rank increases, self-image rank also increases

On its own, Rho does not have a good strength interpretation

But Rho2 is a PRE measure For this example, Rho2 = (0.86)2 = 0.74 Therefore, we would make 74% fewer errors if we

used the rank of jogging to predict the rank on self-image compared to if we ignored the rank on jogging

In addition to strength and direction, a hypothesis test of Rho can also indicate if the two variables share a relationship in the population, or if the two variables are significantly related

Hypothesis Test of Spearman’s Rho:

Step 1: Make Assumptions and Meet Test Requirements

Random sampling Ordinal level of measurement Normal sampling distribution



Step 2: State the Null Hypothesis Ho: ρs = 0

No relationship exists between the variables in the population

H1: ρs ≠ 0 A relationship exists between the variables in

the population



Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling distribution = Student’s t Alpha = 0.05 (two-tailed) Degrees of freedom = N-2 = 8 t(critical) = ±2.306



Step 4: Compute the Test Statistic



Step 5: Make a Decision and Interpret the Results t(obtained) = 4.77 t(critical) = ±2.306 t(obtained) falls in the critical region, so

reject Ho

Jogging and self-image are related in the population from which the sample was drawn


Chapter 14Association Between Variables

Measured at the Interval-Ratio Level

This Presentation

Scattergrams Graphs that display relationships between two interval-ratio

variables Regression Coefficients and the Regression Line

Regression line summarizes the linear relationship between X and Y

Regression coefficients predict scores on Y from scores on X

Pearson’s r Preferred measure of association for two interval-ratio

variables Coefficient of determination: r2

Correlation matrix

Scattergrams

Scattergrams have two dimensions: The X (independent) variable is arrayed along the

horizontal axis The Y (dependent) variable is arrayed along the

vertical axis Each dot on a scattergram is a case The dot is placed at the intersection of the case’s

scores on X and Y

Scattergrams

A regression line, which summarizes the linear relationship between X and Y, is added to the graph “Eyeball” a straight line that connects all of the

dots or comes as close as possible to connecting all of the dots

To be more precise: calculate the conditional mean of Y for each value of X, plot those values, and connect the dots

Inspection of a scattergram should always be the first step in assessing the relationship between two interval-ratio level variables

Scattergrams

Linearity A key assumption of scattergrams and regression

analysis is that X and Y share a linear relationship In a linear relationship the dots of a scattergram form a

straight line pattern

Linear Relationship: Example

Scattergrams

Linearity In a nonlinear relationship the dots do not form a straight

line pattern

Scattergrams

Three Questions Does a relationship exist?

A relationship exists if the conditional means of Y change across values of X

As long as the regression line lies at an angle to the X axis (and is not parallel to the X axis), we can conclude that a relationship exists between the two variables

ScattergramsThree Questions How strong is the relationship?

Strength of the relationship is determined by the spread of the dots around the regression line

In a perfect association, all dots fall on the regression line

In a stronger association, the dots fall close (are clustered tightly around) the regression line

In a weaker association, the dots are spread out relatively far from the regression line

Scattergrams

Three Questions What is the direction of the relationship? (Direction of

association is determined by the angle of the regression line)

What is the Direction of the Relationship?

ScattergramsBased on this scattergram for percent college educated (X) and voter turnout (Y) on election day for 50 states:Does a relationship exist? How strong is the relationship?What is the direction of the relationship? Is the relationship linear?

Scattergrams

Does a relationship exist? The regression line falls at an angle to the X axis (it is not

parallel), therefore we can conclude that an association exists between voter turnout and college education

Scattergrams

How strong is the relationship? The greater the extent to which dots are clustered around

the regression line, the stronger the relationship This relationship is weak to moderate in strength

Scattergrams What is the direction of the relationship?

Positive: Regression line rises from lower-left to upper-right Negative: Regression line falls from upper-left to lower-right This is a positive relationship: As percent college educated increases, voter

turnout increases

Scattergrams

Is the relationship linear? The conditional means on Y form a straight line, as

demonstrated by the regression line Therefore, the relationship is linear

Pearson’s r

Pearson’s r is a measure of association for interval-ratio level variables

Pearson’s r can indicate the direction of association, but it does not have an acceptable strength interpretation

But, by squaring r, we obtain a PRE measure called the coefficient of determination

The coefficient of determination indicates the percentage of the variation in Y that is explained by X

Pearson’s r

Calculate r

Pearson’s r

r = 0.50 r is positive, therefore the relationship between X and Y is

positive As the number of children in dual-career families increases,

husbands’ hours of housework per week also increases r2 = (0.50)2 = 0.25

r2 is 0.25, therefore the number of children in dual-career families explains 25% of the variation in husbands’ hours of housework per week

Pearson’s r

Hypothesis Test of Pearson’s r Step 1: Make Assumptions and Meet Test

Requirements Random sample Interval-ratio level measurement Bivariate normal distributions Linear relationship Homoscedasticity Normal sampling distribution

Pearson’s r

Hypothesis Test of Pearson’s r Step 2: State the Null Hypothesis

Ho: ρ = 0

H1: ρ ≠ 0

Step 3: Select the Sampling Distribution and Establish the Critical Region Sampling distribution = Student’s t Alpha = 0.05 (two-tailed) Degrees of freedom = N-2 = 10 t(critical) = ±2.228

Pearson’s r

Hypothesis Test of Pearson’s r Step 4: Compute Test Statistic

Pearson’s r

Hypothesis Test of Pearson’s r Step 5: Make a Decision and Interpret the Results

t(critical) = ±2.228 t(obtained) = 1.83 t(obtained) does not fall in the critical region, so we

fail to reject Ho

The two variables are not related in the population

Correlation Matrix

A correlation matrix is a table that shows the relationships between all possible pairs of variables

Correlation Matrix Using the matrix below:

What is the correlation between GDP and inequality? Of all the variables correlated with Inequality, which has the

strongest relationship? The weakest?

Documents

Week 12 Chapter 13 – Association between variables measured at the ordinal level & Chapter 14: Association Between Variables Measured at the Interval-Ratio