Week 9: Chapter 15, 17 (and 16) Association Between Variables Measured at the Interval-Ratio Level The Procedure in Steps

Step 1: Make Scattergrams and Regression Lines  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  Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.

Scattergrams  Horizontal X axis - % of population of a state with a college education. Scores range from 15.3% to 34.6% and increase from left to right.

Scattergrams  Vertical (Y) axis is voter turnout. Scores range from 44.1% to 70.4% and increase from bottom to top

Scattergrams: Regression Line  A single straight line that comes as close as possible to all data points.  Indicates strength and direction of the relationship.

Scattergrams: Strength of Regression Line  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: Direction of Regression Line  Positive: regression line rises left to right.  Negative: regression line falls left to right.  This a positive relationship: As % college educated increases, turnout increases.

Scattergrams  Inspection of the scattergram should always be the first step in assessing the correlation between two I-R variables

The Regression Line: Formula  This formula defines the regression line: Y = a + bX Where:  Y = score on the dependent variable  a = the Y intercept or the point where the regression line crosses the Y axis.  b = the slope of the regression line or the amount of change produced in Y by a unit change in X  X = score on the independent variable

Regression Analysis  Before using the formula for the regression line, a and b must be calculated.  Compute b first, use Formula 15.3 (see Healey p. 401):

Regression Analysis  The Y intercept (a) is computed from Formula 15.4 (see Healey p. 402):

Regression Analysis  For the relationship between % college educated and turnout: b (slope) =.42 a (Y intercept)=  A slope of.42 means that turnout increases by.42 (less than half a percent) for every unit increase of 1 in % college educated.  The Y intercept means that the regression line crosses the Y axis at Y =  The regression line here is: Y = X

Exercise: Predicting Y  What turnout would be expected in a state where only 10% of the population was college educated?  What turnout would be expected in a state where 70% of the population was college educated?

Step 2: What is the pattern/direction of the association?  See results of step 1  Focus on the b: a slope of.42 means that turnout increases by.42 (less than half a percent) for every unit increase of 1 in % college educated.  This a positive relationship: As % college educated increases, turnout increases.

Step 3: How Strong is the Relationship?  See results of step 1  The greater the extent to which dots are clustered around the regression line, the stronger the relationship  This relationship between education and voter turnout is weak to moderate in strength  Pearson’s r is a measure of association for I-R variables.

Pearson’s r  Calculation Pearson’s r: Formula 15.5 and 15.6 (see Healey pp ).  For the relationship between % college educated and turnout, r =.32. This relationship is positive and weak to moderate.  As level of education increases, turnout increases.

Step 4: Is the Strength of the Association Significant?  Testing Pearson’s r for significance  See Chapter 15 of Healey (pp ) for five-step model for test to find out whether the strength of the association between the variables is significant or not

Step 5: What is the r 2 ?  The value of r 2 is.10.  Interpretation Percent college educated explains 10% of the variation in turnout 10% of the variance in turnout is explained by education

Step 6: Is there still an Association, if Control Variables are Added?  See Chapter 16 in Healey  See week 10 of this course