Chapter 15 Association Between Variables Measured at the Interval-Ratio Level

Chapter Outline  Introduction  Scattergrams  Regression and Prediction  The Computation of a and b  The Correlation Coefficient (Pearson’s r)

Chapter Outline  Interpreting the Correlation Coefficient: r 2  The Correlation Matrix  Testing Pearson’s r for Significance  Interpreting Statistics: The Correlates of Crime

This Presentation  Scattergrams Graphs that display relationships between two interval-ratio variables.  The Regression Line Summarizes the linear relationship between X and Y. Predicts score on Y from score on X.  Pearson’s r Preferred measure of association for two I-R variables.

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.

Scattergrams  Each dot on a scattergram is a case.  The dot is placed at the intersection of the case’s scores on X and Y.

Scattergra ms  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, using Formula 15.3:

Regression Analysis  The Y intercept (a) is computed from Formula 15.4:

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 =

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?  This is a positive relationship so the value for Y increases as X increases: For X =10, Y = 54.5 For X =70, Y = 79.7

Pearson’s r  Pearson’s r is a measure of association for I-R variables.  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.

Example of Computation  The computation and interpretation of a, b, and Pearson’s r will be illustrated using Problem  The variables are: Voter turnout (Y) Average years of school (X)  The sample is 5 cities. This is only to simplify computations, 5 is much too small a sample for serious research.

Example of Computation  The scores on each variable are displayed in table format: Y = Turnout X = Years of Education CityXY A B C D E13.070

Example of Computation  Sums are needed to compute b, a, and Pearson’s r. XYX2X2 Y2Y2 XY

Interpreting Pearson’s r  An r of 0.98 indicates an extremely strong relationship between years of education and voter turnout for these five cities.  The coefficient of determination is r 2 =.96. Education, by itself, explains 96% of the variation in voter turnout.

Interpreting Pearson’s r  Our first example provides a more realistic value for r. The r between turnout and % college educated for the 50 states was:  r =.32  This is a weak to moderate, positive relationship.  The value of r 2 is.10. Percent college educated explains 10% of the variation in turnout.