Presentation on theme: "Association Between Variables Measured at the Interval-Ratio Level"— Presentation transcript:
1Association Between Variables Measured at the Interval-Ratio Level Chapter 15Association Between Variables Measured at the Interval-Ratio Level
2Chapter Outline Interpreting the Correlation Coefficient: r 2 The Correlation MatrixTesting Pearson’s r for SignificanceInterpreting Statistics: The Correlates of Crime
3Scattergrams Scattergrams have two dimensions: The X (independent) variable is arrayed along the horizontal axis.The Y (dependent) variable is arrayed along the vertical axis.
4Scattergrams Each dot on a scattergram is a case. The dot is placed at the intersection of the case’s scores on X and Y.
5Scattergra msShows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 states.
6ScattergramsHorizontal 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.
7ScattergramsVertical (Y) axis is voter turnout.Scores range from 44.1% to 70.4% and increase from bottom to top
8Scattergrams: Regression Line A single straight line that comes as close as possible to all data points.Indicates strength and direction of the relationship.
9Scattergrams: 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.
10Scattergrams: 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.
11ScattergramsInspection of the scattergram should always be the first step in assessing the correlation between two I-R variables
12The Regression Line: Formula This formula defines the regression line:Y = a + bXWhere:Y = score on the dependent variablea = 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 XX = score on the independent variable
13Regression AnalysisBefore using the formula for the regression line, a and b must be calculated.Compute b first, using Formula 15.3 (we won’t do any calculation for this chapter)
14Regression AnalysisThe Y intercept (a) is computed from Formula 15.4:
15Regression AnalysisFor the relationship between % college educated and turnout:b (slope) = .42a (Y intercept)= 50.03Regression formula: Y = XA 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 =
16Predicting YWhat 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 = (10) = 54.5For X =70, Y = (70) = 79.7
17Pearson correlation coefficient But of course, this is just an estimate of turnout based on % college educated, and many other factors also affect voter turnout.How much of the variation in voter turnout depends on % college educated? The relevant statististic is the coefficient of determination (r squared), but first we need to learn about Pearson’s correlation coefficient (r).
18Pearson’s r Pearson’s r is a measure of association for I-R variables. It varies from -1.0 to +1.0Relationship may be positive (as X increases, Y increases) or negative (as X increases, Y decreases)For the relationship between % college educated and turnout, r =.32.The relationship is positive: as level of education increases, turnout increases.How strong is the relationship? For that we use R squared, but first, let’s look at the calculation process
19Example of Computation The computation and interpretation of a, b, and Pearson’s r will be illustrated using Problem 15.1.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.
20Example of Computation The scores on each variable are displayed in table format:Y = TurnoutX = Years of EducationCityXYA11.955B12.160C12.765D12.868E13.070
21Example of Computation Sums are needed to compute b, a, and Pearson’s r.XYX2Y2XY11.955141.613025654.512.160146.41360072612.765161.294225825.512.868163.844624870.413.070169490091062.5318782.15203743986.4
22Interpreting Pearson’s r An r of 0.98 indicates an extremely strong relationship between average years of education and voter turnout for these five cities.The coefficient of determination is r2 = .96. Knowing education level improves our prediction of voter turnout by 96%. This is a PRE measure (like lambda and gamma)We could also say that education explains 96% of the variation in voter turnout.
23Interpreting 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 = .32This is a weak to moderate, positive relationship.The value of r2 is .10. Percent college educated explains 10% of the variation in turnout.