Presentation on theme: "Chapter 15 Association Between Variables Measured at the Interval-Ratio Level."— Presentation transcript:
Chapter 15 Association Between Variables Measured at the Interval-Ratio Level
Chapter Outline Interpreting the Correlation Coefficient: r 2 The Correlation Matrix Testing Pearson’s r for Significance Interpreting Statistics: The Correlates of Crime
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 (we won’t do any calculation for this chapter)
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)= 50.03 Regression formula: Y = 50.03 +.42 X 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 = 50.03.
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 = 50.3 +.42(10) = 54.5 For X =70, Y = 50.3 +.42(70) = 79.7
Pearson 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).
Pearson’s r Pearson’s r is a measure of association for I-R variables. It varies from -1.0 to +1.0 Relationship 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
Example 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.
Example of Computation The scores on each variable are displayed in table format: Y = Turnout X = Years of Education CityXY A11.955 B12.160 C12.765 D12.868 E13.070
Example of Computation Sums are needed to compute b, a, and Pearson’s r. XYX2X2 Y2Y2 XY 11.955141.613025654.5 12.160146.413600726 12.765161.294225825.5 12.868163.844624870.4 13.0701694900910 62.5318782.15203743986.4
Interpreting 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 r 2 =.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.
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.