Chapter 15 (Ch. 13 in 2nd Can.)

Association Between Variables Measured at the Interval-Ratio Level:

Bivariate Correlation and Regression

Introduction: Scattergrams / Scatterplots

Graphs that display relationships between two interval-ratio variables.

The Regression Line, Slope, and Intercept. The regression line, y=a+bX, summarizes the linear

relationship between X and Y. Predicts the score of Y from a score of X.

b represents the slope of the line. a, called the intercept, is the point on the Y-axis where the

regression line crosses it. Pearson’s r and the Coefficient of Determination (r2)

r is a measure of association for two I-R variables. r2 tells you how much variation in the dependent variable is

explained by the independent variable.

Scattergram / Scatterplot

Has 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 in the

data set. The dot is placed at the intersection of the

case’s scores on X and Y.

Example of a Hypothetical Scattergram Showing the Relationship Between X and Y

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 cities.

Turnout

Scattergram Example (cont.)

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

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

Turnout

Scattergram Example (cont.)

Turnout By % College

43

48

53

58

63

68

73

15 20 25 30 35

% College

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

Turnout

The Regression Line on a Scattergram A single straight line that comes as close as possible to

all data points. “least squares regression line” Indicates strength and direction of the relationship.

Turnout

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

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.

Turnout

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

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.

Turnout

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

Scattergrams and Linearity Inspection of the scattergram should always be the

first step in assessing the correlation between two interval-ratio variables. In addition to assessing the strength and direction, the relationship must also be linear.

Turnout By % College

43

48

53

58

63

68

73

15 17 19 21 23 25 27 29 31 33 35

% College

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 and Prediction

We can use the regression line to find the predicted value of y (symbolized as y’) for values of x.

Once we know the values of the coefficients b and a, we can use the following prediction formula by substituting any value for x to predict y. The predicted level of y can be calculated by:

We can also use the regression formula to accurately plot the regression line on our scattergram.

bxayy )('

Regression Analysis: Healey’s definitional formula for calculating the slope of the line (Formula 15.2 or 13.2 in 2nd Can.)

Note: The numerator is the covariation of x and y (how x and y vary together). The denominator is the sum of the squared deviations around the mean of x

2

xx

yyxxb

Regression Analysis: *computational formula* for b (Formula 13.3 in 2nd) Below is the computational (working) formula to calculate b. It is

a re-arrangement of the theoretical formula and is much easier to calculate!

The slope tells you what the change in Y is, for every unit of X. The sign of the slope coefficient (+/- b) tells you whether the

relationship is positive or negative.

22 )(

))((

XXn

YXXYnb

Regression Analysis

The Y intercept (a) is computed from Healey, Formula 15.3 (or 13.4 in 2nd):

The intercept (a) is the point where the regression line crosses the Y-axis, when X=0.

xbya

Results of a Hypothetical Regression Analysis of the Relationship Shown in the Scattergrams Above: For the relationship between % college educated

and % turnout: Assume b (slope) = .42 Assume a (Y intercept)= 50.03

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.

An example of prediction:

We can use the regression equation y’=a+bx for prediction. For instance, we could ask, what % turnout would be expected in a city where only 10% of the population was college educated?

What % turnout would be expected in a city where 70% of the population was college educated?

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

Calculating the Correlation Coefficient: Formula for Pearson’s r Definitional formula for Pearson’s r:

*Use the computational formula to calculate*:

])(][)([

))((2222 YYnXXn

YXXYnr

22yyxx

yyxxr

Pearson’s r

Like Gamma, r varies from -1.00 to +1.00 Pearson’s r is a measure of association for

Interval-Ratio variables. For the hypothetical relationship between %

college educated and turnout, assume r =.32 This relationship would be positive and weak to

moderate. As level of education increases, % turnout

increases.

The Coefficient of Determination: r2

Total variation in y ( ) is the sum of the explained variation ( ) and the unexplained variation ( )

The explained variation (the portion explained by x) is represented by the formula:

Or, alternatively: r2 = (r)2

2 yy

2' yy

2' yy

yy

yyr

'2

Practical Example using Healey

Problem 15.1 (Problem 13.1 in 2nd Can.) The computation and interpretation of a, b, Pearson’s

r and r2 will be illustrated using a similar example from Healey Problem 15.1 ( % Turnout by Education (Years of Schooling) but with only 5 cases)

The variables are: Voter turnout (Y) is the dependent variable. Average years of school (X) is the independent variable.

The sample is 5 cities. This is only to simplify the calculation. A sample of 5 is

actually very small.

Data from Problem 15.1:

The scores on each variable are displayed in table format: Y = % Turnout X = Years of Education

City X Y

A 11.9 55

B 12.1 60

C 12.7 65

D 12.8 68

E 13.0 70

1. Draw and Interpret the Scattergram:

The relationship between X and Y is linear. Estimate regression line. Relationship is positive and strong.

2. Make a Computational Table:

5.125/5.62/ nXX

Sums (Σ) are needed to compute b, a, and Pearson’s r. As well, the mean of X and Y are needed:

X Y X2 Y2 XY

11.9 55 141.61 3025 654.5

12.1 60 146.41 3600 726

12.7 65 161.29 4225 825.5

12.8 68 163.84 4624 870.4

13.0 70 169 4900 910

∑X = 62.5 ∑Y = 318 ∑X2 =782.15 ∑Y2 = 20374 ∑XY = 3986.4

6.635/318/ nYY

3. Next, calculate b and a….

Calculate slope:

Calculate y-intercept:

22 )(

))((

XXn

YXXYnb

XbYa

Interpret Slope (b), the Intercept (a)

For every unit increase in X, Y increases by 12.67. This means that for 1 additional year of schooling, voter turnout goes up by 12.67%.

This is the point at which the regression line crosses the Y-axis (when X is equal to 0, Y is equal to -94.78)

67.12)5.62()15.782(5

)318)(5.62()4.3986(5

)(

))((222

XXn

YXXYnb

78.94)5.12(67.126.63 XbYa

Find the Regression Line*:

*Note: you can now substitute two values for X and solve for Y to find points to plot the actual regression line on your scattergram.

For prediction:Suppose years of schooling = 10 years…Then, Y = -94.78 + 12.67 (10) = 31.92. We would predict that when average years of education is 10 years, the voter turnout would be 31.92%

)(67.1278.94 XbXaY

4. Pearson’s r

Calculate the correlation coefficient r

])(][)([

))((2222 YYnXXn

YXXYnr

Interpret Pearson’s r

An r of 0.98 indicates an extremely strong relationship between years of education and voter turnout for these five cities (use the table given in Ch. 14 to estimate strength)

2222 )(][)([

))((

YYnXXn

YXXYnr

984.])318()20374(5][)5.62()15.782(5[

)318)(5.62()4.3986(522

5. Find the Coefficient of Determination (r2) and Interpret:

The coefficient of determination is r2 = .968. Education, by itself, explains 96.8% of the variation in voter turnout.

968.)984(.)( 222 rr

6. Testing r for significance:

We can test the relationship between % turnout and years of education (represented by Pearson’s r) for significance using the 5 step model and the following formula:

Degrees of Freedom = N-2

21

2

r

nrtobtained

Step 1: Assumptions There are 3 main assumptions…

1. The dependent and independent are normally distributed. We can test this by looking at the histograms for the two variables.

2. The relationship between X and Y is linear. We can check this by looking at the scattergram.

3. The relationship is homoscedastic. We can test homoscedasticity by looking at the scattergram and observing that the data points form a “roughly symmetrical, cigar-shaped pattern” about the regression line.

If the above 3 assumptions have been met, then we can use linear regression and correlation and test r for significance.

Step 2: Null and Alternate Hypotheses:

Ho: ρ = 0.0

H1: ρ ≠ 0.0 (Note that ρ (rho) is the population parameter, while r is the sample statistic.)

Step 3: Sampling Distribution and Critical Region:

S.D. = t-distribution Alpha = .05 DF = n - 2 = 5 - 2 = 3 tcritical = 3.182

Step 4. Computing the Test Statistic: Use Formula 15.6 in Healey (13.6 in 2nd Can.)

Step 5. Decision and Interpretation:

Tobtained = 9.53 > tcritical = 3.182

Reject Ho. The relationship between % turnout and years of schooling is significant.

53.9)984(.1

25984.

1

222

r

nrtobtained

Always include a brief summary of your results:

There is a very strong, positive relationship between % voter turnout and years of schooling for the five cities. As years of schooling increase, the % of voter turnout goes up. The relationship is significant (t=9.53, df=3, α = .05) . Years of schooling explain 96.8% of the variation in % voter turnout.

Practice Problem

Working with a partner, calculate, interpret and summarize the results for Healey 1st Can. #15.1 (2nd Can. 13.1) for “% Turnout” and “Unemployment” and for “% Turnout” and “Negative Campaigning”.