Download - Chapter 12 Correlation and Regression
Chapter 12Correlation and
Regression
Part III: Additional Hypothesis Tests
Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences
Figure 12.1
Relationship Between Effort and Grade Point Average
Table 12.1
Raw Data
Figure 12.2
Relationship between effort and grade point average
Definitions
Positive (direct) relationships: As X increases, Y increases.
Negative (inverse) relationships: As X increases, Y decreases.
Perfect relationships: All data points fall on the best-fit line.
Imperfect relationships: All data points do not fall on the best-fit line.
Figure 12.3
Types of Linear Relationships
Correlation
A correlation coefficient is a statistic that expresses the degree of the fit of the data to a line and the type of relationship (direct or inverse).
Correlation
Correlation coefficients make no assumptions about the cause-and-effect direction of an X-Y relationship but rather simply measure the degree to which two sets of paired scores vary together in a consistent (linear) manner.
Characteristics of Correlation Coefficients
1. Values always range between –1 and +1.
2. A positive coefficient indicates a direct relationship (positive slope), whereas a negative coefficient (negative slope) indicates an inverse relationship.
Characteristics of Correlation Coefficients
3. A coefficient of zero indicates that there is no relationship between the two variables.
4. A coefficient that is equal to –1.00 or +1.00 indicates that you have a perfect relationship between your variables.
Pearson’s r Formula
Formula for Pearson’s r
r =
n
YY
n
XX
n
YXXY
22
22 )()(
))((
= ))((
))((
YX SSSSn
YXXY
r = XbyexplainedisthatYofvariancetotalofproportion
R2 = The proportion of the variability in Y that is explained by X.
Hours(X) Grade(Y)
Hours(X) 1
Grade(Y) 0.91543 1
Results when you use Microsoft Excel to calculate a Correlation
HOURS GPA
HOURS Pearson Correlation 1.000 .915
Sig. (2-taile) . .000
N 10 10
GPA Pearson Correlation .915 1.000
Sig. (2-tailed) .000 .
N 10 10
** Correlation is significant at the 0.01 level (2-tailed).
Results when you use SPSS to calculate a Correlation
When is it appropriate to use Correlation?
1. You have two variables on an interval or ratio (continuous) scale.
2. The relationship between the two variables is linear (rather than curvilinear, or not fitting a straight line).
3. You wish to describe the strength of the relationship between your two variables.
Linear Regression
Linear regression is a technique that is closely related to correlation.
In regression, we generally assume that the X variable is the predictor variable (number of hours of study effort, in our example) and the Y variable is the criterion variable (GPA).
Linear Regression
Least Squares Regression Line: Y’ = byX + ay where: Y’ = predicted value of Y by = slope of the line that minimizes the errors in predicting Y from X ay = y-intercept of the line that minimizes the errors in predicting Y from X
Linear Regression
Y-intercept: The value of Y when X is equal to zero, which is where the line crosses the Y-axis.
Slope: The change in Y divided by the change in X.
Linear Regression: Formulas for slope and y-intercept
by=
n
XX
n
YXXY
22 )(
))((
ay = Y -by X
where: n = number of pairs of scores ΣXY = Sum of the product of each X and Y pair, or the sum of the cross products.
Figure 12.4
Relationship Between Effort and Grade Point Average
Results if you use Microsoft Excel to calculate a regression on study time-GPA data
Results if you use SPSS to calculate the regression Model Summary
a Predictors: (Constant), HOURSANOVA
a Predictors: (Constant), HOURSb Dependent Variable: GPACoefficients
a Dependent Variable: GPA
Standard Error of the Estimate (SEE)
This is the amount of error around the estimate (the regression line), just like the standard deviation measures error around the mean.
Formula for calculating the Standard Error of the Estimate (SEE)
sxy =
2
))(( 2
n
SSn
YXXY
SSX
Y
When is it appropriate to use Regression?
1. When you have a predictor and a criterion variable on an interval or ratio scale.
2. When the relationship between the two variables is linear.
3. When your data are homoscedastic. This means that the variability around the regression line is uniform for all of the values of X.
Figure 12.5
Example of Data That are Homoscedastic and Non-Homoscedastic
Linear Equation for Regression
Y = a + b(X) + SEE
Multiple Regression
Multiple regression is like the two-way ANOVA because it has more than one predictor variable effect that is assessed at the same time.
Linear Equation for Multiple Regression
Y = a + b1(X1) + b2(X2) + E