1 Chapter 10 Correlation

2  Finding that a relationship exists does not indicate much about the degree of association, or correlation, between two variables. Many relationships are statistically significant; few express perfect correlation.  To illustrate, we know that the level of a degree and income are associated, because the more education an individual has, the higher the base salary.

3 Direction of Correlation  Correlation can often be described with respect to direction as either positive or negative. A positive correlation indicates that respondents getting high scores on the X variable also tend to get high scores on the Y variable.  Conversely, respondents who get low scores on X also tend to get low scores on the Y variable. A negative correlation exists if respondents who obtain high scores on the X variable tend to obtain low scores on the Y variable.  Conversely, respondents achieving low scores on X tend to achieve high scores on Y.

4 Strength of Correlation  Correlations actually vary with respect to their strength. Scatter plot  = scores on any two variables, X and Y

Positive and Negative Correlation 5

6

7 Curvilinear Correlation  For the most part, social researchers seek to establish a straight-line correlation, whether positive or negative. It is important to note, however, that not all relationships between X and Y can be regarded as forming a straight line.  That is, a relationship between X and Y that begins as positive becomes negative; a relationship that starts as negative becomes positive.

Curvilinear Relationships 8

9

10 The Correlation Coefficient  Correlation coefficients range between and as follows:  perfect negative correlation -.60  strong negative correlation -.30  moderate negative correlation -.10  weak negative correlation.00  no correlation +.10  weak positive correlation +.30  moderate positive correlation +.60  strong positive correlation  perfect positive correlation

11 Pearson’s Correlation Coefficient o For example, we might be interested in examining the relationship between one’s attitude towards legalization of prostitution (X) and their attitudes towards legalization of marijuana (Y) Prostitution (x) Marijuana (Y) A12 B65 C43 D33 E21 F74

12 Pearson’s Correlation Coefficient Summary Table ChildXYX2X2 Y2Y2 XY A 12 B 65 C 43 D 33 E 21 F 74 Total

13 Pearson’s Correlation Coefficient Summary Table Completed ChildXYX2X2 Y2Y2 XY A B C D E F Σ Mean of X = 3.83 Mean of Y = 3

Calculating the Correlation Coefficient 14 Using the results from the summary table, calculate the correlation coefficient. ΣX = 23ΣY = 23ΣX 2 = 115ΣY 2 = 64 ΣXY = 83

15 Testing the Significance of Pearson’s r  Pearson's r gives us a precise measure of the strength and direction of the correlation in the sample being studied. If we have taken a random sample from a specified population, we may still seek to determine whether the obtained association between X and Y exists in the population and is not due merely to sampling error. To test the significance of a measure of correlation, we usually set up the null hypothesis that no correlation exists in the population. Can use either a t test or a simplified method using r to assess significance

16 Testing the Significance of Pearson’s r  When we turn to Table F in Appendix C, we find the critical value of r with 4 degrees of freedom (N – 2) and an alpha level of.05 is.8114  Because our calculated r value (.86) exceeds this critical value (.81), we can reject the null hypothesis p = 0. The correlation is statistically significant.

Correlation Steps  Step 1: Create a summary table  Step 2: Find the values of ΣX, ΣY, ΣX 2, ΣY 2, ΣXY, and the mean of X and Y.  Step 3: Insert values from step 2 into the correlation formula.  Step 4: Find the degrees of freedom, alpha, and critical r  Step 5: Compare computed r with critical value of r using Table F 17

18 Requirements for the Use of Pearson’s r Correlation Coefficient  To employ Pearson’s correlation coefficient correctly as a measure of association between X and Y variables, the following requirements must be taken into account:  A straight-line relationship.  Interval data.  Random sampling.  Normally distributed characteristics.

The Importance of Scatter Plots  Computers have made many calculations much faster and easier.  Data peculiarities (such as outliers) may conceal true relationships.  Use scatter plots to help sort through these.  A scatter plot visually displays all the information contained in a correlation coefficient.

Importance of Graphing 20

21

Review  Correlation Strength Direction  Test of significance  Curvilinear correlation  Importance of graphing 22

End Day 1 23

Partial Correlation  Usually, researchers examine more than two variables at a time.  Must consider if a correlation between two measures holds up when controlling for a third variable.  Requires a correlation matrix  Useful statistic for finding spurious variables 24

Correlation Matrix 25 Attitude toward School (X) Grades (Y) Employment (Z) Attitude toward School (X) Grades (Y) Employment (Z)

26 Suppose a consultant for a police department finds a -.40 correlation between performance on a physical fitness test (X) and salary (Y) for a sample of 50 officers. When taking into consideration the years on force (Z), we find a vastly different picture. How do we go about finding the correlation between X and Y when holding Z constant?

How Years on Force (Z) affects correlation 27 Physical Fitness (X) Salary (Y)Years on Force (Z) Physical Fitness (X) Salary (Y) Years on Force (Z) Rxy = -.44 Rxz = -.68 Ryz =.82

Formula 28 Correlations: Rxy = -.44 Rxz = -.68 Ryz =.82 Rxy.z = -.44 – (-.68)(.82) √1-(-.68) 2 *√1-(.82) 2 Rxy.z = +.28 The partial correlation of physical fitness score (X) and salary (Y) while holding constant years on the force (Z) is calculated as follows:

Testing for significance When testing for significance, we use t scores for partial correlations and not Table F. 29

Testing for significance 30 Our calculated T: t =.28(√50-3). √( ) t =.28(√47). √(1-.08) t = 1.92 /.96 t = 2.04 Table T: df = N – 3 df = 50 – 3 df = 47 a =.05 t = 2.021

Steps for Partial Correlation 31  Step 1: Determine Pearson’s r for each of Rxy, Rxz, and Ryz  Step 2: Plug the values from step 1 into the formula to find the partial correlation coefficient  Step 3: Using the partial coefficient, calculate a t score  Step 4: Find the degrees of freedom (N-3) and the table t  Step 5: Compare calculated t versus table t

Chi Square & Strength of Association  Knowing that the result is significant is not enough  Need to know how strong the association between the two is  Phi coefficient  Cramer’s V correlation coefficient 32

Phi Coefficient  33

Cramer’s V A researcher is examining those who participate in a GED program, work skills program, and those who do not and whether or not the individual, once released from prison, was arrested within a 2 year time frame. The researcher found there was a statistically significant difference and found the following results: x 2 = 8.42 N =