1 Chapter 10 Correlation

Positive and Negative Correlation 2

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

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Curvilinear Relationships 5

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7 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

8 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

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

10 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 11 Using the results from the summary table, calculate the correlation coefficient. ΣX = 23ΣY = 23ΣX 2 = 115ΣY 2 = 64 ΣXY = 83

12 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

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 13

Importance of Graphing 14

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Review  Correlation Strength Direction  Test of significance  Curvilinear correlation  Importance of graphing 16

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 17

Correlation Matrix 18 Attitude toward School (X) Grades (Y) Employment (Z) Attitude toward School (X) Grades (Y) Employment (Z)

How Years on Force (Z) affects correlation 19 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 20 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. 21

Testing for significance 22 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

Chi Square & Strength of Association  Knowing that the result is significant is not enough  Only use this when examining two variables and the correlation is significant!!  Need to know how strong the association between the two is  Phi coefficient  Cramer’s V correlation coefficient 23

Phi Coefficient 24

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 =