Upcoming SlideShare
×

# Aron chpt 3 correlation compatability version f2011

4,561 views

Published on

Published in: Health & Medicine, Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

• Be the first to like this

Views
Total views
4,561
On SlideShare
0
From Embeds
0
Number of Embeds
7
Actions
Shares
0
37
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Aron chpt 3 correlation compatability version f2011

1. 1. Aron, Coups, & Aron <ul><li>Chapter 3 </li></ul><ul><li>Correlation and Prediction </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
2. 2. Correlations <ul><li>Can be thought of as a descriptive statistic for the relationship between two variables </li></ul><ul><li>Describes the relationship between two equal-interval numeric variables </li></ul><ul><ul><li>e.g., the correlation between amount of time studying and amount learned </li></ul></ul><ul><ul><li>e.g., the correlation between number of years of education and salary </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
3. 3. Scatter Diagram
4. 4. Graphing a Scatter Diagram <ul><li>To make a scatter diagram: </li></ul><ul><ul><li>Draw the axes and decide which variable goes on which axis. </li></ul></ul><ul><ul><ul><li>The values of one variable go along the horizontal axis and the values of the other variable go along the vertical axis. </li></ul></ul></ul><ul><ul><li>Determine the range of values to use for each variable and mark them on the axes. </li></ul></ul><ul><ul><ul><li>Numbers should go from low to high on each axis starting from where the axes meet . </li></ul></ul></ul><ul><ul><ul><li>Usually your low value on each axis is 0. </li></ul></ul></ul><ul><ul><ul><li>Each axis should continue to the highest value your measure can possibly have. </li></ul></ul></ul><ul><ul><li>Make a dot for each pair of scores. </li></ul></ul><ul><ul><ul><li>Find the place on the horizontal axis for the first pair of scores on the horizontal-axis variable. </li></ul></ul></ul><ul><ul><ul><li>Move up to the height for the score for the first pair of scores on the vertical-axis variable and mark a clear dot. </li></ul></ul></ul><ul><ul><ul><li>Keep going until you have marked a dot for each person. </li></ul></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
5. 5. Linear Correlation <ul><li>A linear correlation </li></ul><ul><ul><li>relationship between two variables that shows up on a scatter diagram as dots roughly approximating a straight line </li></ul></ul>
6. 6. Curvilinear Correlation <ul><li>Curvilinear correlation </li></ul><ul><ul><li>any association between two variables other than a linear correlation </li></ul></ul><ul><ul><li>relationship between two variables that shows up on a scatter diagram as dots following a systematic pattern that is not a straight line </li></ul></ul>
7. 7. No Correlation <ul><li>No correlation </li></ul><ul><ul><li>no systematic relationship between two variables </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
8. 8. Positive and Negative Linear Correlation <ul><li>Positive Correlation </li></ul><ul><ul><li>High scores go with high scores. </li></ul></ul><ul><ul><li>Low scores go with low scores. </li></ul></ul><ul><ul><li>Medium scores go with medium scores. </li></ul></ul><ul><ul><li>When graphed, the line goes up and to the right. </li></ul></ul><ul><ul><ul><li>e.g., level of education achieved and income </li></ul></ul></ul><ul><li>Negative Correlation </li></ul><ul><ul><li>High scores go with low scores . </li></ul></ul><ul><ul><ul><li>e.g., the relationship between fewer hours of </li></ul></ul></ul><ul><ul><ul><li>sleep and higher levels of stress </li></ul></ul></ul><ul><li>Strength of the Correlation </li></ul><ul><ul><li>how close the dots on a scatter diagram fall to a simple straight line </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
9. 9. Importance of Identifying the Pattern of Correlation <ul><li>Use a scatter diagram to examine the pattern, direction, and strength of a correlation. </li></ul><ul><ul><li>First, determine whether it is a linear or curvilinear relationship. </li></ul></ul><ul><ul><li>If linear, look to see if it is a positive or negative correlation. </li></ul></ul><ul><ul><li>Then look to see if the correlation is large, small, or moderate. </li></ul></ul><ul><li>Approximating the direction and strength of a correlation allows you to double check your calculations later. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
10. 10. The Correlation Coefficient <ul><li>A number that gives the exact correlation between two variables </li></ul><ul><ul><li>can tell you both direction and strength of relationship between two variables (X and Y) </li></ul></ul><ul><ul><li>uses Z scores to compare scores on different variables </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
11. 11. The Correlation Coefficient ( r ) <ul><li>The sign of r (Pearson correlation coefficient) tells the general trend of a relationship between two variables. </li></ul><ul><ul><li>+ sign means the correlation is positive. </li></ul></ul><ul><ul><li>- sign means the correlation is negative. </li></ul></ul><ul><li>The value of r ranges from -1 to 1. </li></ul><ul><ul><ul><li>A correlation of 1 or -1 means that the variables are perfectly correlated. </li></ul></ul></ul><ul><ul><ul><li>0 = no correlation </li></ul></ul></ul>
12. 12. Strength of Correlation Coefficients <ul><li>The value of a correlation defines the strength of the correlation regardless of the sign. </li></ul><ul><ul><ul><li>e.g., -.99 is a stronger correlation than .75 </li></ul></ul></ul>Correlation Coefficient Value Strength of Relationship +/- .70-1.00 Strong +/- .30-.69 Moderate +/- .00-.29 None (.00) to Weak
13. 13. Formula for a Correlation Coefficient <ul><li>r = ∑Z x Z y </li></ul><ul><ul><ul><li> N </li></ul></ul></ul><ul><ul><ul><li>Z x = Z score for each person on the X variable </li></ul></ul></ul><ul><ul><ul><li>Z y = Z score for each person on the Y variable </li></ul></ul></ul><ul><ul><ul><li>Z x Z y = cross-product of Z x and Z y </li></ul></ul></ul><ul><ul><ul><li>∑ Z x Z y = sum of the cross-products of the Z scores over all participants in the study </li></ul></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
14. 14. Steps for Figuring the Correlation Coefficient <ul><li>Change all scores to Z scores. </li></ul><ul><ul><li>Figure the mean and the standard deviation of each variable. </li></ul></ul><ul><ul><li>Change each raw score to a Z score. </li></ul></ul><ul><li>Calculate the cross-product of the Z scores for each person. </li></ul><ul><ul><li>Multiply each person’s Z score on one variable by his or her Z score on the other variable. </li></ul></ul><ul><li>Add up the cross-products of the Z scores. </li></ul><ul><li>Divide by the number of people in the study. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
15. 15. Calculating a Correlation Coefficient Number of Hours Slept (X) Level of Mood (Y) Calculate r X Zscore Sleep Y Zscore Mood Cross Product ZXZY 5 -1.23 2 -1.05 1.28 7 0.00 4 0.00 0.00 8 0.61 7 1.57 0.96 6 -0.61 2 -1.05 0.64 6 -0.61 3 -0.52 0.32 10 1.84 6 1.05 1.93 MEAN= 7 MEAN= 4 5.14  ZXZY SD= 1.63 SD= 1.91 r=5.14/6 r=  ZXZY r=.85
16. 16. Issues in Interpreting the Correlation Coefficient <ul><li>Direction of causality </li></ul><ul><ul><li>path of causal effect (e.g., X causes Y) </li></ul></ul><ul><li>You cannot determine the direction of causality just because two variables are correlated. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
17. 17. Reasons Why We cannot Assume Causality <ul><li>Variable X causes variable Y. </li></ul><ul><ul><li>e.g., less sleep causes more stress </li></ul></ul><ul><li>Variable Y causes variable X. </li></ul><ul><ul><li>e.g., more stress causes people to sleep less </li></ul></ul><ul><li>There is a third variable that causes both variable X and variable Y. </li></ul><ul><ul><li>e.g., working longer hours causes both stress and fewer hours of sleep </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
18. 18. Ruling Out Some Possible Directions of Causality <ul><li>Longitudinal Study </li></ul><ul><ul><li>a study where people are measured at two or more points in time </li></ul></ul><ul><ul><ul><li>e.g., evaluating number of hours of sleep at one time point and then evaluating their levels of stress at a later time point </li></ul></ul></ul><ul><li>True Experiment </li></ul><ul><ul><li>a study in which participants are randomly assigned to a particular level of a variable and then measured on another variable </li></ul></ul><ul><ul><ul><li>e.g., exposing individuals to varying amounts of sleep in a laboratory environment and then evaluating their stress levels </li></ul></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
19. 19. The Statistical Significance of a Correlation Coefficient <ul><li>A correlation is statistically significant if it is unlikely that you could have gotten a correlation as big as you did if in fact there was no relationship between variables. </li></ul><ul><ul><li>If the probability (p) is less than some small degree of probability (e.g., 5% or 1%), the correlation is considered statistically significant. </li></ul></ul>
20. 20. Prediction <ul><li>Predictor Variable (X) </li></ul><ul><ul><li>variable being predicted from </li></ul></ul><ul><ul><ul><li>e.g., level of education achieved </li></ul></ul></ul><ul><li>Criterion Variable (Y) </li></ul><ul><ul><li>variable being predicted to </li></ul></ul><ul><ul><ul><li>e.g., income </li></ul></ul></ul><ul><li>If we expect level of education to predict income, the predictor variable would be level of education and the criterion variable would be income. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
21. 21. Prediction Using Z Scores <ul><li>Prediction Model </li></ul><ul><ul><li>A person’s predicted Z score on the criterion variable is found by multiplying the standardized regression coefficient (  ) by that person’s Z score on the predictor variable. </li></ul></ul><ul><li>Formula for the prediction model using Z scores: </li></ul><ul><ul><li>Predicted Z y = (  )(Z x ) </li></ul></ul><ul><ul><li>Predicted Z y = predicted value of the particular person’s Z score on the criterion variable Y </li></ul></ul><ul><ul><li>Z x = particular person’s Z score in the predictor variable X </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
22. 22. Steps for Prediction Using Z Scores <ul><li>Determine the standardized regression coefficient (  ). </li></ul><ul><li>Multiply the standardized regression coefficient (  ) by the person’s Z score on the predictor variable. </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
23. 24. How Are You Doing? <ul><li>So, let’s say that we want to try to predict a person’s oral presentation score based on a known relationship between self-confidence and presentation ability. </li></ul><ul><li>Which is the predictor variable (Zx)? The criterion variable (Zy)? </li></ul><ul><li>If r = .90 and Z x = 2.25 then Z y = ? </li></ul><ul><li>So what? What does this predicted value tell us? </li></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
24. 25. Prediction Using Raw Scores <ul><li>Change the person’s raw score on the predictor variable to a Z score. </li></ul><ul><li>Multiply the standardized regression coefficient (  ) by the person’s Z score on the predictor variable. </li></ul><ul><ul><li>Multiply  by Z x. </li></ul></ul><ul><ul><ul><li>This gives the predicted Z score on the criterion variable. </li></ul></ul></ul><ul><ul><ul><ul><li>Predicted Z y = (  )(Z x ) </li></ul></ul></ul></ul><ul><li>Change the person’s predicted Z score on the criterion variable back to a raw score. </li></ul><ul><ul><li>Predicted Y = (SD y )(Predicted Z y ) + M y </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
25. 26. Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores <ul><li>From the sleep and mood study example, we known the mean for sleep is 7 and the standard deviation is 1.63, and that the mean for happy mood is 4 and the standard deviation is 1.92. </li></ul><ul><li>The correlation between sleep and mood is .85. </li></ul><ul><li>Change the person’s raw score on the predictor variable to a Z score. </li></ul><ul><ul><li>Z x = (X - M x ) / SD x </li></ul></ul><ul><ul><li>(4-7) / 1.63 = -3 / 1.63 = -1.84 </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
26. 27. Example of Prediction Using Raw Scores: Find the Predicted Z Score on the Criterion Variable <ul><li>Multiply the standardized regression coefficient (  ) by the person’s Z score on the predictor variable. </li></ul><ul><ul><li>Multiply  by Zx. </li></ul></ul><ul><ul><ul><li>This gives the predicted Z score on the criterion variable. </li></ul></ul></ul><ul><ul><ul><ul><li>Predicted Z y = (  )(Z x ) = (.85)(-1.84) = -1.56 </li></ul></ul></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
27. 28. Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores <ul><li>Change the person’s predicted Z score on the criterion variable to a raw score. </li></ul><ul><ul><li>Predicted Y = (SD y )(Predicted Z y ) + M y </li></ul></ul><ul><ul><li>Predicted Y = (1.92)(-1.56) + 4 = -3.00 + 4 = 1.00 </li></ul></ul>Copyright © 2011 by Pearson Education, Inc. All rights reserved
28. 29. The Correlation Coefficient and the Proportion of Variance Accounted for <ul><li>Proportion of variance accounted for (r 2 ) </li></ul><ul><ul><li>To compare correlations with each other, you have to square each correlation. </li></ul></ul><ul><ul><li>This number represents the proportion of the total variance in one variable that can be explained by the other variable. </li></ul></ul><ul><ul><li>If you have an r= .2, your r 2 = .04 </li></ul></ul><ul><ul><li>Where, a r= .4, you have an r 2 = .16 </li></ul></ul><ul><ul><li>So, relationship with r = .4 is 4x stronger than r=.2 </li></ul></ul>