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# Aron chpt 3 correlation compatability version f2011

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### Transcript

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