Your SlideShare is downloading. ×
Aron chpt 3 correlation compatability version f2011
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Saving this for later?

Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime - even offline.

Text the download link to your phone

Standard text messaging rates apply

Aron chpt 3 correlation compatability version f2011

4,297
views

Published on

Published in: Health & Medicine, Education

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
4,297
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
29
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Aron, Coups, & Aron
    • Chapter 3
    • Correlation and Prediction
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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?
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 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
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 28. Example of Prediction Using Raw Scores: Change Raw Scores to Z Scores
    • Change the person’s predicted Z score on the criterion variable to a raw score.
      • Predicted Y = (SD y )(Predicted Z y ) + M y
      • Predicted Y = (1.92)(-1.56) + 4 = -3.00 + 4 = 1.00
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 29. The Correlation Coefficient and the Proportion of Variance Accounted for
    • Proportion of variance accounted for (r 2 )
      • To compare correlations with each other, you have to square each correlation.
      • This number represents the proportion of the total variance in one variable that can be explained by the other variable.
      • If you have an r= .2, your r 2 = .04
      • Where, a r= .4, you have an r 2 = .16
      • So, relationship with r = .4 is 4x stronger than r=.2