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Understanding Correlation and Regression Coefficients

This brief overview covers interpreting correlations, positive and negative correlations, zero correlation, correlation coefficient, effect size, linear regression, prediction modeling, and calculating regression coefficients. Learn about the strength and direction of relationships in data analysis.

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Understanding Correlation and Regression Coefficients

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  1. Correlation and Regression A BRIEF overview

  2. Correlation Coefficients • Continuous IV & DV • or dichotomous variables (code as 0-1) • mean interpreted as proportion • Pearson product moment correlation coefficient range -1.0 to +1.0

  3. Interpreting Correlations • 1.0, + or - indicates perfect relationship • 0 correlations = no association between the variables • in between - varying degrees of relatedness • r2 as proportion of variance shared by two variables • which is X and Y doesn’t matter

  4. Positive Correlation • regression line is the line of best fit • With a 1.0 correlation, all points fall exactly on the line • 1.0 correlation does not mean values identical • the difference between them is identical

  5. Negative Correlation • If r=-1.0 all points fall directly on the regression line • slopes downward from left to right • sign of the correlation tells us the direction of relationship • number tells us the size or magnitude

  6. Zero correlation • no relationship between the variables • a positive or negative correlation gives us predictive power

  7. Direction and degree

  8. Direction and degree (cont.)

  9. Direction and degree (cont.)

  10. Correlation Coefficient • r = Pearson Product-Moment Correlation Coefficient • zx = z score for variable x • zy = z score for variable y • N = number of paired X-Y values • Definitional formula (below)

  11. Raw score formula

  12. Interpreting correlation coefficients • comprehensive description of relationship • direction and strength • need adequate number of pairs • more than 30 or so • same for sample or population • population parameter is Rho (ρ) • scatterplots and r • more tightly clustered around line=higher correlation

  13. Examples of correlations • -1.0 negative limit • -.80 relationship between juvenile street crime and socioeconomic level • .43 manual dexterity and assembly line performance • .60 height and weight • 1.0 positive limit

  14. Describing r’s • Effect size index-Cohen’s guidelines: • Small – r = .10, Medium – r = .30, Large – r = .50 • Very high = .80 or more • Strong = .60 - .80 • Moderate = .40 - .60 • Low = .20 - .40 • Very low = .20 or less • small correlations can be very important

  15. Correlation as causation??

  16. Nonlinearity and range restriction • if relationship doesn't follow a linear pattern Pearson r useless • r is based on a straight line function • if variability of one or both variables is restricted the maximum value of r decreases

  17. Linear vs. curvilinear relationships

  18. Linear vs. curvilinear (cont.)

  19. Range restriction

  20. Range restriction (cont.)

  21. Understanding r

  22. Simple linear regression • enables us to make a “best” prediction of the value of a variable given our knowledge of the relationship with another variable • generate a line that minimizes the squared distances of the points in the plot • no other line will produce smaller residuals or errors of estimation • least squares property

  23. Regression line • The line will have the form Y'=A+BX • Where: Y' = predicted value of Y • A = Y intercept of the line • B = slope of the line • X = score of X we are using to predict Y

  24. Ordering of variables • which variable is designated as X and which is Y makes a difference • different coefficients result if we flip them • generally if you can designate one as the dependent on some logical grounds that one is Y

  25. Moving to prediction • statistically significant relationship between college entrance exam scores and GPA • how can we use entrance scores to predict GPA?

  26. Best-fitting line (cont.)

  27. Best-fitting line (cont.)

  28. Calculating the slope (b) • N=number of pairs of scores, rest of the terms are the sums of the X, Y, X2, Y2, and XY columns we’re already familiar with

  29. Calculating Y-intercept (a) • b = slope of the regression line • the mean of the Y values • the mean of the X values

  30. Let’s make up a small example • SAT – GPA correlation • How high is it generally? • Start with a scatter plot • Enter points that reflect the relationship we think exists • Translate into values • Calculate r & regression coefficients

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