 Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.

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Correlation and Regression A BRIEF overview

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

Interpreting Correlations l 1.0, + or - indicates perfect relationship l 0 correlations = no association between the variables l in between - varying degrees of relatedness l r 2 as proportion of variance shared by two variables l which is X and Y doesn’t matter

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

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

Zero correlation l no relationship between the variables l a positive or negative correlation gives us predictive power

Direction and degree

Direction and degree (cont.)

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

Raw score formula

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

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

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

Correlation as causation??

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

Linear vs. curvilinear relationships

Linear vs. curvilinear (cont.)

Range restriction

Range restriction (cont.)

Understanding r

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

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

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

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

Best-fitting line (cont.)

Calculating the slope (b) l N=number of pairs of scores, rest of the terms are the sums of the X, Y, X 2, Y 2, and XY columns we’re already familiar with

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

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

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