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Education 793 Class Notes Joint Distributions and Correlation 1 October 2003

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Today’s Agenda Class and lab announcements Your questions? Joint distributions Correlation analysis to regression

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Joint Distributions In correlational studies, the researcher is interested in questions about the relationship between two or more variables. How are scores on one variable associated with scores on another variable? A joint distribution is a distribution in which pairs of scores for each subject are recorded.

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Graphical Representation Scatterplots of the (x,y)’s. SticiGui: Scatterplots and Association Definition: Correlation - a measure of the strength of association between two variables.

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Pearson-Product Correlation: Measure of Association An index showing the degree to which two distributions that show a linear relationship in the scatterplot are associated Values range from –1 to +1, with 0 indicating no relationship The average crossproduct of the standard scores of two variables Computed as:

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Important Properties Will underestimate curvilinear relationships As homogenity increases, correlation coefficient tends to decrease Size of sample does not affect size of correlation coefficient Positive Associations mean that as X increases Y increases and negative association means that as X increases Y decreases Correlation is just the standardized version of the covariance (does not depend on magnitude of sd y and sd x

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Individual Contributions to r Mean of x = 27.50; s = 17.08 Mean of y = 31.25; s = 18.87 ++ +- -- -+

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Visualizing Correlations Plot APlot BPlot CPlot D Plot EPlot FPlot GPlot H

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Squared Correlation Coefficient or Coefficient of Determination Coefficient of Determination tells you how much (percent) of the variance in one set of scores is accounted for by knowing the other set of scores. Shared Variance =shared variance / total variance

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Restricting Range N = 255 R =.63 A

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The Impact of Restricted Range N = 43 R =.17 BC N = 4 R =.10

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Correlation and Causality Correlation does not equal causation The higher the absolute value of a correlation, the stronger the relationship between two variables. Strength, though, does not explain the source of the relationship

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Causal Interpretation Logical possibility Symbolic representation Causal Explanation 1. A B A causes B 2. A B B causes A 3. A C B C causes both A and B 4. D C A D A D causes C which, in turn causes A D causes A directly

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Extending Correlation to Regression Goal: To predict values of our dependent variable based on values of our independent variable(s) and our knowledge of the underlying relationship (measured by Pearson's r) Requirements: Have data appropriate for computing r Be willing to specify nature of relationship (IV DV)

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Extending the Correlation

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Aptitude and Performance

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Creating the Prediction Equation

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Calculating Y-hat

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Predicting the DV

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Residuals

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Standard Error of Estimate A natural extension of the standard deviation –Deviations from the mean predicted value –Squared –Summed –Divide by N (or N-2 when estimating parameters) –This is an estimate or the error made when estimating y from x.

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Formula for SE of Estimate An alternative formula: Since r 2 xy =proportion of variance in y predictable from x, 1- r 2 xy is the proportion that is NOT predictable from x. Hence, the error.

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For Next Week Chapter 8 p. 211-225 Chapter 10 p. 249-271

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