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LECTURE 10: CORRELATION AS DESCRIPTION AND INFERENCE Overview of Correlation and Regression A Simple Model Weight (X1) ------------- + --------------- Body Fat (Y) Height (X2) ------------- + --------------- Correlation “r” = Strength of the relation of weight and fat (+.61) Sig “r” = Probability the r in sample from pop with r of zero (.000) Regression “b” = Slope line describing weight and fat (. 18) Sig “b” = Probability the b in sample from pop with b of zero (.001) A Plot of Height and Body Fat Correlation “r” = Strength of the relation of height and fat (-.025) Sig “r” = Probability the r in sample from pop with r of zero (.69) Regression “b” = Slope line describing height and fat (-.70) Sig “b” = Probability the b in sample from pop with b of zero (.75) Note: * ‘Variation’ in each variable * How scatter plots are most useful when using measures of continuous variables (interval and ratio, yet sometimes ordinal * The utility in thinking in terms of dependent and independent variables * The ability to look for “goofy” data points in plots * Correlation provides information on the strength of a relation * Regression provides information on the “line” summarizing the relation Usually the “line” is a straight line. Thus, linear regression Other relations can be examined * Correlations allow one to compare apples and oranges For example, for the above model the correlation between X1 Y is .613 The correlation between X2 and Y is -.025 Therefore, Weight (X1) has the strongest association with Body Fat (Y) Correlation in More Detail In performing a correlation one obtains two types of information 1. Substantive information on the relation between two variables e.g., an “r” of .613 for the correlation between weight and body fat (X1 & Y) 2. Inferential information to test the null hypothesis that the true correlation in the population from which the sample was drawn is zero ( e.g., a probability of drawing a sample with an “r greater than.613 or less than -.613 from a population in which the true “r” is zero. R2 and Variance Explained Idea of variance explained Correlation Matrix The Limitations of tests of statistical significance Substantive and Statistical Significance are two different things “For large sample sizes, even very small correlation coefficients will have small observed significance levels. Statistically significant doesn’t mean important or useful.” Spearman Rank Order correlation For ordinal data (Order in a race and time training) (Birth order and I.Q. test) Partial Correlation Coefficients Other examples Ice Cream, Summer, Crime Comments on Lab 10