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(3) Correlational Studies Relationship Between Variables Scatterplots I. Descriptive Statistics (3) Correlational studies Correlation Relationship between variables Studies in which two or more variables are Scatterplots (Scatter Diagrams) measured to find the direction and degree to Measuring Correlation: which they covary. • The Correlation Coefficient: r Covary: Two variables covary when a change in one variable is related to a consistent change in the other variable Relationship between variables Scatterplots (Scatter Diagrams) Postive relationship Bivariate distribution: A relationship between two variables in which, as the value of one variable increases, the value of the other A distribution in which two scores are obtained variable tends to increase also from each subject Negative relationship A relationship between two variables in which, as the value of one variable increases, the value of the other variable tends to decrease Scatterplot: No relationship A graph of a bivariate distribution in which the Lack of relationship. No regularity among the pairs of X variable is plotted on the horizontal axis and values of the variables the Y variable is plotted on the vertical axis Scatterplots (Scatter Diagrams) Scatterplots (Scatter Diagrams) Postive correlation Negative correlation 1 Scatterplots (Scatter Diagrams) Scatterplots (Scatter Diagrams) No correlation Linear relationship A relationship between two variables that can be described by a straight line Curvilinear relationship A relationship between two variables that can be described best with a curved line Measuring Correlation Measuring Correlation Correlation Coefficient Pearson Correlation_ _ Coefficient r = (x - X)(y - Y) / (n s s ) A number between –1 and 1 that describes the ∑ x y r = ∑ z z / n (z score formula) relationship between pairs of variables x y r = (n ∑xy - ∑x ∑y) / [√(n ∑x2 – (∑x)2 ) √(n ∑y2 – (∑y)2 )] Pearson Correlation Coefficient (computational formula) A statistic, symbolized by r, that indicates the What does r mean? degree of linear relationship between two r is a type of mean: the mean of the products of paired z variables measured at the interval or ratio level scores -1 ≤ r ≤ 1 Based on a measure of covariation: Cross Products The value of r is a measure of how well a straight line describes the cluster of dots in a scatterplot Measuring Correlation I. Descriptive Statistics (4) Regression But keep in mind that, … a high correlation Regression Line and Predictive Errors does not mean that there is a cause-effect Least Squares Regression Line relation! Least Squares Regression Equation Experimentation is needed! Standard Error of prediction What is r2? 2 Regression: Building on Regression Line and Correlation Predictive errors Prediction (regression) vs. relation (correlation) When a bivariate distribution shows a linear relationship, it (Simple) linear regression: is sometimes useful to try to Statistical tool used to predict scores on the dependent predict X from Y using a variable from scores on (one) independent variable regression line. This line is conceived as an approximation to the cloud of data observed in the scatterplot. Regression Line and Equation for a line Predictive errors Slope: The amount that Y is predicted to increase for an Predictive error: It is the difference, for each X, between increase of 1 in X. the observed corresponding Y and the value of the Y- Y-intercept: the predicted value for Y when X is 0 (point at coordinate. which the line intercepts the y-axis) The position of the regression line should minimize the total predictive error. y = 2x + 5 Least Squares Regression line Least Squares equation least squares regression line: The least squares regression equation minimizes the total the prediction line that minimizes the total of all squared prediction squared predictive error errors for known Y scores in It has the form: y = mx + n the original correlation analysis. Yˆ = bX + a The slope is: b = r(Sy/Sx_) _ The Y-intercept is: a = Y – bX Assumptions: Linearity and Homoscedasticity 3 Least Squares equation Standard error of prediction The least squares regression equation minimizes the total How to determine the amount of error associated of all squared prediction errors for known Y scores in the with these predictions? original correlation analysis. Standard error of prediction (or Standard error of Yˆ = bX + a the estimate): The slope is: b = r(Sy/S_x) _ The Y-intercept is: a = Y – bX A statistic that indicates the typical distance between a regression line and the actual data points Where do the values for a and b come from? And what do they mean? (see pdf notes) Standard error of prediction What is r2? The squared correlation coefficient (r2): Standard error of prediction (or Standard error of is the proportion of total variance in one the estimate): variable that is predictable from its relationship S = S (1 – r2) y|x y √ with the other variable It is a (rough) measure of the average amount of predictive error by which known Y values deviate from provides a measure of the worth of least predicted Yˆ values squares as predictors It reflects the degree to which the points diverge from the regression line. 2 It reflects the accuracy of the prediction. r = (SStotal – SSerror ) / SStotal 4 .
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