chapter 17: introduction to regression
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Chapter 17: Introduction to Regression. Introduction to Linear Regression. The Pearson correlation measures the degree to which a set of data points form a straight line relationship. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 17: Introduction to Regression
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Introduction to Linear Regression
• The Pearson correlation measures the degree to which a set of data points form a straight line relationship.
• Regression is a statistical procedure that determines the equation for the straight line that best fits a specific set of data.
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Introduction to Linear Regression (cont.)
• Any straight line can be represented by an equation of the form Y = bX + a, where b and a are constants.
• The value of b is called the slope constant and determines the direction and degree to which the line is tilted.
• The value of a is called the Y-intercept and determines the point where the line crosses the Y-axis.
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Introduction to Linear Regression (cont.)
• How well a set of data points fits a straight line can be measured by calculating the distance between the data points and the line.
• The total error between the data points and the line is obtained by squaring each distance and then summing the squared values.
• The regression equation is designed to produce the minimum sum of squared errors.
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Introduction to Linear Regression (cont.)
The equation for the regression line is
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Introduction to Linear Regression (cont.)
• The ability of the regression equation to accurately predict the Y values is measured by first computing the proportion of the Y-score variability that is predicted by the regression equation and the proportion that is not predicted.
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Introduction to Linear Regression (cont.)
• The unpredicted variability can be used to compute the standard error of estimate which is a measure of the average distance between the actual Y values and the predicted Y values.
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Introduction to Linear Regression (cont.)
• Finally, the overall significance of the regression equation can be evaluated by computing an F-ratio.
• A significant F-ratio indicates that the equation predicts a significant portion of the variability in the Y scores (more than would be expected by chance alone).
• To compute the F-ratio, you first calculate a variance or MS for the predicted variability and for the unpredicted variability:
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Introduction to Linear Regression (cont.)
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Introduction to Multiple Regression with Two Predictor Variables
• In the same way that linear regression produces an equation that uses values of X to predict values of Y, multiple regression produces an equation that uses two different variables (X1 and X2) to predict values of Y.
• The equation is determined by a least squared error solution that minimizes the squared distances between the actual Y values and the predicted Y values.
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Introduction to Multiple Regression with Two Predictor Variables (cont.)• For two predictor variables, the general form of
the multiple regression equation is:
Ŷ= b1X1 + b2X2 + a
• The ability of the multiple regression equation to accurately predict the Y values is measured by first computing the proportion of the Y-score variability that is predicted by the regression equation and the proportion that is not predicted.
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Introduction to Multiple Regression with Two Predictor Variables (cont.)
As with linear regression, the unpredicted variability (SS and df) can be used to compute a standard error of estimate that measures the standard distance between the actual Y values and the predicted values.
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Introduction to Multiple Regression with Two Predictor Variables (cont.)• In addition, the overall significance of the
multiple regression equation can be evaluated with an F-ratio:
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Partial Correlation
• A partial correlation measures the relationship between two variables (X and Y) while eliminating the influence of a third variable (Z).
• Partial correlations are used to reveal the real, underlying relationship between two variables when researchers suspect that the apparent relation may be distorted by a third variable.
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Partial Correlation (cont.)
• For example, there probably is no underlying relationship between weight and mathematics skill for elementary school children.
• However, both of these variables are positively related to age: Older children weigh more and, because they have spent more years in school, have higher mathematics skills.
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Partial Correlation (cont.)
• As a result, weight and mathematics skill will show a positive correlation for a sample of children that includes several different ages.
• A partial correlation between weight and mathematics skill, holding age constant, would eliminate the influence of age and show the true correlation which is near zero.
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