PSY 1950Regression

November 10, 2008

Definition• Simple linear regression

– Models the linear relationship between one predictor variable and one outcome variable

– e.g., predicting income based upon age

• Multiple linear regression– Models the linear relationship between more than one predictor variables and one outcome variable

– e.g., predicting income based upon age and sex

• Lingo– Independent/dependent, predictor/outcome

History• Astronomical predictions: method of least squares– Piazzi (1801) spotted Ceres, made 22 observations over 41 days, got sick, lost Ceres

– Gauss: "... for it is now clearly shown that the orbit of a heavenly body may be determined quite nearly from good observations embracing only a few days; and this without any hypothetical assumption.”

• Genetics: Regression to the mean– Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute, 15, 246–263.

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Lines• Mathematically, a line is defined by its slope and intercept– Slope is change in Y per change in X

– Intercept is the points at which the line crosses the Y-axis, i.e., Y when X = 0

• Y = bX + a– b is slope– a is intercept

Which Lines is Best?

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Residuals• Residuals are

– Errors in prediction– Difference between expected values (under your model) and observed values (in your dataset)

Y = 0.063X + 131.59

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Minimizing Residuals• Can define the best fit line by summing– Absolute residuals (Method of Least Absolute Deviations)

– Squared residuals (Method of Least Squares)

Which is Better?• Method of Least Squares

– Not robust– Stable (line doesn’t “jump” with small changes in X)

– Only one solution (unique line for each dataset)

• Method of Least Absolute Deviations– Robust– Unstable (line does “jump” with small changes in X)

– Multiple solutions (sometimes)• http://www.math.wpi.edu/Course_Materials/SAS/lablets/7.3/7.3c/lab73c.html

Multiple Solutions• Any line within the “green zone” produces the same summed residuals via the method of least absolute deviations

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Method of (Ordinary) Least Squares-1.02738397 0.34691735

y = 0.063x + 131.59

R2 = 0.355

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Regression Coefficients• Slope

• Intercept

Standardized Coefficients

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Y = 0.5958X

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Regression Line Passes Through (MX, MY)

Correlation and Regression • Statistical distinction based on nature of the variables– In correlation, both X and Y are random– In regression, X is fixed and Y is random

• Practical distinction based on interest of researcher– With correlation, the researcher asks: What is the strength (and direction) of the linear relationship between X and Y

– With regression, the research asks the above and/or: How do I predict Y given X?

Goodness of Fit• The regression equation does not reveal how well your data fit your model– e.g., in the below, both sets of data produce the same regression equation

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Standard Error of Estimate• The standard residual

• Why df = n - 2?– To determine regression equation (and thus the residuals), we need to estimate two population parameters• Slope and intercept OR• Mean of X and mean of Y

– A regression with n = 2 has no df

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Coefficicent of Determination (r2)

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Partitioning Sums of Squares

Partitioning Sums of Squares

Testing the Model

# predictors

n minus # model parametersn minus (1 + # predictors)

Online Applets• Explaining variance

– http://www.duxbury.com/authors/mcclellandg/tiein/johnson/reg.htm

• Leverage– http://www.stat.sc.edu/~west/javahtml/Regression.html

• Distribution of slopes/intercepts– http://lstat.kuleuven.be/java/version2.0/Applet003.html