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Lecture for Multiple Regression HSPM J716
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Data
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Fertilizer-Rain (two X variables) chart
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Yield-Fertilizer Projection of 3-D graph, seen looking across the Fertilizer axis.
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Yield-Rain Projection looking across Rain axis
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Simple regression of Yield vs. Fertilizer Like fitting a plane that is horizontal (no slope) in the Rain direction. No slope in the rain direction means that Rain is assumed to have no effect.
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Simple regression of Yield vs. Fertilizer
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Simple regression prediction
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Simple regression of Yield vs. Rain Like fitting a plane that is horizontal (no slope) in the Fertilizer direction. No slope in the Fertilizer direction means that Fertilizer is assumed to have no effect.
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Simple regression of Yield vs. Rain
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Multiple regression on flat paper Like fitting a grid on the Yield-fertilizer graph The Rain lines all have to have the same slope. The Rain lines have to be equidistant. – Linear assumption is why. Minimize the sum of squares of distances from each point to the regression line that corresponds to that point’s rain amount.
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Multiple regression with prediction
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Simple regression with prediction
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Collinearity Two of your X variables are correlated with each other = One of your X variables can be well predicted from another X variable Multicollinearity – one of your X variables is predictable from a linear combination of other X variables
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Trivial example of … … Collinearity – W = α + β I H I + β F H F – Two of your X variables measure the same thing, like height in inches and height in feet. … Multicollinearity – Y = α + β 1 X 1 + β 2 X 2 + β 3 X 3 + β T X T – Some X variables are the scores for each question on a test. Another X variable is the total score on the test.
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Collinearity … is why you need multiple regression But too much collinearity makes separation of causes impossible
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Collinearity example
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Collinearity or just insignificant? When t-tests show insignificance, an F-test can tell you – Examples: Achievement vs. socio-econ status and schooling Determinants of supervisor rating
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F-test
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