Lecture for Multiple Regression HSPM J716

Data

Fertilizer-Rain (two X variables) chart

Yield-Fertilizer Projection of 3-D graph, seen looking across the Fertilizer axis.

Yield-Rain Projection looking across Rain axis

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.

Simple regression of Yield vs. Fertilizer

Simple regression prediction

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.

Simple regression of Yield vs. Rain

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.

Multiple regression with prediction

Simple regression with prediction

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

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.

Collinearity … is why you need multiple regression But too much collinearity makes separation of causes impossible

Collinearity example

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

F-test