Lecture20 - Lecture 20: Multiple Linear Regression Simple...

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Lecture 20: Multiple Linear Regression Simple linear regression: determines the best least squares equation to predict 1 Y-variable from a single predictor variable, X. Multiple linear regression: finds the best least squares equation that best predicts 1 Y-variable from multiple predictor variables, X 1, X 2, …, X k Sources of Information : Triola et al.: Chapter 9-5 Motulsky: Chapter 31 Sokal & Rohlf: Chapter 16
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Uses of Multiple Linear Regression (MLR) There are two main purposes of MLR: 1. PREDICTION: • To devise an equation that will enable a better prediction of a dependent variable Y than would be possible by any single independent variable X. Example : to predict photosynthetic rate as a function of light intensity, nutrient concentration, atmospheric [CO 2 ], soil moisture, etc. • Typically, one aims for the subset of all possible predictor variables that predicts a significant and appreciable proportion of the variance of Y, trading off adequacy of prediction against the cost of measuring more predictor variables.
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Uses of Multiple Linear Regression (MLR) 2. EXPLORATION: • To explore relationships among multiple X variables to find out which of the X variables significantly influence the Y variable. • It can be instructive to analyze huge data-sets that contain lots of variables to see which ones matter. Warning : You should only look at X-variables that have a plausible link to the Y-variable!
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The Multiple Regression Model A multiple regression equation expresses a linear relationship between a dependent variable (Y) and two or more independent variables (X 1 , X 2 , … , X k ) Ŷ = a + b 1 X 1 + b 2 X 2 + … + b k X k (for sample data) Y = α + β 1 X 1 + β 2 X 2 + … + β k X k + ε (for the statistical population) The b ’s are called a partial regression coefficients . For example, b 1 denotes the coefficient of Y on variable X 1 that one would expect if all the other X variables in the equation were held constant (i.e., after accounting for the other X variables). Each X can be: • a measured variable, • a transformation of a measured variable, or • a discrete variable (e.g., gender, entered as a dummy variable female = 1, male = 0)
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Assumptions of Multiple Regression Random sample : Your sample is randomly sampled from, or at least representative of, the statistical population. Linearity : Increasing a X variable by one unit changes (increases or decreases) Y by the same amount at all values of X. Æ Scatterplots of Y with each predictor (X) should show linear relationship [also look for outliers] No interaction among predictor variables : Increasing a X variable by one unit changes (increases or decreases) Y by a certain amount, regardless of the values of the other X variables. Æ Violated if the X variables are correlated
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Lecture20 - Lecture 20: Multiple Linear Regression Simple...

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