14a_AnCova

# 14a_AnCova - Statistical Techniques II EXST7015 Analysis of Covariance 14a_AnCova 1 Simple Linear Regression Regression is usually done as a least

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Statistical Techniques II EXST7015 Analysis of Covariance 14a_AnCova 1

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Simple Linear Regression Regression is usually done as a least squares technique applied to QUANTITATIVE VARIABLES. ANOVA is the analysis of categorical (class, indicator, group) variables, there are no quantitative "X" variables as in regression, but this is still a least squares technique. 14a_AnCova 2
Analysis of Covariance (AnCova) It stands to reason that if Regression uses the least squares technique to fit quantitative variables, and ANOVA uses the same approach to fit qualitative variables, that we should be able to put both together into a single analysis. We will call this Analysis of Covariance. 14a_AnCova 3

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AnCova (continued) There are actually two conceptual approaches, Multisource regression - adding class variables to a regression Analysis of Covariance - adding quantitative variables to an ANOVA We will talk primarily about Multisource regression. 14a_AnCova 4
AnCova (continued) With multisource regression we start with a regression and ask, would the addition of an indicator or class variable improve the model? Adding a class variable to a regression gives each group it's own intercept. We may further ask, does each group need it's own slope? This can be fitted with an interaction of the quantitative (X) variable and the group variable. 14a_AnCova 5

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AnCova (continued) Adding a class variable fits a separate intercept to each group. X Y 14a_AnCova 6
AnCova (continued) Adding an interaction fits a separate slope to each group. X Y 14a_AnCova 7

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AnCova (continued) How do they do that? For a simple linear regression we start with, Y i = b 0 + b 1 X 1i + e i Now add an indicator variable. In our example we will add just one, but it could be several. We will call our indicator variable X 2i , but it is a variable with values of 0 or 1. 14a_AnCova 8
AnCova (continued) Y i = b 0 + b 1 X 1i + b 2 X 2i + e i When X 2i = 0 we get Y i = b 0 + b 1 X 1i + b 2 0 + e i , which reduces to Y i = b 0 + b 1 X 1i + e i , a simple linear model And when X 2i = 1 we have Y i = b 0 + b 1 X 1i + b 2 1 + e i , which simplifies to Y i = (b 0 +b 2 ) + b 1 X 1i + e i , a simple linear model with intercept (b 0 +b 2 ) 14a_AnCova 9

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AnCova (continued) Two lines with intercepts of b 0 and (b 0 +b 2 ). Note that b 2 is a difference or adjustment. X Y b 0 (b 0 +b 2 ) 14a_AnCova 10
AnCova (continued) Adding an interaction between the quantitative variable (X 1i ) and the indicator variable (X 2i ) will fit separate slopes. With just one indicator variable for two classes the model is Y i = b 0 + b 1 X 1i + b 2 X 2i + b 3 X 1i X 2i + e i 14a_AnCova 11

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AnCova (continued) Y i = b 0 + b 1 X 1i + b 2 X 2i + b 3 X 1i X 2i + e i When X 2i = 0 we get Y i = b 0 + b 1 X 1i + b 2 0 + b 3 X 1i 0 + e i , which reduces to Y i = b 0 + b 1 X 1i + e i , a simple linear model For X 2i =1, then Y i =b 0 +b 1 X 1i +b 2 1 +b 3 X 1i 1 +e i , simplifies to Y i = (b 0 +b 2 ) + (b 1 +b 3 )X 1i + e i , a simple linear model with intercept (b 0 +b 2 ) and slope equal to (b 1 +b 3 ) 14a_AnCova 12
AnCova (continued) Two lines with intercepts of b 0 and (b 0 +b 2 ).

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## This note was uploaded on 12/29/2011 for the course EXST 7087 taught by Professor Wang,j during the Fall '08 term at LSU.

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14a_AnCova - Statistical Techniques II EXST7015 Analysis of Covariance 14a_AnCova 1 Simple Linear Regression Regression is usually done as a least

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