r_lin_regression - Using R for Linear Regression In the...

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Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional entries (all current as of version R-2.4.1). Sample texts from an R session are highlighted with gray shading. Suppose we prepare a calibration curve using four external standards and a reference, obtaining the data shown here: > conc [1] 0 10 20 30 40 50 > signal [1] 4 22 44 60 82 The expected model for the data is signal = β o + β 1 ×conc where β o is the theoretical y-intercept and β 1 is the theoretical slope. The goal of a linear regression is to find the best estimates for β o and β 1 by minimizing the residual error between the experimental and predicted signal. The final model is signal = b o + b 1 ×conc + e where b o and b 1 are the estimates for β o and β 1 and e is the residual error. Defining Models in R To complete a linear regression using R it is first necessary to understand the syntax for defining models. Let’s assume that the dependent variable being modeled is Y and that A, B and C are independent variables that might affect Y. The general format for a linear 1 model is response ~ op1 term1 op2 term 2 op3 term3… 1 When discussing models, the term ‘linear’ does not mean a straight-line. Instead, a linear model contains additive terms, each containing a single multiplicative parameter; thus, the equations y = β 0 + β 1 x y = β 0 + β 1 x 1 + β 2 x 2 y = β 0 + β 11 x 2 y = β 0 + β 1 x 1 + β 2 log(x 2 ) are linear models. The equation y = α x
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where term is an object or a sequence of objects and op is an operator, such as a + or a , that indicates how the term that follows is to be included in the model. The table below provides some useful examples. Note that the mathematical symbols used to define models do not have their normal meanings! Syntax Model Comments Y ~ A Y = β o + β 1 A Straight-line with an implicit y- intercept Y ~ -1 + A Y = β 1 A Straight-line with no y-intercept; that is, a fit forced through (0,0) Y ~ A + I(A^2) Y = β o + β 1 A + β 2 A 2 Polynomial model; note that the identity function I( ) allows terms in the model to include normal mathematical symbols. Y ~ A + B Y = β o + β 1 A + β 2 B A first-order model in A and B without interaction terms. Y ~ A:B Y = β o + β 1 AB A model containing only first-order interactions between A and B. Y ~ A*B Y = β o + β 1 A + β 2 B + β 3 AB A full first-order model with a term; an equivalent code is Y ~ A + B + A:B. Y ~ (A + B + C)^2 Y = β o + β 1 A + β 2 B + β 3 C + β 4 AB + β 5 AC + β 6 AC A model including all first-order effects and interactions up to the n th order, where n is given by ( )^n . An equivalent code in this case is
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r_lin_regression - Using R for Linear Regression In the...

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