Nonlinear regression, multiple regression, testing model fit, choosing predictors

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Colorado State University
STAT 301
Brett Hunter

30 November Nonlinear Regression If we have a curved pattern on a scatterplot, it would not make sense to model it linearly, but that doesn’t mean there is no relationship between X and Y. One way to model the relationship is to transform the explanatory variable i.e., using log (x) or √ x in regression instead of x Model: y = β +0β 1 √ x + ε More common way – polynomial regression i.e., fit data to y = 0 + β1x + β 2 + ε for a quadratic function Can do polynomial regression using X , for k ≥ 1, but we want to include all lower order terms of X in the model as well. y So if we were to fi a 5 degree polynomial, we would use = b0+ b 1 + b x2+ b x 3 3 b 4 + b x5 5 Multiple Regression Sometimes Y is strongly related to several explanatory variables while not having a strong relationship with any one single explanatory variable. If we have k predictors, our model is iid y = β 0 β x1 1β x 2 2 + β x + εk k i N (0, σ 2 for k ≥ 1, integer We fit β 0 b ,0β 1 b ,1etc., where b , 0 ,1… b ark the values that minimize [ y− b +b x+…+b x 2] [(y−̂ y) ] ∑ ( ( 0 1 k k) = ∑ = SSE Testing the fit of the model Is the “overall fit” of our model good? We perform a model utility test. H : β = β = … = β = 0 0 1 2 k H A At least one β i 0 for i = 1, …, k n−(k+1) ¿ ¿
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