Lab 6 Further Regression Analysis

Math 411 Lab 6: Further Regression Analysis 1 Overview In this lab, we’ll examine nonlinear and multiple regression techniques. We will model the relationship of a dependent variable to one or more independent variables. For nonlinear regression analysis, we will develop a polynomial regression equation of the form ? = 𝛽 0 + 𝛽 1 ? + 𝛽 2 ? 2 + ⋯ + 𝛽 𝑘 ? 𝑘 + 𝑒 that quantifies the relationship of the two variables ? and ? . For multiple regression analysis, we will develop a multiple regression equation of the form ? = 𝛽 0 + 𝛽 1 ? 1 + 𝛽 2 ? 2 + ⋯ + 𝛽 𝑘 ? 𝑘 + 𝑒 that quantifies the relationship of the variables ? 1 , ? 2 , … , ? 𝑘 , and ? . For the case of two independent variables, ? 1 and ? 2 , the first-order model without interaction is ? = 𝛽 0 + 𝛽 1 ? 1 + 𝛽 2 ? 2 + 𝑒 whereas the model with first-order predictors and interaction is ? = 𝛽 0 + 𝛽 1 ? 1 + 𝛽 2 ? 2 + 𝛽 3 ? 1 ? 2 + 𝑒 Complete all of the exercises for your lab report. As always, be sure to include all relevant information. For this lab, you should include your R code/scripts along with the output. 2 Polynomial Regression The R commands for computing the coefficients of the quadratic regression equation are as follows: model <- lm ( y ~ x + I(x^2) , data = input_data ) summary ( model) And the R commands for computing the coefficients of the cubic regression equation are as follows: model <- lm ( y ~ x + I(x^2) + I(x^3) , data = input_data ) summary ( model) Exercises: 1. The following data consisting of ? = fermentation time in days and ? = glucose concentration in g/L for a particular blend of malt liquor is derived from an article ‘Improving Fermentation and Productivity with Reverse Osmosis’. ? (days) ? (g/L) 1 74 2 54 3 52 4 51 5 52 6 53 7 58 8 71

a. Using a scatterplot, verify that the data is consistent with the choice of a quadratic regression model. b. Compute the estimated quadratic regression equation. c. Predict the value of glucose concentration for a fermentation time of 6 days and compute the corresponding residual. d. Compute the estimated quadratic regression equation for the centered model. e. Predict the value of glucose concentration for a fermentation time of 6 days and compute the corresponding residual for the centered model. 2. The following data consisting of ? = temperature difference in °𝐾 and ? = energy output in W was provided in the article ‘Comparison of Energy and Energy Efficiency for Solar Box and Parabolic Cookers’. ? ( °𝐾 ) ? (W) 23.20 3.78 23.50 4.12 23.52 4.24 24.30 5.35 25.10 5.87 26.20 6.02 27.40 6.12 28.10 6.41 29.30 6.62 30.60 6.43 31.50 6.13 32.01 5.92 32.63 5.64 33.23 5.45 33.62 5.21 34.18 4.98 35.43 4.65 35.62 4.50 36.16 4.34 36.23 4.03 36.89 3.92 37.90 3.65 39.10 3.02 41.66 2.89 a. Using a scatterplot, verify that the data is consistent with the choice of a cubic regression model. b. Compute the estimated cubic regression equation. c. Also, compute the estimated linear and quadratic regression equations. d. Does the cubic predictor appear to provide useful information about ? above that provided by the linear and quadratic predictors? 3 Multiple Regression Analysis The R commands for computing the coefficients of the multiple regression equation are as follows: model <- lm ( y ~ x1 + x2 , data = input_data ) summary ( model) To see which predictor variables are significant, examine the coefficients table as follows: summary ( model )$ coefficient