Multiple Linear Regression Analysis: Models and Applications, Slides of Data Analysis & Statistical Methods

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11: Multiple Linear

Regression

EM 7: Engineering Data Analysis

Second Semester, 2019-

Pamantasan ng Lungsod ng Valenzuela

Introduction

• In most research problems where regression analysis is applied, more than one independent variable is needed in the regression model. As a response, a multiple regression model is needed. When this model is linear in the coefficients, it is called a multiple linear regression model.
• For the case of independent variables, , the mean of

is given by the multiple linear regression model

Introduction

• Similar least squares techniques can also be applied for estimating, say, powers and products of independent variables. For example, when the experimenter may believe that the means

are more appropriately described as polynomial regression model

And the estimated response is from the polynomial regression equation

Introduction

• Polynomial models are linear model. Nonlinear models happen when parameters occur nonlinearly, regardless of how the independent variables enter the model. An example of a nonlinear model is the exponential relationship

Whose response is estimated by the regression equation

Estimating the Coefficients In using the concept of least squares to arrive the estimates ଴ ଵ ௞ the expression

Differentiating SSE with respect to the estimators and equating to zero, we arrive to generate set of normal equations for multiple linear regression. (^7) DEFINITION: The multiple linear regression (SLR) model is given by ௜ ଴ ଵ ଵ௜ ଶ ଶ௜ ௞ ௞௜ ௜ or ௜ 𝒊 ௜ ଴ ଵ ଵ௜ ଶ ଶ௜ ௞ ௞௜ ௜ where (^) ௜ and (^) ௜ are the random error and residual, respectively, associated with the response (^) ௜ and fitted value (^) 𝒊.

Estimating the Coefficients DEFINITION: The normal estimation equations for multiple regression is composed of set: ଴ ଵ ଵ௜ ଶ ଶ௜ ௞ ௞௜ ௜

These equations can be solved for for any appropriate method for solving systems of linear equations.

10 The solution of the set of estimating equation, based from normal estimation equations for MLR, yields the unique estimates, ଴ ଵ ଶ ଷ Therefore, the regression equation is ଵ ଶ ଷ For 50% humidity, a temperature of 76 degrees F, and a barometric pressure of 29.30, the estimated amount of nitrous oxide emitted is

Estimating the Coefficients Now suppose we wish to fit the polynomial equation

to the n pairs of observations. Each observation satisfies the equation or where is the degree of the polynomial and and are the random error and residual associated with the response and fitted value.

Example Solving the normal equations, Therefore, When , the estimate of

is

Example The data below represent the percent of impurities that resulted for various temperatures and sterilizing times during a reaction associated with the manufacturing of a certain beverage. Estimate the regression coefficients in the polynomial model