AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

1
AAEC 4302ADVANCED STATISTICAL METHODS IN
AGRICULTURAL RESEARCH

• Chapter 7 (Part 1)
• Theory and Application of the
• Multiple Regression Model

2
Introduction

• When we have one independent variable (X) we
  called it a Simple Regression Model.
• When there is more than one independent variable,
  it is called Multiple Regression.

3
Introduction

• We postulate that cotton yields are a function
  of
• Irrigation water applied
• Phosphorus fertilizer applied
• We can write this mathematically as Y
  B0B1X1B2X2ui

4
Introduction

• Y B0B1X1B2X2ui
• Y Cotton Yield (lint in lbs/ac)
• X1 Irrigation Water Use (in/ac)
• X2 Phosphorus Fertilizer use (lbs/ac)

5
Introduction

• The multiple regression model aims to and must
  include all of the independent variables X1, X2,
  X3, , Xk that are believed to affect Y.
• Their values are taken as given It is critical
  that, although X1, X2, X3, , Xk are believed to
  affect Y, and
• Y does not affect the values taken by them.

6
Introduction

• The multiple regression model is given by
• Yi B0B1X1iB2X2i B3X3iBkXki ui
• where i1,,n represents the observations
• k is the total number of independent variables in
  the model
• B0, B1,,Bk are the parameters to be estimated
• ui is the population error term, with the same
  properties as in the simple regression model.

7
The Model

• As before
• EYi B0B1X1iB2X2i B3X3iBkXki
• Yi EYi ui,
• systematic unsystematic
• the systematic (explainable) and unsystematic
  (random) components of Yi

8
The Model

• The model to be estimated, therefore, is
• And the corresponding prediction of Yi

9
Model Estimation

• Also as before, the parameters of the multiple
  regression model (B0, B1, B2, B3,,Bk) are
  estimated by minimizing SSR,
• that is, the sum of the squares of the residuals
  (ei), or differences between the values of Y
  observed in the sample, and the regression line
  (i.e. the OLS method)

10
Model Estimation

• As before, the formulas to estimate the
  regression model parameters that would make the
  SSR as small as possible are obtained by taking
  derivatives

11
Model Estimation

• Specifically, the k1 partial derivatives of the
  just discussed SSR function, with respect to
  , are taken and set
  equal to zero.
• This results in a system of k1 linear equations
  with k1 unknowns (the ßs).

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Model Estimation

• Solving this systems for the unknowns, yields the
  formulas for calculating ,
  which depend on the Yi and the X1i, X2i, X3i, ,
  Xki values in the sample.
• The formulas for the case where there are only
  two independent variables (X1 and X2) in the
  model are given in the next slide.

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Model Estimation
Page 134
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Y
X2
Regression surface (plane) EY BoB1X1B2X2
Ui
X2 slope measured by B2
Bo
X1 slope measured by B1
X1
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Model Estimation

• Notice that when there are two independent
  variables in the model, the formula for
  calculating is not the same as when there is
  only one independent variable.
• Therefore, the estimated value of this parameter
  will be different if X2 is not included in the
  model.

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Model Estimation

• In general, only a model that is estimated
  including all of the (independent) variables that
  affected the values taken by Y in the sample will
  produce correct parameter estimates.
• Only then will the formulas for estimating these
  parameters be unbiased.

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Interpretation of the Coefficients

• The intercept B0 estimates the value of Y when
  all of the independent variables in the model
  take a value of zero
• which may not be empirically relevant or even
  correct in some cases.

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Interpretation of the Coefficients

• In a strictly linear model, B1, B2,..., Bk are
  slope coefficients that measure the unit change
  in Y when the corresponding X (X1, X2,..., Xk)
  changes by one unit and the values of all of the
  other independent variables remain constant at
  any given level (it does not matter which level).

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The Models Goodness of Fit

• The key same measure of goodness of fit is used
  in the case of the multiple regression model
• The only difference is in the calculation of the
  eis, which now equal

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The Models Goodness of Fit

• The interpretation and everything else is the
  same as in the case of the simple linear
  regression model
• The SER is also calculated as before, but using
  the eis above and dividing by n-k-1

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The Models Goodness of Fit

• A disadvantage of R2 as a measure of a models
  goodness of fit is that it tends to increases in
  value as independent variables are added into the
  model, even if those variables cant be
  statistically shown to affect Y. Why?
• This happens because, when estimating the models
  coefficients by OLS, any new independent variable
  would likely allow for a smaller SSR

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The Models Goodness of Fit

• An increase in the R2 as a result of adding an
  independent variable to the model does not mean
  that the expanded model is better, or that that
  variable really affects Y (in the population)

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The Models Goodness of Fit

• The adjusted R2 denoted by R2 is a better measure
  to assess whether the addition of an independent
  variable increases the ability of the model to
  predict the dependent variable Y.

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The Models Goodness of Fit

• R2 is always less than the R2, unless R2 1
• Unfortunately, R2 lacks the same straightforward
  interpretation as R2 under unusual
  circumstances, it can even be negative.
• It is only useful to assess whether an
  independent variable should be added to the model.

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The Specification Question

• Any variable that is suspected to directly affect
  Y, and that did not hold a constant value
  throughout the sample, should be included in the
  model.
• Excluding such a variable would likely cause the
  estimates of the remaining parameters to be
  incorrect i.e. the formulas for estimating
  those parameters would be biased.

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The Specification Question

• The consequences of including irrelevant
  variables in the model are less serious
• if in doubt, this is preferred
• If a variable only affects Y indirectly, through
  another independent variable in the model, it
  should not be included in the model.