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Fitting Regression Models. It is usually interesting to build a model to relate the response to process variables for prediction, process optimization, and process control A regression model is a mathematical model which fits to a set of sample data to approximate the exact appropriate relation

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fitting regression models
Fitting Regression Models
  • It is usually interesting to build a model to relate the response to process variables for prediction, process optimization, and process control
  • A regression model is a mathematical model which fits to a set of sample data to approximate the exact appropriate relation
  • Low-order polynomials are widely used
  • There is a strong “interaction” between the design of experiments and regression analysis
  • Regression analysis is often applied to unplanned experiments
linear regression models
Linear Regression Models
  • In general, the response variable y may be related to k regressor variables by a multiple linear (first order) regression model

y = bo + b1x1 + b2x2 +  + bkxk + e

  • Models of more complex forms can be analyzed similarly. E.g.,

y = bo + b1x1 + b2x2 + b12x1x2 + e =>

y = bo + b1x1 + b2x2 + b3x3 + e

  • Any regression model that is linear in the parameters (b values) is a linear regression model, regardless of the shape of the response surface
  • The variables have to be quantitative
model fitting estimating parameters
Model Fitting – Estimating Parameters
  • The method of least squares is typically used
  • Assuming that the error term e are uncorrelated random variables
  • The data can be expressed as
  • The model equation is

yi = bo + b1xi1 + b2xi2 +  + bkxik + ei

  • Least square method chooses the b´s so that the sum of the squares of the errors ei is minimized
model fitting estimating parameters1
Model Fitting – Estimating Parameters
  • The least squares function is
  • The least squares estimators ( ) must satisfy
          • and
  • or
model fitting estimating parameters2
Model Fitting – Estimating Parameters
  • Or in the matrix notation
  • y = Xb + e
  • The least squares estimators of b are
  • The fitted model and residuals are
  • Example 10-1:
    • Response: viscosity of a polymer (y)
    • Variables: reaction temperature (x1), catalyst feed rate (x2)
  • The model
  • y = bo + b1x1 + b2x2 + e
fitting regression models in designed experiments
Fitting Regression Models in Designed Experiments
  • Example 10-2: regression analysis of a 23 factorial design
    • Response: yield of a process
    • Variables: temperature, pressure, and concentration
    • Single replicate with 4 center points
fitting regression models in designed experiments1
Fitting Regression Models in Designed Experiments
  • A main effects only model
  • y = bo + b1x1 + b2x2 + b3x3 + e
fitting regression models in designed experiments2
Fitting Regression Models in Designed Experiments
  • The regression coefficients are exactly one-half of the effect estimates in a 2k design
  • Because the factorial designs are orthogonal designs, the off-diagonal elements in X´X are zero, or X´X is diagonal
  • Regression method is useful when the experiment (or data) is not perfect
  • Regression analysis of data with missing observations. Example 10-3: assuming run 8 of the observations in Example 10-2 was missing. Fit the main effect model using the remaining observations
  • y = bo + b1x1 + b2x2 + b3x3 + e
example 10 3
Example 10-3

Original model

fitting regression models in designed experiments3
Fitting Regression Models in Designed Experiments
  • Regression analysis of experiments with inaccurate factor levels
  • Example 10-4: assuming the process variables are not at their exact assumed values in Example 10-2. Fit the main effect model y = bo + b1x1 + b2x2 + b3x3 + e
example 10 4
Example 10-4

Original model

fitting regression models in designed experiments4
Fitting Regression Models in Designed Experiments
  • Regression analysis can be used to de-alias interactions in a fractional factorial using fewer than a full fold-over fraction in a resolution III design
  • Example 10-5: consider Example 8-1, assume effects A, B, C, D, and AB+CD were large – de-alias AB+CD using fewer than 8 additional runs. Consider the model
  • y = bo + b1x1 + b2x2 + b3x3 + b4x4 + b12x1x2 + b34x3x4 + e
slide16

The X matrix for the model is

  • Adding one run from the alternate fraction to the original 8 runs, the X matrix becomes
slide17

Hypothesis Testing in Multiple Regression

  • Measuring the usefulness of the model
  • Test for significance of regression – determine if there is a linear relationship between the response y and a subset of the regressor variables x1, x2, , xk.
  • Testing hypothesis
  • Ho: b1 = b2 = = bk = 0
  • H1: bj 0 for at least one j
  • Analysis of variance
  • SST = SSR + SSE
slide18

Hypothesis Testing in Multiple Regression

  • If Fo exceeds Fa,k,n-k-1, the null hypothesis is rejected
slide20

Hypothesis Testing in Multiple Regression

  • Testing individual and group of coefficients – determine if one or a group of regressor variables should be included in the model
  • Testing hypothesis (for an individual regression coefficient)
  • Ho: bj = 0
  • H1: bj 0
  • if Ho is not rejected, then xj can be deleted from the model.
  • Ho is rejected if |to| > ta/2,n-k-1
slide21

The contribution of a particular variable, or a group of variables can be quantified using sums of squares

  • Confidence intervals on individual regression coefficients
  • Example 10-7
slide22

Prediction of new response observations

  • The future observation yo at a point (xo1, xo2, ,xok) with x’o =[1, xo1, xo2, ,xok]
  • Regression model diagnostics
  • Testing for lack of fit
  • Sections 10-7, 8