Linear Regression Analysis 5E Montgomery, Peck & Vining 1 Chapter 13 Generalized Linear Models.

Linear Regression Analysis 5E Montgomery, Peck & Vining

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Chapter 13Generalized Linear Models


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Generalized Linear Models

• Traditional applications of linear models, such as DOX and multiple linear regression, assume that the response variable is – Normally distributed– Constant variance– Independent

• There are many situations where these assumptions are inappropriate– The response is either binary (0,1), or a count– The response is continuous, but nonnormal


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Some Approaches to These Problems• Data transformation

– Induce approximate normality– Stabilize variance– Simplify model form

• Weighted least squares– Often used to stabilize variance

• Generalized linear models (GLM)– Approach is about 25-30 years old, unifies linear and

nonlinear regression models– Response distribution is a member of the exponential family

(normal, exponential, gamma, binomial, Poisson)


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Generalized Linear Models• Original applications were in biopharmaceutical sciences • Lots of recent interest in GLMs in industrial statistics• GLMs are simple models; include linear regression and

OLS as a special case• Parameter estimation is by maximum likelihood

(assume that the response distribution is known)• Inference on parameters is based on large-sample or

asymptotic theory• We will consider logistic regression, Poisson regression,

then the GLM


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References

• Montgomery, D. C., Peck, E. A5, and Vining, G. G. (2012), Introduction to Linear Regression Analysis, 4th Edition, Wiley, New York (see Chapter 14)

• Myers, R. H., Montgomery, D. C., Vining, G. G. and Robinson, T.J. (2010), Generalized Linear Models with Applications in Engineering and the Sciences, 2nd edition, Wiley, New York

• Hosmer, D. W. and Lemeshow, S. (2000), Applied Logistic Regression, 2nd Edition, Wiley, New York

• Lewis, S. L., Montgomery, D. C., and Myers, R. H. (2001), “Confidence Interval Coverage for Designed Experiments Analyzed with GLMs”, Journal of Quality Technology 33, pp. 279-292

• Lewis, S. L., Montgomery, D. C., and Myers, R. H. (2001), “Examples of Designed Experiments with Nonnormal Responses”, Journal of Quality Technology 33, pp. 265-278

• Myers, R. H. and Montgomery, D. C. (1997), “A Tutorial on Generalized Linear Models”, Journal of Quality Technology 29, pp. 274-291


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Binary Response Variables

• The outcome ( or response, or endpoint) values 0, 1 can represent “success” and “failure”

• Occurs often in the biopharmaceutical field; dose-response studies, bioassays, clinical trials

• Industrial applications include failure analysis, fatigue testing, reliability testing

• For example, functional electrical testing on a semiconductor can yield: – “success” in which case the device works– “failure” due to a short, an open, or some other failure mode


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• Possible model:

• The response yi is a Bernoulli random variable

01

1,2,...,

0 or 1

k

i j ij i i ij i

i ny x

y

x

2

( 1) with 0 1

( 0) 1

( )

( ) (1 )i

i i i

i i

i i i i

i y i i

P y

P y

E y

Var y

x


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Problems With This Model

• The error terms take on only two values, so they can’t possibly be normally distributed

• The variance of the observations is a function of the mean (see previous slide)

• A linear response function could result in predicted values that fall outside the 0, 1 range, and this is impossible because

0 ( ) 1i i i iE y x


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Binary Response Variables – The Challenger Data

Temperature at Launch

At Least One O-ring Failure

Temperature at Launch

At Least One O-ring Failure

53 1 70 1

56 1 70 1

57 1 72 0

63 0 73 0

66 0 75 0

67 0 75 1

67 0 76 0

67 0 76 0

68 0 78 0

69 0 79 0

70 0 80 0

70 1 81 0

Data for space shuttle launches and static tests prior to the launch of Challenger


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• There is a lot of empirical evidence that the response function should be nonlinear; an “S” shape is quite logical

• See the scatter plot of the Challenger data

• The logistic response function is a common choice

exp( 1( )

1 exp( 1 exp(E y

x

x x


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The Logistic Response Function

• The logistic response function can be easily linearized. Let:

• Define

• This is called the logit transformation

and ( )E y x

ln1


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Logistic Regression Model

• Model:

• The model parameters are estimated by the method of maximum likelihood (MLE)

( )

where

( )

exp(

1 exp(

i i i

i i

i

i

y E y

E y

x

x


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A Logistic Regression Model for the Challenger Data (Using Minitab)

Binary Logistic Regression: O-Ring Fail versus Temperature

Link Function: Logit

Response Information

Variable Value Count

O-Ring F 1 7 (Event)

0 17

Total 24

Logistic Regression Table

Odds 95% CI

Predictor Coef SE Coef Z P Ratio Lower Upper

Constant 10.875 5.703 1.91 0.057

Temperat -0.17132 0.08344 -2.05 0.040 0.84 0.72 0.99

Log-Likelihood = -11.515


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A Logistic Regression Model for the Challenger Data

Test that all slopes are zero: G = 5.944, DF = 1, P-Value = 0.015

Goodness-of-Fit Tests

Method Chi-Square DF P

Pearson 14.049 15 0.522

Deviance 15.759 15 0.398

Hosmer-Lemeshow 11.834 8 0.159

exp(10.875 0.17132 )ˆ

1 exp(10.875 0.17132 )

xy

x


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Note that the fitted function has been extended down to 31 deg F, the temperature at which Challenger was launched


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Maximum Likelihood Estimation in Logistic Regression

• The distribution of each observation yi is

• The likelihood function is

• We usually work with the log-likelihood:

1( ) (1 ) , 1,2,...,i iy yi i i if y i n

1

1

( , ( ) (1 )i i

n ny y

i i i ii i

L f y

y

1 11

ln ( , ln ( ) ln ln(1 )1

n n ni

i i i ii ii i

L f y y

y


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• The maximum likelihood estimators (MLEs) of the model parameters are those values that maximize the likelihood (or log-likelihood) function

• ML has been around since the first part of the previous century

• Often gives estimators that are intuitively pleasing• MLEs have nice properties; unbiased (for large

samples), minimum variance (or nearly so), and they have an approximate normal distribution when n is large


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• If we have ni trials at each observation, we can write the log-likelihood as

• The derivative of the log-likelihood is 1

ln ( , ln[1 exp(n

i ii

L n

y X y x

1

1

ln ( ,exp(

1 exp(

because )

ni

i ii i

n

i i ii

i i i

nL

n

n

yX y x x

x

X y x

X y X


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• Setting this last result to zero gives the maximum likelihood score equations

• These equations look easy to solve…we’ve actually seen them before in linear regression:

( X y 0

1

results from OLS or ML with normal errors

Since ,

ˆ ˆ, and ) (OLS or the normal-theory MLE)

y X

X y 0

X X y X y X 0

X X X y X X X y


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• Solving the ML score equations in logistic regression isn’t quite as easy, because

• Logistic regression is a nonlinear model• It turns out that the solution is actually fairly easy, and is based on

iteratively reweighted least squares or IRLS (see Appendix for details)

• An iterative procedure is necessary because parameter estimates must be updated from an initial “guess” through several steps

• Weights are necessary because the variance of the observations is not constant

• The weights are functions of the unknown parameters

, 1, 2,...,1 exp(

ii

i

ni n

x


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Interpretation of the Parameters in Logistic Regression

• The log-odds at x is

• The log-odds at x + 1 is

• The difference in the log-odds is

0 1

ˆ( ) ˆ ˆˆ( ) lnˆ1 ( )

xx x

x

0 1

ˆ( 1) ˆ ˆˆ( 1) ln ( 1)ˆ1 ( 1)

xx x

x

1̂ˆ ˆ( 1) ( )x x


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Interpretation of the Parameters in Logistic Regression

• The odds ratio is found by taking antilogs:

• The odds ratio is interpreted as the estimated increase in the probability of “success” associated with a one-unit increase in the value of the predictor variable

1̂1ˆ xR

x

OddsO e

Odds


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Odds Ratio for the Challenger Data

This implies that every decrease of one degree in temperature increases the odds of O-ring failure by about 1/0.84 = 1.19 or 19 percent

The temperature at Challenger launch was 22 degrees below the lowest observed launch temperature, so now

This results in an increase in the odds of failure of 1/0.0231 = 43.34, or about 4200 percent!! There’s a big extrapolation here, but if you knew this prior to launch, what decision would you have made?

0.17132ˆ 0.84RO e

22( 0.17132)ˆ 0.0231RO e


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Inference on the Model Parameters


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Inference on the Model Parameters

See slide 15; Minitab calls this “G”.


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Testing Goodness of Fit


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Pearson chi-square goodness-of-fit statistic:


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The Hosmer-Lemeshow goodness-of-fit statistic:


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Refer to slide 15 for the Minitab output showing all three goodness-of-fit statistics for the Challenger data


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Likelihood Inference on the Model Parameters

• Deviance can also be used to test hypotheses about subsets of the model parameters (analogous to the extra SS method)

• Procedure:1 2 2 2

0 2

1 2

1 1

, with parameters, has parameters

This full model has deviance (

:

:

The reduced model is , with deviance ( )

The difference in deviance between the full and reduce

p r

H

H

X X

0

0

X

1 1

1 0

1 0

d models is

( | ) ( ) ( with degrees of freedom

( | ) has a chi-square distribution under :

Large values of ( | ) imply that : should be rejected

r

H

H

0

0


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Inference on the Model Parameters• Tests on individual model coefficients can also be done using

Wald inference• Uses the result that the MLEs have an approximate normal

distribution, so the distribution of

is standard normal if the true value of the parameter is zero. Some computer programs report the square of Z (which is chi-square), and others calculate the P-value using the t distribution See slide 14 for the Wald test on the temperature parameter for the Challenger data

0

ˆ

ˆ( )Z

se


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Another Logistic Regression Example: The Pneumoconiosis Data

• A 1959 article in Biometrics reported the data:


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The fitted model:


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Diagnostic Checking


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Consider Fitting a More Complex Model


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A More Complex Model

Is the expanded model useful? The Wald test on the term (Years)2 indicates that the term is probably unnecessary.

Consider the difference in deviance:

( (

( | ) ( (

with 1 df (chi-square P-value = 0.0961)

Compare the P-values for the Wald and deviance tests


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Other models for binary response data

Logit model

Probit model

Complimentary log-log model


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More than two categorical outcomes


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Poisson Regression• Consider now the case where the response is a count of

some relatively rare event:– Defects in a unit of product

– Software bugs

– Particulate matter or some pollutant in the environment

– Number of Atlantic hurricanes

• We wish to model the relationship between the count response and one or more regressor or predictor variables

• A logical model for the count response is the Poisson distribution

( ) , 0,1,..., and 0!

yef y y

y


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Poisson Regression• Poisson regression is another case where the response

variance is related to the mean; in fact, in the Poisson distribution

• The Poisson regression model is

• We assume that there is a function g that relates the mean of the response to a linear predictor

( ) and ( )E y Var y

( ) , 1, 2,...,i i i i iy E y i n

0 1

( )

...i i

i i k ik

i

g

x x

x


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Poisson Regression

• The function g is called a link function• The relationship between the mean of the response

distribution and the linear predictor is

• Choice of the link function:– Identity link

– Log link (very logical for the Poisson-no negative predicted values)

1 1( ) (i i ig g x

1

( ) ln( )

( i

i i i

i i

g

g e

x

x

x


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Poisson Regression• The usual form of the Poisson regression model is

• This is a special case of the GLM; Poisson response and a log link

• Parameter estimation in Poisson regression is essentially equivalent to logistic regression; maximum likelihood, implemented by IRLS

• Wald (large sample) and Deviance (likelihood-based) based inference is carried out the same way as in the logistic regression model

, 1, 2,...,ii iy e i n x


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An Example of Poisson Regression

• The aircraft damage data

• Response y = the number of locations where damage was inflicted on the aircraft

• Regressors:

1

2

3

0 = A-4 type of aircraft

1 = A-6

bomb load (tons)

total months of crew experience

x

x

x


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The table contains data from 30 strike missions

There is a lot of multicollinearity in this data; the A-6 has a two-man crew and is capable of carrying a heavier bomb load

All three regressors tend to increase monotonically


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Based on the full model, we can remove x3

However, when x3 is removed, x1 (type of aircraft) is no longer significant – this is not shown, but easily verified

This is probably multicollinearity at work

Note the Type 1 and Type 3 analyses for each variable

Note also that the P-values for the Wald tests and the Type 3 analysis (based on deviance) don’t agree


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Let’s consider all of the subset regression models:

Deleting either x1 or x2 results in a two-variable model that is worse than the full model

Removing x3 gives a model equivalent to the full model, but as noted before, x1 is insignificant

One of the single-variable models (x2) is equivalent to the full model


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The one-variable model with x2 displays no lack of fit (Deviance/df = 1.1791)

The prediction equation is

21.6491 0.2282ˆ xy e


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Another Example Involving Poisson Regression

•The mine fracture data•The response is a count of the number of fractures in the mine•The regressors are:

1

2

3

4

inner burden thickness (feet)

Percent extraction of the lower

previously mined seam

Lower seam height (feet)

Time in years that mine has been open

x

x

x

x


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The * indicates the best model of a specific subset size

Note that the addition of a term cannot increase the deviance (promoting the analog between deviance and the “usual” residual sum of squares)

To compare the model with only x1, x2, and x4 to the full model, evaluate the difference in deviance:

38.03 - 37.86 = 0.17

with 1 df. This is not significant.


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There is no indication of lack of fit: deviance/df = 0.9508

The final model is:

1 2 43.721 0.0015 0.0627 0.0317ˆ x x xy e


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The Generalized Linear Model• Poisson and logistic regression are two special cases of the GLM:

– Binomial response with a logistic link

– Poisson response with a log link

• In the GLM, the response distribution must be a member of the exponential family:

• This includes the binomial, Poisson, normal, inverse normal, exponential, and gamma distributions

( , , ) exp{[ ( )] / ( ) ( , )}

scale parameter

natural location parameter(s)

i i i i i i

i

f y y b a h y


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The Generalized Linear Model• The relationship between the mean of the response

distribution and the linear predictor is determined by the link function

• The canonical link is specified when

• The canonical link depends on the choice of the response distribution

1 1( ) (i i ig g x

i i


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Canonical Links for the GLM


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Links for the GLM

• You do not have to use the canonical link, it just simplifies some of the mathematics

• In fact, the log (non-canonical) link is very often used with the exponential and gamma distributions, especially when the response variable is nonnegative

• Other links can be based on the power family (as in power family transformations), or the complimentary log-log function


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Parameter Estimation and Inference in the GLM

• Estimation is by maximum likelihood (and IRLS); for the canonical link the score function is

• For the case of a non-canonical link,

• Wald inference and deviance-based inference is conducted just as in logistic and Poisson regression

( X y 0

(

( / )i idiag d d X y 0


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This is “classical data”; analyzed by many.

y = cycles to failure, x1 = cycle length, x2 = amplitude, x3 = load

The experimental design is a 33 factorial

Most analysts begin by fitting a full quadratic model using ordinary least squares


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DESIGN-EXPERT PlotCycles

LambdaCurrent = 1Best = -0.19Low C.I. = -0.54High C.I. = 0.22

Recommend transform:Log (Lambda = 0)

Lambda

Ln

(Re

sid

ua

lSS

)

Box-Cox Plot for Power Transforms

12.18

14.27

16.37

18.46

20.56

-3 -2 -1 0 1 2 3

Design-Expert V6 was used to analyze the data

A log transform is suggested


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Response: Cycles Transform: Natural log Constant: 0.000 ANOVA for Response Surface Linear ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 22.32 3 7.44 213.50 < 0.0001A 12.47 1 12.47 357.87 < 0.0001B 7.11 1 7.11 204.04 < 0.0001C 2.74 1 2.74 78.57 < 0.0001Residual 0.80 23 0.035Cor Total 23.12 26

Std. Dev. 0.19 R-Squared 0.9653Mean 6.34 Adj R-Squared 0.9608C.V. 2.95 Pred R-Squared 0.9520

PRESS 1.11 Adeq Precision 51.520

Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low High

Intercept 6.34 1 0.036 6.26 6.41

A-A 0.83 1 0.044 0.74 0.92 B-B -0.63 1 0.044 -0.72 -0.54 C-C -0.39 1 0.044 -0.48 -0.30

The Final Model is First-Order:

1 2 36.34 0.83 0.63 0.39ˆ x x xy e


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DESIGN-EXPERT Plot

Ln(Cycles)Design Points

X = A: AY = B: B

Actual FactorC: C = -1.00

Ln(Cycles)

A: A

B: B

-1.00 -0.50 0.00 0.50 1.00

-1.00

-0.50

0.00

0.50

1.00

5.84934

6.33631

6.82328

7.31025

7.89149

DESIGN-EXPERT Plot

Ln(Cycles)X = A: AY = B: B

Actual FactorC: C = -1.00

193.529

1043.84

1894.16

2744.48

3594.79

C

ycle

s

-1.00

-0.50

0.00

0.50

1.00

-1.00

-0.50

0.00

0.50

1.00

A: A B: B

Contour plot (log cycles) & response surface (cycles)


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A GLM for the Worsted Yarn Data

• We selected a gamma response distribution with a log link

• The resulting GLM (from SAS) is

• Model is adequate; little difference between GLM & OLS

• Contour plots (predictions) very similar

1 2 36.3489 0.8425 0.6313 0.3851ˆ x x xy e


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The SAS PROC GENMOD output for the worsted yarn experiment, assuming a first-order model in the linear predictor

Scaled deviance divided by df is the appropriate lack of fit measure in the gamma response situation


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Comparison of the OLS and GLM Models


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A GLM for the Worsted Yarn Data

• Confidence intervals on the mean response are uniformly shorter from the GLM than from least squares

• See Lewis, S. L., Montgomery, D. C., and Myers, R. H. (2001), “Confidence Interval Coverage for Designed Experiments Analyzed with GLMs”, JQT, 33, pp. 279-292

• While point estimates are very similar, the GLM provides better precision of estimation


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Residual Analysis in the GLM• Analysis of residuals is important in any model-fitting

procedure

• The ordinary or raw residuals are not the best choice for the GLM, because the approximate normality and constant variance assumptions are not satisfied

• Typically, deviance residuals are employed for model adequacy checking in the GLM.

• The deviance residuals are the square roots of the contribution to the deviance from each observation, multiplied by the sign of the corresponding raw residual:

ˆ( )iD i i ir d sign y y


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Deviance Residuals:• Logistic regression:

• Poisson regression:

ˆ

1 ( / )ln ( )

ˆ ˆ1

1ˆ

1 i

i i ii i i i

i i i

i

y y nd y n y

n

e

x

ˆ

ˆln ( )i

i

ii i i

yd y y e

e

x

x


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Deviance Residual Plots

• Deviance residuals behave much like ordinary residual in normal-theory linear models

• Normal probability plot is appropriate• Plot versus fitted values, usually transformed to

the constant-information scale:

1

ˆNormal responses,

ˆBinomial responses, 2sin ( )

ˆPoisson responses, 2

ˆGamma responses, 2 ln( )

i

i

i

i

y

y

y

y


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Deviance Residual Plots for the Worsted Yarn Experiment


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Overdispersion

• Occurs occasionally with Poisson or binomial data• The variance of the response is greater than one would

anticipate based on the choice of response distribution• For example, in the Poisson distribution, we expect the

variance to be approximately equal to the mean – if the observed variance is greater, this indicates overdispersion

• Diagnosis – if deviance/df greatly exceeds unity, overdispersion may be present

• There may be other reasons for deviance/df to be large, such as a poorly specified model, missing regressors, etc (the same things that cause the mean square for error to be inflated in ordinary least squares modeling)


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Overdispersion• Most direct way to model overdispersion is with a

multiplicative dispersion parameter, say , where

• A logical estimate for is deviance/df• Unless overdispersion is accounted for, the standard errors

will be too small.• The adjustment consists of multiplying the standard errors

by

( ) (1 ), binomial

( ) , Poisson

Var y

Var y

deviance/df


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The Wave-Soldering Experiment

• Response is the number of defects• Seven design variables:

– A = prebake condition

– B = flux denisty

– C = conveyor speed

– D = preheat condition

– E = cooling time

– F = ultrasonic solder agitator

– G = solder temperature


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One observation has been discarded, as it was suspected to be an outlier

This is a resolution IV design


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5 of 7 main effects significant; AC, AD, BC, and BD also significant

Overdispersion is a possible problem, as deviance/df is large

Overdispersion causes standard errors to be underestimated, and this could lead to identifying too many effects as significant

/

4.234 2.0577

deviance df


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After adjusting for overdispersion, fewer effects are significant

C, G, AC, and BD the important factors, assuming a 5% significance level

Note that the standard errors are larger than they were before, having been multiplied by

/

4.234 2.0577

deviance df


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The Edited Model for the

Wave-Soldering Experiment


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Generalized Linear Models

• The GLM is a unification of linear and nonlinear models that can accommodate a wide variety of response distributions

• Can be used with both regression models and designed experiments

• Computer implementations in Minitab, JMP, SAS (PROC GENMOD), S-Plus

• Logistic regression available in many basic packages• GLMs are a useful alternative to data transformation, and

should always be considered when data transformations are not entirely satisfactory

• Unlike data transformations, GLMs directly attack the unequal variance problem and use the maximum likelihood approach to account for the form of the response distribution

Linear Regression Analysis 5E Montgomery, Peck & Vining 1 Chapter 13 Generalized Linear Models.

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Transcript of Linear Regression Analysis 5E Montgomery, Peck & Vining 1 Chapter 13 Generalized Linear Models.