Chapter 13 Generalized Linear Models

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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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Binary Response Variables

• Possible model
• The response yi is a Bernoulli random variable

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Problems With This Model

• The error terms take on only two values, so they
  cant 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

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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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Binary Response Variables

• 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

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The Logistic Response Function

• The logistic response function can be easily
  linearized. Let
• Define
• This is called the logit transformation

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Logistic Regression Model

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

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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
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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

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Maximum Likelihood Estimation in Logistic
Regression

• 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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Maximum Likelihood Estimation in Logistic
Regression

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

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Maximum Likelihood Estimation in Logistic
Regression

• Setting this last result to zero gives the
  maximum likelihood score equations
• These equations look easy to solveweve actually
  seen them before in linear regression

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Maximum Likelihood Estimation in Logistic
Regression

• Solving the ML score equations in logistic
  regression isnt 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

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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

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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

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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!!
• Theres a big extrapolation here, but if you
  knew this prior to launch, what decision would
  you have made?

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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

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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

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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
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

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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

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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)

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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

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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

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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) dont agree
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Lets 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
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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

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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
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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

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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

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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

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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 V6 was used to analyze the data A
log transform is suggested
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The Final Model is First-Order
Response Cycles Transform Natural
log Constant 0.000 ANOVA for Response
Surface Linear Model Analysis of variance table
Partial sum of squares Sum of Mean F Source
Squares DF Square Value Prob gt
F Model 22.32 3 7.44 213.50 lt 0.0001 A 12.47 1 1
2.47 357.87 lt 0.0001 B 7.11 1 7.11 204.04 lt
0.0001 C 2.74 1 2.74 78.57 lt 0.0001 Residual 0.8
0 23 0.035 Cor Total 23.12 26 Std.
Dev. 0.19 R-Squared 0.9653 Mean 6.34 Adj
R-Squared 0.9608 C.V. 2.95 Pred
R-Squared 0.9520 PRESS 1.11 Adeq
Precision 51.520 Coefficient Standard 95
CI 95 CI Factor 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
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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

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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

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Deviance Residuals

• Logistic regression
• Poisson regression

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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

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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

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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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The Wave-Soldering Experiment
One observation has been discarded, as it was
suspected to be an outlier This is a resolution
IV design
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The Wave-Soldering Experiment
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
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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
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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