Transcript VT PowerPoint Template

Generalized Linear Models
2010 LISA Short Course Series
Mark Seiss, Dept. of Statistics
April 6, 2010
Presentation Outline
1. Introduction to Generalized Linear Models
2. Binary Response Data - Logistic Regression Model
3. Teaching Method Example
4. Count Response Data - Poisson Regression Model
5. Mining Example
6. Open Discussion
Reference Material



Categorical Data Analysis – Alan Agresti
Contemporary Statistical Models for Plant and Soil
Sciences – Oliver Schabenberger and F.J. Pierce
Presentation and Data from Examples

www.lisa.stat.vt.edu
Generalized Linear Models
• Generalized linear models (GLM) extend ordinary
regression to non-normal response distributions.
• Response distribution must come from the Exponential
Family of Distributions
• Includes Normal, Bernoulli, Binomial, Poisson, Gamma, etc.
• 3 Components
• Random –
Identifies response Y and its probability distribution
• Systematic – Explanatory variables in a linear predictor
function (Xβ)
• Link function – Invertible function (g(.)) that links the mean of the
response (E[Yi]=μi) to the systematic component.
Generalized Linear Models
• Model
•
gi     j xij
for i =1 to n
j
• Equivalently,
j= 1 to p


i  g 1    j xij 

j

Generalized Linear Models
• Why do we use GLM’s?
• Linear regression assumes that the response is distributed
normally
• GLM’s allow us to analyze the linear relationship
between predictor variables and the mean of the response
variable when it is not reasonable to assume the data is
distributed normally.
Generalized Linear Models
• Connection Between GLM’s and Multiple Linear
Regression
• Multiple linear regression is a special case of the GLM
• Response is normally distributed with variance σ2
• Identity link function μi = g(μi) = xiTβ
Generalized Linear Models
• Predictor Variables
• Two Types: Continuous and Categorical
• Continuous Predictor Variables
• Examples – Time, Grade Point Average, Test Score, etc.
• Coded with one parameter – βixi
• Categorical Predictor Variables
•
•
•
•
Examples – Sex, Political Affiliation, Marital Status, etc.
Actual value assigned to Category not important
Ex) Sex - Male/Female, M/F, 1/2, 0/1, etc.
Coded Differently than continuous variables
Generalized Linear Models
• Predictor Variables cont.
• Consider a categorical predictor variable with L
categories
• One category selected as reference category
• Assignment of reference category is arbitrary
• Some suggest assign category with most observations
• Variable represented by L-1 dummy variables
• Model Identifiability
Generalized Linear Models
• Predictor Variables cont.
• Two types of coding
• Dummy Coding (Used in R)
• xk =
• xk =
1
0
0
If predictor variable is equal to category k
Otherwise
For all k if predictor variable equals category i
• Effect Coding (Used in JMP)
• xk =
• xk =
1 If predictor variable is equal to category k
0 Otherwise
-1 For all k if predictor variable equals category i
Generalized Linear Models
• Model Evaluation - -2 Log Likelihood
• Specified by the random component of the GLM model
• For independent observations, the likelihood is the
product of the probability distribution functions of the
observations.
• -2 Log likelihood is -2 times the log of the likelihood
function
 n
 n
 2LogL  2 log  f ( yi )     2 log f ( yi )
 i 1
 i 1
• -2 Log likelihood is used due to its distributional
properties – Chi-square
Generalized Linear Models
• Saturated Model
• Contains a separate indicator parameter for each
observation
• Perfect fit μi = yi
• Not useful since there is no data reduction
• i.e. number of parameters equals number of observations
• Maximum achievable log likelihood (minimum
-2 Log L) – baseline for comparison to other model fits
Generalized Linear Models
• Deviance
• Let L(β|y) =
Maximum of the log likelihood for the
model
L(y|y) = Maximum of the log likelihood for the
saturated model
• Deviance = D(β) = -2 [L(β|y) - L(y|y)]
Generalized Linear Models
• Deviance cont.
 2Log(ˆ | y)
 2Log( y | y)
 2Log(ˆ0 | y)
  
D ˆ0  D ˆ

D ˆ
Model Chi-Square
 
D ˆ0
Generalized Linear Models
• Deviance cont.
• Lack of Fit test
• Likelihood Ratio Statistic for testing the null hypothesis that the
model is a good alternative to the saturated model
• Has an asymptotic chi-squared distribution with N – p degrees of
freedom, where p is the number of parameters in the model.
• Also allows for the comparison of one model to another
using the likelihood ratio test.
Generalized Linear Models
• Nested Models
• Model 1 - Model with p predictor variables
{X1, X2…,Xp} and vector of fitted values μ1
• Model 2 - Model with q<p predictor variables
{X1, X2,…,Xq} and vector of fitted values μ2
• Model 2 is nested within Model 1 if all predictor
variables found in Model 2 are included in Model 1.
• i.e. the set of predictor variables in Model 2 are a subset of the
set of predictor variables in Model 1
Generalized Linear Models
• Nested Models
• Model 2 is a special case of Model 1 - all the coefficients
corresponding to Xp+1, Xp+2, Xp+3,….,Xq are equal to zero
g(u) = 0  1 X1 + …+ p X p + 0  X p1 + 0  X p2 +… 0  X q
Generalized Linear Models
• Likelihood Ratio Test
• Null Hypothesis for Nested Models: The predictor
variables in Model 1 that are not found in Model 2 are
not significant to the model fit.
• Alternate Hypothesis for Nested Models - The predictor
variables in Model 1 that are not found in Model 2 are
significant to the model fit.
Generalized Linear Models
• Likelihood Ratio Test
• Likelihood Ratio Statistic =-2L(y,u2) - (-2L(y,u1))
= D(y,μ2) - D(y, μ1)
Difference of the deviances of the two models
• Always D(y,μ2) > D(y,μ1) implies LRT > 0
• LRT is distributed Chi-Squared with p-q degrees of
freedom
• Later, the Likelihood Ratio Test will be used to test the
significance of variables in Logistic and Poisson
regression models.
Generalized Linear Models
• Theoretical Example of Likelihood Ratio Test
• 3 predictor variables – 1 Continuous (X1), 1 Categorical
with 4 Categories (X2, X3, X4), 1 Categorical with 1
Category (X5)
• Model 1 - predictor variables {X1, X2, X3, X4, X5}
• Model 2 - predictor variables {X1, X5}
• Null Hypothesis – Variables with 4 categories is not
significant to the model (β2 = β3 = β4 = 0)
• Alternate Hypothesis - Variable with 4 categories is
significant
Generalized Linear Models
• Theoretical Example of Likelihood Ratio Test Cont.
• Likelihood Ratio Statistic = D(y,μ2) - D(y, μ1)
• Difference of the deviance statistics from the two models
• Equivalently, the difference of the -2 Log L from the two models
• Chi-Squared Distribution with 5-2=3 degrees of freedom
Generalized Linear Models
• Model Selection
• 2 Goals: Complex enough to fit the data well
Simple to interpret, does not overfit the data
• Study the effect of each predictor on the response Y
• Continuous Predictor – Graph Y versus X
• Discrete Predictor - Contingency Table of Mean of Y (μy)
versus categories of X
• Unbalance Data – Few responses of one type
• Guideline – 10 outcomes of each type for each X terms
• Example – Y=1 for only 30 observations out of 1000
Model should contain no more than 3 X terms
Generalized Linear Models
• Model Selection cont.
• Multicollinearity
• Correlations among predictors resulting in an increase in
variance
• Reduces the significance value of the variable
• Occurs when several predictor variables are used in the model
• Affects sign, size, and significance of parameter estimates
• Determining Model Fit
• Other criteria besides significance tests (i.e. Likelihood Ratio
Test) can be used to select a model
Generalized Linear Models
• Model Selection cont.
• Determining Model Fit cont.
• Akaike Information Criterion (AIC)
– Penalizes model for having many parameters
– AIC = Deviance+2*p where p is the number of parameters
in model
• Bayesian Information Criterion (BIC)
– BIC = -2 Log L + ln(n)*p where p is the number of
parameters in model and n is the number of observations
– Also known as the Schwartz Information Criterion (SIC)
Generalized Linear Models
• Model Selection cont.
• Selection Algorithms
• Best subset –
Tests all combinations of predictor variables
to find best subset
• Algorithmic – Forward, Backward and Stepwise
Procedures
Generalized Linear Models
• Stepwise Selection
• Idea: Combination of forward and backward selection
• Forward Step then backward step
• Step One:
• Step Two:
• Step Three:
• Step Four:
Fit each predictor variable as a single predictor
variable and determine fit
Select variable that produces best fit and add to
model
Add each predictor variable one at a time to the
model and determine fit
Select variable that produces best fit and add to
the model
Generalized Linear Models
• Stepwise Selection Cont.
• Step Five:
Delete each variable in the model one at a time
and determine fit
• Step Six:
Remove variable that produces best fit when
deleted
• Step Seven:
Return to Step Two
• Loop until no variables added or deleted improve the fit.
Generalized Linear Models
• Outlier Detection
• Studentized Residual Plot and Deviance Residual Plots
• Plot against predicted values
• Looking for “sore thumbs”, values much larger than those for
other observations
Generalized Linear Models
• Summary
• Setup of the Generalized Linear Model
• Continuous and Categorical Predictor Variables
• Log Likelihood
• Deviance and Likelihood Ratio Test
• Test lack of fit of the model
• Test the significance of a predictor variable or set of predictor
variables in the model.
• Model Selection
• Outlier Detection
Generalized Linear Models
• Questions/Comments
Logistic Regression
• Consider a binary response variable.
• Variable with two outcomes
• One outcome represented by a 1 and the other
represented by a 0
• Examples:
Does the person have a disease? Yes or No
Outcome of a baseball game?
Win or loss
Logistic Regression
• Teaching Method Data Set
• Found in Aldrich and Nelson (Sage Publications, 1984)
• Researcher would like to examine the effect of a new teaching
method – Personalized System of Instruction (PSI)
• Response variable is whether the student received an A in a statistics
class (1 = yes, 0 = no)
• Other data collected:
• GPA of the student
• Score on test entering knowledge of statistics (TUCE)
Logistic Regression
• Consider the linear probability model
T


E Yi  P(Yi  0 | xi )   ( xi )  xi 
where
yi =
xi =
p=
response for observation i
1x(p+1) matrix of covariates for
observation i
number of covariates
Logistic Regression
• GLM with binomial random component and identity link
g(μ) = μ
• Issues:
• π(Xi) can take on values less than 0 or greater than 1
• Predicted probability for some subjects fall outside of the [0,1]
range.
Logistic Regression
• Consider the logistic regression model
 
 
exp xi 
EYi   P(Yi  0 | xi )   ( xi ) 
T
1  exp xi 
T
  xi  
  xi T 

 logit xi   log
 1   xi  
• GLM with binomial random component and logit link
g(μ) = logit(μ)
• Range of values for π(Xi) is 0 to 1
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio
• The odds ratio is a statistic that measures the odds of an
event compared to the odds of another event.
• Say the probability of Event 1 is π1 and the probability of
Event 2 is π2 . Then the odds ratio of Event 1 to Event 2
is:
Odds( 1 ) 1 11
Odds_ Ratio 
 2
Odds( 2 )
1 2
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
• Value of Odds Ratio range from 0 to Infinity
• Value between 0 and 1 indicate the odds of Event 2 are
greater
• Value between 1 and infinity indicate odds of Event 1 are
greater
• Value equal to 1 indicates events are equally likely
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
• Link to Logistic Regression :
Log (Odds _ Ratio )  Log ( 11 1 )  Log ( 12 2 )  Logit ( 1 )  Logit ( 2 )
• Thus the odds ratio between two events is
Odds_ Ratio  exp{Logit( 2 )  Logit(1 )}
• Note: One should take caution when interpreting
parameter estimates
• Multicollinearity can change the sign, size, and significance of
parameters
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
• Consider Event 1 is Y=0 given X and Event 2 is Y=0
given X+1
• From our logistic regression model
Log(Odds_ Ratio)  Logit( P(Y  0 | X  1))  Logit( P(Y  0 | X ))
 (   ( X  1))  (  X )  
• Thus the ratio of the odds of Y=0 for X and X+1 is
Odds_ Ratio  exp( )
Logistic Regression
• Interpretation for a Continuous Predictor Variable
• Consider the following JMP output:
Parameter Estimates
Term
Estimate
Intercept
11.8320025
GPA
-2.8261125
TUCE
-0.0951577
PSI[0]
1.18934379
Std Error
4.7161552
1.262941
0.1415542
0.5322821
ChiSquare
6.29
5.01
0.45
4.99
Prob>ChiSq
0.0121*
0.0252*
0.5014
0.0255*
Interpretation of the Parameter Estimate:
Exp{-2.8261125} = 0.0592 =
Odds ratio between the odds at x+1 and
odds at x for all x
The ratio of the odds of NOT getting an A between a person with a 3.0
gpa and 2.0 gpa is equal to 0.0592 or in other words the odds of the
person with the 3.0 is 0.0592 times the odds of the person with the 2.0.
Equivalently, the odds of getting an A for a person with a 3.0 gpa is
equal to 1/0.0592=16.8919 times the odds of getting an A for a person
with a 2.0 gpa.
Logistic Regression
• Single Categorical Predictor Variable
• Consider the following JMP output:
Parameter Estimates
Term
Estimate
Intercept
11.8320025
GPA
-2.8261125
TUCE
-0.0951577
PSI[0]
1.18934379
Std Error
4.7161552
1.262941
0.1415542
0.5322821
ChiSquare
6.29
5.01
0.45
4.99
Prob>ChiSq
0.0121*
0.0252*
0.5014
0.0255*
Interpretation of the Parameter Estimate:
Exp{2*1.1893} = 10.78 = Odds ratio between the odds of NOT getting
an A for a student that was not subject to the teaching method and the
odds of NOT getting an A for a student that was subject to the teaching
method.
The odds of getting an A without the teaching method is 1/10.78=0.0927
times the odds of getting an A with the teaching method.
Logistic Regression
• ROC Curve
• Receiver Operating Curve
• Sensitivity –
Proportion of positive cases (Y=1) that
were classified as a positive case by the
model
P( yˆ  1 | y  1)
• Specificity -
Proportion of negative cases (Y=0) that
were classified as a negative case by the
model
P( yˆ  0 | y  0)
Logistic Regression
• ROC Curve Cont.
• Cutoff Value - Selected probability where all cases in
which predicted probabilities are above the cutoff are
classified as positive (Y=1) and all cases in which the
predicted probabilities are below the cutoff are classified
as negative (Y=0)
• 0.5 cutoff is commonly used
• ROC Curve – Plot of the sensitivity versus one minus the
specificity for various cutoff values
• False positives (1-specificity) on the x-axis and True positives
(sensitivity) on the y-axis
Logistic Regression
• ROC Curve Cont.
• Measure the area under the ROC curve
• Poor fit – area under the ROC curve approximately equal to 0.5
• Good fit – area under the ROC curve approximately equal to 1.0
Logistic Regression
• Teaching Method Example
Logistic Regression
• Summary
• Introduction to the Logistic Regression Model
• Interpretation of the Parameter Estimates β – Odds Ratio
• ROC Curves
• Teaching Method Example
Logistic Regression
• Questions/Comments
Poisson Regression
• Consider a count response variable.
• Response variable is the number of occurrences in a
given time frame.
• Outcomes equal to 0, 1, 2, ….
• Examples:
Number of penalties during a football game.
Number of customers shop at a store on a given day.
Number of car accidents at an intersection.
Poisson Regression
• Mining Data Set
• Found in Myers (1990)
• Response of interest is the number of fractures that occur
in upper seam mines in the coal fields of the Appalachian
region of western Virginia
• Want to determine if fractures is a function of the
material in the land and mining area
• Four possible regressors
•
•
•
•
Inner burden thickness
Percent extraction of the lower previously mined seam
Lower seam height
Years the mine has been open
Poisson Regression
• Mining Data Set Cont.
• Coal Mine Seam
Poisson Regression
• Mining Data Set Cont.
• Coal Mine Upper and Lower Seams
• Prevalence of overburden fracturing may lead to collapse
Poisson Regression
• Consider the model
EYi   i  xi 
T
where
Yi =
xi =
p=
μi =
Response for observation i
1x(p+1) matrix of covariates for
observation i
Number of covariates
Expected number of events given xi
• GLM with Poisson random component and identity link
g(μ) = μ
• Issue: Predicted values range from -∞ to +∞
Poisson Regression
• Consider the Poisson log-linear model
 
EYi | xi   i  exp xi 
T

logi   xi 
T
• GLM with Poisson random component and log link
g(μ) = log(μ)
• Predicted response values fall between 0 and +∞
• In the case of a single predictor, An increase of one unit
of x results an increase of exp(β) in μ
Poisson Regression
• Continuous Predictor Variable
• Consider the JMP output
Term
Intercept
Thickness
Pct_Extraction
Height
Age
Estimate
-3.59309
-0.001407
0.0623458
-0.00208
-0.030813
Std Error
1.0256877
0.0008358
0.0122863
0.0050662
0.0162649
L-R ChiSquare Prob>ChiSq
14.113702
0.0002*
3.166542
0.0752
31.951118
<.0001*
0.174671
0.6760
3.8944386
0.0484*
Lower CL
-5.69524
-0.003162
0.0392379
-0.012874
-0.064181
Upper CL
-1.660388
0.0001349
0.0875323
0.0070806
-0.000202
Interpretation of the parameter estimate:
Exp{-0.0308} = .9697 = multiplicative effect on the
expected number of fractures for an increase of 1 in the
years the mine has been opened
Poisson Regression
• Overdispersion for Poisson Regression Models
• More variability in the response than the model allows
• For Yi~Poisson(λi), E [Yi] = Var [Yi] = λi
• The variance of the response is much larger than the
mean.
• Consequences: Parameter estimates are still consistent
Standard errors are inconsistent
• Detection: D(β)/n-p
• Large if overdispersion is present
Poisson Regression
• Overdispersion for Poisson Regression Models Cont.
• Remedies
1. Change linear predictor – XTβ
– Add or subtract regressors, transform regressors,
add interaction terms, etc.
2. Change link function – g(XTβ)
3. Change Random Component
– Use Negative Binomial Distribution
Poisson Regression
• Mining Example
Poisson Regression
• Summary
• Introduction to the Poisson Regression Model
• Interpretation of β
• Overdispersion
• Mining Example
Poisson Regression
• Questions/Comments
Generalized Linear Models
• Open Discussion