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1
www.lisa.stat.vt.edu
Analyzing Non-Normal Data with
Generalized Linear Models
2010 LISA Short Course Series
Sai Wang, Dept. of Statistics
Presentation Outline
1. Introduction to Generalized Linear Models
2. Binary Response Data - Logistic Regression Model
Ex. Teaching Method
3. Count Response Data - Poisson Regression Model
Ex. Mining Example
4. Non-parametric Tests
3
Normal: continuous, symmetric, mean μ and var σ2
Bernoulli: 0 or 1, mean p and var p(1-p)
special case of Binomial
Poisson: non-negative integer, 0, 1, 2, …, mean λ var λ
# of events in a fixed time interval
4
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.
5
Generalized Linear Models
• Model
• gi   0 
 x
j ij
for i =1 to n, where n is # of obs
j
j= 1 to k, where k is # of predictors



• Equivalently, i  g 1   0 
• In matrixform:
j

gμ   Xβ
 1 
1 x11
 
1 x
2
21



μ     , X   



  n 1 
1 xn 1,1
  n 
1 xn1
n1
6

  j xij 

x1k 
 β0 
 β 

x2 k 
 1 

  ,β    



 xn 1,k 1 
 βk 1 
 βk 

xnk  n p
p1
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.
7
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
8
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 – βjxj
• 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
9
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
10
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 reference category
• Effect Coding (Used in JMP)
• xk =
• xk =
11
1 If predictor variable is equal to category k
0 Otherwise
-1 For all k if predictor variable is reference category
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
12
Generalized Linear Models
• Saturated Model (Perfect Fit 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
13
Generalized Linear Models
• Deviance
• Let L(β|y) =
Maximum of the log likelihood for a
proposed model
L(y|y) = Maximum of the log likelihood for the
saturated model
• Deviance = D(β) = -2 [L(β|y) - L(y|y)]
14
Generalized Linear Models
• Deviance cont.
 2Log(ˆ | y)
 2Log( y | y)
  
D ˆ0  D ˆ

D ˆ
Model Chi-Square
 
D ˆ0
15
 2Log(ˆ0 | y)
Generalized Linear Models
• Deviance cont.
• Lack of Fit test
• Likelihood Ratio Statistic for testing the null hypothes is 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.
16
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
17
Generalized Linear Models
• Nested Models
• Model 2 is a special case of Model 1 - all the coefficients
corresponding to Xq+1, Xq+2, Xq+3,….,Xp are equal to zero
g(u) = 0  1 X1 + …+ q X q + 0  X q1 + 0  X q2 +… 0  X p
18
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.
19
Generalized Linear Models
• Likelihood Ratio Test
• Likelihood Ratio Statistic =-2L(y, μ2) - (-2L(y, μ1))
= 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.
20
Generalized Linear Models
• Theoretical Example of Likelihood Ratio Test
• 3 predictor variables – 1 Continuous (X1: GPA), 1
Categorical with 4 Categories (X2, X3, X4, Year in
college), 1 Categorical with 2 Category (X5: Sex)
• 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
21
Generalized Linear Models
• Theoretical Example of Likelihood Ratio Test Cont.
• Likelihood Ratio Test 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
22
Generalized Linear Models
• Model Comparison
• Determining Model Fit cont.
• Akaike Information Criterion (AIC)
– Penalizes model for having many parameters
– AIC = -2 Log L +2*p where p is the number of parameters
in model, small is better
• 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
– Usually stronger penalization for additional parameter than
AIC
23
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 Comparison
24
Generalized Linear Models
• Questions/Comments
25
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
26
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)
27
Logistic Regression
• Consider the linear probability model
EYi   P(Yi  1 | xi )  pi  xi 
T
where
28
yi = response for observation i
xi = 1 x p vector of covariates for
observation i
p = 1+k, number of parameters
Logistic Regression
• GLM with binomial random component and identity link
g(μ) = μ
• Issues:
• pi can take on values less than 0 or greater than 1
• Predicted probability for some subjects fall outside of the [0,1]
range.
29
Logistic Regression
• Consider the logistic regression model
 
 
T
exp x i β
EYi   P(Yi  1 | xi )  pi 
T
1  exp xi β
 pi 
  x i Tβ

 logit pi   log
 1  pi 
• GLM with binomial random component and logit link
g(μ) = logit(μ)
• Range of values for pi is 0 to 1
30
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio
• Odds: fraction of Prob(event)=p vs Prob(not event)=1-p
p
O
1 p
• The odds ratio is a statistic that measures the odds of an
event compared to the odds of another event.
Ex. Say the probability of Event 1 is p1 and the
probability of Event 2 is p2. Then the odds ratio of Event
2 to Event 1 is:
O
OR21  2 
O1
31
p2
p1
1 p2
1 p1
, or O2  OR21  O1
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
O
OR 21  2 
O1
p2
p1
1 p 2
1 p1
, or O 2  OR 21  O1
suppose OR 21  10
3
3
p1  .75  , O1   3
4
1
 O 2  OR 21  O1  30 
30
30
, p2 
 .97, amountof changeis .97 - .75  .22
1
31
1
1
p1  .50  , O1   1
2
1
 O 2  OR 21  O1  10 
32
10
10
, p2 
 .91, amountof changeis .91- .50  .41
1
11
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 1 are
greater
• Value between 1 and infinity indicate odds of Event 2 are
greater
• Value equal to 1 indicates events are equally likely
33
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
• Link to Logistic Regression :
Log (OR 21 )  Log ( 1pp2 2 )  Log ( 1p1p1 )  Logit ( p2 )  Logit ( p1 )
• Thus the odds ratio of event 2 to event 1 is
OR21  expLogit ( p2 )Logit ( p1 )
• Note: One should take caution when interpreting
parameter estimates
• Multicollinearity can change the sign, size, and significance of
parameters
34
Logistic Regression
• Interpretation of Coefficient β – Odds Ratio Cont.
• Consider Event 1 is Y=1 given X (prob=p1) and Event 2
is Y=1 given X+1 (prob=p2)
• From our logistic regression model
Log(OR21 )  Logit( P2 )  Logit( P1 )
 (0  1 ( X  1))  (0  1 X )  1
• Thus the odds ratio of Y=1 for per unit increase in X is
OR21  exp1
35
Logistic Regression
• Interpretation for a Continuous Predictor Variable
• Consider the following JMP output:
Parameter Estimates
Term
Estimate
Intercept
-11.832
GPA
2.8261126
TUCE
0.0951577
PSI[0]
-1.189344
Std Error
4.7161554
1.2629411
0.1415542
0.5322821
L-R ChiSquare Prob>ChiSq
9.9102818
0.0016*
6.7842138
0.0092*
0.4738788
0.4912
6.2036976
0.0127*
Lower CL
-23.38402
0.6391582
-0.170202
-2.40494
Upper CL
-3.975928
5.7567314
0.4050175
-0.239233
Interpretation of the Parameter Estimate:
exp2.8261125 = 16.8797 = Odds ratio between the odds at x+1 and
odds at x for any gpa score
The ratio of the odds of getting an A between a person with a 3.0 gpa
and 2.0 gpa is equal to 16.8797 or in other words the odds of the person
with the 3.0 is 16.8797 times the odds of the person with the 2.0.
Equivalently, the odds of NOT getting an A for a person with a 3.0 gpa
is equal to 1/16.8797 =0.0592 times the odds of NOT getting an A for a
person with a 2.0 gpa.
36
Logistic Regression
• Single Categorical Predictor Variable
• Consider the following JMP output:
Parameter Estimates
Term
Estimate
Intercept
-11.832
GPA
2.8261126
TUCE
0.0951577
PSI[0]
-1.189344
Std Error
4.7161554
1.2629411
0.1415542
0.5322821
L-R ChiSquare Prob>ChiSq
9.9102818
0.0016*
6.7842138
0.0092*
0.4738788
0.4912
6.2036976
0.0127*
Lower CL
-23.38402
0.6391582
-0.170202
-2.40494
Upper CL
-3.975928
5.7567314
0.4050175
-0.239233
Interpretation of the Parameter Estimate:
exp 2*-1.1893 = 0.0928 = Odds ratio between the odds of getting an A for a
student that was not subject to the teaching method and for a student that
was subject to the teaching method.
The odds of NOT getting an A without the teaching method is
1/0.0928=10.7898 times the odds of NOT getting an A with the teaching
method.
37
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)
38
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
39
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
40
Logistic Regression
• Summary
• Introduction to the Logistic Regression Model
• Interpretation of the Parameter Estimates β – Odds Ratio
• ROC Curves
• Teaching Method Example
41
Logistic Regression
• Questions/Comments
42
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.
43
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 predictors
•
•
•
•
44
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
45
Poisson Regression
• Mining Data Set Cont.
• Coal Mine Upper and Lower Seams
• Prevalence of overburden fracturing may lead to collapse
46
Poisson Regression
• Consider the model
EYi   i  xi β
T
where
Yi =
xi =
p=
μi =
Response for observation i
1x(k+1) vector of covariates for
observation i
Number of covariates
Expected number of events given xi
• GLM with Normal random component and identity link
g(μ) = μ
• Issue: Predicted values range from -∞ to +∞
47
Poisson Regression
• Consider the Poisson log-linear model
 
EYi | xi   i  exp xi β
T

logi   xi β
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 by one unit
β
in x results an multiple of exp 1 in μ
2
T
T
log2   log1   x 2 β  x1 β,
 expx βx β
1
T
2
48
T
1
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
49
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
50
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
51
Poisson Regression
• Summary
• Introduction to the Poisson Regression Model
• Interpretation of β
• Overdispersion
• Mining Example
52
Poisson Regression
• Questions/Comments
53
Non-parametric Tests
• Mann–Whitney U test or Wilcoxon rank-sum test
• Alternative to 2-sample T test for comparing
measurements in two samples of indep obs
• Measurement is not interval, distribution is unclear
• Rather than using original values, test statistic based on
ranks
• Pros: no normality assumption, robust to outliers
• Cons: less powerful than t-test if normality holds
54
Non-parametric Tests
• Kruskal–Wallis test
• Alternative to ANOVA for comparing >2 groups
• To compare measurements in >2 samples of indep obs
• It is an extension of the Mann–Whitney U test to 3 or
more groups.
• If Kruskal-Wallis test is significant, perform pair-wise
multiple-comparisons using Mann–Whitney U with
adjusted significance level
55
Non-parametric Models
• Non-parametric Models
• Objective is to find a unknown non-linear relationship
between a pair of random variables X and Y
• Different from parametric models in that the model
structure is not specified a priori but is instead
determined from data.
• ‘Non-parametric’ does not imply absolute absence of
parameters
56
Non-parametric Models
• Ex. Kernel Regression
• Estimation based on localized weighting average
57
Non-parametric Models
• Brief introductions:
http://en.wikipedia.org/wiki/Non-parametric
• Future LISA short course on Non-parametric methods?
58

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