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Analyzing Non-Normal Data with Generalized Linear Models 2010 LISA Short Course Series

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### Analyzing Non-Normal Data with Generalized Linear Models2010 LISA Short Course Series

Sai Wang, Dept. of Statistics

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

Normal: continuous, symmetric, mean μ and varσ2

Bernoulli: 0 or 1, mean p and var p(1-p)

special case of Binomial

Poisson: non-negative integer, 0, 1, 2, …, mean λvarλ

# of events in a fixed time interval

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

for i =1 to n, where n is # of obs

j= 1 to k, where k is # of predictors

Equivalently,

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 – 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

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 = 1 If predictor variable is equal to category k

0 Otherwise

xk = 0 For all k if reference category

Effect Coding (Used in JMP)

xk = 1 If predictor variable is equal to category k

0 Otherwise

xk = -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

-2 Log likelihood is used due to its distributional properties – Chi-square

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

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)]

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.

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 Xq+1, Xq+2, Xq+3,….,Xpare equal to zero

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, 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.

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

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

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

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

Generalized Linear Models

Questions/Comments

Generalized Linear Models

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

where 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.

Logistic Regression

Consider the logistic regression model

GLM with binomial random component and logit link g(μ) = logit(μ)

Range of values for pi is 0 to 1

Logistic Regression

Interpretation of Coefficient β – Odds Ratio

Odds: fraction of Prob(event)=p vsProb(not event)=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 p1and the probability of Event 2 is p2. Then the odds ratio of Event 2 to Event 1 is:

Logistic Regression

Interpretation of Coefficient β – Odds Ratio Cont.

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

Logistic Regression

Interpretation of Coefficient β – Odds Ratio Cont.

Link to Logistic Regression :

Thus the odds ratio of event 2 to event 1 is

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=1 given X (prob=p1) and Event 2 is Y=1 given X+1 (prob=p2)

From our logistic regression model

Thus the odds ratio of Y=1 for per unit increase in X is

Logistic Regression

Interpretation for a Continuous Predictor Variable

Consider the following JMP output:

Parameter Estimates

Term Estimate Std Error L-R ChiSquareProb>ChiSq Lower CL Upper CL

Intercept -11.832 4.7161554 9.9102818 0.0016* -23.38402 -3.975928

GPA 2.8261126 1.2629411 6.7842138 0.0092* 0.6391582 5.7567314

TUCE 0.0951577 0.1415542 0.4738788 0.4912 -0.170202 0.4050175

PSI[0] -1.189344 0.5322821 6.2036976 0.0127* -2.40494 -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.

Logistic Regression

Single Categorical Predictor Variable

Consider the following JMP output:

Parameter Estimates

Term Estimate Std Error L-R ChiSquareProb>ChiSq Lower CL Upper CL

Intercept -11.832 4.7161554 9.9102818 0.0016* -23.38402 -3.975928

GPA 2.8261126 1.2629411 6.7842138 0.0092* 0.6391582 5.7567314

TUCE 0.0951577 0.1415542 0.4738788 0.4912 -0.170202 0.4050175

PSI[0] -1.189344 0.5322821 6.2036976 0.0127* -2.40494 -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.

I

Logistic Regression

ROC Curve

Receiver Operating Curve

Sensitivity – Proportion of positive cases (Y=1) that were classified as a positive case by the model

Specificity - Proportion of negative cases (Y=0) that were classified as a negative case by the model

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

Summary

Introduction to the Logistic Regression Model

Interpretation of the Parameter Estimates β – Odds Ratio

ROC Curves

Teaching Method Example

Logistic Regression

Questions/Comments

Logistic 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 predictors

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 Upper and Lower Seams

Prevalence of overburden fracturing may lead to collapse

Poisson Regression

Consider the model

where Yi = Response for observation i

xi = 1x(k+1) vector of covariates for observation i

p = Number of covariates

μi = Expected number of events given xi

GLM with Normal random component and identity link g(μ) = μ

Issue: Predicted values range from -∞ to +∞

Poisson Regression

Consider the Poisson log-linear model

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 in x results an multiple of in μ

Poisson Regression

Continuous Predictor Variable

Consider the JMP output

Term Estimate Std Error L-R ChiSquareProb>ChiSq Lower CL Upper CL

Intercept -3.59309 1.0256877 14.113702 0.0002* -5.69524 -1.660388

Thickness -0.001407 0.0008358 3.166542 0.0752 -0.003162 0.0001349

Pct_Extraction 0.0623458 0.0122863 31.951118 <.0001* 0.0392379 0.0875323

Height -0.00208 0.0050662 0.174671 0.6760 -0.012874 0.0070806 Age -0.030813 0.0162649 3.8944386 0.0484* -0.064181 -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

Change linear predictor – XTβ

Add or subtract regressors, transform regressors, add interaction terms, etc.

Change link function – g(XTβ)

Change Random Component

Use Negative Binomial Distribution

Poisson Regression

Summary

Introduction to the Poisson Regression Model

Interpretation of β

Overdispersion

Mining Example

Poisson Regression

Questions/Comments

Poisson Regression

Mann–Whitney U test or Wilcoxon rank-sum test

Alternative to 2-sample T test for comparing measurements in two samples of indepobs

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

Non-parametric Tests

Kruskal–Wallis test

Alternative to ANOVA for comparing >2 groups

To compare measurements in >2 samples of indepobs

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

Non-parametric Tests

- 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

- Ex. Kernel Regression
- Estimation based on localized weighting average

- Brief introductions: http://en.wikipedia.org/wiki/Non-parametric
- Future LISA short course on Non-parametric methods?

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