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Laboratory for Interdisciplinary Statistical Analysis LISA helps VT researchers benefit from the use of Statistics Experimental Design • Data Analysis • Interpreting Results Grant Proposals • Software (R, SAS, JMP, SPSS...) Walk-In Consulting Collaboration Monday—Friday* 12-2PM for questions requiring <30 mins From our website request a meeting for personalized statistical advice *Mon—Thurs during the summer Great advice right now: Meet with LISA before collecting your data Short Courses Designed to help graduate students apply statistics in their research All services are FREE for VT researchers. We assist with research—not class projects or homework. 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 • gi 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 gμ Xβ 1 1 x11 1 x 2 21 μ , X n 1 1 xn 1,1 n 1 xn1 n1 6 j xij x1k β0 β x2 k 1 ,β xn 1,k 1 βk 1 βk xnk n p p1 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 q1 + 0 X q2 +… 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 EYi 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 EYi 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 ( 1pp2 2 ) Log ( 1p1p1 ) 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 exp1 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 EYi 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 EYi | xi i exp xi β T logi 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 log2 log1 x 2 β x1 β, expx βx 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