HSRP 734: Advanced Statistical Methods June 5, 2008

HSRP 734: HSRP 734: Advanced Statistical Advanced Statistical

MethodsMethodsJune 5, 2008June 5, 2008

Introduction

• Categorical data analysis

– multinomial

– 2x2 and RxC analysis

– 2x2xK, RxCxK analysis

• Stratified analysis (CMH) considers the problem of controlling for other variable

Introduction

• Need to extend to scientific questions of higher dimension.

• When the number of potential covariates increases, traditional methods of contingency table analysis become limited

• One alternative approach to stratified analyses is the development of regression models that incorporate covariates and interactions among variables.

Introduction

• Logistic regression is a form of regression analysis in which the outcome variable is binary or dichotomous

• General theory: analysis of variance (ANOVA) and logistic regression all are special cases of General Linear Model (GLM)

OBJECTIVES

• To describe what simple and multiple logistic regression is and how to perform

• To describe maximum likelihood techniques to fit logistic regression models

• To describe Likelihood ratio and Wald tests

OBJECTIVES

• To describe how to interpret odds ratios for logistic regression with categorical and continuous predictors

• To describe how to estimate and interpret predicted probabilities from logistic models

• To describe how to do the above 5 using SAS Enterprise

What is Logistic Regression?

• In a nutshell:

A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables.

Used when the research method is focused on whether or not an event occurred, rather than when it occurred (time course information is not used).


• What is the “Logistic” component?

Instead of modeling the outcome, Y, directly, the method models the log odds(Y) using the logistic function.


• What is the “Regression” component?

Methods used to quantify association between an outcome and predictor variables. Could be used to build predictive models as a function of predictors.


0 20 40 60 80Age (yrs.)

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20 30 40 50 60 70Age

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DWhat is Logistic Regression?

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00 d

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Age (yrs.)

Fig 1. Logistic regression curves for the three drug combinations. The dashed reference line represents the probability of DLT of .33. The estimated MTD can be obtained as the value on the horizontal axis that coincides with a vertical line drawn through the point where the dashed line intersects the logistic curve. Taken from “Parallel Phase I Studies of Daunorubicin Given With Cytarabine and Etoposide With or Without the Multidrug Resistance Modulator PSC-833 in Previously Untreated Patients 60 Years of Age or Older With Acute Myeloid Leukemia: Results of Cancer and Leukemia Group B Study 9420” Journal of Clinical Oncology, Vol 17, Issue 9 (September), 1999: 283. http://www.jco.org/cgi/content/full/17/9/2831

http://www.jco.org/cgi/content/full/17/9/2831

What can we use Logistic Regression for?

• To estimate adjusted prevalence rates, adjusted for potential confounders (sociodemographic or clinical characteristics)

• To estimate the effect of a treatment on a dichotomous outcome, adjusted for other covariates

• Explore how well characteristics predict a categorical outcome

History of Logistic Regression

• Logistic function was invented in the 19th century to describe the growth of populations and the course of autocatalytic chemical reactions.

• Quetelet and Verhulst

• Population growth was described easiest by exponential growth but led to impossible values


• Logistic function was the solution to a differential equation that was examined from trying to dampen exponential population growth models.


• Published in 3 different papers around the 1840’s. The first paper showed how the logistic models agreed very well with the actual course of the populations of France, Belgium, Essex, and Russia for periods up to the early 1830’s.

( ) log(1 )

pLOGIT p zp

exp( )1 exp( )

zpz

exp( ) ln

(1 ) 1 expzpLOGIT p z p

p z

The Logistic Curve

z (log odds)

p (p

roba

bilit

y)

Logistic Regression

• Simple logistic regression = logistic regression with 1 predictor variable

• Multiple logistic regression = logistic regression with multiple predictor variables

• Multiple logistic regression = Multivariable logistic regression = Multivariate logistic regression

The Logistic Regression Model

0 1 1 2 2 K K

0 1 1 2 2 K K

Logistic Regression:

P Yln

1-P Y

Linear Regression:

Y

X X X

X X X


0 1 1 2 2 K K

P Yln

1-P YX X X

predictor variables

YP1

YPln is the log(odds) of the outcome.

dichotomous outcome


0 1 1 2 2 K K

P Yln

1-P YX X X

intercept

YP1


model coefficients

Logistic Regression uses Odds Ratios

• Does not model the outcome directly, which leads to effect estimates quantified by means (i.e., differences in means)

• Estimates of effect are instead quantified by “Odds Ratios”

Relationship between Odds & Probability

Probability eventOdds event =

1-Probability event

Odds eventProbability event

1+Odds event

The Odds Ratio

Definition of Odds Ratio: Ratio of two odds estimates.

So, if Pr(response | trt) = 0.40 and Pr(response | placebo) = 0.20

Then:

0.40Odds response| trt group 0.6671 0.40

25.020.01

20.0group placebo |responseOdds

0.667 OR Trt vs. Placebo 2.670.25

Interpretation of the Odds Ratio

•Example cont’d:

Outcome = response, 67.2OR plb trt vs.

Then, the odds of a response in the treatment group were estimated to be 2.67 times the odds of having a response in the placebo group.

Alternatively, the odds of having a response were 167% higher in the treatment group than in the placebo group.

Odds Ratio vs. Relative Risk

• An Odds Ratio of 2.67 for trt. vs. placebo does NOT mean that the outcome is 2.67 times as LIKELY to occur.

• It DOES mean that the ODDS of the outcome occurring are 2.67 times as high for trt. vs. placebo.

Odds Ratio vs. Relative Risk

• The Odds Ratio is NOT mathematically equivalent to the Relative Risk (Risk Ratio)

• However, for “rare” events, the Odds ratio can approximate the Relative risk (RR)

1-P response | trtOR=RR

1-P response | plb

Maximum Likelihood

Idea of Maximum Likelihood

• Flipped a fair coin 10 times: T, H, H, T, T, H, H, T, H, H

• What is the Pr(Heads) given the data? 1/100? 1/5? 1/2? 6/10?

• Did you do the home experiment?

T, H, H, T, T, H, H, T, H, H

• What is the Pr(Heads) given the data? • Most reasonable data-based estimate would be

6/10.

• In fact,

is the ML estimator of p.

flips of # totalheads of #ˆ

NXp

Maximum Likelihood

• The method of maximum likelihood estimation chooses values for parameter estimates (regression coefficients) which make the observed data “maximally likely.”

• Standard errors are obtained as a by-product of the maximization process


0 1 1 2 2 K K

P Yln

1-P YX X X

intercept

YP1


model coefficients

Maximum Likelihood

• We want to choose β’s that maximizes the probability of observing the data we have:

N

iiNN yyyyyyyL

12121 PrPrPrPr,,,Pr

Assumption: independent y’s

Maximum Likelihood

• Define p = Pr(y=1). Then for dichotomous outcome => Pr(y=0) = 1-Pr(y=1) = 1-p. Then:

yy ppy 11Pr

ppp

ppp

11)0Pr( 0,yFor

1)1Pr( 1,yFor

10

01

• So, given that Pr(y) = py(1-p)1-y :

N

i

yi

yi

N

ii

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1

1

1Pr

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ii

y

i

i

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y

i

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i

i

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• Taking the logarithm of both sides:

i

ii

i

ii p

ppyL 1ln

1lnln

Can you see why?

0 1 1 2 2 K K

P Yln

1-P YX X X

Remember that:

ix

• Substituting in using logistic regression model:

• Now we choose values of β that make this equation as large as possible.

• Maximizing the lnL => maximizes L

• Maximizing involves derivatives & iteration

i

iii

i xxyL exp1lnln

Maximum Likelihood

• The method of maximum likelihood estimation chooses values for parameter estimates which make the observed data “maximally likely.”

• ML estimators have great properties:– Unbiased (estimate true β’s)

– Asymptotically efficient (narrow CI’s)

– Asymptotically Normally distributed (can calculate CI’s and Test Statistics using familiar Z formulas)

Estimating a Logistic Regression Model

Steps:

• Observe data on outcome, Y, and charactersitiscs X1, X2, …, XK

• Estimate model coefficients using ML

• Perform inference: calculate confidence intervals, odds ratios, etc.


0 1 1 2 2 K K

P Yln

1-P YX X X

predictor variables

YP1


dichotomous outcome


0 1 1 2 2 K K

P Yln

1-P YX X X

intercept

YP1


model coefficients

Form for Predicted Probabilities

0 1 1 2 2 K K

0 1 1 2 2 K K

0 1 1 2 2 K K

P Yln

1-P Y

expP Y

1 exp

X X X

X X XX X X

In this latter form, the logistic regression model directly relates the probability of Y to the predictor variables.


0 1 1 2 2 K K

0 1 1 2 2 K K

Logistic Regression:

P Yln

1-P Y

Linear Regression:

Y

X X X

X X X

Why not use linear regression for dichotomous outcomes?

• If we model Y directly and Y is dichotomous, this necessarily violates the linear regression assumptions (homoscedasticity)

• One of the more intuitive reasons not to is that will end up with predicted Y’s other than 0 or 1 (possibly more extreme than 0 or 1).

Assumptions in logistic regression

• Assumptions in logistic regression

– Yi are from Bernoulli or binomial (ni, i) distribution

– Yi are independent

– Log odds P(Yi = 1) or logit P(Yi = 1) is a linear function of covariates

• Relationships among probability, odds and log odds

Measure Min Max NamePr(Y=1) 0 1 prob

0 ∞ odds

-∞ ∞ log odds

)1Pr(1

)1Pr(logY

Y

)1Pr(1)1Pr(

YY

Commonality between linear and logistic regression

• Operating on the logit scale allows a linear model that is similar to linear regression to be applied

• Both linear and logistic regression are apart of the family of Generalized Linear Models (GLM)

Logistic Regresion is a General Linear Model (GLM)

• Family of regression models that use the same general framework

• Outcome variable determines choice of model

Outcome GLM Model Continuous Linear regression Dichotomous Logistic regression Counts Poisson regression

Logistic Regression Models are estimated by Maximum Likelihood

• Using this estimation gives model coefficient estimates that are asymptotically consistent, efficient, and normally distributed.

• Thus, a 95% Confidence Interval for is given by:

K

2

,

KK z SE

L U

Logistic Regression Models are estimated by Maximum Likelihood

• The Odds Ratio for the kth model coefficient is:

• We can also get a 95% CI for the OR from:

KOR = exp

L U ,

where L , U is a 95% CI for K

e e

The Logistic Regression ModelExample:

In Assisted Reproduction Technology (ART) clinics, one of the main outcomes is clinical pregnancy.

There is much empirical evidence that the candidate mother’s age is a significant factor that affects the chances of pregnancy success.

A recent study examined the effect of the mother’s age, along with clinical characteristics, on the odds of pregnancy success on the first ART attempt.


Age13.067.2exp1

Age13.067.2exppregnancyPr

Age13.067.2pregnancyPr1

pregnancyPrln



pregnancyPrln

Q1. What is the effect of Age on Pregnancy?

88.013.0expOR Age

This implies that for every 1 yr. increase in age, the odds of pregnancy decrease by 12%.

A. The


Q2. What is the predicted probability of a 25 yr. old having pregnancy success with first ART attempt?


Age13.067.2exp1

Age13.067.2exppregnancyPr


pregnancyPrln


Q2. What is the predicted probability of a 25 yr. old having pregnancy success with first ART attempt?

A. From this model, a 25 yr. old has about a 36% chance of pregnancy success.

0.359

2513.067.2exp12513.067.2exppregnancyPr

Hypothesis testing

• Usually interested in testing

• Two types of tests we’ll discuss:

1. Likelihood Ratio test

2. Wald test

0:0 KH

Likelihood Ratio test

• Idea is to compare the (log) Likelihood of two models to test

• Two models:1. Full model = with predictor included2. Reduced model = without predictor

• Then,

0:0 KH

0.05)for 84.3 Critical 1;df (here

model fullin parameters extra of # df with ~

ˆln2ˆln2ˆˆ

ln2

21

2

FullReducedFull

Reduced

LLL

L

Wald test

• Idea is to use large sample Z statistic from a single model to test:

• Critical Z value for =0.05 is 1.96 (two-sided)

0:0 KH

1 ,0~ whereˆ

Here,ˆ

NZSE

ZK

K

Hypothesis testing

• As the sample size gets larger and larger, the Wald test will approximate the Likelihood ratio test.

• The LR test is preferred but Wald test is common

• Why? Not to scald the Wald but…

Predictive ability of Logistic regression

• Generalized R-squared statistics controversial

• ROC curve plots Sensitivity vs. 1-Specificity based on fitted model

• c statistic = Area under ROC curve commonly used to summarize predictive ability of model

SAS Enterprise:

chd.sas7bdat

Logistic Regression

• Motivating example

• Consider the problem of exploring how the risk for coronary heart disease (CHD) changes as a function of age.

• How would you test whether age is associated with CHD?

• What does a scatter plot of CHD and age look like?

20 30 40 50 60Age Group

0.0

0.2

0.4

0.6

0.8

Pro

port

ion

of C

HD

Logistic regression

Placebo"" vs.Drug" New" e th takingpersonsfor CHD of ratio odds 1 e

Taking the exp of β1 gives the odds ratio:

Placebo takingCHD of odds

Drug takingCHD of oddsexp 1

Logistic Regression

• We can add multiple predictor variables in modeling the log odds of getting CHD:

Agei = Person i’s age in years

)45()Pr(

)Pr(log 210

i

i

i AgeDrugCHDno

CHD

iPerson for Placebo if0,

Drug if1,Drug i

Interpretation for Drug

• Interpretation of coefficient 1 when there are more than one variable

– log odds ratio when the other variables are held constant

– e.g., log odds ratio between having CHD with and without drug, adjusting for age

Interpretation for Age

• Interpretation of coefficient 1 when there are more than one variable

• Interpretation of coefficient 1 for continuous covariate

– e.g., log odds ratio for 1 year of change in age (unit difference in covariate), adjusting for drug

• Maximum Likelihood Estimates:

• Conclusion:Strong evidence that the odds of CHD is associated with the drug.

?efor CI %95 ?for CI %95

?ˆ

?? ?ˆ

1

1

1

1

1

ˆ

1

ˆ

ˆ1

SEz

SEe

Likelihood Ratio test

• Idea is to compare the (log) Likelihood of two models to test

• Two models:1. Full model = with predictor included2. Reduced model = without predictor

• Then,

0:0 KH

0.05)for 84.3 Critical 1;df (here

model fullin parameters extra of # df with ~

ˆln2ˆln2ˆˆ

ln2

21

2

FullReducedFull

Reduced

LLL

L

Suggested exercises

• Read Kleinbaum Chapter 1,2,3 Detailed Outline

• Chapter 1 in Kleinbaum & KleinPractice Exercises (can check answers)

• No need to hand in

Looking ahead

• HW 3: Due June 12th

• Next 2 classes: Model Building, Diagnostics & Extensions

• Review & Exam 1

HSRP 734: Advanced Statistical Methods June 5, 2008

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