PH6415 Review PH6415 Review QuestionsQuestions

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Question 1Question 1

A journal article reports a 95%CI for the A journal article reports a 95%CI for the relative risk (RR) of an event (treatment relative risk (RR) of an event (treatment versus control as (0.55, 0.97). What can be versus control as (0.55, 0.97). What can be said of the p-value associated with testing said of the p-value associated with testing Ho: RR=1 vs. Ha: RR not equal 1?Ho: RR=1 vs. Ha: RR not equal 1?

The p-value is < 0.01.The p-value is < 0.01. The p-value is < 0.05.The p-value is < 0.05. The p-value is > 0.05The p-value is > 0.05 No statement can be said about the p-No statement can be said about the p-

value.value.

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Question 2Question 2

If S (t) is the survival function and t is in If S (t) is the survival function and t is in years what is the meaning of S(3) .years what is the meaning of S(3) .

The probability of dying at year 3.The probability of dying at year 3. The probability of surviving to year 3.The probability of surviving to year 3. The probability of dying by year 3 The probability of dying by year 3 The hazard of dying at year 3.The hazard of dying at year 3.

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Question 3Question 3

In In logistic regressionlogistic regression with a continuous with a continuous variable age what is the meaning of variable age what is the meaning of 11 ? ?

The difference in log odds between two The difference in log odds between two persons 1 year apart in agepersons 1 year apart in age

The relative odds between two persons The relative odds between two persons 1 year apart in age1 year apart in age

The difference in probabilities between The difference in probabilities between two persons 1 year apart in agetwo persons 1 year apart in age

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Question 4Question 4

If the probability of developing diabetes is If the probability of developing diabetes is 0.20 among Hispanics and 0.15 among 0.20 among Hispanics and 0.15 among whites, what is the whites, what is the relative oddsrelative odds (Hispanics v white) of developing (Hispanics v white) of developing diabetes. diabetes.

1.421.42 0.700.70 0.750.75 1.331.33

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Question 5Question 5

Suppose the Suppose the logistic regressionlogistic regression model: log model: log odds = odds = 00 + + 11XX11 + + 22XX22 + +33XX11*X*X22 where X where X11 is is an indicator for treatment and Xan indicator for treatment and X22 is an is an indicator for male gender. The indicator for male gender. The relative oddsrelative odds (treatment versus no treatment) for women (treatment versus no treatment) for women is: is:

exp(exp(11)) exp(exp(22)) exp(exp(11 + + 33)) Exp( Exp( 11 - - 33))

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Question 6Question 6

The probability and odds of an event will The probability and odds of an event will be nearly equal if: be nearly equal if:

The probability of the event is smallThe probability of the event is small The probability of the event is largeThe probability of the event is large The probability of the event is 0.50The probability of the event is 0.50

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Cox Proportional Hazards Cox Proportional Hazards Regression in SAS Regression in SAS

A ReviewA ReviewGoto:Goto:www.biostat.umn.edu/~susant/PH6415DATA.htmlwww.biostat.umn.edu/~susant/PH6415DATA.html CC• Read uis.readme file on website.Read uis.readme file on website.• Input data from uis (SAS data set)Input data from uis (SAS data set)

• Use Use Proc Lifetest Proc Lifetest to plot the to plot the Kaplan-Meier CurveKaplan-Meier Curve for for each categorical predictor separately. Look to see if the each categorical predictor separately. Look to see if the survival curves are approximately parallel and if there survival curves are approximately parallel and if there appears to be a difference in survival. appears to be a difference in survival.

• Use Use Proc PHREGProc PHREG to with model containing age, number to with model containing age, number of previous drug treatments, treatment and site.of previous drug treatments, treatment and site.

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Cox Proportional Hazards Cox Proportional Hazards Regression in SAS Regression in SAS

A ReviewA Review CC• Consider the interaction between age and site. Is this Consider the interaction between age and site. Is this

interaction significant? interaction significant?

• Consider the Consider the final model of age, number of previous drug final model of age, number of previous drug treatments, site and age_sitetreatments, site and age_site..

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Questions About the Questions About the Survival CurvesSurvival Curves

• What does the log-rank test of equality across strata What does the log-rank test of equality across strata indicate for the survival curves of the short and long indicate for the survival curves of the short and long treatment programs?treatment programs?

• What does the log-rank test of equality across strata What does the log-rank test of equality across strata indicate for the survival curves of the two different sites? indicate for the survival curves of the two different sites? Why might the p-value for the log-rank test be inflated?Why might the p-value for the log-rank test be inflated?

• What does the log-rank test of equality across strata What does the log-rank test of equality across strata indicate for the three combinations of heroine and indicate for the three combinations of heroine and cocaine use? Do the curves overlap? cocaine use? Do the curves overlap?

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Questions about Survival Questions about Survival DataData

• What is the median time to relapse for those at site A? What is the median time to relapse for those at site A? What is the median time to relapse for those at site B?What is the median time to relapse for those at site B?

• How many people relapsed at site A? What percent of How many people relapsed at site A? What percent of site A relapsed? How many people relapse at site B? site A relapsed? How many people relapse at site B? What percent of site B relapsed?What percent of site B relapsed?

• When did the first person relapse at site A? When did When did the first person relapse at site A? When did the first person relapse at site B?the first person relapse at site B?

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Questions about CensoringQuestions about Censoring

• What percent of people where censored in the long What percent of people where censored in the long treatment program compared to the short treatment?treatment program compared to the short treatment?

• For both treatment groups, does that censoring appear For both treatment groups, does that censoring appear to be patients who do not relapse or patients who are to be patients who do not relapse or patients who are loss to follow-up?loss to follow-up?

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Questions about Questions about parameters in Cox parameters in Cox

Proportional HazardsProportional Hazards• What is the relative risk of relapse for a one unit increase What is the relative risk of relapse for a one unit increase

in previous drug treatments if all other variables are held in previous drug treatments if all other variables are held constant? This represents a ________ percent increase constant? This represents a ________ percent increase in rate of relapse.in rate of relapse.

• If treatment length is altered from short(trt =0) to long If treatment length is altered from short(trt =0) to long (trt=1), while holding all other variables constant, the rate (trt=1), while holding all other variables constant, the rate of relapse decreases by ______ percent. (RR of of relapse decreases by ______ percent. (RR of trt=1/trt=0).trt=1/trt=0).

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Considering Interactions in Considering Interactions in Cox Proportional HazardsCox Proportional Hazards

• What is the relative risk of relapse for a person who is 30 What is the relative risk of relapse for a person who is 30 compared to 25 if they are at site A (site=0) with all other compared to 25 if they are at site A (site=0) with all other variables held constant? This translates to a _____ variables held constant? This translates to a _____ percent decrease in rate of relapse.percent decrease in rate of relapse.

• What is the relative risk of relapse for a person who is What is the relative risk of relapse for a person who is 30 compared to 25 if they are at site B (site=1) with all 30 compared to 25 if they are at site B (site=1) with all other variables held constant? This translates to a other variables held constant? This translates to a _____ percent decrease in relapse. _____ percent decrease in relapse.

• Is this difference in rate of relapse for a five year Is this difference in rate of relapse for a five year increase in age between the two sites significant?increase in age between the two sites significant?

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Logistic Regression ReviewLogistic Regression Review

Can age, educational level and gender Can age, educational level and gender (female=1) predict the odds that (female=1) predict the odds that someone votes for a particular candidate? someone votes for a particular candidate? Let Let = proportion of voters who vote for = proportion of voters who vote for candidate “Superman”.candidate “Superman”.

Model:Model:

3322110)1

log( xxx

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Logistic Regression ReviewLogistic Regression Review

The following is a sample of logistic The following is a sample of logistic output:output:

dfdf bb SE(b)SE(b) XX22 P-P-valuevalue

InterceptIntercept 11 0.1120.112 .3481.3481 .103.103 .748.748

AgeAge 11 0.0020.002 .0032.0032 ..376..376 .540.540

EducationEducation 11 --0.0100.010

.0184.0184 .299.299 .585.585

GenderGender 11 0.4280.428 .1040.1040 16.916.955

.001.001

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Questions for LogisticQuestions for Logistic

What is the equation of the estimated What is the equation of the estimated Log(odds)?Log(odds)?

What do we predict the odds to be for a 35 What do we predict the odds to be for a 35 year-old male with 16 years of school?year-old male with 16 years of school?

What is the probability a 35 year-old male What is the probability a 35 year-old male with 16 years of school will vote for with 16 years of school will vote for “Superman”?“Superman”?

What is the odds a woman will vote for What is the odds a woman will vote for “Superman” compared to a man (all other “Superman” compared to a man (all other covariates held fixed)?covariates held fixed)?

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TOMHS ExampleTOMHS Example

Question: Does the effect of active blood Question: Does the effect of active blood pressure treatment on CVD differ for pressure treatment on CVD differ for young versus older persons?young versus older persons?

Looking at an Looking at an interaction effect (effect interaction effect (effect modification)modification)

Compare Compare Odds CVD (treatment/placebo) in Odds CVD (treatment/placebo) in

younger patientsyounger patients Odds CVD (treatment/placebo) in older Odds CVD (treatment/placebo) in older

patientspatients

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Logistic Model For InteractionLogistic Model For Interaction

X1 = 1 for X1 = 1 for active treatmentactive treatment and 0 for and 0 for placeboplacebo

X2 = 1 for X2 = 1 for age age ≥≥ 55 55 and 0 for and 0 for age < 55age < 55

X3 = X1 * X2X3 = X1 * X2

So, X3 = 1 for active treatment and age > 55So, X3 = 1 for active treatment and age > 55

X3 = all other combinations.X3 = all other combinations.

21322110)1

log( xxxx

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Logistic Model For InteractionLogistic Model For Interaction

X1 = 1 for active treatment and 0 for placeboX1 = 1 for active treatment and 0 for placebo

X2 = 1 for age X2 = 1 for age ≥≥ 55 and 0 for age < 55 55 and 0 for age < 55

X3 = X1 * X2X3 = X1 * X2

21322110)1

log( xxxx

Log Odds (placebo, young) = 0

Log Odds (active, young) = 0 + 1

Log Odds (placebo, old) = 0 + 2

Log Odds (active, old) = 0 + 1 + 2 +3

Dif = 1; exp(1) is odds (A v P) for young

Dif = 1 + ; exp(1 + 3 ) is odds (A v P) for old

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Log Odds (placebo, young) = 0

Log Odds (active, young) = 0 + 1

Log Odds (placebo, old) = 0 + 2

Log Odds (active, old) = 0 + 1 + 2 +3

exp(1) is odds (A v P) for young

exp(1 + 3 ) is odds (A v P) for old

What does 3 Mean?

=Odds (A v P) for Old exp(1 + 3)

Odds (A v P) for Young exp (1)exp (3)=

A ratio of ratios!!

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Interaction HypothesisInteraction Hypothesis

Q: Does the effect of active treatment on CVD differ Q: Does the effect of active treatment on CVD differ for young versus older persons?for young versus older persons?

Ho: Ho: 33 = 0 = 0

Ha: Ha: 33 ≠ 0≠ 0

Test in SAS just like any other Test in SAS just like any other coefficientcoefficient

21322110)1

log( xxxx

The LOGISTIC Procedure

Analysis of Maximum Likelihood Estimates

Standard WaldParameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -1.6843 0.2566 43.0730 <.0001active 1 -0.8806 0.3301 7.1180 0.0076old 1 0.0850 0.3549 0.0573 0.8108active_old 1 0.7771 0.4395 3.1261 0.0770

Odds Ratio Estimates

Point 95% WaldEffect Estimate Confidence Limits

active 0.415 0.217 0.792old 1.089 0.543 2.183active_old 2.175 0.919 5.147

b1b2b3

Odds CVD (A v P) for younger patients = exp(b1) = 0.415

Odds CVD (A v P) for older patients = exp(b1 + b3) = exp(-0.11) = 0.90

2.175 = 0.90/.415

Ratio of Odds Ratios

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In patients < age 55 the CVD risk was 58% lower in the active treatment (OR: 0.42) – Exp(b1)

For patients over 55 years of age the CVD risk was only 10% lower (OR:.90). - Exp(b1+b3)

The test for interaction between treatment and age approached significance (p=.07).

Description of Findings