Dichotomous and Dichotomous and survival outcomessurvival outcomes

Brian Healy, PhDBrian Healy, PhD

Comments from previous Comments from previous classclass

Book suggestion/practice problemsBook suggestion/practice problems– Fundamentals of BiostatisticsFundamentals of Biostatistics by Bernard Rosner by Bernard Rosner– Available in CountwayAvailable in Countway

How do I know when I need help?How do I know when I need help?– When you think your project is more complicated When you think your project is more complicated

than we have discussedthan we have discussed Correlated observationsCorrelated observations Skewed dataSkewed data

– This class designed to help you do basic analysis, This class designed to help you do basic analysis, but also help you communicate with a statisticianbut also help you communicate with a statistician

ObjectivesObjectives Dichotomous outcomeDichotomous outcome

– Chi-square testChi-square test– Logistic regressionLogistic regression

Survival analysisSurvival analysis– Log-rank testLog-rank test

Quick asideQuick aside What is the most common proportion What is the most common proportion

to see in the news?to see in the news?– Political pollingPolitical polling

Most polls look like this:Most polls look like this:– 50% of people support Scott Brown50% of people support Scott Brown– 45% of people support Martha Coakley45% of people support Martha Coakley– Margin of error +/- 3% Margin of error +/- 3%

What does the margin of error tell us?What does the margin of error tell us?– The plausible values for the true proportion The plausible values for the true proportion

accounting for sampling variability (chance)accounting for sampling variability (chance) What does the margin of error not tell What does the margin of error not tell

us?us?– Sample designSample design

Who was sampled?Who was sampled? How was sampling done?How was sampling done?

– Was there any missing data?Was there any missing data?– Were all people treated the same?Were all people treated the same?

Margin of error Margin of error

Statement regarding Statement regarding accuracyaccuracy

For confidence interval:For confidence interval:– We are 95% confident that the true parameter value lies We are 95% confident that the true parameter value lies

within our confidence boundswithin our confidence bounds For polling (from For polling (from

http://www.pollingreport.com/sampling.htm)http://www.pollingreport.com/sampling.htm)– ““In theory, with a sample of this size, one can say with In theory, with a sample of this size, one can say with

95 percent certainty that the results have a statistical 95 percent certainty that the results have a statistical precision of plus or minus __ percentage points of what precision of plus or minus __ percentage points of what they would be if the entire adult population had been they would be if the entire adult population had been polled with complete accuracy. Unfortunately, there are polled with complete accuracy. Unfortunately, there are several other possible sources of error in all polls or several other possible sources of error in all polls or surveys that are probably more serious than theoretical surveys that are probably more serious than theoretical calculations of sampling error. They include refusals to calculations of sampling error. They include refusals to be interviewed (non-response), question wording and be interviewed (non-response), question wording and question order, interviewer bias, weighting by question order, interviewer bias, weighting by demographic control data, and screening (e.g., for likely demographic control data, and screening (e.g., for likely voters). It is difficult or impossible to quantify the errors voters). It is difficult or impossible to quantify the errors that may result from these factors.”that may result from these factors.”

ReviewReview Steps for hypothesis testSteps for hypothesis test

– How do we set up a null hypothesis?How do we set up a null hypothesis? Choosing the right testChoosing the right test

– Continuous outcome/dichotomous Continuous outcome/dichotomous predictor: Two sample t-testpredictor: Two sample t-test

– Continuous outcome/categorical Continuous outcome/categorical predictor: ANOVApredictor: ANOVA

– Continuous outcome/continuous Continuous outcome/continuous predictor: Correlation or regressionpredictor: Correlation or regression

Types of analysis-Types of analysis-independent independent samplessamples

OutcomeOutcome ExplanatoryExplanatory AnalysisAnalysisContinuousContinuous DichotomousDichotomous t-test, Wilcoxon t-test, Wilcoxon

test, linear regtest, linear regContinuousContinuous CategoricalCategorical ANOVA, linear ANOVA, linear

regressionregressionContinuousContinuous ContinuousContinuous Correlation, Correlation,

linear regressionlinear regressionDichotomousDichotomous DichotomousDichotomous Chi-square test, Chi-square test,

logistic logistic regressionregression

DichotomousDichotomous ContinuousContinuous Logistic Logistic regressionregression

Time to eventTime to event DichotomousDichotomous Log-rank testLog-rank test

Dichotomous outcomeDichotomous outcome Sustained disease progression in MS is Sustained disease progression in MS is

often defined as a one-unit increase on often defined as a one-unit increase on EDSS that lasts for at least six monthsEDSS that lasts for at least six months

This is a common outcome in clinical This is a common outcome in clinical trials and observational studiestrials and observational studies

Patients are often classified as Patients are often classified as progressed or not progressed, which is progressed or not progressed, which is a dichotomous outcomea dichotomous outcome

ExampleExample MS is known to have a genetic MS is known to have a genetic

componentcomponent Several single nucleotide Several single nucleotide

polymorphisms have been associated polymorphisms have been associated with susceptibility to MSwith susceptibility to MS

Question: Do patients with Question: Do patients with susceptibility SNPs experience more susceptibility SNPs experience more sustained progression than patients sustained progression than patients without susceptibility SNPs?without susceptibility SNPs?

DataData Initially, we will focus on presence vs. Initially, we will focus on presence vs.

absence of SNPsabsence of SNPs Among our 190 treated patients, 74 had Among our 190 treated patients, 74 had

the SNP and 116 did notthe SNP and 116 did not– 12 patients with the SNP experienced 12 patients with the SNP experienced

sustained progressionsustained progression

– 13 patients without the SNP experienced 13 patients without the SNP experienced sustained progressionsustained progression

162.07412ˆ SNPp

112.011613ˆ SNPp

Contingency tableContingency table A common way to look at this data is a 2x2 A common way to look at this data is a 2x2

tabletable Does the SNP have an effect on whether or Does the SNP have an effect on whether or

not patients progress?not patients progress?SNP+SNP+ SNP-SNP- TotalTotal

ProgProg 1212 1313 2525No progNo prog 6262 103103 165165TotalTotal 7474 116116 190190

QuestionQuestion In our analysis, we assume that the In our analysis, we assume that the

margins are setmargins are set Under the null hypothesis of no Under the null hypothesis of no

relationship between the two relationship between the two variables, what would we expect the variables, what would we expect the values in the table be?values in the table be?

ExampleExample As an example, use this tableAs an example, use this table

SNP+SNP+ SNP-SNP- TotalTotalProgProg 100100No progNo prog 100100TotalTotal 5050 150150 200200

50*100/200=2550*100/200=25

150*100/200=75150*100/200=75

Expected tableExpected table Expected table for our analysisExpected table for our analysis

SNP+SNP+ SNP-SNP- TotalTotalProgProg 2525No progNo prog 165165TotalTotal 7474 116116 190190

25*74/190=9.73

116*165/ 190=100.7

165*74/190=64.3

25*116/190=15.3

How different is our observed data compared to the expected table?

Does our data show an Does our data show an effect?effect?

To test for an association between To test for an association between the outcome and the predictor, we the outcome and the predictor, we would like to know if our observed would like to know if our observed table was different from the table was different from the expected table under the null expected table under the null hypothesishypothesis

How could we investigate if our table How could we investigate if our table was different?was different?

cells i

ii

EEO 2

This quantity has a chi-square distributionIf it is large, it implies a large difference from the expected

Critical information for Critical information for 22

For 1 degree of freedom, cut-off for For 1 degree of freedom, cut-off for =0.05 is 3.84=0.05 is 3.84– For normal distribution, this is 1.96For normal distribution, this is 1.96– Note 1.96Note 1.9622=3.84=3.84

Inherently, two-sided since it is Inherently, two-sided since it is squaredsquared

Has problems with small cell countsHas problems with small cell counts– Fix: Fisher’s exact testFix: Fisher’s exact test

Chi-square distributionChi-square distribution

Area=0.05

X2=3.84

Hypothesis test with Hypothesis test with 22

1)1) HH00: No association between SNP and : No association between SNP and progressionprogression

2)2) Dichotomous outcome, dichotomous Dichotomous outcome, dichotomous predictorpredictor

33 22 test test4)4) Summary statistic: Summary statistic: 22=0.99=0.995)5) p-value=0.32p-value=0.326)6) Since the p-value is greater than 0.05, we fail Since the p-value is greater than 0.05, we fail

to reject the null hypothesis to reject the null hypothesis 7)7) We conclude that there is no significant We conclude that there is no significant

association between SNP and progressionassociation between SNP and progression

2 statistic

p-value

Question: Why 1 degree of Question: Why 1 degree of freedom?freedom?

We used a We used a 2 2 distribution with 1 degree of distribution with 1 degree of freedom, but there are 4 numbers. Why?freedom, but there are 4 numbers. Why?– For our analysis, we assume that the margins For our analysis, we assume that the margins

are fixed.are fixed.– If we pick one number in the table, the rest of If we pick one number in the table, the rest of

the numbers are knownthe numbers are known

SNP+SNP+ SNP-SNP- TotalTotalProgProg 2525No ProgNo Prog 165165TotalTotal 7474 116116 190190

3 2271 94

Estimated effectEstimated effect When you compare two groups with a When you compare two groups with a

dichotomous outcome, there are dichotomous outcome, there are three common ways to show the three common ways to show the difference between the groupsdifference between the groups– Risk differenceRisk difference

Prob of diseaseProb of diseaseGroup 1Group 1-Prob of disease-Prob of diseaseGroup 2Group 2

– Relative risk/risk ratioRelative risk/risk ratio Prob of diseaseProb of diseaseGroup 1Group 1/Prob of disease/Prob of diseaseGroup 2Group 2

– Odds ratioOdds ratio

Odds ratioOdds ratio Odds:Odds:

Odds ratio: Odds ratio:

– Under the null, what is the OR?Under the null, what is the OR?

ppOdds

1

ExposureDiseasePExposureDiseasePExposureDiseasePExposureDiseaseP

OddsOdds

ORExposure

Exposure

|(1)|(|(1)|(

ExposureExposureDiseaseDisease YY NN TotalTotalYY aa bb nn11

NN cc dd nn22

TotalTotal mm11 mm22 NN

bcad

dbca

OddsOdds

OR

dbOddsOdds

ca

caccaa

EDPEDPOddsOdds

caaEDP

ED

ED

EDExposureDisease

EDExposureDisease

|

|

||

|| )/()/(

)|(1)|(

)|(

This is the estimate of the odds ratio from a cohort study

ExposureExposureDiseaseDisease YY NN TotalTotalYY aa bb nn11

NN cc dd nn22

TotalTotal mm11 mm22 NN

bcad

dcba

OddsOdds

OR

dcOddsOdds

baOddsOdds

baaDEP

DE

DE

DEDiseaseExposure

DEDiseaseExposure

|

|

||

||

)|( This is the estimate of the odds ratio from a case-control study

Amazing!!Amazing!! Estimated odds ratio from each kind of Estimated odds ratio from each kind of

study ends up being the same thing!!!study ends up being the same thing!!! Therefore, we can complete a case Therefore, we can complete a case

control study and get an estimate that control study and get an estimate that we really care about, which is the we really care about, which is the effect of the exposure on the diseaseeffect of the exposure on the disease

This relationship is one reason why the This relationship is one reason why the odds ratio is so commonly reported odds ratio is so commonly reported

Logistic regressionLogistic regression

Types of analysis-Types of analysis-independent independent samplessamples

OutcomeOutcome ExplanatoryExplanatory AnalysisAnalysisContinuousContinuous DichotomousDichotomous t-test, Wilcoxon t-test, Wilcoxon

test, linear regtest, linear regContinuousContinuous CategoricalCategorical ANOVA, linear ANOVA, linear

regressionregressionContinuousContinuous ContinuousContinuous Correlation, Correlation,

linear regressionlinear regressionDichotomousDichotomous DichotomousDichotomous Chi-square test, Chi-square test,

logistic logistic regressionregression

DichotomousDichotomous ContinuousContinuous Logistic Logistic regressionregression

Time to eventTime to event DichotomousDichotomous Log-rank testLog-rank test

Linear regressionLinear regression When we fit linear regression, we When we fit linear regression, we

used indicator variables to represent used indicator variables to represent dichotomous predictorsdichotomous predictors– Ex. Effect of genderEx. Effect of gender– Gender=0 if FemaleGender=0 if Female– Gender=1 if MaleGender=1 if Male

What is the interpretation of What is the interpretation of 11??

ii eGenderY 10

OutcomeOutcome What if the outcome is dichotomous?What if the outcome is dichotomous?

– ProgressionProgression– Y=0 if no progressionY=0 if no progression– Y=1 if progressionY=1 if progression

Can we just use linear regression Can we just use linear regression with 0/1 as the outcome?with 0/1 as the outcome?

Can we fit a line to this data?

Is there another measure?0

.2.4

.6.8

1P

rogr

essi

on

10 20 30 40 50Age

Better outcomeBetter outcome Rather than investigating the 0/1 Rather than investigating the 0/1

value, we focus our attention on the value, we focus our attention on the probability of the eventprobability of the event

Therefore, we could use the following Therefore, we could use the following regression equationregression equation

Is there anything wrong with this Is there anything wrong with this function?function?

xpi *10

Technical aside-ProbabilitiesTechnical aside-Probabilities Probabilities are required to be Probabilities are required to be

between 0 and 1between 0 and 1– Does the present equation impose this Does the present equation impose this

restriction?restriction?– NoNo

We would like a similar equation, but We would like a similar equation, but with the restriction that 0<=p<=1with the restriction that 0<=p<=1

One optionOne option

xpi *10

x

x

i eep *

*

10

10

1

Logistic regressionLogistic regression The previous function is quite complex The previous function is quite complex

to deal with, but we can transform the to deal with, but we can transform the equationequation

Note that the right side of the equation Note that the right side of the equation looks EXACTLY like our normal looks EXACTLY like our normal regressionregression

xOddspp

epp

eep

x

x

x

*)ln(1

ln

1

1

10

*

*

*

10

10

10

Parameter interpretation-Parameter interpretation-reviewreview

Let’s think about the following linear Let’s think about the following linear regression model for the effect of age regression model for the effect of age on BPFon BPF

In linear regression, the meaning of In linear regression, the meaning of 11 in in this model is that for a one unit increase this model is that for a one unit increase in age the mean BPF goes up by in age the mean BPF goes up by 11..

The meaning of The meaning of is the mean value of is the mean value of BPF when age=0BPF when age=0

iii ageageBPFE *)|( 10

Parameter interpretationParameter interpretation How does this change for our logistic How does this change for our logistic

model?model?– Not at all!!!Not at all!!!

Logistic model:Logistic model: The meaning of The meaning of 11 in this model is that in this model is that

for a one unit increase in age, the for a one unit increase in age, the ln(Odds) goes up by ln(Odds) goes up by 11

The meaning of The meaning of is the value of is the value of ln(Odds) when age=0ln(Odds) when age=0

ii

i agepp

*1

ln 10

ResultsResults When we fit our data, the parameter When we fit our data, the parameter

estimates were estimates were

For a one unit increase in age, the For a one unit increase in age, the estimated log(Odds) increases by 0.086estimated log(Odds) increases by 0.086

Is this a statistically significant Is this a statistically significant increase?increase?

ii

i agepp *086.058.4ˆ1ˆ

ln

Hypothesis testHypothesis test If there was no effect of age on the If there was no effect of age on the

probability of progression, what would the probability of progression, what would the value of value of 11 equal? equal?

How could we test the hypothesis that How could we test the hypothesis that there is no effect?there is no effect?– HH00: : 11=0=0

Need an estimate of the variance of the Need an estimate of the variance of the estimated estimated 11, but this is provided by STATA, but this is provided by STATA

Assume approximate normalityAssume approximate normality

Hypothesis testHypothesis test1)1) HH00: : 11=0=02)2) Dichotomous outcome with continuous Dichotomous outcome with continuous

predictorpredictor3)3) Logistic regressionLogistic regression4)4) Summary statistic: z=1.99Summary statistic: z=1.995)5) p-value=0.047p-value=0.0476)6) Since the p-value is less than 0.05, we Since the p-value is less than 0.05, we

reject the null hypothesis reject the null hypothesis 7)7) We conclude that there is a significant We conclude that there is a significant

effect of age at symptom onset on effect of age at symptom onset on probability of progressionprobability of progression

Estimated intercept coefficient

p-value for H0: 0=0

p-value for H0: 1=0Estimated coefficient for age

0.2

.4.6

.81

Pro

gres

sion

10 20 30 40 50Age

event2yr Pr(event2yr)

ConclusionsConclusions Logistic regression allows us to Logistic regression allows us to

investigate the relationship between a investigate the relationship between a continuous predictor and a continuous predictor and a dichotomous outcomedichotomous outcome

Interpretation of coefficients is the Interpretation of coefficients is the same as linear regression, but on the same as linear regression, but on the log(odds) scalelog(odds) scale

We can calculate the predicted We can calculate the predicted probability just like we could calculate probability just like we could calculate the predicted mean valuethe predicted mean value

Survival analysisSurvival analysis

Types of analysis-independent Types of analysis-independent samplessamples

OutcomeOutcome ExplanatoryExplanatory AnalysisAnalysisContinuousContinuous DichotomousDichotomous t-test, Wilcoxon t-test, Wilcoxon

testtestContinuousContinuous CategoricalCategorical ANOVA, linear ANOVA, linear

regressionregressionContinuousContinuous ContinuousContinuous Correlation, Correlation,

linear regressionlinear regressionDichotomousDichotomous DichotomousDichotomous Chi-square test, Chi-square test,

logistic logistic regressionregression

DichotomousDichotomous ContinuousContinuous Logistic Logistic regressionregression

Time to eventTime to event DichotomousDichotomous Log-rank testLog-rank test

ExampleExample An important marker of disease activity in An important marker of disease activity in

MS is the occurrence of a relapseMS is the occurrence of a relapse– This is the presence of new symptoms that This is the presence of new symptoms that

lasts for at least 24 hourslasts for at least 24 hours Many clinical trials in MS have Many clinical trials in MS have

demonstrated that treatments increase demonstrated that treatments increase the time until the next relapsethe time until the next relapse– How does the time to next relapse look in the How does the time to next relapse look in the

clinic?clinic? What is the distribution of survival times?What is the distribution of survival times?

Kaplan-Meier curveKaplan-Meier curveEach drop in the curve represents an event

Survival dataSurvival data To create this curve, patients placed on To create this curve, patients placed on

treatment were followed and the time of the treatment were followed and the time of the first relapse on treatment was recordedfirst relapse on treatment was recorded– Survival timeSurvival time

If everyone had an event, some of the If everyone had an event, some of the methods we have already learned could be methods we have already learned could be appliedapplied

Often, not everyone has event Often, not everyone has event – Loss to follow-upLoss to follow-up– End of studyEnd of study

CensoringCensoring The patients who did not have the event The patients who did not have the event

are considered censored are considered censored – We know that they survived a specific We know that they survived a specific

amount of time, but do not know the exact amount of time, but do not know the exact time of the eventtime of the event

– We believe that the event would have We believe that the event would have happened if we observed them long enough happened if we observed them long enough

These patients provide some information, These patients provide some information, but not complete informationbut not complete information

CensoringCensoring How could we account for censoring?How could we account for censoring?

– Ignore it and say event occurred at time of Ignore it and say event occurred at time of censoringcensoring Incorrect because this is almost certainly not trueIncorrect because this is almost certainly not true

– Remove patient from analysisRemove patient from analysis Potential bias and loss of powerPotential bias and loss of power

– Survival analysisSurvival analysis Our objective is to estimate the survival Our objective is to estimate the survival

distribution for patients in the presence of distribution for patients in the presence of censoringcensoring

Comparison of survival Comparison of survival curvecurve

One important aspect of survival One important aspect of survival analysis is the comparison of survival analysis is the comparison of survival curvescurves

Null hypothesis: survival curve in Null hypothesis: survival curve in group 1 is the same as survival curve group 1 is the same as survival curve in group 2in group 2

Method: log-rank testMethod: log-rank test

ExampleExampleUntreatedUntreatedPatientPatient TimeTime11 3322 8+8+33 151544 27+27+55 323266 464677 494988 515199 55+55+1010 7070

TreatedTreatedPatientPatient TimeTime11 303022 383833 52+52+44 585855 666666 73+73+77 777788 898999 107+107+

0.00

0.25

0.50

0.75

1.00

0 20 40 60 80 100analysis time

group = 0 group = 1

Kaplan-Meier survival estimates

Technical aside-Log-rank Technical aside-Log-rank testtest

To compare survival curves, a log-rank To compare survival curves, a log-rank test creates 2x2 tables at each event test creates 2x2 tables at each event time and combines across the tablestime and combines across the tables– Similar to MH-testSimilar to MH-test

Provides a Provides a 22 statistic with 1 degree of statistic with 1 degree of

freedom (for a two sample freedom (for a two sample comparison) and a p-valuecomparison) and a p-value

Same procedure for hypothesis testingSame procedure for hypothesis testing

Hypothesis testHypothesis test1)1) HH00: Survival distribution in group 1 = : Survival distribution in group 1 =

survival distribution in group 2survival distribution in group 22)2) Time to event outcome, dichotomous Time to event outcome, dichotomous

predictorpredictor3)3) Log rank testLog rank test4)4) Summary statistic: Summary statistic: 22=4.4=4.45)5) p-value=0.036p-value=0.0366)6) Since the p-value is less than 0.05, we Since the p-value is less than 0.05, we

reject the null hypothesisreject the null hypothesis7)7) We conclude that there is a significant We conclude that there is a significant

difference in the survival time in the treated difference in the survival time in the treated compared to untreatedcompared to untreated

p-value

What we learnedWhat we learned Chi-square testChi-square test Logistic regressionLogistic regression Survival analysisSurvival analysis