hazards for the heart and a pim for the soul -...

Hazards for the heart and a PIM for the soul

Thomas Alexander GerdsBiostatistics, University of Copenhagen

15 October 2015

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Outline

Life time risks

Analogy between living and cycling

PIM

Statistics for researchers with hypertension

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Motivating example

Suppose a 40 year old and a 80 year old person both need to decidefor or against preventive cardiovascular therapy and changes oflifestyle. Suppose further that for both persons the predicted risk ofa cardiovascular event within the next 10 years is 12%.

How can both predictions be valid?

One plausible explanation is that the 40 year old person has riskfactors that the 80 year old person does not have.

Another plausible explanation is that the 80 year old person has amuch higher risk to die due to non-cardiovascular causes within thenext 10 years than the 40 year old person.

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Documented confusion

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A new paradox?

What could be the reason for this apparently contradictory result?

One plausible explanation is that the two calculations are based ontwo di�erent cohorts and corresponding data sets and thus analysesare not consistent because of random variation and di�erentprevalences in the background populations.

Another plausible explanation is that the statistical model whichpredicts the 10-year risk of ASCVD wrongly treatednon-cardiovascular mortality in the same way as right censored.

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Analogy between living and cycling

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Speed and hazard

Speed is the rate at which an object covers distance.

I A fast-moving object has a high speed and covers a relativelylarge distance in a given amount of time

I A slow-moving object covers a relatively small amount ofdistance in the same amount of time.

Hazard rate is the speed at which a person gets a disease or dies.

I An exposed person has a high hazard rate and will get diseasedwith a relatively large probability within a given time period.

I A non-exposed person has a low hazard rate and will getdiseased with a relatively small probability within the sametime period.

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Speed and duration

If we know the (average) speed of a cyclist on the road from placeA to place B we can calculate the (expected) duration that thecyclist would need to cycle from A to B.

The speed changes along the road according to gradient and otherthings, the duration is a function of the speed and the length of theroad.

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Hazard and absolute risk

If we know the (average) hazard rate of a person in a time intervalbetween date A and date B we can calculate the (expected)probability that the person would have a stroke between the datesA and B.

The hazard rate changes over time according to predisposition,exposure and disease, the absolute risk of stroke is a function of thehazard rate and the length of the time interval.

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Computing duration

Suppose the speed of a cyclist is piece-wise constant on the roadbetween A and B:

Section Distance (km) Speed (km/h) Duration (min)

A �> a 10 15 40.0a �> b 20 20 60.0b �> B 20 22 54.5

I Total distance: 50 km

I Total duration: 154.5 minutes

Conclusion: The cyclist needs 154.5 minutes to cycle from A to B.

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Computing absolute risk

Suppose the stroke hazard rate of a patient is piece-wise constantin the time period between A and B:

Interval Duration (mth) Hazard rate (%/mth) Abs. risk (%)

A �> a 10 1.5 13.9a �> b 20 2.0 33.0b �> B 20 2.2 35.6

I Total duration: 50 months

I Total absolute risk: 62.8%

calculation: 1- (1-0.139)(1-0.33)(1-0.356) = 0.628

Conclusion: The patient will experience a stroke with a probabilityof 62.8% in the period between date A and date B.

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E�ect of doping

Suppose the same cyclist as before knows a way to increase thespeed by 10%:

Section Distance (km) Speed (km/h) Duration (min)

A �> a 10 16.5 36.4a �> b 20 22.0 54.5b �> B 20 24.2 49.6

I Total duration without doping: 154.5 minutes

I Total duration with doping: 140.5 minutes

Conclusion: With doping the cyclist needs 14 minutes (9.1%) lessto cycle from A to B.

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E�ect of aging

Suppose the same patient as before is one year older and thisincreases the hazard rate of stroke by 10% (hazard ratio=1.1):

Interval Duration (mth) Hazard rate (%/mth) Abs. risk (%)

A �> a 10 1.65 15.2a �> b 20 2.20 35.6b �> B 20 2.42 38.4

I Total absolute risk previous age: 62.8%

I Total absolute risk one year older: 66.4%

calculation: 1- (1-0.152)(1-0.356)(1-0.384) = 0.664

Conclusion: Each year of age increases the risk of stroke in theperiod between date A and date B by 3.6 %.

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Relative Risks for Stroke by Age, Sex, and Population Basedon Follow-Up of 18 European Populations in the

MORGAM ProjectKjell Asplund, MD, PhD; Juha Karvanen, DSc(Tech); Simona Giampaoli, MD;

Pekka Jousilahti, MD, PhD; Matti Niemelä, MD; Grazyna Broda, MD; Giancarlo Cesana, MD;Jean Dallongeville, MD; Pierre Ducimetriere, MD; Alun Evans, MD; Jean Ferrières, MD;

Bernadette Haas, MD; Torben Jorgensen, MD; Abdonas Tamosiunas, MD; Diego Vanuzzo, MD;Per-Gunnar Wiklund, MD, PhD; John Yarnell, MD; Kari Kuulasmaa, PhD; Sangita Kulathinal, PhD;

for the MORGAM Project

Background and Purpose—Within the framework of the MOnica Risk, Genetics, Archiving and Monograph (MORGAM)Project, the variations in impact of classical risk factors of stroke by population, sex, and age were analyzed.

Methods—Follow-up data were collected in 43 cohorts in 18 populations in 8 European countries surveyed forcardiovascular risk factors. In 93 695 persons aged 19 to 77 years and free of major cardiovascular disease at baseline,total observation years were 1 234 252 and the number of stroke events analyzed was 3142. Hazard ratios werecalculated by Cox regression analyses.

Results—Each year of age increased the risk of stroke (fatal and nonfatal together) by 9% (95% CI, 9% to 10%) in menand by 10% (9% to 10%) in women. A 10-mm Hg increase in systolic blood pressure involved a similar increase in riskin men (28%; 24% to 32%) and women (25%; 20% to 29%). Smoking conferred a similar excess risk in women (104%;78% to 133%) and in men (82%; 66% to 100%). The effect of increasing body mass index was very modest. Higherhigh-density lipoprotein cholesterol levels decreased the risk of stroke more in women (hazard ratio per mmol/L0.58; 0.49 to 0.68) than in men (0.80; 0.69 to 0.92). The impact of the individual risk factors differed somewhatbetween countries/regions with high blood pressure being particularly important in central Europe (Poland andLithuania).

Conclusions—Age, sex, and region-specific estimates of relative risks for stroke conferred by classical risk factors invarious regions of Europe are provided. From a public health perspective, an important lesson is that smoking confersa high risk for stroke across Europe. (Stroke. 2009;40:2319-2326.)

Key Words: blood pressure � cholesterol � cohort studies � smoking � stroke risk factors

The appraisal of stroke risk in populations or individuals isbased on the recognition that all cardiovascular disordersare multifactorial in nature. The most widely used risk scorefor stroke was developed within the framework of TheFramingham Heart Study, the original version being pub-lished in 1971 with later refinements based on longer follow-up, addition of more predictors, and using more sophisticated

statistical techniques.1 The Framingham stroke risk score hasbeen used extensively when international and national guide-lines for cardiovascular prevention have been developed.Other stroke prediction scores have been developed later, forinstance based on data from a large population-based US cohortof elderly men and women (the Cardiovascular Health Study),2

a large number of cohorts within the Systematic COronary Risk

Received January 15, 2009; final revision received February 23, 2009; accepted March 12, 2009.From the Department of Public Health and Clinical Medicine (K.A.), Medical Unit, Umeå University Hospital, Umeå, Sweden; the National Institute

for Health and Welfare (J.K., P.J., M.N., K.K.), Helsinki, Finland; the National Centre for Epidemiology, Surveillance and Health Promotion (S.G.),Istituto Superiore di Sanità, Rome, Italy; the Department of Cardiovascular Epidemiology and Prevention (G.B.), National Institute of Cardiology,Warsaw, Poland; Dipartimento de Medicina (G.C.), Prevenzione e Biotecnologie Sanitarie, Università degli Studi Milano-Bicocca, Monza, Italy; theDepartment of Epidemiology and Public Health (J.D.), Pasteur Institute of Lille, Lille, France; the National Institute of Health and Medical Research(U258; P.D.), Paris, France; the Department of Epidemiology and Public Health (A.E., J.Y.), The Queen’s University of Belfast, Belfast, UK; theDepartment of Epidemiology (J.F.), Faculty of Medicine, Toulouse–Purpan, Toulouse, France; the Department of Epidemiology and Public Health (B.H.),Louis Pasteur University, Faculty of Medicine, Strasbourg, France; the Research Centre for Prevention and Health (T.J.), Capital Region, Denmark;Kaunas University of Medicine (A.T.), Institute of Cardiology, Kaunas, Lithuania; Centro di Prevenzione Cardiovasculare (D.V.), Agenmzia Regionaledella Sanità Friuli Venezia Giulia, Udine, Italy; the Department of Medicine (P.-G.W.), Umeå University Hospital, Umeå, Sweden; and the Indic Societyfor Education and Development (INSEED; S.K.), Nashik, India.

Correspondence to Kjell Asplund, MD, PhD, Riks-Stroke, Medicine, Department of Public Health and Clinical Medicine, University Hospital,SE-70185 Umeå, Sweden. E-mail [email protected]

© 2009 American Heart Association, Inc.

Stroke is available at http://stroke.ahajournals.org DOI: 10.1161/STROKEAHA.109.547869

2319 by guest on May 15, 2014http://stroke.ahajournals.org/Downloaded from

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MORGAM project (Asplund et al.)

Discussion

Error: a hazard ratio is not arelative risk

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Probabilistic index models

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Multiple Cox regression

Survival setting

I survival time: T and covariates: X , Z .

I conditional survival function

S(t|X ,Z ) = P(T > t|X ,Z )

Strati�ed Cox proportional hazards model:

dS(t | X ,Z ) = −S(t − |X ,Z ) eβX hZ (t)dt

I hZ is a zoo of baseline hazard functions.

I exp(β) = hazard ratio(s)

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The probabilistic index

There is a one-to-one relation between the hazard ratio and theprobabilistic index:

P(Ti < Tj | Xi ,Xj , z , z) =1

1+ eβ(Xj−Xi )

I A hazard ratio of eβ=1 corresponds to a probabilistic index of50%.

I The results of Asplund et al., male HR=1.09; CI-95%: (1.09;1.10) can be reformulated as:

The probability that a male patient who is one year older than an

otherwise similar patient will have a longer lifetime is 47.8%,

CI-95%: (47.6; 47.8).

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UseR!

library(Publish) ## install_github('tagteam/Publish')

f

The probabilistic index regression model

This class of regression models1 includes the Wilcoxon rank test asa special case (only one binary covariate). PIM:

P(Yi < Yj |Xi ,Xj) +1

2P(Yi = Yj |Xi ,Xj) = m(β,Xi ,Xj).

Special case: time to event outcome T and risk prediction modelπ(t,Xi ) ≈ risk of event for subject i until time t.

P(Ti < Tj |π(t,Xi ), π(t,Xj)) = βI{π(t,Xi ) > π(t,Xj)}

Expect β > 0 but truncation and further research needed.

1Thas et al. (2012) Probabilistic index models JRSSB. 74: 623�671.21 / 35

Closely related: c-index for prediction

Harrel's c-index:

CHarrel = P(π(t,Xi ) > π(t,Xj)|Ti < Tj)

Truncated c-index:

C(t) = P(π(t,Xi ) > π(t,Xj)|Ti < Tj ,Ti ≤ t)

Retrospective interpretation, only proportional hazards models!

Time-dependent area under the ROC curve (with competing riskssee Blanche et al.2)

AUC(t) = P(π(t,Xi ) > π(t,Xj)|Ti ≤ t,Tj > t)

2StatMed 2013, 32(30):5381-9722 / 35

The novel marker question

Suppose a new marker Z is available in addition to conventionalrisk factors X.

A common question is if the new model (X,Z) predicts better thanthe old model (X).

I Reclassi�cation scatterplot

I Di�erence of AUCs

I Di�erence of Brier scores

A signi�cant hazard ratio of Z is not good enough

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Multi-center study on 24h bloodpressure measurements

Outcome:

Cardiovascular mortality 10 years after24h SABP measurements.

Conventional factors:

age, sex, smoking, drinking, cholesterol,

history of cardiovascular disease,

diabetes, ongoing hypertension

treatment

Novel marker question:

I X = daytime SABP andconventional factors

I Z = nighttime SABP

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Hour of measurement

Am

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(mm

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10 20 24 06 10 20

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Copenhagen id: 10111 Cycle 1Cycle 2

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Statistics for researchers with

hypertension

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Meta average hazard ratios across 6 studies

*P

Reclassi�cation scatterplot

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Predicted risk (daytime and nighttime SABP)

Pre

dict

ed r

isk

(day

time

SA

BP

)

0 %

20 %

40 %

60 %

80 %

100 %

0 % 20 % 40 % 60 % 80 % 100 %

Predicted 10 year risk of cardiovascular mortality

27 / 35

Retrospective analysis of added value of nighttime SABP

10 year risk prediction Daytime SABP

10 year risk prediction Nighttime SABP

10 year risk prediction Daytime+Nighttime SABP

85.56

−0.14 [−0.52;0.24]

0.11 [−0.08;0.29]

0.47

0.26

AUC [95% CI] P−value

−3 −2 −1 0 1 2 3

●

●

●

Cardiovascular mortality Difference in AUC (%)

Model(daytime SABP)

better

Model(daytime + nighttime SABP)

better

28 / 35

Reclassi�cation analysis conditional on outcome

For each subject we can compute the di�erence between thepredicted 10-year risk of CVD mortality:

θi = π(t,Xi ,Zi )− π(t,Xi )

Aim:θi > 0|Ti ≤ t,Ei = 1 and θi ≤ 0|Ti ≤ t,Ei 6= 1

The quantiles of the conditional distribution of θi given outcomeafter t-years can be estimated using Bayes formula (and IPCW):

P(θ > q|T ≤ t,E = 1) = P(T ≤ t,E = 1|θ > q)P(θ > q)P(T ≤ t,E = 1)

29 / 35

Reclassi�cation analysis conditional on outcome

−10.0 −5.0 −2.5 0.0 2.5 5.0 10.0

Event−free86.0 %

Non−cardiovascularmortality

8.0 %

Cardiovascularmortality

5.9 %

Any100.0 %

Outcome after 10 years

Difference (%)Predicted risk

(daytime SABP) larger

Predicted risk(daytime + nighttime SABP)

larger

Predicted 10 year risk of cardiovascular mortality

30 / 35

Summary and conclusions

I Statisticians cannot solve a clinical problem

I Doctors should not convince statisticians that HR is easier tounderstand when it is called RR.

I A signi�cant HR does not guarantee improved predictionperformance. Cox is a PIM.

I Non-cardiovascular mortality and generally competing riskshave to be considered both for modelling and for estimation ofmodel performance

31 / 35

Competing risk

Speed = 0

Censored

Speed = ?

32 / 35

Life time risksAnalogy between living and cyclingPIMStatistics for researchers with hypertension

hazards for the heart and a pim for the soul -...

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