Causality and causalinference in survivalanalysis

NORBIS course, OsloFriday, December 8, 2017Odd O. AalenDept. of Biostatistics, Institute of Basic Medical SciencesUniversity of Oslo

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New England Journal of Medicine, Editorial, Jan. 6, 2000, p. 42-49� The eleven most important developments in

medicine in the past millennium� Elucidation of human anatomy and physiology� Discovery of cells and their substructures� Elucidation of the chemistry of life� Application of statistics to medicine� Development of anesthesia� Discovery of the relation of microbes to disease� Elucidation of inheritance and genetics� Knowledge of the immune system� Development of body imaging� Discovery of antimicrobial agents� Development of molecular pharmacotherapy

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� From NEJM: The origin of modern epidemiology:

� 1854, when John Snow demonstrated the transmission of cholera from contaminated water

� The majority of people who got ill used the Broad Street Pump in London's Golden Square

� He removed the pump handle from the polluted well and the spread of the disease stopped.

The original Broad Street pump© Wellcome Library, London 4

� From NEJM:� Earliest clinical trial in 1747� Scurvy (Serious disease:

Magellan lost 80% of his men from scurvy)

� James Lind treated 12 scorbutic ship passengers on a British navy ship with cider, an elixir of vitriol, vinegar, sea water, oranges and lemon

� Those who got oranges and lemon did not get ill

� Supply of lemon juice eliminated scurvy from the navy

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So, it is all about causality� Statistics is important because it is conceived as

contributing to a causal understanding which is needed in prevention and treatment of disease.

� Statistics can indicate causality even in the absence of a mechanistic understanding. � Treatment of scurvy far ahead of the knowledge of

vitamin C� John Snow: 20 years ahead of Pasteur

� Going to modern times next: Causality and statistics – a happy couple?

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Modern breakthrough based on statistics:Sleeping position influences risk of crib death

Crib deaths in Norway 1986-2003

Num

ber

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Sudden infant death syndrome (SIDS)

� The risk of SIDS is strongly increased (RR up to 13) when the infant is sleeping on its stomach compared to sleeping on its back.

� This is simple because� An intervention could be conceived and was easy to

carry out in practice� The effect was immediate� The effect was very strong

� None of these conditions normally hold in epidemiology

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Lancet 2010; 376: 1741–50

Proximal colon

Distal colon

Questions

� Why is causality important in medicine and other fields?

� How can statistics say something about causality?

� What is causality?

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What is causality?

� The simplest formulation: it is about intervention� What is the effect of an action?

� This is the focus of Judea Pearl’s famous book� Seeing vs. doing

� Pearl implies that statistics per se says nothing about interventions

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Do-operator:P(y | do(x), z)

Interventions in medicine

� Medical doctors, psychologists and otherhealth personell constantly intervene:�Giving treatment, medications, advice …

� Statistics is expected to be a tool for validating such actions, to prove or disprove their validity

� This cannot be done without taking thecausal issue seriously

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Seeing vs doing (Pearl)

� Pearl makes a fundamental distinction between seeing and doing. Causality is about doing, while most statistical data is about seeing

� Seeing and doing may coincide in experimentsbecause of the ability to control the setting. The “big” experiment in medicine is the randomized clinical trial where the effect of doing is directly visible

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Registries contain data on seeing only, and not doing� We want to say something about the effect of

intervention. BUT: the registry only contains a description of what has happened, there is no information about what could have happened if one acted differently. Therefore, you can’t (directly) say anything about the effect of intervention

� This is the case for observational data in general� Still, causal inference can help us if we collect enough

data and the right type of data…

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Can statistics really be useful?

� Interventions, or actions, are directedtowards individuals

� while statistics is about groups of different people

� How can this be reconciled?� This is the issue of causal inference� Dangers are lurking everywhere. They

have weird names: confounders, colliders14

Graphical models

� Graphical models with arrows and boxes are common. However, Judea Pearl has lifted them to a new level

� A number of rules for evaluating graphs can be defined

� These are applicable in practice as shall be demonstrated

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Directed acyclic graph – DAG

� Graph with arrows, where you never return to the same node

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A

U

Y

L

Outcome

Confounder

Treatment /Risk factor

Mediator

Collider: where two or more arrows meet

Statistical association� If A and Y are associated, then

this is compatible with four different types of causal relationship:

� Direct causation

� Reverse causation

� Confounding

� Collider effect (selection)

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A Y

A Y

A Y

U

A Y

S

Examples:

Smoking/Lung cancer

Heart disease /cancer

The following rules decide whether a path is open or closed

1. A path with colliding arrows is closed (→←). If there are no colliders the path is open.

2. To conditon on a non-collider closes thepath.

3. To condition on a collider (or descendantof a collider) opens the path

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What do we mean by “to condition on”� We mean e.g. to include a variable in the

regression.�To include a confounder is usually ok�To include a collider is dangerous

� However, a collider may not be avoided if it represents inherent selection in the data

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Keep causal paths open and non-causal paths closed� Example (Hein Stigum): red arrow is causal, black path

is not causal (backdoor path). Conditioning on age (or obesity) blocks the back-door path

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Vitamin E

Age

Birth defects

Obesity

Exercise (Hein Stigum)

� We want the causal effect of physical activity on CHD (coronary heart disease). What should we adjust for?

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Birth defects. Adjustment for confounder?Source: Hernán et al, Amer. J. Epidem. 2002, 155, 176-184

� When estimating the effect of E on D, shall you adjust for C?� No, one should not adjust for a collider.� Case-control study on folic acid supplementation and neural tube

defects. Adjusted OR: 0.80 (0.62, 1.21), non-adjusted OR 0.65 (0.46,0.94)

When is a DAG causal?Two views

� Robins and Hernán: A DAG is causal when1. Lack of an arrow can be interpreted as lack of direct

causal effect2. All common causes, even if unmeasured, of any pair

of variables on the graph are themselves on the graph

� Note: this requires a concept of direct cause

� Pearl: A DAG becomes causal if intervening on a node has the effect of removing all arrows into the node while the DAG is otherwise unchanged

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Causal issues in survival analysis

� Start with the Kaplan-Meier survival plots.� Surely, if we have a randomized trial, they are

causal?� Well, we have the issue of censoring. The

Kaplan-Meier plots is supposed to tell us whatwe would observe in the absence of censoring. Non-trivial causal issue� Is now being studied by Kjetil Røysland, Vanessa

Didelez and others

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Cox regression

� The standard method in thousands of clinicaltrials in cancer, heart disease etc.

� Has been of immense importance as the centralorganizing tool of a large and difficult field.

� Before the Cox model came in 1972, theanalysis of censored survival data was in shambles. One did not know what to do.

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But, what about causality?� We consider a randomized trial. A marginal structural Cox

model is by definition a type of causal model. But this is not the whole story.

� There is an unavoidable selection effect. Central concept: frailty (Vaupel, Hougaard and many others). This is unobserved differences between individuals

� Figures showing how frailty distribution changes over time for high risk and low risk group:

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Low risk High risk

Frailty is a causal issue

� Consider a treatment X with two values 0 and 1. Let Z be an uknown factor (frailty). The counterfactual hazardrates for the two groups are

� Causal hazard ratio (relative risk) is r(t)

� This model has e.g. been used to explain the obesityparadox (Stensrud, Valberg, Røysland, Aalen: Epidemiology 2017)

� Example: Obesity is a risk factor for developing a condition, like heart failure but once this condition has occurred obesity appears as a protective factor for survival.

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Frailty selection is well known� Assume a constant causal hazard ratio: r(t)= r

� Observed hazard ratio:

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• The relative risk may declineand even cross over due to artefacts.

• This may not reflect anychange in risk at theindividual level.

• This is a causal issue if youare interested in whathappens to individuals whentime passes.

• See related comments in Hernan (2010): “The hazard of hazard ratios”

Another view of the same issue- conditioning on a collider

� Assume that X and Z are independent (randomization). Conditioning on survival, S, will make X and Z dependent due to collider effect.� Exception: X and Z act additively on the hazard rate, the collider

does not have any effect, that is no bias

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Treatment: Outcome

Frailty

Implication for the Cox model

� Consider a randomized clinical survival study with a real difference between treatments. If there is an unknown variable influencing the hazard, then only the first factor in the likelihood gives a randomized comparison. All others are biased. (Aalen, Cook, Røysland, 2015)

� But no bias under the null hypothesis� The Cox model is a model for the observed hazard ratio,

and not the causal one30

Cox partiallikelihood:

Illustration(Valberg)

Perfectlyfitting Cox model

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Seeing vs. doing

� Comparing two treatment groups by hazard ratio. End of trial, should all patients be put on the default treatment?

� Assume hazard ratio equals 1. Apparently no difference in treatment effect. So, ok to change?

� Surprise! The intervention yields a sharp change in the event rate in the group that changes treatment.

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0 1 2 3 4 5Time

0.5

1.0

1.5

Hazard ratio

Intervene

Questions

� What is the purpose of randomization?� Does testing the treatment effect with the

Cox model yield a causally valid answer?� Why is it not clear that the Cox model

estimates a causal effect even in randomized trials

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Survival paradoxes

� There are a number of well known paradoxes in epidemiology.

� Obesity paradox denotes cases where obesity is a risk factor for developing a condition, like heart failure, kidney failure or myocardial infarction, but once this condition has occurred obesity appears as a protective factor for survival.

� Cholesterol paradox: low cholesterol is associated with a poor prognosis in heart failure and in myocardial infarction.

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Obesity paradox

� 5070 references in Google Scholar� Apparent “obesity paradox” seen in diabetes,

coronary artery disease, heart failure, peripheral arterial disease, hypertension, chronic obstructive pulmonary disease, lung cancer, esophageal adenocarcinoma, myocardial infarction, coronary revascularization, acute coronary syndromes, and among hemodialysis patients. Preston and Stokes, Epidemiology, 2014

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Examples from the literature

� Obesity increases mortality in women, but reduces mortality in diabetic women. Lajous et al, Epidemiology, 2014

� Obesity reduces mortality among heart failure patients. Banack and Kaufman, Epidemiology, 2013

� Smoking appears to protect against Alzheimer’s disease. Part of the effect is due to survival paradox. Chang, Alzheimer Dis Assoc Disord, 2012

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Why these paradoxes

� There would be a mix of explanations�Some biological, I do not look at those�Some statistical, only those are considered

here

� So, how can biases be produced? The typical explanation concentrates on collider effects

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Questions

� Why do we denote findings like this as paradoxes?

� Can you imagine a reason for these paradoxes?

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Remember

� When we condition on a collider we may activate a non-causal pathway, thereby creating bias.

� We’ll come back to the obesity paradoxes in a moment, but first …

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One collider is ageThe survival of the fittest� Age selection is unavoidable. People die. If you

concentrate on an elderly population it is selected. People who are vulnerable to high cholesterol, obesity, smoking etc. may have died. The robust ones survive

� So, you may measure something in a young person and it may have a causal effect. Then the causal effect dissipates with age and is substituted by a selection effect

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� Gulsvik et al (2012) demonstrate that high serum cholesterol appears as a "protective" factor with respect to general mortality at old age, something which could be an artefact, especially since statin treatments reduce disease incidence of cardiovascular disease in the elderly (Savarese et al, 2013). A strongly reduced risk with age was also seen for smoking in Gulsvik et al. (2012).

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102-Year-Old Woman Quits Smoking (But Not for Her H ealth)

Collider effects and survival

Survival bias due to age

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Survived to time t

Randomizedtreatment

Survived to time t+dt

Con-founder

Questions

� In what sense is age a collider?� Why may the apparent protective effect of

high cholesterol in old age be a selection effect?

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Banack and Kaufman: Epidemiology, 2013

Obesity paradox in heart failure

� For patients with heart failure: The odds ratio of mortality was 0.79 (0.70–0.88) for overweight and 0.65 (0.57–0.74) for obese

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We need computation as well

� The collider effects just tell us that there might be something to worry about, but not whether the bias is large or small, or even nonexistent. (There do exist models where colliders have no effect at all)

� This requires a mathematical model where we explicitly model individual variation

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Risk model� Model for the first event (like diabetes etc)

�Hazard rate equals rate due to special risk factor (e.g. obesity) plus model with built-in risk variation Z

� By doing some mathematics, we can calculate the rate of mortality after the first event has occurred. It turns out to be smaller for those with the special risk than for those without

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Risk function when the event occurs

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Starting distribution

Special risk

No special risk

Lower curve: with special risk factorUpper curve: without

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Question

� What is the collider in the heart failure example?

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Mediation� Can we understand mechanisms

by using statistics?� Path analysis (Wright, 1921)

introduced the idea of direct, indirect and total effects and presented a simple calculus for these effects based on linear regression models.

� A lot of recent and sophisticated development of these ideas in the causal inference literature

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OutcomeTreatment

Mediator

How much of the effect passes directly from treatment to outcome (direct effect) and how much passes through the mediator (indirect effect)?

Mediation: Cholesterol treatment

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Mediation and confounding. Can we estimate the direct effect of statin on coronary heart disease?Confounders between mediator an outcome may give a false impression of an increased indirect effect

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Example

� Psychotherapy: It is well documented that therapeutic alliance appears to be a mediator. However, there is an obvious possibility of confounding effects

� Which type of effects could that be?

Psycho-therapy

Outcome

Therapeuticalliance

Confounder

You ignore causal inference at your own peril

� Example: Hormone replacement therapy (HRT) at menopause used to be viewed as a good idea because it was supposed to improve the health and well being ofthe woman. In fact, HRT was dangerous, increasing risk of breast cancer, heart disease and stroke

� But the epidemiological studies were wrongly interpreteddue to a lack of causal understanding, see Hernan et al, (2008). Randomized trials gave the correct answer

� Frailty selection is a relevant issue (Stensrud et al, 2017)� Today, the attitude to use of HRT is very cautious

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Summary

� Causal inference is a large and complex area� Various types of survival bias are ever present� Causal inference more and more becomes the

norm of analysis and presentation in an international epidemiological setting

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Some references

� Lange, T. and Hansen, J. V, (2011). Direct and Indirect Effects in a Survival Context. Epidemiology, 22(4):575-581. The electronic supplement contains some information on software

� Røysland, K., Jon Michael Gran et al. (2011). Analyzing direct and indirect effects of treatment using dynamic path analysis applied to data from the Swiss HIV Cohort Study. Statistics in Medicine. DOI: 10.1002/sim.4324

� Pearl J. Causality: Models, Reasoning, and Inference, 2nd edn. Cambridge University Press: Cambridge, 2009.

� Hernán MA, Robins JM. Instruments for causal inference: an epidemiologist’s dream? Epidemiology 2006; 17:360–372.

� Gran JM et al. A sequential Cox approach for estimating the causal effect of treatment in the presence of time dependent confounding. Statistics in Medicine 2010; 29:2757–2768

� Robins JM, Hernan MA, Brumback B. Marginal structural models and causal inference in epidemiology. Epidemiology 2000; 11(5):550–560.