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Goldsmith’s teachers lecture
2008
Medical statistics
Sandra Eldridge
Professor of biostatistics
Aims
• To introduce medical statistics
• To highlight (without much detail) various different mathematical/statistical contributions
• To give examples of where medical statistics has contributed to society
Statistics - definition
Statistics is a mathematical science
pertaining to the collection, analysis,
interpretation or explanation, and
presentation of data. It is applicable to a
wide variety of academic disciplines,
from the natural and social sciences to
the humanities, and to government and
business.
Structure of lecture
• Historical– Interpretation and explanation – John Graunt
– Collection – Smoking and lung cancer, folic acid and neural tube defects
– Analysis and presentation – Hypothesis tests, relative risk, confidence intervals
• More recent developments– Analysis, interpretation and presentation – Forest
plots
– Interpretation and presentation – Funnel plots
– Bayesian approaches
– *** Design – Cluster randomised trials ***
Natural and Political Observations
Made upon the Bills of Mortality (1662)
John Graunt’s pioneering work
• used analysis of the mortality bills
• to create a system to warn of the onset
and spread of bubonic plague
• work resulted in the first statistically-based
estimation of the London’s population
collection, analysis, interpretation or
explanation, and presentation
Analysis and interpretationOf more than a quarter of a million deaths only 392 were assigned to the Pox
Forasmuch as by the ordinary discourse of the world it seems a great part of men have, at one time or other, had some species of this disease, I wondering why so few died of it, especially because I could not take that to be so harmless, whereof so many complained very fiercely; upon enquiry, I found that those who died of it out of the hospitals (especially that of Kingsland, and the Lock in Southwark) were returned of ulcers and sores. And in brief, I found, that all mentioned to die of the French Pox were returned by the clerks of St Giles' and St Martin's in the Fields only, in which places I understood that most of the vilest and most miserable houses of uncleanness were: from whence I concluded, that only hated persons, and suoh, whose very noses were eaten off were- reported by the searchers to have died of this too frequent malady
Mortality bills = routinely collected data
(largely legal rationale so estates could be
disposed of etc)
Q. How collected? Quality of data?
Sometimes need to collect data to answer
specific question
Q. How should data be collected?
Richard Doll (doctor) and Austin
Bradford Hill (statistician)
Is there a relationship between
smoking and lung cancer?
First study: Those who did have lung cancer and those who did not – how many smoked in each group? (1940s)
For patients aged 45 to 74, the relative risk of the disease in men and women combined was estimated to be 6, 19, 26, 49, and 65 when the number of cigarettes smoked per day was 3, 10, 20, 35, and 60.Richard Doll and A. Bradford Hill Br Med J. 1950 September 30; 2(4682): 739–748. Smoking and Carcinoma of the Lung
Interpretation in terms of events
and probabilitiesA = event (getting lung cancer)
B = event (not a smoker)
B1 = event (smoke 3 cigarettes a day)
B2 = event (smoke 10 cigarettes a day)
etc
• A and B are not independent
• B, B1 etc mutually exclusive
• P(A/B1) = 6P(A/B)
Issues
• Criticism from RA Fisher (father of
statistics): how do we know that smoking
came first?
• Second study: Followed up British doctors
from 1950
• Large numbers - 34,000 doctors – more
precise estimates (confidence intervals)
Can folic acid reduce neural tube
defects (e.g. spina bifida)?
• Study design = randomised controlled trial
– Trial = As much like an experiment as possible
– Randomised = allocated to groups on basis of chance
e.g. tossing a coin (ensures fair comparison)
– Controlled = a comparison group
• Hypothesis testing
• Large: 1817 women, 33 centres, 7 countries
Folic acid and neural tube defects
• Data collection – whether neural tube defect or not
– 1% NTD for women given folic acid, 3.5% for women not given folic acid
• Parameter = ρ = relative risk = 1%/3.5% = 0.28
• Null hypothesis: ρ = 0
• Alternative hypothesis: ρ # 0
• Two sided hypothesis test
Presentation
“72% protective effect (relative risk 0.28, 95%
confidence interval 0.12-0.71)”
• Confidence interval = range within which 95% certain
that ‘true’ value lies
• Uncertainty - ‘true’ value not the same as value in
sample
• The central limit theorem: the sum of a large number of
independent and identically-distributed random variables will be
approximately normally distributed if the random variables have a
finite variance
• Random variable = value of outcome for individual patient
Lancet. 1991 Jul 20;338(8760):131-7.
Presentation of results –
importance of confidence intervals
• Effect size: e.g. relative risk
• Confidence interval: measure of uncertainty
Overemphasis on hypothesis testing has detracted from more useful approaches to interpreting study results, such as estimation and confidence intervals. In medical studies investigators are usually interested in determining the size of difference of a measured outcome between groups, rather than a simple indication of whether or not it is statistically significant. Confidence intervals present a range of values, on the basis of the sample data, in which the population value for such a difference may lie. (Gardner & Altman, British Medical Journal 1986)
So far….
• Collection, interpretation, explanation
– Graunt (information bias), Doll & Hill, Fisher
(co-response)
• Presentation
– Estimates and confidence intervals
• Analysis
– Summary values e.g. relative risk, hypothesis
tests, confidence intervals
Analysis based on….
• Understanding probability
• Properties of combinations of random
variables
• Properties of normal distribution
• Understanding hypothesis testing
Example
• Length of stay in hospital after operation
• Two different forms of aftercare
• Q: Does one form of aftercare have significantly
shorter stay in hospital than the other?
– Parameter of interest? Who of interest to? Design of
study? How is parameter produced from underlying
random variables? Likely distribution of random
variables?
– How many individuals to sample? What happens if
not enough individuals?
– Null hypothesis? Alternative hypothesis?
More recent developments
• Analysis, interpretation, presentation -
forest plots and funnel plots
• Bayesian approaches (increasingly
common in many branches of medical
statistics) - diagnosis
• Collection - Cluster randomised trials
Several studies looking at the same
thing
• Each study may be relatively inconclusive because of too much uncertainty (too small)
• Statistical (mathematical) method of combining and presenting results from several studies
• Can indicate more robust results
Copyright ©2007 BMJ Publishing Group Ltd.
Eurich, D. T et al. BMJ 2007;335:497
Pooled odds ratio for thiazolidinediones compared with other treatments for all cause mortality
Proportions dying
in each groupOdds ratio
Relies on the
logarithm of the
odds ratio being
approximately
normally
distributed
Forest plot
Odds ratio
• Odds= prob. of event happening/prob of event not happening
• In treatment group = (168/818) / ((818-168)/818)
168/650
• In control group = 1192/3508
• Odds ratio = (168/650) / (1192/3508)
(168x3508) / (650x1192)
0.76
• Indicates mortality less in thiazolidinediones group
• 95% confidence interval = 0.63 to 0.91
• Indicates 95% certain that ‘true’ value in this interval
• Interpretation = thiazolidinediones almost certainly reduce mortality
Comparing institutions, individual
doctors and identifying outliers
• What’s the problem?
– Lots of variables important
– Random variation
– Random variation greater for smaller
units or institutions
• Way of presenting the values for units
so that this is taken into account
Funnel plot
Bayes theorem and diagnosisIf D+ represents disease and S=symptom
P(D+|S) = P(D+) * P(S|D+) / P(S)
P(D+|S) represents probability of person
with particular symptom having disease
• Diagnosis
• Risk scores dependent on symptoms
Increase in chronic conditions
Trials to evaluate methods for
improving the management,
treatment, symptoms, quality of
life of those suffering from
Different from evaluating a drug
Clinical trials
• Example already – folic acid supplementation
trial
– Women recruited and randomly allocated to groups
Individually randomised trials
Intervention Control
But sometimes need more complicated design
Cluster randomised trials –
intervention (treatment) aimed at
whole cluster
Intervention Control
Understanding main issue in
cluster randomised trials
• Example 1: 1000 balls in large bag, some
blue some yellow – how many do you
need to draw out to estimate proportion of
yellow balls?
• Example 2: Balls placed in smaller bags of
5; there are no ‘mixed colour’ bags – how
many small bags would you need to draw
out to estimate proportion of yellow balls?
Main statistical issue
• Non-independence of characteristics for
individuals within clusters
– Get less information from a bunch of clusters
than from same number of individuals
selected from whole population
– Need larger sample sizes (but not quite as
bad as example of blue and yellow balls!)
– Need analysis to take account of greater
uncertainty in estimates (wider confidence
intervals)
Example 1
• Does an asthma liaison nurse reduce
emergency contact with health services for
those with asthma?
• Reduced the percentage of participants
attending with acute asthma (58% v 68%;
odds ratio 0.62, 95% CI 0.38 to 1.01)
Example 2
• Does extra help for GPs to screen for TB
mean they pick up more cases?
• Intervention practices showed increases in
the diagnosis of active tuberculosis cases
in primary care compared with control
practices (47% vs 34%; odds ratio 1.68,
95% CI 1.05-2.68)
Many trials are of interventions that
show only very small effects
How can we design better
interventions?
Developing mathematical models of
often complex interventions based on
probability trees
Conclusion
• As much about collection, interpretation
and presentation as calculation
• Concepts of random variables, probability
feed into analyses
• Making sense out of uncertainty
• Changing techniques as times change
