David Salsburg AP Statistics Reading Daytona Beach Florida June 16 2011 Harvey was wrong William Harvey circulation of the blood 1628 Bishop of Chichester Harvey was wrong because he used experimentation and It is well known that Nature abhors experimentation and will purp ID: 302866
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Slide1
Statistics and Experimentation
David
Salsburg
AP Statistics Reading
Daytona Beach, Florida
June
16, 2011Slide2
Harvey was wrong
William Harvey, circulation of the blood, 1628
Bishop
of Chichester:Harvey was wrong because he used experimentation and, “It is well known that Nature abhors experimentation and will purposely do things wrong if you attempt to experiment.”Slide3
An experiment that “went wrong”
The
Lanarkshire
Milk Study (1929) Question: Does Pasteurization take the “good” out of the milk? How do you measure the “good” in milk? Measure weight gain in children as a surrogate.
Yule:
“In our lust for measurement, we frequently measure that which we can, rather than that which we wish to measure, and forget that there is a difference.”Measures of “intelligence”Slide4
An experiment that “went wrong”
If children were to be used, which children
?
in schoolWhere?Easily available in London or Manchester, but too heterogeneous a population, too much variability in socioeconomic factors.Lanarkshire County, Scotland, population 300,000,evenly
divided into small factory towns
and rural
communities.How many children?Neyman-Pearson concept of power not yet published.Slide5
Final Design
2
0,000
children, 200-400 per school, several grades5,000 randomly assigned an extra daily ration of raw milk 5,000 randomly assigned an extra daily ration of Pasteurized milk
1
0,000
randomly assigned to no extra milk—controlsStudy ran from Feb-June, 1930, the children weighed at the beginning of the study and at the end.Slide6
Results
Average
weight gain for children on raw milk almost exactly the same as average weight gain for children on Pasteurized
milk.Average weight gain for children kept as controls (no extra milk) three times the average weight gain of the two other groups.No loss of “good” in milk (as measured by weight gain) when pasteurizedBest not to give children any extra milk, raw or pasteurized!Slide7
An experiment that “went wrong”
Royal Commission sent to investigate
William Sealy Gossett (“Student”)
chairmanConclusion: The teachers had been told to “randomly assign” but many of them took pity on the sickly and poor students and assigned them the extra milk.Slide8
HOW DO YOU “RANDOMIZE”?
Can you do it with haphazard choice by
humans?
Problem of digit preferenceCan you let “nature” do it?Toxicological studies of mice.Slide9
How did R.A. Fisher randomize?
Last two digits of populations of English towns in the 1921 census.
A table of 7500 two digit numbers arranged in blocks of 25. First block of 25
: 03 47 43 73 86 36 96 47 36 61 46 98 63 71 62
33
26 16 80 45
60 11 14 10 95Slide10
Rand Corporation book of 1 million random digits
Martin
Gardiner
(Scientific American): “This is the quintessential book of the Twentieth Century. Not only was no book produced like this in previous centuries, no one would have ever conceived of a book like this in previous centuries.”Slide11
How do you use a table of random numbers?
You do not start at the beginning. Otherwise, all randomizations would be the same.
You do not begin haphazardly (at random?). Books tend to have broken binding so haphazard openings often are at the same page.Slide12
RAND book preface
You
open the book haphazardly and pick a point to start
haphazardly.You pick out three digits, two digits, two more digits, and one digit.You go to the page indicated by the three digits, the line indicated by the first of the two digits, the column indicated by the second of the two digits. Then you proceed up and to the left (at the top of the page) if the final single digit is odd—or down and to the right if it is even.Slide13
Applying this method to Fisher’s 6-page table
I open it haphazardly (to page 2) and pick a spot haphazardly, yielding the following
sequence
2, 12, 23, 6I go to page 2, line 12, column 23, and go left and up from there. This yields the sequence: 67, 96, 57, 88, 30, 22, 23, 51, 14, 40, 24, 96
,…
Slide14
Comparing Three Treatments
Suppose I have three treatments, A, B, and C, to be applied to blocks of
three
A, B, C / A, B, C / A,…I append the sequence of numbers to this sequence of symbols A-67, B-96, C-57/ A-88,B-30,C-22/ A-23, B-51, C-14/…
I
reorder the symbols A
, B, C within each block following the order of the random numbers CAB/CBA/CAB/BAC…Slide15
Modern Methods
Use computer algorithm to generate a pseudo-random sequence.
Most
popular method, congruence generator: X(i+1) = res( AX(i) + B | C)A,B,C are mutually prime.The congruence generator cycles after K values, but K is a function of X(1), A, B, and C and can be calculated
.Slide16
Philosophical question
Can a pseudo-random number generator produce truly “random” numbers?
Fisher: Foolish question. All that is needed is that all possible treatment assignments be equally probable.Slide17
Can we do “better” than random?
NAACP and jury lists in Texas counties (
1960s)
Knut-Vik designs “Student” (1932) showed that Knut-Vik designs produce biased (downwards) estimates of the residual variance.Fisher (1935) random assignment produces the least variance of all unbiased designs.Slide18
A study that did “work”
Women’s Health Initiative Study of aspirin
vrs
placebo to prevent heart attacks or cardiovascular death in women. (March, 2005, New England J. of Medicine)Question: Does low dose aspirin prevent cardiovascular problems for women as it does for men?All but one of prior studies had used only men.
Consistent
finding: 81 mg aspirin a day reduces the incidence of non-fatal heart attacks by app. 30% and the incidence of cardiovascular related death by app. 20%.
One study that did use women as well as men enrolled 214 women, reduced incidence of cardiovascular related death by 9% (not statistically significant).Slide19
New Study
Large number of women (39,876) because incidence of cardiovascular events lower in women than in men.
Higher daily dose of aspirin (100 mg)
Longer follow-up (10 years vrs 5 in men’s studies)Single predefined
end-point:
Stroke, MI, or cardiovascular related death.
Problems with the end-point:Equivocal symptoms when patients arrive in emergency roomsDeath certificates unreliable
What happens if a patient has multiple events over the 10 year period?Solution: Set up elaborate check-list to “define” the events of interest. Choose only the first such event in a patient’s record to count.Slide20
Results
477
women on aspirin had a cardiovascular event
522 women on placebo had a cardiovascular eventp-value of the comparison—0.13Slide21
Confidence Bounds
Neyman’s
original
definition (1934)“On the two different aspects of the representative method,” J. Royal Statistical Society, vol. 97, pp. 558-625.
The paper establishes the fundamental ideas of survey sampling. It was used by the statisticians in the U.S. Bureau of Labor Statistics to establish the
first
surveys of unemployment. An
appendix establishes the fundamental ideas of confidence intervals.A Confidence interval on a parameter θ is a set of hypotheses about the value of θ that cannot be rejected by the data.Slide22
Other definitions of a confidence interval
Bayesian
The
expected coverage of the computed confidence interval is 0.95 regardless of the prior distribution on θ.Frequentist (derived by Neyman to meet Harold Hottelling’s
criticism of the Bayesian
definition)
95% of all confidence intervals computed this way will contain the true value of θ.Anscombe:
“What has the statistician’s long run probability of error to do with whether this patient should be given this treatment?”Slide23
Women’s Health Initiative study
T
hey
computed 95% confidence bounds on the ratio Prob{event|aspirin}/Prob{event|placebo}
95% C.I. = [0.80, 1.03]
Interpretation: Use of low dose aspirin in women might reduce incidence of cardiovascular events by as much as 20%
(or increase it by as much as 3%)Slide24
Can the study be repeated with more subjects and greater statistical power
?
Modern clinical studies cost more than
$10,000 per patient.100,000 subject study would cost > $1 billion.Slide25
Conclusion?
L.J. Cohen, philosopher at Oxford University
Critic of the use of statistical models in science.
One can never come to a certain conclusion with statistical models alone.To reach a scientific conclusion, it is necessary to bring in information external to the experimental study.(Cohen’s solution is to replace hypothesis testing with modal valued logic, a system of symbolic logic that denies the law of the excluded middle.)Slide26
Ignoring Cohen’s solution: What
information exists from outside this trial
?
The pharmacological mechanism of low dose aspirin is firmly established and is not gender related in experimental animals.The cost of a false positive is small. Aspirin is cheap. Low doses of aspirin are very safe for most people.
The cost of a false negative, if the use of low dose aspirin decreases CV events by 20%, is immense.
Conclusion
: Women should be given daily low doses of aspirin to prevent cardiovascular events.Slide27
Was it worth doing the study?
Side note: All the male studies and this women’s study of low dose aspirin have shown a consistent 8-fold increase in the incidence of hemorrhagic stroke for patients on aspirin—the comparison sometimes reaching statistical significance.