Gender Bias Simpsons Paradox and Causal Inference
15 déc. 2019 suggesting gender bias in UC Berkeley admission process. 10. Page 12. Which departments were guilty? UC Berkeley's graduate ...
Estimating Average Proportional Changes in Large Sparse Data
Ratio of ratios. (compare totals). Easy to calculate very simple. Simpson's. Paradox. Average of logs. Intuitive (logs are for proportions). Sparse data.
CS70: Jean Walrand: Lecture 37. Simpsons Paradox Bertrands
Bertrand's Paradox. ? Confirmation Bias. ? Thinking fast and slow. ? The Problem with Statistics. ? What's next? Simpson's Paradox.
Simpsons paradox: A statisticians case study
variables confounded by a third.3-5 The Berkeley case has been discussed widely; we will revisit those admission data to illustrate Simpson's paradox.
Simpsons paradox - Wikipedia the free encyclopedia
29 jui. 2012 Simpson's Paradox and the Yule-Simpson effect. [11]. Contents. 1 Examples. 1.1 Kidney stone treatment. 1.2 Berkeley gender bias case.
Lecture 35
Wikipedia gives a good explanation of Simpson's paradox by another example. A real word example is the Berkeley sex bias case. In 1973 U.C. Berkeley was sued
Research Agenda Ryan Copus My main research interest is in
We argue that judicial politics researchers have been inattentive to the threat of Simpson's Paradox – the possibility that estimated effects disappear or
Causality Confounding
https://www.econstor.eu/bitstream/10419/238838/1/ier-v12-i1-p050-074.pdf
Simpsons Paradox: A Data Set and Discrimination Case Study
teach a range of statistical concepts including Simpson's paradox. Berkeley was sued for bias against women who had applied for admission to graduate ...
Simpsons paradox
22 mar. 2017 The name Simpson's paradox was introduced by Colin R. Blyth in 1972.[12]. Contents. 1 Examples. 1.1 UC Berkeley gender bias.
Simpson Paradox - Berkeley Math Circle
Simpson Paradox - Berkeley Math Circle
Simpson's paradox - Wikipedia the free encyclopedia
Jun 29 2012 · Simpson's paradox shows us an extreme example of the importance of including dataabout possible confounding variables when attempting to calculate causal relations Precise criteria for selecting a set of "confounding variables" (i e variables that yieldcorrect causal relationships if included in the analysis) is given in Pearl[2] using
Table of Outline Lecture 23 - Johns Hopkins Bloomberg School
Simpson’s paradox Berkeley data Confounding Weighting Mantel/Haenszel estimator Discussion Marginally white defendants received the death penalty a greater percentage of time than black defendants Across white and black victims black defendant’s received the death penalty a greater percentage of time than white defendants
Should Simpson's paradox be conflated with causal interaction?
Simpson’s Paradox should not be conflated with causalinteraction, however. What is distinctive of the paradox is not thatthe probabilistic relationship reverses upon partitioning, but ratherthat it reverses in allof the resulting subpopulations. Debate 2: Average Effects
What is the probability of Simpson's paradox?
A paper by Pavlides and Perlman presents a proof, due to Hadjicostas, that in a random 2 × 2 × 2 table with uniform distribution, Simpson's paradox will occur with a probability of exactly 1?60.
Which linear regression model illustrates Simpson's paradox for bivariate Cardinal data?
Figure 4:A linear regression model thatillustrates Simpson’s Paradox for bivariate cardinal data. Eachcluster of values corresponds to a single person (repeatedmeasurement). A similar example is presented in Figure 4, adapted from Kievit, Frankenhuis, Waldorp, and Borsboom (2013).
What is Fitelson's confirmation-theoretic explanation of Simpson's paradox?
Fitelson’s confirmation-theoretic explanation of Simpson’sParadox is that reasoners are not attentive to the difference betweenthe suppositional and conjunctive readings of confirmation statementswhen considering the evidential relevance of learning anindividual’s gender.
Gender Bias, Simpson's Paradox and Causal
InferenceLisa Goldberg
December 15, 2019
Berkeley Math Circle
Julia Hall Bowman Robinson (December 8,
1919{July 30, 1985) was an American mathematician noted for her
contributions to the elds of computability theory and computational complexity theory|most notably in decision problems. Her work on Hilbert's 10th problem (now known as Matiyasevich's theorem or the MRDP theorem) played a crucial role in its ultimate resolution. 2 As a graduate student, Julia was employed as a teaching assistant with the Department of Mathematics and later as a statistics lab assistant by Jerzy Neyman in the Berkeley Statistical Laboratory, where her work resulted in her rst published paper, \A Note onExact Sequential Analysis."
3A nepotism rule prevented Julia from teaching
in the mathematics department since she was married to Professor Raphael M. Robinson. So she stayed in the statistics department despite wanting to teach calculus. Raphael retired in 1971. In1976, after her election to the National Academy of Sciences, Julia
was oered a professorship in the UC Berkeley mathematics department. 4Graduate admissions at UC Berkeley
in 1973 Was there gender bias in the 1973 graduate admissions pro- cess?29% of admitted students were female.34% of applicants were female.
Men Women
Applicants Admitted Applicants AdmittedTotal 8442 44% 4321 35%The overall acceptance rate was 41%.
5 How might a statistician decide if the data indicate gender bias?Observed DataMen Women
Admitted 3738 1494
Denied 4704 2827Total 8442 4321
6 7 How might a statistician decide if the data indicate gender bias? Use a test statistic : a quantity derived from sample data. It is important that the distribution of the test statistic b ekno wn(o r approximately known) under a ssumptions .Adistribution is a collection of outcomes and their lik elihoods. An example of a distribution is a fair coin: outcomes are heads and tails; likelihoods are 50-50.8 Bickel et al. used expectations based on the overall acceptance rate of approximately 41% to generate a test statisticExpected DataMen Women
Admitted 3460.7 1771.3
Denied 4981.3 2549.7Total 8442 4321
9 The test statistic is called \Pearson chi-squared"Expected Data
Men Women
Admitted 3460.7 1771.3
Denied 4981.3 2549.7Observed minus Expected Data
Men Women
Admitted 277.3 -277.3
Denied -277.3 277.3
2=4X n=1(OnEn)2E n= 110:8Underassumptions , the probability that2is 110.8 or greater by pure chance is6:51026......suggesting gender bias in UC Berkeley admission process. 10Which departments were guilty?
UC Berkeley's graduate admissions processes are conducted by individual departments...... so a dean set out to determine the source of the bias... ...but did not nd it.Number of
Departments Result16 No women applied or no one was rejected4 Biased toward men
6 Biased toward women
75 No bias11
Simpson's paradox, or the Yule-Simpson eect
When looking at the statistical scores of groups, these scores may change, depending on whether the groups are looked at one by one, or if they are combined into a larger group. 12Causal inference and Simpson's
paradoxThe Book of Why
In 2018, Judea Pearl and Dana Mackenzie publishedThe Book of Why, a historically-grounded, accessible, colorful treatment of causal inference and statistics.The book is about a framework
fo rextracting cause-and-e ect relationships from data .13Why do we need a book about this?
Correlation
is the w orkho rseof statistics. It measures the tendency of two random quantities to move together.14Why do we need a book about this?
But correlation is symmetric in its arguments...
(X;Y)P n(XnX)(YnY) P n(XnX)2P n(YnY)21=2 ...so on its own, it can't implyX!YorY!X. 15Why do we need a book about this?
16 Still, researchers infer cause and eect from correlation all the time 17Simpson's reversal
In elementary school, we learn that summing numerators and denominators isnotthe way to add fractions.In fact, it is possible that: a=b > c=dande=f > g=h while (a+e)=(b+f)<(c+g)=(d+h):I'll illustrate with a simple example.
18A new miracle drug
19Does the new miracle drug prevent heart attacks?
Treatment Control
Heart attack 11 13
Healthy 49 47Total 60 60
20Does the new miracle drug prevent heart attacks?
All Treatment Control
Heart attack 11 13
Healthy 49 47Group 1
Heart attack 3 1
Healthy 37 19
Group 2
Heart attack 8 12
Healthy 12 2821
Does a new miracle drug prevent heart attacks?
All Treatment Control
Heart attack 11 13
Healthy 49 47
Percent healthy
8278 Group 1
Heart attack 3 1
Healthy 37 19
Percent healthy 93
95Group 2
Heart attack 8 12
Healthy 12 28
Percent healthy 60
70 22Simpson's reversal: percent healthy rates
control(1)>treatment(1) and control(2)>treatment(2)19=20>37=40 and 28=40>12=20
while control(total)Is treatment recommended?
The data seem to show that treatment was eective overall but damaging to each subgroup. Pearl and others argue that with with information about the nature of the groups, a causal model can tell us whether to trust the aggregated or disaggregated data. 24Age Suppose Group 1 contains younger patients and Group 2 contains older patients Both the treatment and the outcome depend on age (suppose younger patients are more open to trying the drug).X = DrugC = AgeY = Heart attack25 Age
All Treatment Control
82 78Younger
9395
Older 60
70 X = DrugC = AgeY = Heart attackConditioning on age, a
confounder, is necessary, so the subset-specic results provide the proper recommendation: no treatment .26Blood pressure
Suppose the drug operates, in part, by lowering blood pressure, which is a mediator of the drug's eect.X = DrugM = Blood pressureY = Heart attack27Blood pressure
All Treatment Control
8278 Lower
93 95Higher
60 70X = DrugM = Blood pressureY = Heart attackConditioning on blood pressure
would disable one of the drug's causal paths. The aggregate results provide the proper recommendation: t reatment .28Graduate admissions at UC Berkeley
in 1973Pearl's assessment of the 1973 graduate admission conundrumX = GenderM = DepartmentY = AdmissionWe have seen before that conditioning on a mediator is
incorrect if we want to estimate the total eect of one variable on another. But in a case of discrimination, according to the court, it is not the total eect but the direct eect that matters.|Judea Pearl29Bickel, Hammell and O'Connell's assessment
In their study, these authors examine the admissions data in detail. ...and nd that an important assumption underlying the statistical test they applied is not satised for the aggregate applicant pool. 30Assumptions underlying the Pearson chi-squared test
Assumption 1:
In any given discipline male and female applicants do not dier in respect of their intelligence, skill, qualications, promise, or other attribute deemed legitimately pertinent to their acceptance as student.It is precisely this assumption that makes the study of "sex bias" meaningful, for if we did not hold it any dierences in acceptance of applicants by sex could be attributed to dierences in their qualications, promise as scholars, and so on.|P.J. Bickel, E.A. Hammell, J.W. O'Connell 31What conclusions did Bickel, Hammell and O'Connell draw? Assumption 2:The sex ratios of applicants to the various elds of graduate study are not importantly associated (or correlated) with any other factors in admission. 32
Records from the largest departments
DepartmentMenWomen
Applicants AdmittedApplicants Admitted
825 62%108 82%
560 63%25 68%
325 37%593 34%
417 33%375 35%
191 28%393 24%
373 6%341 7%
33Records from the largest departments
Acceptance rates were relatively high in the largest departments, and they were higher for women than for men...DepartmentMenWomenApplicants AdmittedApplicants Admitted
82562% 10882%
56063% 2568%
325 37%593 34%
417 33%375 35%
191 28%393 24%
373 6%341 7%
34Records from the largest departments
...but women were severely underrepresented in the applicant pools.DepartmentMenWomenApplicants AdmittedApplicants Admitted
82562% 10882%
56063% 2568%
325 37%593 34%
417 33%375 35%
191 28%393 24%
373 6%341 7%
35Relationships among acceptance rates, percentage of female applicants and size of applicant pool.36 A statistical test is no more valid than its assumptions Assumption 2:The sex ratios of applicants to the various elds of graduate study are not importantly associated with any other factors in admission.The demonstrated falsity of this assumption invalidates the results
of the Pearson chi-squared test on the aggregate applicant pool.After further analysis taking account of the tendency of women to
apply to departments with lower acceptance rates, Bickel, Hammell and O'Connell concluded that there was no evidence of bias against women. However...37Final words in the 1975 paper
Women are shunted by their socialization and education toward elds of graduate study that are generally more crowded, less productive of completed degrees, and less well funded, and that frequently oer poorer professional employment prospects.|P.J. Bickel, E.A. Hammell, J.W. O'Connell 38Final words from me
Thoughtfully applied, mathematical tools can provide insight into complex, societal problems. But these societal problems will be solved only when individuals take responsibility. 39The Julia Robinson Mathematics
Festival
Julia lives on to inspire young mathematicians
In 2007, Nancy Blachman founded the Julia Robinson Mathematics Festival (JRMF), which sponsors locally organized mathematics events targeting K12 students. The events are designed to introduce students to the richness and beauty of mathematics in a collaborative and non-competitive forum. 40Thank you Berkeley Math Circle
and thank you Zvezda 41References
Bickel, P. J., Hammel, E. A. & O'Connell, J. (1975), `Sex bias in graduate admissions: Data from Berkeley',Science187, 398{403.
Moore, C. C. (2007),Mathematics at Berkeley: A History, A.K.Peters, Ltd.
Pearl, J. & Mackenzie, D. (2018),The Book of Why, Basic Books. Simpson, E. H. (1951), `The interpretation of contingency tables', Journal of the Royal Statistical Society, Series B13, 238{241. Yule, U. (1903), `Notes on the theory of association of attributes in statistics',Biometrika2(2), 121{134. 42quotesdbs_dbs14.pdfusesText_20
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