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*Interesting Statistical PhenomenonVA San Diego Addictions Seminar 4/16/08 Kevin Cummins

*DefinitionsParadox: a statement that is seemingly contradictory or opposed to common sense and yet is perhaps trueFallacy: a false or mistaken ideaPrinciple: a comprehensive and fundamental law, doctrine, or assumption

*OutlineObjectiveSimpsons ParadoxWill Rogers Paradox Lords Paradox Berksons ParadoxMonte Hall Paradox Others

*ObjectiveCreate awareness of several statistical issues that might arise during your research

*OutlineObjectiveSimpsons ParadoxWill Rogers ParadoxLords Paradox Berksons ParadoxMonte Hall ParadoxOthers

*Simpsons ParadoxOccurs when the relationship between two categorical variables is reversed after a third variables is introduced.

The relationship between two variables differs within subgroups compared to that observed for the aggregated data.

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*Alaska Airlines America West.11.05.17.14.08.29

*Simpsons Paradox: Remedies/ResponsesStudy Design Use Experiments Collect appropriate covariate data Know the Research SystemCollect appropriate covariate dataAnalytically introduce conditionals (i.e. moderators/covariates)Use appropriate interpretations

*OutlineObjectiveSimpsons Fallacy Will Rogers Paradox Lords ParadoxBerksons ParadoxMonte Hall ParadoxOthers

*Will Rogers Paradox

When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states.

*The Will Rogers Paradox (WRP) is obtained when moving an element from one set to another set the average values of both sets increase:

The effect will occur when both of these conditions are met:

1. The element being moved is below average for its current set.

2. The element being moved is above the current average of the set it is entering.

*WRP: Effect Shifting One Observation on the Mean

*WRP: Health Insurance ExampleThe 1997 migration moved lower cost PPO subjects into the HMOPPO No Longer FreeYoung et al. 1999

19961997HMO$98/Subscriber$119/SubscriberPPO$126/Subscriber$142/Subscriber

*Will Rogers: Remedies/ResponsesKnow Your SystemIn This Case:Statistically Adjust for Baseline Costs

*OutlineObjectiveSimpsons FallacyWill Rogers ParadoxLords Paradox Berksons Paradox Monte Hall ParadoxOthers

*Lords ParadoxSituation where change score analysis and ANCOVA yield apparently conflicting results

*A Simplified ExampleAssessment of a supplemental educational program (tutoring)

10 students from different schools

5 schools opted into the programs (free-choice)

Pre and post assessments given

No random/sampling/measurement error (simplified)

*Two StatisticiansStatistician OneCalculates difference scores for each groupChange scores the same for both groups

Statistician TwoAdjust for initial scoreFinds group differences

*Two StatisticiansPaired t-TestStatistician One

Data: group 1 vs. group 2

t = -0.002, df = 299, p-value = 0.99

ANCOVAStatistician Two

Coefficients: Value Pr(>|t|) (Intercept) 15.00.00 Pre 0.5 0.00 Group 20.0 0.00

*Statisticians AssumptionsStatistician one assumes that in the absence of any differential treatment effect the two groups despite different baselines would show equivalent changes Statistician two assumes that in the absence of any differential treatment effect the change of the groups as a whole is the same as the change within groups Both of these causal assumptions are untestable

*Lords Paradox: Remedies/ResponsesEstablish causal/system assumptions Use the best descriptive statementUse and report multiple approaches (Wright 2006)Know that ANCOVA has greater powerGraph your data

*OutlineObjectiveSimpsons FallacyWill Rogers ParadoxLords PrincipleBerksons ParadoxMonte Hall Paradox Others

*Berksons ParadoxAn association reported from a hospital-based case-control study can be distortedIf cases and controls experience differential hospital admission rates with respect to the suspected causal factor

*Typical Berkson ExampleExample from Roberts et al. 1978

Investigated the relationship between circulatory and respiratory disease.

Sampled the general population and hospital populations.

*OR = 3.9 [95% CI: 1.4-10.9]Circulatory Disease

*Circulatory DiseaseOR = 1.3 [95% CI: 0.9-2.3]

*Real Berkson ExampleExample from Berkson 1946Hypothetical example exploring the relationship between diabetes and cholecystitisNo greater admission rate for subjects with multiple conditionsDifferent rates of admission for cases and controls

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*Different Conditional Rates Not Required

*Berksons: Remedies/ResponsesThere is no analytical mitigationLimit conclusionsConsider alternative study design

*OutlineObjectiveSimpsons FallacyWill Rogers ParadoxLords PrincipleBerksons ParadoxMonte Hall Paradox Others

*Monty Hall Paradox

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*Lindleys ParadoxStandard Sampling Theory VS. Bayesian TheoryUnder some circumstances strong evidence against the null hypothesis doesnt result in the null being rejected

*Benfords LawOnes are the most common leading digit in most data. Notice that if a data entry (base 10) begins with a 1, the entry has to be at most doubled to have a first significant digit of 2. However, if a data entry begins with a 9, it only has to be increased by, at most, 11% to change the first significant digit into a 1.

*ReviewBig Picture Use care to interpret observational studies Know your systemConditional Responses Simpsons Lords Will Rogers Perspective Problems Berksons Monte Hall

*Doctor Tyrano, Look for a Covariate!

*************************valid estimates of effect require the sample of observations to be random equivalent /rep of the population***********stripplot var1, over(var5) xlabel(0 (1) 1)

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