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2 Interesting Statistical Phenomenon VA San Diego Addictions Seminar 4/16/08 Kevin Cummins

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3 Definitions Paradox: a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true Fallacy: a false or mistaken idea Principle: a comprehensive and fundamental law, doctrine, or assumption

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4 Outline Objective Simpson’s Paradox Will Roger’s Paradox Lord’s Paradox Berkson’s Paradox Monte Hall Paradox Others

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5 Objective Create awareness of several statistical issues that might arise during your research

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6 Outline Objective Simpson’s Paradox Will Roger’s Paradox Lord’s Paradox Berkson’s Paradox Monte Hall Paradox Others

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7 Simpson’s Paradox Occurs 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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8 DelayedOn Time Alaska Airline 178 13% 1,338 88% America West 661 10% 5,804 90% Which Airline Should You Fly?

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10 Alaska Airlines America West.11.05.17.14.08.29

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11 Simpson’s Paradox: Remedies/Responses Study Design Use Experiments Collect appropriate covariate data Know the Research System Collect appropriate covariate data Analytically introduce conditionals (i.e. moderators/covariates) Use appropriate interpretations

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12 Outline Objective Simpson’s Fallacy Will Roger’s Paradox Lord’s Paradox Berkson’s Paradox Monte Hall Paradox Others

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13 Will Roger’s Paradox “When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states.”

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14 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.

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15 WRP: Effect Shifting One Observation on the Mean

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16 WRP: Health Insurance Example 19961997 HMO$98/ Subscriber $119/ Subscriber PPO$126/ Subscriber $142/ Subscriber The 1997 migration moved lower cost PPO subjects into the HMO PPO No Longer Free Young et al. 1999

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17 Will Rogers: Remedies/Responses Know Your System In This Case: Statistically Adjust for Baseline Costs

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18 Outline Objective Simpson’s Fallacy Will Roger’s Paradox Lord’s Paradox Berkson’s Paradox Monte Hall Paradox Others

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19 Lord’s Paradox Situation where change score analysis and ANCOVA yield apparently conflicting results

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20 A Simplified Example Assessment 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)

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21 Two Statisticians Statistician One Calculates difference scores for each group Change scores the same for both groups Statistician Two Adjust for initial score Finds group differences

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23 Two Statisticians Paired t-Test Statistician One Data: group 1 vs. group 2 t = -0.002, df = 299, p-value = 0.99 ANCOVA Statistician Two Coefficients: Value Pr(>|t|) (Intercept) 15.00.00 Pre 0.5 0.00 Group 20.0 0.00

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24 Statistician’s Assumptions Statistician 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

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26 Lord’s Paradox: Remedies/Responses Establish causal/system assumptions Use the best descriptive statement Use and report multiple approaches (Wright 2006) Know that ANCOVA has greater power Graph your data

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27 Outline Objective Simpson’s Fallacy Will Roger’s Paradox Lord’s Principle Berkson’s Paradox Monte Hall Paradox Others

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28 Berkson’s Paradox An association reported from a hospital- based case-control study can be distorted If cases and controls experience differential hospital admission rates with respect to the suspected causal factor

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29 Typical Berkson Example Example from Roberts et al. 1978 Investigated the relationship between circulatory and respiratory disease. Sampled the general population and hospital populations.

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

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31 Circulatory Disease OR = 1.3 [95% CI: 0.9-2.3]

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32 Real Berkson Example Example from Berkson 1946 Hypothetical example exploring the relationship between diabetes and cholecystitis No greater admission rate for subjects with multiple conditions Different rates of admission for cases and controls

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34 Different Conditional Rates Not Required

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35 Berkson’s: Remedies/Responses –There is no analytical mitigation –Limit conclusions –Consider alternative study design

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36 Outline Objective Simpson’s Fallacy Will Roger’s Paradox Lord’s Principle Berkson’s Paradox Monte Hall Paradox Others

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37 Monty Hall Paradox

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39 Lindley’s Paradox Standard Sampling Theory VS. Bayesian Theory Under some circumstances strong evidence against the null hypothesis doesn’t result in the null being rejected

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40 Benford’s Law Ones 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.

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41 Review Big Picture Use care to interpret observational studies Know your system Conditional Responses Simpson’s Lord’s Will Roger’s Perspective Problems Berkson’s Monte Hall

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42 Doctor Tyrano, Look for a Covariate!

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