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What To Do About the Multiple Comparisons Problem? Peter Z. Schochet February 2008

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Overview of Presentation Background Suggested testing guidelines Background Suggested testing guidelines 2

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Background

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Overview of the Problem Multiple hypothesis tests are often conducted in impact studies –Outcomes –Subgroups –Treatment groups Standard testing methods could yield: – Spurious significant impacts – Incorrect policy conclusions Multiple hypothesis tests are often conducted in impact studies –Outcomes –Subgroups –Treatment groups Standard testing methods could yield: – Spurious significant impacts – Incorrect policy conclusions 4

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Assume a Classical Hypothesis Testing Framework True impacts are fixed for the study population Test H 0j : Impact j = 0 Reject H 0j if p-value of t-test < =.05 Chance of finding a spurious impact is 5 percent for each test alone True impacts are fixed for the study population Test H 0j : Impact j = 0 Reject H 0j if p-value of t-test < =.05 Chance of finding a spurious impact is 5 percent for each test alone 5

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But Suppose No True Impacts and the Tests Are Considered Together Probability 1 t-test Number of Tests a Is Statistically Significant 1.05 5.23 10.40 20.64 50.92 a Assumes independent tests 6

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Impact Findings Can Be Misrepresented Publishing bias A focus on stars Publishing bias A focus on stars 7

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Adjustment Procedures Lower Levels for Individual Tests Control the combined error rate Many available methods: –Bonferroni: Compare p-values to (.05 / # of tests) –Fishers LSD, Holm (1979), Sidak (1967), Scheffe (1959), Hochberg (1988), Rom (1990), Tukey (1953) –Resampling methods (Westfall and Young 1993) –Benjamini-Hochberg (1995) Control the combined error rate Many available methods: –Bonferroni: Compare p-values to (.05 / # of tests) –Fishers LSD, Holm (1979), Sidak (1967), Scheffe (1959), Hochberg (1988), Rom (1990), Tukey (1953) –Resampling methods (Westfall and Young 1993) –Benjamini-Hochberg (1995) 8

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These Methods Reduce Statistical Power- The Chances of Finding Real Effects Simulated Statistical Power a Number of Tests Unadjusted Bonferroni 5.80.59 10.80.50 20.80.41 50.80.31 a Assumes 1,000 treatments and 1,000 controls, 20 percent of all null hypotheses are true, and independent tests 9

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Big Debate on Whether To Use Adjustment Procedures What is the proper balance between Type I and Type II errors? 10

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To Adjust or Not To Adjust?

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February, July, December 2007 Advisory Panel Meetings Held at IES Chairs: Phoebe Cottingham, IES Rob Hollister, Swarthmore Rebecca Maynard, U. of PA Chairs: Phoebe Cottingham, IES Rob Hollister, Swarthmore Rebecca Maynard, U. of PA Participants: Steve Bell, Abt Howard Bloom, MDRC John Burghardt, MPR Mark Dynarski, MPR Andrew Gelman, Columbia David Judkins, Westat Jeff Kling, Brookings David Myers, AIR Larry Orr, Abt Peter Schochet, MPR 12

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Basic Principles for a Testing Strategy

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The Multiplicity Problem Should Not Be Ignored Erroneous conclusions can result otherwise But need a strategy that balances Type I and II errors Erroneous conclusions can result otherwise But need a strategy that balances Type I and II errors 14

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Limiting the Number of Outcomes and Subgroups Can Help But not always possible or desirable Need flexible strategy for confirmatory and exploratory analyses But not always possible or desirable Need flexible strategy for confirmatory and exploratory analyses 15

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Problem Should Be Addressed by First Structuring the Data Structure will depend on the research questions Adjustments should not be conducted blindly across all contrasts Structure will depend on the research questions Adjustments should not be conducted blindly across all contrasts 16

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Suggested Testing Guidelines

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The Plan Must Be Specified Up Front Rigor requires that the strategy be documented prior to data analysis 18

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Delineate Separate Outcome Domains Based on a conceptual framework that relates the intervention to the outcomes Represent key clusters of constructs Domain items are likely to measure the same underlying trait –Test scores –Teacher practices –School attendance Based on a conceptual framework that relates the intervention to the outcomes Represent key clusters of constructs Domain items are likely to measure the same underlying trait –Test scores –Teacher practices –School attendance 19

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Testing Strategy: Both Confirmatory and Exploratory Components Confirmatory component –Addresses central study hypotheses –Must adjust for multiple comparisons –Must be specified in advance Exploratory component –Identify impacts or relationships for future study –Findings should be regarded as preliminary Confirmatory component –Addresses central study hypotheses –Must adjust for multiple comparisons –Must be specified in advance Exploratory component –Identify impacts or relationships for future study –Findings should be regarded as preliminary 20

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Confirmatory Analysis Has Two Potential Parts 1. Domain-specific analysis 2. Between-domain analysis 1. Domain-specific analysis 2. Between-domain analysis 21

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Domain-Specific Analysis

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Test Impacts for Outcomes as a Group Create a composite domain outcome –Weighted average of standardized outcomes Simple average Index Latent factor Conduct a t-test on the composite Create a composite domain outcome –Weighted average of standardized outcomes Simple average Index Latent factor Conduct a t-test on the composite 23

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What About Tests for Individual Domain Outcomes? If impact on composite is significant –Test impacts for individual domain outcomes without multiplicity corrections –Use only for interpretation If impact on composite is not significant –Further tests are not warranted If impact on composite is significant –Test impacts for individual domain outcomes without multiplicity corrections –Use only for interpretation If impact on composite is not significant –Further tests are not warranted 24

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Between-Domain Analysis

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Applicable If Studies Require Summative Evidence of Impacts Constructing unified composites may not make sense –Domains measure different latent traits Test domain composites individually using adjustment procedures Constructing unified composites may not make sense –Domains measure different latent traits Test domain composites individually using adjustment procedures 26

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Testing Strategy Will Depend on the Research Questions Are impacts significant in all domains? –No adjustments are needed Are impacts significant in any domain? –Adjustments are needed Are impacts significant in all domains? –No adjustments are needed Are impacts significant in any domain? –Adjustments are needed 27

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Other Situations That Require Multiplicity Adjustments 1. Designs with multiple treatment groups –Apply Tukey-Kramer, Dunnett, or resampling methods to domain composites 2. Subgroup analyses that are part of the confirmatory analysis –Conduct F-tests for differences across subgroup impacts 1. Designs with multiple treatment groups –Apply Tukey-Kramer, Dunnett, or resampling methods to domain composites 2. Subgroup analyses that are part of the confirmatory analysis –Conduct F-tests for differences across subgroup impacts 28

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Statistical Power Studies must be designed to have sufficient statistical power for all confirmatory analyses –Includes subgroup analyses Studies must be designed to have sufficient statistical power for all confirmatory analyses –Includes subgroup analyses 29

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Reporting Must Link to the Study Protocols Qualify confirmatory and exploratory analysis findings in reports –No one way to present adjusted and unadjusted p-values –Confidence intervals may be helpful –Emphasize confirmatory analysis results in the executive summary Qualify confirmatory and exploratory analysis findings in reports –No one way to present adjusted and unadjusted p-values –Confidence intervals may be helpful –Emphasize confirmatory analysis results in the executive summary 30

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Testing Approach Summary Pre-specify plan in the study protocols Structure the data –Delineate outcome domains Confirmatory analysis –Within and between domains Exploratory analysis Qualify findings appropriately Pre-specify plan in the study protocols Structure the data –Delineate outcome domains Confirmatory analysis –Within and between domains Exploratory analysis Qualify findings appropriately 31

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