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4/13/2012 Interpreting Heterogeneity in Meta Analyses of Clinical Trials Steven A. Julious University of Sheffield Steven A. Julious 1 Introduction Outline how to investigate heterogeneity Give statistical test Highlight graphical

  1. 4/13/2012 Interpreting Heterogeneity in Meta Analyses of Clinical Trials Steven A. Julious University of Sheffield Steven A. Julious 1 Introduction • Outline how to investigate heterogeneity • Give statistical test • Highlight graphical methods • Use worked examples throughout Steven A. Julious 2 1

  2. 4/13/2012 Assessing Statistical Assumptions Steven A. Julious 3 Statistical Tests • The test of homogeneity of the treatment effect across studies is based on the statistic k � 2 ( ) ˆ ˆ Q = θ − θ w i i = i 1 Steven A. Julious 4 2

  3. 4/13/2012 The Residuals from a Meta Analysis • The weighted residuals from fitting a fixed-effects meta-analysis are given by ˆ ˆ = ( θ − θ ) q w i i i • Under the standard fixed-effects meta-analysis the standardised weighted residuals are ˆ ˆ θ − θ ( ) w i i ′ = q i k � − 1 w w i i i = 1 5 Steven A. Julious 5 Assessing the Residuals • The standardised residuals should follow a standard Normal distribution • Under the standard random-effects meta-analysis the weighted residuals and standardised weighted residuals would be ˆ w θ calculated from same equations by replacing and with i * w ˆ * θ and i Steven A. Julious 6 3

  4. 4/13/2012 Assessing the Residuals q ′ • A Normal probability plot of the q or may be used to i i check the distributional assumptions of the meta-analysis model Steven A. Julious Steven A. Julious 7 7 Multi-Centre Trials v Meta Analysis • A multi-centre trial may be analysed in the same way as a meta-analysis • That is by treating centres as studies. • Multi-centre trials can be analysed using an overall model which includes the terms centre and treatment. • Homogeneity of treatment effects across centres can be undertaken by fitting a treatment by centre interaction to obtain an estimate of treatment effect with its variance for each centre. • The standardised residual can be calculated replacing the w’s with the inverse of the variance. Steven A. Julious 8 8 4

  5. 4/13/2012 Worked Example Paroxetine in Adults Steven A. Julious 9 Meta analysis of 23 Paroxetine Trials Steven A. Julious 10 5

  6. 4/13/2012 Paroxetine Trials • The treatment effect is measured by the difference in mean HAMD between Paroxetine and placebo. • There seems to be some evidence of heterogeneity in treatment effect though most of the heterogeneity seems to be in the smaller trials • The fixed effect meta-analysis gave an overall mean difference of 2.91 with a 95% confidence interval of (2.66 to 3.17). • The test of homogeneity gives a P-value of <0.001 (chi- squared of 170.23 on 22 degrees of freedom). • The overall mean difference from the random effects meta- analysis is 3.36 with 95% confidence interval of (2.59 to 4.12) Steven A. Julious 11 Steven A. Julious 11 Fixed Effects Steven A. Julious 12 6

  7. 4/13/2012 q ′ Histogram of i 13 q ′ Normality Probability Plot for i Estimating the Mean and Variance from the Data 14 7

  8. 4/13/2012 q ′ Normal Probability Plot for Setting i the Mean and Variance to be 0 and 1 Respectively 15 Assessment of Homogeneity of Treatment Effects with Normality Assumption of Fixed Effects Meta-analysis Normal probability plot estimating the mean and variance from the data Histogram Normal probability plot setting the mean and variance to be 0 and 1 respectively 16 8

  9. 4/13/2012 Random Effects Steven A. Julious 17 q ′ Histogram of i 18 9

  10. 4/13/2012 Normality Probability Plot for q ′ i Estimating the Mean and Variance from the Data 19 q ′ Normal Probability Plot for setting i the Mean and Variance to be 0 and 1 Respectively Steven A. Julious 20 10

  11. 4/13/2012 Assessment of Homogeneity of Treatment Rffects with Normality Assumption of Random Effects meta-Analysis Normal probability plot estimating the mean and variance from the data Histogram Normal probability plot setting the mean and variance to be 0 and 1 respectively 21 Summary of Results so Far • There seems to be deviation from the assumptions for a fixed effects meta analysis although the data do seem to take a Normal form • The assumptions seem to hold better for a random effects meta analysis • If a fixed effects analysis was the primary analysis then the results would need to be interpreted with care • The diagnostics so far although easy to produce in most statistics packages but have limitations • Could an assessment of outliers better be made? Steven A. Julious 22 22 11

  12. 4/13/2012 Confidence Bands for the Normal Probability Plots • Confidence bands for the Normal probability plots can be constructed to assist in interpretation • For the Normal probability line confidence bands can be calculate – Using Normal approximation (using the Friendly macro) – Using simulation • For the observed data a confidence band can be calculated – Using bootstrapping Steven A. Julious 23 Simulation Envelopes 1. Simulate a value for the estimated treatment effect from w − 1 study i from a N(0, ), for i = 1,…, k . i q ′ 2. Perform the fixed-effects meta-analysis and calculate the i q ′ 3. Order the from smallest to largest i 4. Repeat 1. to 3. a number of times, for example 1,000 5. For each i, calculate the 2.5 th and 97.5 percentile of the 1,000 values, to give the 95% lower and upper bounds for all points Steven A. Julious 24 24 12

  13. 4/13/2012 Bootstrapping 1. From the observed data pairs ( θ , w − 1 ), i = 1,…,k, take a i i random sample with replacement of size k 2. Perform the fixed-effects meta-analysis and calculate q ′ the i q ′ 3. Order the from smallest to largest i 4. Repeat 1. to 3. a number of times, for example 1,000. 5. For each i, calculate the 2.5 th and 97.5 percentile of the 1,000 values, to give the 95% lower and upper bounds for all points 25 Steven A. Julious 25 Fixed Effects Steven A. Julious 26 13

  14. 4/13/2012 Normal Approximation, Estimating the Expected Values with the Mean and Variance from the Data Steven A. Julious 27 Normal Approximation, Estimating the Expected Values from N(0,1) Steven A. Julious 28 14

  15. 4/13/2012 Simulated Envelopes Steven A. Julious 29 Bootstrap Envelopes Steven A. Julious 30 15

  16. 4/13/2012 Comparison of Plots Normal using data Normal using N(0,1) Simulation using N(0,1) Simulation 31 Random Effects Steven A. Julious 32 16

  17. 4/13/2012 Normal Approximation, Estimating the Expected Values with the Mean and Variance from the Data Steven A. Julious 33 Normal Approximation, Estimating the Expected Values from N(0,1) 34 17

  18. 4/13/2012 Simulated Envelopes An outlying study? 35 Bootstrap Envelopes 36 18

  19. 4/13/2012 Comparison of Plots Normal using N(0,1) Normal using data Bootstrapping Simulation using N(0,1) Steven A. Julious 37 Summary of Results so Far (Again) • The additional analyses confirm the original analysis • The bands are wider at the extremes of the range • For this example bootstrapping has wide bands at one end • Allows for an assessment of outliers Steven A. Julious 38 38 19

  20. 4/13/2012 What if the Treatment Effects Differ Study to Study? • Find explanation – Investigate baseline imbalances – Look at subgroups • Are there any possible explanations? Steven A. Julious 39 Summary of Meta Analysis • Gave overview of how to investigate heterogeneity in a meta analysis • Recommended not to rely on statistical tests to assess heterogeneity • Described graphical approaches to assess heterogeneity Steven A. Julious 40 20

  21. 4/13/2012 Steven A. Julious 41 21

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