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Meta Analysis
Zhezhen Jin
A statistical technique for summarizing the results of several Karl Pearson (1904) Averaged correlations for studies of the effectiveness of inoculation for typhoid fever R. A. Fisher (1944) When a number of quite independent tests of significance have been made, it sometimes happens that although few or none can be claimed individually as significant, yet the aggregate gives an impression that the probabilities are on the whole lower than would often have been obtained by chance Source of the idea of cumulating probability values W. G. Cochran (1953) Discusses a method of averaging means across independent studies Laid-out much of the statistical foundation that modern meta-analysis is built upon (e.g., inverse variance weighting and homogeneity testing) Cochrane Collaboration started in 1993 (Evidence based health Team work: based on protocol and use summary statistical “The Cochrane Collaboration is an international network of more than 28,000 dedicated people from over 100 countries. We work together to help healthcare practitioners, policy-makers, patients, their advocates and carers, make well-informed decisions about health care, by preparing, updating, and promoting the accessibility of Cochrane Reviews over 5,000 so far, published online in the Cochrane Database of Systematic Reviews, part of The Cochrane Library. We also prepare the largest collection of records of randomised controlled trials in the world, called CENTRAL, published as part of The Cochrane Library.” Driven by the evidence-based medicine movement and the Resolves controversy between conflicting findings Provides reliable basis for decision making 4. Study selection (inclusion/exclusion criteria) Example: Marfan’s syndrome
It is genetic disorder of the connective tissues Usually tall, with long limb, leg, thin fingers Major causes of morbidity and mortality: cardiovascular complication of aortic dissection and rupture In late 1960s, blood pressure lowing medication improves survival of general patients with acute dissection of aortic aneurysms.
With the observation: Blood pressure lowering drug used to treat patients with aortic root dilatation related to Marfan’s syndrome.
Beta blocker therapy: make heart beats more slowly and with less force, thereby reducing blood pressure, and also help blood vessels Several small studies available, using echocardiography to measure Clinically, beta-blocker therapy is used routinely for Marfan’s Problem: No convincing evidence of long-term outcome.
• Observational studies ((e.g. case control, non-randomized cohorts, cross-sectional prevalence studies, etc.) • Combination of randomized and observational studies • Individual patient data studies Outcome Measures
Create a summary statistic that is comparable across all studies: 1. Binary data: alive/dead, diseased/non-diseased, Risk difference, relative risk (risk ratio), odds ratio 2. Continuous data: weight loss, blood pressure Mean difference, standardized mean difference, z-statistic, 3. Survival data: time to death, time to recurrence, time to 4. Ordinal data (ordered categorical data): disease severity, Binary data
Risk difference, relative risk (risk ratio), odds ratio where qT = 1 − p , n treated and control patients, and a, b, c, d denote the number of Relative risk and odds ratio both use logarithmic scales Continuous data
Required for each group: mean, standard deviation, sample size (nT − 1)s2 + (nC − 1)s2 2. Effect size (standardized mean difference) Difference of means divided by the variability of the measures Survival data
Time to event arise whenever subjects were followed over time until Problem: not everyone has an event, censoring HR: ratio of the risk of having an event at any given time in treatment group over the the risk of an event in the control group.
Analysis with Cox proportional hazards models Estimates from different studies are different Two possibilities: sampling error (homogeneous), true variation Cochran’s Chi-square test and I2-test, plots Suppose there are K studies with summary statistics θi, astatistical test for the homogeneity H0 : θ1 = θ2 = · · · = θK = θ H1 : At least one θi is different , the χ2 distribution with K − 1 degrees of freedom.
Some comments on Cochran’s Q statistic Power of the test might be very low due to small number of studies When sample sizes in each study are very large, then H0 may be rejected even when the individual effect size estimates do not The likelihood of design flaws in primary studies and publication biases makes the interpretation of test complex.
Higgins and Thompson’s I2
I2 = 100(Q − (K − 1))/Q the proportion of total variability explained by heterogeneity Values < 25% be thought to be ‘low’ The effect size: has two values, estimate and its standard error In usual statistics: only one measure is available Notation: (yi, si), var(yi)=s2i Fixed effects model and random effects model All studies share common true effect size Factors that could impact on the true effect size are the same Observed effect sizes vary among studies only because a random The random sampling error can be estimated where ϵi ∼ N (0, s2) for i = 1, 2, · · · , K.
Approximate 100(1 − α)% confidence interval for θ is: Random Effects Model (DerSimonian and Laird model)
Studies are a random sample of a hypothetical population of Two sources of variation: the between and within study variance.
yi = θ + δi + ϵi where δi ∼ N (0, τ 2) and ϵi ∼ N (0, s2) for i = 1, 2, · · · , K. The τ 2 is The random effects model will usually generate a confidence interval as wide or wider than that using the fixed effect model.
Results from a random effects model will usually be more conservative (there are exceptions).
If τ 2 is known, the pooled estimate of θ is given by where Wi(τ ) = 1/(s2 + τ 2).
Approximate 100(1 − α)% confidence interval for θ is: As we said, τ 2 is unknown, and need to be estimated.
Under normal error assumptions, maximum likelihood estimate or restricted maximum likelihood estimate can be used.
Random effects models account for heterogeneity between studies, Two approaches can address the issue: subgroup analyses and Subgroup analyses: focus factors that are possibly different across studies, such as patient characteristics, study conduct Meta-regression: include covariates to the fixed and mixed effects models. Used to estimate the impact/influence of categorical and/or continuous covariates (moderators) on effect sizes or to predict effect sizes in studies with specific characteristics. A ratio of 10:1 (studies to covariates) is recommended Publication Bias
One major concern of meta-analysis is publication bias: 1. If missing studies are random, failure to include these studies will result in less information, wider confidence intervals, and 2. If missing studies are systematically different from available 1. Large studies are likely to be published regardless of statistical 2. Moderately-sized studies are at risk for being lost, only those with significant results might be published 3. Small studies are at the greatest risk of being lost, studies with small sample sizes only very large effects are likely to be significant and those with small and moderate effects are likely How to assess publication bias and how to adjust it? Funnel plot: is a scatter plot of sample size or other measure of precision on the y-axis versus the estimated effect size on the x-axis.
In the absence of publication bias the studies will be distributed symmetrically about the combined effect size In the presence of bias, the plot may become skewed, the bottom of the plot would tend to show a higher concentration of studies on one side of the mean than the other This would reflect the fact that smaller studies (which appear toward the bottom) are more likely to be published if they have larger than average effects, which makes them more likely to meet the criterion for statistical significance It is informal visual method, and a useful funnel plot needs a range Different people might interpret the same plot differently Skewed funnel plot might be caused by factors other than Formal tests
Rank correlation test (Begg and Mazumdar): yw) = s2 − 1/ Then the rank correlation test is based on statistic Eggers linear regression method, quantifies the bias captured by the funnel plot using the actual values of the effect sizes and their In the Egger test, the standardized effect (effect size divided by standard error) is regressed on precision (inverse of standard error) Then fit weighted linear regression either with weight 1/si orunweighted and the equation α is used to measure asymmetry.
If it is significantly different from 0, then it is concluded that there A negative α indicates that smaller studies are associated with Small studies generally have a precision close to zero, due to their large standard error In the absence of bias such studies would be associated with small standardized effects and large studies associated with large standardized effects This would create a regression line whose intercept approaches the If the intercept deviates from this expectation, publication bias This would occur when small studies are disproportionately If the publication bias is suspected, may model the selection process into the model for bias correction. One possibility is to view the problem as a missing data problem and assume that the studies are missing with probabilities that are a function of their lack of statistical significance. For example,  0, if z ≤ 1.96 pi(z) =  1, if z > 1.96 Treat pi(z) as missing probability and carry out analysis.
Meta regression
Two types of regression models are possible: fixed effects meta-regression and random effects mete-regression model yi = θ + XT β + ϵi yi = θ + XT β + δi + ϵi where δi ∼ N (0, τ 2) and ϵi ∼ N (0, s2).
Challenging issues Studies do not report same measures. For example, the measure for variation might be ‘Not significant’ or ‘P < 0.05’ Individual-patients meta-analysis (recent and still ongoing) Example: Prevention of fractures after organ transplantation Organ transplantation: Heart, kidney, lung, liver Treatment: bisphosphonates, active metabolites of vitamin D Available studies: small, no definitive clinical trial To assess differences in bone fracture among treated and untreated Data sources and Searches
Cochrane Controlled Clinical Trials Register Unpublished abstracts for various meetings Keywords: transplant, osteoporosis, bone loss, fracture, transplant and calcitriol, transplant and bisphosphonates, transplant and Study selection
Randomized clinical trials, treatment and control groups, fracture Eligible treatment: oral or IV bisphosphonates (alendronate, risedronate, pamidronate, ibandronate, zoledronic acid) or active vitamin D analogs (calcitriol, calcidiol, 1α-hydroxyvitamin D) No restriction on sample size or specific dose of bisphosphonate or Studies with historical controls were excluded.
Bone marrow transplants were excluded.
Treatments for bone loss prevention: hormone replacement therapy, calcitonin, or resistence exercise were excluded Study design, methods, subjects, interventions, fracture, and bone Primary outcome: vertebral or nonvertebral fracture sustained Radiographs of the thoracic and lumbar spine (LS) at baseline and Secondary outcome: BMD measured by dual-energy x-ray absorptiometry in grams per square centimeter at the LS and Method of randomization, presence/absence of double-blinding, Studies were scored between 0 and 5, with 5 as the highest quality.
685 abstracts: 607 eliminated, 42 duplicate 36 remained, and among them, 28 published 9 were not adequately randomized, 8 were excluded due to delay on treatment and no response from the authors Included: 11 studies with 780 participants Prophylactic use of lidocaine after heart attack Source: Hine, L.K., Laird, N., Hewitt, P. and Chalmers, T.C.
(1989) Meta-analytic evidence against prophylactic use of lidocaine in Myocardial Infarction, Archives of Internal Medicine, 149,
Sodium fuoride (NaF) with sodium monoflouorophosphate (SMFP) dentrifrices for the purpose of reducing dental decay.
The outcome: decayed, missing or filled teeth score (DMFS) Source: Johnson, M.F. (1993) Comparative efficacy of NaF and SMFP dentrifrices in caries prevention: a meta-analytic overview, Caries Res., 27, 328-336.

Source: http://www.math.nankai.edu.cn/~mqliu/teach/JinZhezhen-nankai2013-meta626.pdf