統合分析 ( Meta-Analysis) 張 玉 坤 淡江大學數學系 教授 國防醫學院護理系 兼任教授

What is meta-analysis? The National Library of Medicine define meta-analysis as a “quantitative method of combining the results of independent studies (usually drawn from the published literature) and synthesizing summaries and conclusions which may be used to evaluate therapeutic effectiveness, plan new studies, etc., with application chiefly in the areas of research and medicine.” Meta-analysis may be broadly defined as the quantitative review and synthesis of the results of related but independent studies.

Examples: Antipsychotic Combinations vs. Monotherapy in Schizophrenia Shenqi fuzheng for NSCLC Care management for Type 2 diabetes in the United States

Note: Conducting a meta-analysis is conceptually no different than conducting primary research. It is multidisciplinary and therefore requires a research team comprising several experts: (a) A subject-matter specialist (e.g. psychiatrist, cardiologist); (b) A biostatistician to help in the design and analytic aspects of the research; (c) A group of subject-matter specialists to aid in judging the relevance of the retrieved documents

The objectives of a meta-analysis include: (1)increasing power to detect an overall treatment effect, (2)estimation of the degree of benefit associated with a particular study treatment, (NNT or NNH) (3)assessment of the amount of variability between studies, (4)identification of study characteristics associated with particularly effective treatments.

LANCET.pdf from Stefan

LANCET.pdf

Q: What is NNT (Number Needed to Treat) ? or NNH (Number Needed to Harm) ?

95% Confidence Interval for NNT (or NNH) 先算出 95% Confidence Interval for :

NNT&NNH

Figure1-1 LANCET.pdf Figure1-1.excel

Figure1-2.excel Figure1-2.dta Forest Plot ?! LANCET.pdf

Figure 2-1 Figure 2-1.dta LANCET.pdf

Figure1.dta

Figure 2-2 Figure 2-2.dta LANCET.pdf

The objectives of a meta-analysis include: (1)increasing power to detect an overall treatment effect, (2)estimation of the degree of benefit (Effect Size) associated with a particular study treatment, (3)assessment of the amount of variability between studies (Homogeneity Test), (4)identification of study characteristics associated with particularly effective treatments.

常用的 Meta-Analyses: 二項式數據 (Binary Data) (a) Relative Risk (RR, 相對危險度 ) (b) Odds Ratio (OR, 優勢比 ) (c) Rate Difference (RD, 率差 ) 常態分佈數據 (Normally Distributed Data) Fixed Effect Method: Random Effect Method: Meta Regression:

試驗組對照組 Total 正反應 aibim 1i =(ai+bi) 負反應 cidim 2i =(ci+di) Totaln 1i =(ai+ci)n 2i =(bi+di)Ni Notations and Terminologies: 二項式數據 (Binary Data) (i.e. 收錄 K 個獨立臨床實驗 ) Fixed Effect Method:

二項式數據 (Binary Data) Pti = ai/(ai+ci) : Treatment Response Rate of i th Clinical Trial ; Pci = bi/(bi+di) : Control Response Rate of i th Clinical Trial; RRi = Pti / Pci ( 第 i 個臨床實驗的相對危險度 ) (a) Relative Risk (RR, 相對危險度 ) 試驗組對照組 Total 正反應 aibim 1i =(ai+bi) 負反應 cidim 2i =(ci+di) Totaln 1i =(ai+ci)n 2i =(bi+di)Ni Note: (i.e. 抽樣分佈通常呈現偏常態分佈 )

為校正此偏常態分佈現象, 對 RRi 取自然對數值, i.e. RRi = Pti / Pci ( 兩邊取自然對數 ) ln RRi = ln(Pti) – ln(Pci) 可使得轉換後之 ln RRi 的抽樣分佈較為近似常態分佈, 且其標準誤 (Standard Error; SE) 可由下式計算 : 換言之, By δ-method

(1) 如何利用所有的 RRi or ln(RRi) 來 估 計共同的 RR (i.e. Pooled RR)? Ans: 採用加權平均 (Weighted Average) 法。 Q: 如何選取 Weight? Guideline: 1. 樣本數越大， Weight 越重。 2. 變異數越大， Weight 越輕。

(1) 如何利用所有的 RRi or ln(RRi) 來 估 計共同的 RR (i.e. Pooled RR)? Ans: 其中 稱為加權 (Weight). 因此, ln(RR Pooled ) 的標準誤 (SE) 為 : RR.Excel

(2) 如何求取 RR Pooled 的 95% 信賴區間 (CI)? RR Pooled 的 95% 信賴區間 (95% C.I.) 為 : 其中 ̥ RR.Excel

(3) Testing H * 0 : RR=1 vs. H 1 : or H * 0 : ln(RR)=0 vs. H 1 : Reject H * 0 if Under H* 0 RR.Excel

(4) RR i 同質性檢定 (Test of Homogeneity) i.e. Testing 檢驗 K 個獨立研究結果的 RR i 是否相同的 Testing Statistics, Q RR, 為 : Under H 0, Q RR 遵從自由度為 K-1 的卡方分佈̥ 換言之, Reject H 0 if Q RR > RR.Excel

Example: Chlorpheniramine 對 Drowsiness 影響的 8 個臨床實驗結果 RR.Excel Results: 1. P(Z RR > ) < , i.e. Chlorpheniramine 對 Drowsiness 的影響比 Placebo 明顯大很多 ; 2. 8 個獨立臨床實驗具有相同之相對危險度 (RR). RR.dta

常用的 Meta-Analyses: 二項式數據 (Binary Data) (a) Relative Risk (RR, 相對危險度 ) (b) Odds Ratio (OR, 優勢比 ) (c) Rate Difference (RD, 率差 ) 常態分佈數據 (Normally Distributed Data) Fixed Effect Method: Random Effect Method: Meat regression:

(b) Odds Ratio (OR, 優勢比 ) 優勢 ( Odds) = 則, 且 因此, 優勢比 (Odds Ratio), OR i = 試驗組對照組 Total 正反應 aibim 1i =(ai+bi) 負反應 cidim 2i =(ci+di) Totaln 1i =(ai+ci)n 2i =(bi+di)Ni Y=1 Y=0

What is Odds ( 勝算, 涉險, …)? where Y=1 or 0 and p = Pr( Y=1). ( 若以 Y=1 代表正面事件發生, 如 : 痊癒, 中獎, … 等. 則 Odds 可解讀為勝算 ; 若 Y=1 代表負面事 件發生, 如 : 死亡, 失敗, … 等. 則 Odds 可解讀為 涉險 or 風險 ) CHD StatusWhiteBlackHispanicOther Present ( p i )5 (0.2)20 (0.67)15 (0.6)10 (0.5) Absent ( 1-p i )20 (0.8)10 (0.33)10 (0.4)10 (0.5) Total Odds ( p i / 1-p i )1/42/13/21

Ex. 比較種族間 CHD 罹患率的差異 CHD StatusWhiteBlackHispanicOther Present Absent2010 Total Odds Ratio log e (O.R.)

Note: 如何利用所有的 ORi or ln(ORi) 來 估 計共同的 OR (i.e. Pooled OR)? Ans: 採用加權平均 (Weighted Average) 法。 Q: 如何選取 Weight? Guidelines: 1. 樣本數越大， Weight 越重。 2. 變異數越大， Weight 越輕。

(1) Woolf ’s Logit estimate of Odds Ratio 其中 OR i = aidi/bici 是第 i 個獨立臨床實驗的 優勢比, 且. 而 V(ln OR i ) 為 ln OR i 的變異數, i.e.. 換言之, OR W 的 95% 信賴區間 (95% C.I.) 為 : OR.Excel

(2) OR W 的顯著性檢驗 : i.e. Testing H 0 : OR W = OR 0 (e.g. OR 0 =1) 因為, Under H 0, 因此,under H 0, i.e. Chi-Square with 1 degree of freedom. OR.Excel

(3) OR i 的同質性 (Common OR) 檢驗 : i.e. Testing 對於 K 個獨立研究結果的 OR i 是否相同的 Testing Statistics, Q W, 為 : i.e. Under H 0, Q W 近似地遵從自由度為 K-1 的卡方分佈 Test for Homogeneity! OR.Excel

Example: Chlorpheniramine 對 Drowsiness 影響的 8 個臨床實驗結果 OR.Excel Results: 1. P(U W > ) < , i.e. 與對照組 相比, Chlorpheniramine 顯著影響 Drowsiness ; 2. 8 個獨立臨床實驗具有相同之優勢比. OR70.dta OR.dta

常用的 Meta-Analyses: 二項式數據 (Binary Data) (a) Relative Risk (RR, 相對危險度 ) (b) Odds Ratio (OR, 優勢比 ) (c) Rate Difference (RD, 率差 ) 常態分佈數據 (Normally Distributed Data) Fixed Effect Method: Random Effect Method: Meta Regression:

(c) Rate Difference (RD, 率差 ) 率差又稱為危險差 (Risk Difference). 試驗組對照組 Total 正反應 aibim 1i =(ai+bi) 負反應 cidim 2i =(ci+di) Totaln 1i =(ai+ci)n 2i =(bi+di)Ni Pti = ai/(ai+ci) : Treatment Response Rate of i th Clinical Trial ; Pci = bi/(bi+di) : Control Response Rate of i th Clinical Trial; RDi = Pti – Pci 且 (i.e. 收錄 K 個獨立臨床實驗 ) Example

Note: 如何利用所有的 RDi 來估 計 共同的 RD (i.e. Pooled RD)? Ans: 採用加權平均 (Weighted Average) 法。 Q: 如何選取 Weight? Guidelines: 1. 樣本數越大， Weight 越重。 2. 變異數越大， Weight 越輕。 Note:

(1) 如何利用所有的 RDi 來估 計共同的 RD (i.e. Pooled RD)? 其中. 故 所以, RD.Excel

Note: RD 的 95% 信賴區間 (95% C.I.) 為 : (2) RD 的顯著性檢驗, i.e. Testing H 0 : RD = 0 Under H 0,, i.e. Reject H 0 if. RD.Excel

(3) RD 的同質性 (Homogeneity) 檢驗, i.e. Testing H 0 : 對於 K 個獨立研究結果的 RD i 是否相同的顯著 性檢驗統計量, Q RD, 為 : or i.e. 當實驗組和對照組樣本量較大時, under H 0, Q RD 近似地遵從自由度為 K-1 的卡方分佈. RD.Excel

Example: Chlorpheniramine 對 Drowsiness 影響的 8 個臨床實驗結果 RD.Excel Results: 1.P(Z RD >4.737) < , i.e. 與對照組相比, Chlorpheniramine 顯著影響 Drowsiness ; 2.P(Q RD >11.169) =0.134 i.e. 8 個獨立臨床實驗具有相同之率差. RD.dta

Example: Thrombolytic therapy in acute myocardial infarction (meta-analysis) ReferencesYearaiciNoTrbidiNoCoNi Fletcher Dewar Lippschutz European European Heikinhermo Italian Australian Frankfurt Gonnsen NHLBI SMIT Brochier Euro Collab Frank Valere Klein UK-Collab Australian Australian OR70.dta

labbe ai ci bi di, xlabel(0, 0.1, 0.2, 0.3, 0.4, 0.5) ylabel(0, 0.1, 0.2, 0.3, 0.4, 0.5) psize(50) For binary data, a L’Abb′e plot (L’Abb′e et al. 1987) plots the event rates in control and experimental groups by study. OR70.dta userguide

OR70.dta

常態分佈數據 (Normally Distributed Data) 連續型的資料且吻合常態分佈的情形在臨床 試驗也是常見的類型 假設有 K 個獨立臨床試驗, 每個試驗分實驗組 (T) 與對照組 (C), 且 療效評估以效益值 (Effect Size) 呈現 ( 又稱為 Standardized difference between two means)

Hedges (1981) 提出下列估計統計量 : Hedges & Olkin (1985) 又提出下列具不偏性 (Unbiased Estimator) 之估計統計量 : 且 用來估計 的變異數. ES.Excel

Example: 8 個臨床研究探討 Flouride Varnishes (Duraphat) 是否對 兒童的恆齒 (Permanent Teeth) 具有保護作用 ?( 測量值為蛀牙增長, Caries Increment) ES.Excel

(1) 估計 Pooled ES 及其標準誤 (SE): 且 (2) ES 的 95% 信賴區間 (95% C.I.) 為 : ES.Excel

(3) ES 的顯著性檢驗, i.e. Testing H 0 : Effect Size = 0 Under H 0,, i.e. Reject H 0 if ES.Excel

(4) ES 的同質性檢驗, i.e. Testing H 0 : Under H 0,, i.e. Reject H 0 if. ES.Excel

Example: 8 個臨床研究探討 Flouride Varnishes (Duraphat) 是否對 兒童的恆齒 (Permanent Teeth) 具有保護作用 ?( 測量值為蛀牙增長, Caries Increment) Results: 1. ES < 0; 2. 95% C.I. (ESL, ESU)=( , ) ; 3. U ES = (p-value < ) 4. 但是, P(Q ES > ) ~ , i.e. ESi’s 不全相同 ( 存在 Heterogeneity). ES.Excel ES.dta

Note: 上述所提方法 (Fixed Effect Method) 皆假設各獨立 臨床試驗的測量值, e.g. RR’s, OR’s, RD’s, ES’s, 之間不存在異質性 (Heterogeneity) 。換言之，僅考 慮 Within-study Variance 而忽略 Between-study Variance 對試驗效果的影響, 如 : Pooled RR, Pooled OR, Pooled ES 。

Q: 若各獨立臨床試驗的測量值之間確實存在異質 性， i.e. Reject the hypothesis of homogeneity ，是 否有較適當之統計方法可採用 ? Ans.: (A) Random Effect Model – when the potential different characteristics of the study were unknown or (not available) (B) Meta-regression Model -- when the potential different characteristics of the study were know and available (presented in the published paper)

(A) Random Effect Model 假設  i, i = 1, …, K, 為第 i 個臨床試驗的實驗效 果，如 : RD i, OR i 等，滿足  i ~ N (μ ， τ 2 ) 。 換言之，實驗效果 其中 δ i 代表實驗效果的異質性源自於不同臨 床試驗的群體特徵之差異 (Ethnic Differences) 或不同試驗 Follow-up 的時間差異所造成， τ 2 代表異質性的嚴重度。 Reference: DerSimonian R, Larid N., “Meta-analysis in clinical trials”, Controlled Clinical Trials 1986; V 7; p

根據同質性檢驗統計量 Q 和加權 W i 可以 直接估計 Among-study Variance, τ 2 : 令 其中 則 Pooled 實驗效果  * 估計量為 : 且 Figure 2-1

 * 的 95% 信賴區間為 :  * 的有效性檢驗統計量為 : i.e. Reject H 0 if or i.e. Reject H 0 if Figure 2-1

Note: 隨機效應模式估計 ( 檢驗 ) 結果較保守。 3. 各獨立臨床試驗間不存在異質性則採固 定效應模式，反之，採隨機效應模式。

Example: 8 個臨床研究探討 Flouride Varnishes (Duraphat) 是否對 兒童的恆齒 (Permanent Teeth) 具有保護作用 ?( 測量值為蛀牙增長, Caries Increment) Results: 1. ES < 0; 2. 95% C.I. (ESL, ESU)=( , ) ; 3. U ES = (p-value < ) 4. 但是, P(Q ES > ) ~ , i.e. ESi’s 不全相同 ( 存在 Heterogeneity). ES.Excel ES.dta

Example: Efficacy analysis: number of patients with no clinically significant response (Figure 2) P(Q RD > ) = , i.e. 13 個獨立臨床實驗 之率差 (RDi) 值有顯著差異, i.e. 存在 Heterogeneity. 試驗間的變異數, τ 2, 之估計值為 : Figure 2-1

Q: 是否有現成統計軟體可供使用 ? Ans: Yes! STATA and CMA Figure 2-1.dtaFigure1-1.dta

Example: OR70.dta

Q: Can we use STATA to analyze this ES? Yes! ES.Excel ES.dta

ES.Excel

Random Effect ES.Excel

Data 2 From SIM ReferencesNTNT YTYT STST NCNC YCYC SCSC Example 2: ES2.Excel

Fixed Effect ES2.Excel ES2.dta

ES2.Excel

Random Effect ES2.Excel ES2.dta

ES2.Excel

Note: 1.There are a variety of potential problems such as publication and selection bias. It is well-recognized that negative results are not often published. 2.Outside funded studies yield stronger positive results than non-funded studies. 3.Blinded randomized results are generally less positive than unblinded randomized results, which it turn are less positive than non-randomized results. 4.Soft endpoints generally yield stronger results than hard endpoints. Soft endpoints permit more flexibility and subjectivity in the reporting of results.

Funnel plots are simple graphical displays of a measure of study size on the vertical axis against intervention or treatment effect on the horizontal axis. The name "funnel plot" is based on the fact that the precision in the estimation of the underlying intervention or treatment effect will increase as the size of component studies increases. Results from small studies will therefore scatter more widely, with the spread narrowing among larger studies. In the absence of bias, the plot will resemble a symmetrical inverted funnel. If there is bias, for example, because smaller studies showing no statistically significant effects remain unpublished, then such publication bias will lead to an asymmetrical appearance of the funnel plot. It should be noted that although funnel plots have traditionally been used to examine evidence for publication bias, funnel-plot asymmetry may reflect other types of bias or even result from the true intervention or treatment effect differing between small and large studies. They should, thus, be seen as displaying the evidence for "small study effects" in general rather than publication bias in particular. These issues are discussed by Egger et al. (1997) and Sterne, Egger, and Davey Smith (2001).

(B) Meta-Regression Meta-regressions are similar in essence to simple regressions, in which an outcome variable is predicted according to the values of one or more explanatory variables. In meta-regression, the outcome variable is the effect estimate (for example, a mean difference, a risk difference, a log odds ratio or a log risk ratio). The explanatory variables are characteristics of studies that might influence the size of intervention effect. These are often called ‘potential effect modifiers’ or covariates. Note: Meta-regression should generally not be considered when there are fewer than ten studies in a meta-analysis.

研究目的：ＢＷ Gestation, Smoking (n > 2)

Y: Dependent Variable X: Independent Variable(s) Predicted (fitted) values

(B) Meta-Regression Meta-regressions usually differ from simple regressions in two ways: (1)larger studies have more influence on the relationship than smaller studies, since studies are weighted by the precision of their respective effect estimate. (2)Sometimes we should allow for the residual heterogeneity among intervention effects not modelled by the explanatory variables. This gives rise to the term ‘random-effects meta-regression’, since the extra variability is incorporated in the same way as in a random- effects meta-analysis.

(B) Meta-Regression The regression coefficient obtained from a meta- regression analysis will describe how the outcome variable (the intervention effect) changes with a unit increase in the explanatory variable (the potential effect modifier). The statistical significance of the regression coefficient is a test of whether there is a linear relationship between intervention effect and the explanatory variable. If the intervention effect is a ratio measure, the log- transformed value of the intervention effect should always be used in the regression model, and the exponential of the regression coefficient will give an estimate of the relative change in intervention effect with a unit increase in the explanatory variable.

(B) Meta-Regression Meta-regression can also be used to investigate differences for categorical explanatory. If there are J categories of a particular explanatory, we usually use J – 1 dummy variables (which can only take values of 0 or 1) and use them simultaneously enter into/remove from the model in the meta-regression model (as in standard linear regression modelling). The regression coefficients will estimate how the intervention effect in each subcategory differs from a nominated reference subcategory. The p-value of each regression coefficient will indicate whether this difference is statistically significant. Note: Meta-regression may be performed using the ‘metareg’ macro available for the STATA statistical package.

(B) Meta-Regression Generally, three types of models can be distinguished in the literature on meta- regression: 1.simple regression, 2.fixed effect meta-regression and 3.random effects meta-regression.

(1)Simple regression: The model can be specified as Where Y i is the effect size in study i and  0 is the overall effect size. The variable X ji specify different characteristics of the study, j=1,…,p and i=1,…,K.  specifies the between study variation. Note that this model does not allow specification of within study variation.

(2) Fixed-effect meta-regression Fixed-effect meta-regression assumes that: Here is the variance of the effect size in study j. Fixed effect meta-regression ignores between study variation, i.e. test for homogeneity is not significant. As a result, parameter estimates are biased if between study variation can not be ignored. Furthermore, generalizations to the population are not possible.

(3) Random effects meta-regression Random effects meta-regression rests on the assumption that : Here is the variance of the effect size in study j. Between study variance estimated using common estimation procedures for random effects models (restricted maximum likelihood (REML) estimators).

Example: a meta-analysis of 28 randomized controlled trials of cholesterol-lowering interventions for reducing risk of ischemic heart disease (IHD). The outcome event was death from IHD or nonfatal myocardial infarction. The measure of effect size is the odds ratio. With the following known potential different characteristics: 1.Intervention methods (different drugs and dietary) 2.Different eligibility criteria 3.Reduction in cholesterol varied among trials cholesterol.dta

Example STATA 11.0 RNFLMetaRegression.dta logOR4metareg.dta cholesterol.dta

twoway (lfitci roci rochati [aweight = Wi]) (scatter roci rochati [aweight = Wi], mcolor(black) msize(medsmall) msymbol(circle_hollow))

References (Funnel Plots): Egger, M., G. Davey Smith, M. Schneider, and C. Minder Bias in meta- analysis detected by a simple, graphical test. British Medical Journal 315: Sterne, J. A. C., M. Egger, and G. Davey Smith Investigating and dealing with publication and other biases in meta-analysis. British Medical Journal 323: Sterne, J. A. C. and M. Egger Funnel plots for detecting bias in meta- analysis: guidelines on choice of axis. Journal of Clinical Epidemiology 54:

References (Meta-Analysis): 葉明功 & 白璐 (1998), “ 統合分析 —(I) 基本統計方法 ”, 醫學 研究 (J. Med. Sci.), V18 (5); p Sharon-Lise T. Normand (1999), “Tutorial in Biostatistics Meta-Analysis: Formulating, Evaluating, Combining, and Reporting”, Statistics in Medicine, V18; p

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