Plan of the Session. Session I: Basic Methods of Meta-Analysis Short Introduction to R. Why Conduct a Meta-Analysis? What is a Meta-Analysis?

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Session I: Basic Methods of Meta-Analysis

Short Introduction to R

James Carpenter1, Ulrike Krahn2,3, Gerta R¨ucker4, Guido Schwarzer4

1_{London School of Hygiene and Tropical Medicine & MRC Clinical Trials Unit, London, UK} 2_{Institute of Medical Biostatistics, Epidemiology and Informatics, Mainz, Germany} 3_{Institute of Medical Informatics, Biometry and Epidemiology, Duisburg-Essen, Germany}

4_{Institute for Medical Biometry and Statistics, Freiburg, Germany}

[email protected]

IBC Short Course Florence, 6 July 2014

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Plan of the Session

At the end of this session theaim is that you should understand

I the basic principles of meta-analysis;

I how R works.

Theobjectivesare that you are able to:

I use the help system and read data into R;

I conduct a meta-analysis with binary outcomes;

I summarize the main result of a meta-analysis to a medical audience.

Carpenter/Krahn/R¨ucker/Schwarzer Session I: Introduction to Meta-Analysis Florence, 6 July 2014 2

Introduction R in Action Meta-Analysis Binary Data Summary References

What is a Meta-Analysis?

See Egger and Smith (1997), BMJ Definitions:

I Term “Meta”:

Implies occuring later, more comprehensive, new but related discipline which critically deals with original discipline

I Term “Meta-Analysis”:

Statistically combining and analysing data from separate studies

I Term “Systematic Review”:

Any type of review utilising strategies to avoid bias Meta-analysis:

I May or may not be part of a systematic review

I Medical Subject Heading (MeSH) in Medline

Why Conduct a Meta-Analysis?

Egger and Smith (1997), BMJ

I More objective appraisal of evidence which may lead to resolution of

uncertainty and disagreement

I Reduce probability of false negative results and thus prevent undue

delays in introduction of effective treatments into practice

I Heterogeneity between study results may be explored (and sometimes

explained)

I Allows testing of a priori hypotheses regarding treatment effects in

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What is R?

I General purpose statistical package (http://www.r-project.org/)

I Based on statistical programming language S (→S-PLUS)

I Almost 20 years old, actively developed and maintained

I Available for Windows, Linux, Unix, Mac OS

I Released under the GNU General Public License (GPL) version 2

(or any later version)

I Licence costs:

0 e

/ 0$

I R can be used in regulated clinical trial environments

(http://www.r-project.org/doc/R-FDA.pdf)

I More than 5000 add-on packages available on CRAN

(http://cran.at.r-project.org/)

I Short introductions / reviews of add-on packages inThe R Journal

(http://journal.r-project.org/) – successor of R News

I Mailing lists: http://stat.ethz.ch/mailman/listinfo/(R-help)

R – Used as an overgrown calculator

> 2 + 2 [1] 4 > 4 -+ 1 [1] 3 > exp(1) [1] 2.718282 > x = 2.25 > x + x [1] 4.5

R – Working with vectors

> 1:10 [1] 1 2 3 4 5 6 7 8 9 10 > c(1:4, 6:10) [1] 1 2 3 4 6 7 8 9 10 > y = c(1,4,9,16) > sqrt(y) [1] 1 2 3 4 > sqrt(y)[4] [1] 4

> y^2 # same result: y**2

[1] 1 16 81 256

R – Calculate mean and standard deviance

> x = 1:10 > sum(x) / length(x) [1] 5.5 > mean(x) [1] 5.5 > sqrt(sum((x-mean(x))^2)/(length(x)-1)) [1] 3.02765 > sd(x) [1] 3.02765

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R – Missing values

I NA: Not Available > x = c(1:5, NA) > mean(x) [1] NA > # Getting help:

> help(mean) # Show documentation on R command mean > ?mean # Show documentation on R command mean > help.start() # Show documentation in webbrowser > help("+") # Show help on arithmetic operator + > ?"+" # Show help on arithmetic operator + > mean(x, na.rm=TRUE)

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Example: Aggressive Non-Hodgkin Lymphoma

Greb et al. (2008), Cochrane Database Syst Rev1, CD004024:

I Cochrane Review including 15 randomised controlled trials (RCTs)

I Adult patients with aggressive non-Hodgkin lymphoma

I First line treatment with high-dose chemotherapy (HDCT) versus

conventional chemotherapy

I Primary outcome:

Overall survival (14 RCTs, 2444 patients)

I Secondary outcome:

Complete response (14 RCTs, 2126 patients)

Aggressive Non-Hodgkin Lymphoma – Forestplot

Study De Souza Gianni Gisselbrecht Haioun Intragumtornchai Kaiser Kluin−Nelemans Martelli Martelli 2003 Milpied Rodriguez 2003 Santini Verdonck Vitolo 0.5 1 2 Hazard Ratio

Favours HDCT Favours control

HR 0.92 0.52 1.45 0.96 0.64 1.08 1.23 0.69 1.01 0.64 1.34 0.81 1.40 1.41 95%−CI [0.45; 1.89] [0.24; 1.11] [1.08; 1.93] [0.71; 1.30] [0.30; 1.36] [0.75; 1.55] [0.72; 2.08] [0.29; 1.65] [0.59; 1.73] [0.40; 1.05] [0.68; 2.65] [0.48; 1.37] [0.73; 2.67] [0.82; 2.41]

Meta-Analysis – Calculate a Weighted Mean

Weighted mean of estimated treatment effects in individual studies (Fleiss, 1993): ˆ θ= PK k=1wk·ˆθk PK k=1wk

I Estimated treatment effectθˆ_k in studyk (k = 1, ...,K)

I Weight wk correspond to information of study k

I Methods of meta-analysis differ in definition of weights

(especially fixed effect and random effects model)

> args(weighted.mean) function (x, w, ...) NULL

> weighted.mean(1:2, c(0.2, 0.8)) [1] 1.8

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Fixed effect model – Inverse Variance Method

Fixed effect model:

ˆ

θk =θ+σˆkk, ki.i.d.∼ N(0,1), k = 1, . . . ,K (1)

Maximum likelihood estimate under model (1) for given (ˆθ_k,σˆ_k): ˆ θF = PK k=1θˆk/σˆ2k PK k=11/σˆ2k = PK k=1wkˆθk PK k=1wk with weights w_k = 1/σˆ2 k. Estimated variance of ˆθ_F: d Var (ˆθ_F) = 1 PK k=1wk = 1 PK k=11/σˆ2k

(1-α) confidence interval for ˆθF:

ˆ

θF ± z1−α₂ S.E.(ˆθF) with S.E.(ˆθF) =

q d

Var(ˆθ_F)

Fixed Effect Model – Graphical Presentation

Odds ratio 0.1 0.2 0.5 1 2 5 10 Study 1 Study 2 Study 3 Study 4 Study 5 True effect

Aggressive Non-Hodgkin Lymphoma – Overall Survival

Hazard log(HR) SE(log(HR)) wk = Study Ratio (HR) 95%-CI (=ˆθk) (=σˆk) 1/σˆ2_k De Souza 0.92 0.45 - 1.89 -0.0807 0.3672 7.42 Gianni 0.52 0.24 - 1.11 -0.6527 0.3850 6.75 Gisselbrecht 1.45 1.08 - 1.93 0.3683 0.1487 45.22 Intragumtornchai 0.96 0.71 - 1.30 -0.0387 0.1529 42.77 Kaiser 0.64 0.30 - 1.36 -0.4480 0.3852 6.74 Kluin-Nelemans 1.08 0.75 - 1.55 0.0761 0.1834 29.73 Martelli 1996 1.23 0.72 - 2.08 0.2031 0.2697 13.75 Martelli 2003 0.69 0.29 - 1.65 -0.3773 0.4473 5.00 Milpied 1.01 0.59 - 1.73 0.0087 0.2748 13.24 Rodriguez 2003 0.64 0.40 - 1.05 -0.4402 0.2481 16.25 Santini 1998 1.34 0.68 - 2.65 0.2921 0.3482 8.25 Santini-2 0.81 0.48 - 1.37 -0.2141 0.2697 13.75 Verdonck 1.40 0.73 - 2.67 0.3368 0.3290 9.24 Vitolo 1.41 0.82 - 2.41 0.3412 0.2749 13.23 Carpenter/Krahn/R¨ucker/Schwarzer Session I: Introduction to Meta-Analysis Florence, 6 July 2014 15

R packages for meta-analysis on CRAN

I rmeta (Lumley, 2012)

I Fixed and random effects meta-analysis (Mantel-Haenszel, Peto, DerSimonian-Laird)

I metafor (Viechtbauer, 2010)

I Tests for funnel plot asymmetry / Trim and fill method

I General linear (mixed-effects) model approach for meta-regression I Multivariate meta-analysis

I meta (Schwarzer, 2007)

I Tests for funnel plot asymmetry / Trim and fill method I Import data from RevMan 5 / Link to R package metafor

I mvmeta (Gasparrini, 2014)

I Multivariate meta-analysis and meta-regression on multiple outcomes

I metasens (Schwarzer et al., 2014; Carpenter et al., 2009)

I Advanced methods to model and adjust for bias in meta-analysis I Add-on package to R packagemeta/ replaces R packagecopas

I netmeta (R¨ucker et al., 2014)

I Network meta-analysis

I Add-on package to R packagemeta

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R package meta

Function Comment

metabin Meta-analysis of binary outcome data

metacont Meta-analysis of continuous outcome data

metagen Generic inverse variance meta-analysis

metacor Meta-analysis of correlations

metainc Meta-analysis of incidence rates

metaprop Meta-analysis of single proportions

read.rm5 Import RevMan 5 data files (csv-files with special format)

metacr Meta-analysis of outcome data from Cochrane review

forest Forest plot

funnel Plot to assess funnel plot asymmetry

metabias Test for funnel plot asymmetry

trimfill Trim and fill method for meta-analysis

metareg Meta-regression (wrapper function to R package metafor)

... Cumulative meta-analysis / Influence analysis in meta-analysis

Meta-Analysis of Overall Survival

> os = read.csv("hd-os.txt", as.is=TRUE) > library(meta) > m1 = metagen(logHR, selogHR, + sm="HR", data=os, studlab=study, + comb.random=FALSE) > str(m1) List of 62 $ TE : num [1:14] -0.0807 -0.6527 0.3683 -0.0387 -0.448 ... $ seTE : num [1:14] 0.367 0.385 0.149 0.153 0.385 ...

$ studlab : chr [1:14] "De Souza" "Gianni" "Gisselbrecht" "Haioun" ... $ w.fixed : num [1:14] 7.42 6.75 45.22 42.77 6.74 ... $ w.random : num [1:14] 6.18 5.71 20.34 19.83 5.7 ... $ TE.fixed : num 0.0441 $ seTE.fixed : num 0.0657 $ lower.fixed : num -0.0848 $ upper.fixed : num 0.173 $ zval.fixed : num 0.671 $ pval.fixed : num 0.502

Meta-Analysis of Survival Data – Print Function

> class(m1)

[1] "metagen" "meta"

> m1 # Calls R function print.meta

HR 95%-CI %W(fixed) De Souza 0.9225 [0.4491; 1.8946] 3.21 Gianni 0.5206 [0.2448; 1.1073] 2.92 Gisselbrecht 1.4453 [1.0799; 1.9343] 19.55 Haioun 0.9620 [0.7129; 1.2982] 18.49 Intragumtornchai 0.6389 [0.3003; 1.3593] 2.91 Kaiser 1.0791 [0.7533; 1.5458] 12.85 Kluin-Nelemans 1.2252 [0.7222; 2.0786] 5.94 Martelli 0.6857 [0.2854; 1.6477] 2.16 Martelli 2003 1.0087 [0.5887; 1.7286] 5.72 Milpied 0.6439 [0.3960; 1.0471] 7.02 Rodriguez 2003 1.3392 [0.6768; 2.6500] 3.57 Santini 0.8073 [0.4758; 1.3696] 5.94 Verdonck 1.4005 [0.7349; 2.6688] 3.99

Meta-Analysis of Overall Survival – Summary Function

> summary(m1) # Calls summary.meta and print.summary.meta

Number of studies combined: k=14

HR 95%-CI z p.value

Fixed effect model 1.0451 [0.9187; 1.1888] 0.6709 0.5023 Quantifying heterogeneity:

tau^2 = 0.0270; H = 1.19 [1; 1.64]; I^2 = 29.9% [0%; 63%] Test of heterogeneity:

Q d.f. p.value

18.55 13 0.1379

Details on meta-analytical method: - Inverse variance method

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Overall Survival – Forestplot

> forest(m1, hetstat=FALSE) # Calls function forest.meta Study

Fixed effect model

De Souza Gianni Gisselbrecht Haioun Intragumtornchai Kaiser Kluin−Nelemans Martelli Martelli 2003 Milpied Rodriguez 2003 Santini Verdonck Vitolo TE −0.08 −0.65 0.37 −0.04 −0.45 0.08 0.20 −0.38 0.01 −0.44 0.29 −0.21 0.34 0.34 seTE 0.3672 0.3850 0.1487 0.1529 0.3852 0.1834 0.2697 0.4473 0.2748 0.2481 0.3482 0.2697 0.3290 0.2749 0.5 1 2 Hazard Ratio _HR 1.05 0.92 0.52 1.45 0.96 0.64 1.08 1.23 0.69 1.01 0.64 1.34 0.81 1.40 1.41 95%−CI [0.92; 1.19] [0.45; 1.89] [0.24; 1.11] [1.08; 1.93] [0.71; 1.30] [0.30; 1.36] [0.75; 1.55] [0.72; 2.08] [0.29; 1.65] [0.59; 1.73] [0.40; 1.05] [0.68; 2.65] [0.48; 1.37] [0.73; 2.67] [0.82; 2.41] W(fixed) 100% 3.2% 2.9% 19.5% 18.5% 2.9% 12.9% 5.9% 2.2% 5.7% 7.0% 3.6% 5.9% 4.0% 5.7%

Example: Aggressive Non-Hodgkin Lymphoma

Greb et al. (2008), Cochrane Database Syst Rev1, CD004024:

I Cochrane Review including 15 randomised controlled trials (RCTs)

I Adult patients with aggressive non-Hodgkin lymphoma

I First line treatment with high-dose chemotherapy (HDCT) versus

conventional chemotherapy

I Primary outcome:

Overall survival (14 RCTs, 2444 patients)

I Secondary outcome:

Complete response (14 RCTs, 2126 patients)

Aggressive Non-Hodgkin Lymphoma – Complete Response

HDCT Control Study

Events Total Events Total

De Souza 14 28 10 26 Gianni 46 48 35 50 Gisselbrecht 119 189 116 181 Intragumtornchai 10 23 9 25 Kaiser 110 158 97 154 Kluin-Nelemans 67 98 56 96 Martelli 1996 3 22 4 27 Martelli 2003 57 75 51 75 Milpied 74 98 56 99 Rodriguez 2003 39 55 30 53 Santini 1998 46 63 34 61 Santini-2 80 117 71 106 Verdonck 25 38 26 35 Vitolo 35 60 46 66

Milpied Study – Complete Response (CR)

CR no CR

HDCT 74 (a) 24 (b) 98 (a+b =nT)

Control 56 (c) 43 (d) 99 (c+d =n_C)

130 (a+c) 67 (b+d) 197 (n)

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Binary Data – Effect Measures

Let

I p_T: Experimental event rate ˆp_T =a/(a+b)

I p_C: Control event rate pˆ_C =c/(c +d)

Risk Ratio φ: φ=pT pC ˆ φ= pˆT ˆ pC Odds ratio ψ: ψ= p_T 1−p_T ! pC 1−pC ! =φ× 1−pC 1−pT ˆ ψ= a d b c (2) Risk Difference η: η=p_T −p_C ˆη= ˆp_T−pˆ_C

Binary Data – Effect Measures

> cr = read.csv("hd-cr.txt", as.is=TRUE) > library(meta)

> mil = metabin(crHDCT, nHDCT, crControl, nControl,

+ sm="OR", data=cr, studlab=study,

+ subset=study=="Milpied")

> round(exp(mil$TE), 2)

[1] 2.37

Binary Effect Measures – Confidence Interval

Large sample variance estimates (Fleiss, 1993):

d Var(log ˆφ) = 1 a+ 1 c − 1 a+b − 1 c +d d Var(log ˆψ) = 1 a+ 1 b + 1 c + 1 d (3) d Var(ηˆ) = a b (a+b)3 + c d (c+d)3

(1−α)-confidence interval (on log scale for risk ratio and odds ratio):

ˆ

θ ± z₁−α₂ S.E.(ˆθ)

with standard error S.E.(ˆθ) =

q d

Var(ˆθ).

Binary Effect Measures – Confidence Interval

Large sample variance estimates (Fleiss, 1993):

d Var(log ˆφ) = 1 a+ 0.5 + 1 c + 0.5− 1 a+b+ 0.5 − 1 c+d + 0.5 d Var(log ˆψ) = 1 a+ 0.5 + 1 b+ 0.5 + 1 c + 0.5 + 1 d + 0.5 d Var(ˆη) = (a+ 0.5) (b+ 0.5) (a+b+ 1)3 + (c + 0.5) (d + 0.5) (c+d + 1)3

Add 0.5 if any cell counts are zero (Gart and Zweifel, 1967; Pettigrew et al., 1986)

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Binary Effect Measures – Confidence Interval

> mil = metabin(crHDCT, nHDCT, crControl, nControl,

+ subset=study=="Milpied")

> # Print variance estimate > mil$seTE^2

[1] 0.09629314

> ## Print confidence interval > print(mil, digits=2)

OR 95%-CI z p.value

2.37 [1.29; 4.35] 2.78 0.0055 Details:

- Inverse variance method

Aggressive Non-Hodgkin Lymphoma – Forestplot

Study De Souza Gianni Gisselbrecht Intragumtornchai Kaiser Kluin−Nelemans Martelli Martelli 2003 Milpied Rodriguez 2003 Santini Santini−2 Verdonck Vitolo Events 14 46 119 10 110 67 3 57 74 39 46 80 25 35 Total 28 48 189 23 158 98 22 75 98 55 63 117 38 60 HDCT Events 10 35 116 9 97 56 4 51 56 30 34 71 26 46 Total 26 50 181 25 154 96 27 75 99 53 61 106 35 66 Control 0.1 0.5 1 2 10 Odds Ratio

Favours control Favours HDCT OR 1.60 9.86 0.95 1.37 1.35 1.54 0.91 1.49 2.37 1.87 2.15 1.07 0.67 0.61 95%−CI [0.54; 4.73] [2.11; 45.96] [0.62; 1.45] [0.43; 4.36] [0.84; 2.16] [0.86; 2.78] [0.18; 4.57] [0.73; 3.06] [1.29; 4.35] [0.84; 4.14] [1.01; 4.56] [0.61; 1.87] [0.24; 1.83] [0.29; 1.27]

Naive Pooling – Fictitious Example

CR no CR pˆT ˆpC RR [95%-CI]c HDCT 4 56 Study 1 Control 11 139 6.7% 7.3% 0.91 [0.30; 2.74] HDCT 40 140 Study 2 Control 12 38 22.2% 24.0% 0.93 [0.53; 1.63] HDCT 44 196 Study 1&2 Control 23 177 18.3% 11.5% 1.59 [1.00; 2.55] Appropriate meta-analysis 0.92 [0.56; 1.52]

Inverse Variance Method – Odds ratio – Definition

Overall odds ratio ˆψIV (Fleiss, 1993):

ˆ ψIV = exp                 K P k=1 wk ·log ˆψk K P k=1 wk                 (4) I Study index: k= 1, ...,K I Weights: wk = 1 .

dVar(log ˆψ_k) (→fixed effect model) I See formulae (2) and (3) for definition of ˆψ_k anddVar(log ˆψ_k) I Analogous for risk ratio as effect measure: log ˆφk

I For risk difference: ˆηk (without exp function in equation (4))

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Meta-Analysis of CR – Inverse Variance Method

> cr = read.csv("hd-cr.txt", as.is=TRUE)

> library(meta)

> m2 = metabin(crHDCT, nHDCT, crControl, nControl,

+ comb.random=FALSE, method="Inverse")

> summary(m2)

Number of studies combined: k=14

OR 95%-CI z p.value

Fixed effect model 1.3228 [1.0999; 1.5909] 2.9713 0.003 Quantifying heterogeneity:

tau^2 = 0.0897; H = 1.3 [1; 1.78]; I^2 = 41% [0%; 68.6%] Test of heterogeneity:

Q d.f. p.value

22.03 13 0.0549

Forest Plot – CR – Inverse Variance Method

> forest(m2, hetstat=FALSE, text.fixed="IV estimate")

Study IV estimate De Souza Gianni Gisselbrecht Intragumtornchai Kaiser Kluin−Nelemans Martelli Martelli 2003 Milpied Rodriguez 2003 Santini Santini−2 Verdonck Vitolo Events 14 46 119 10 110 67 3 57 74 39 46 80 25 35 Total 1072 28 48 189 23 158 98 22 75 98 55 63 117 38 60 Experimental Events 10 35 116 9 97 56 4 51 56 30 34 71 26 46 Total 1054 26 50 181 25 154 96 27 75 99 53 61 106 35 66 Control 0.1 0.5 1 2 10 Odds Ratio OR 1.32 1.60 9.86 0.95 1.37 1.35 1.54 0.91 1.49 2.37 1.87 2.15 1.07 0.67 0.61 95%−CI [1.10; 1.59] [0.54; 4.73] [2.11; 45.96] [0.62; 1.45] [0.43; 4.36] [0.84; 2.16] [0.86; 2.78] [0.18; 4.57] [0.73; 3.06] [1.29; 4.35] [0.84; 4.14] [1.01; 4.56] [0.61; 1.87] [0.24; 1.83] [0.29; 1.27] W(fixed) 100% 2.9% 1.4% 19.0% 2.5% 15.3% 9.8% 1.3% 6.6% 9.2% 5.4% 6.0% 10.8% 3.3% 6.3%

Mantel-Haenszel Method – Odds ratio – Definition

Mantel and Haenszel (1959):

I Estimator for common odds ratio in stratified case-control study

I Can be used in meta-analysis of RCTs

I Fixed effect method

Mantel-Haenszel odds ratio ˆψMH:

ˆ ψMH = k P k=1 w_k ·ψˆ_k k P k=1 wk (5) I Weights: wk = bkck n_k

Meta-Analysis of CR – Mantel-Haenszel Method

> cr = read.csv("hd-cr.txt", as.is=TRUE)

> library(meta)

> m3 = metabin(crHDCT, nHDCT, crControl, nControl,

+ comb.random=FALSE, method="MH")

> # Same result (Mantel-Haenszel method is default) > m3 = metabin(crHDCT, nHDCT, crControl, nControl,

+ comb.random=FALSE)

> # Same result (use of R function update.meta) > m3 = update(m2, method="MH")

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Forest Plot – CR – Mantel-Haenszel Method

> forest(m3, hetstat=FALSE, text.fixed="MH estimate")

Study MH estimate De Souza Gianni Gisselbrecht Intragumtornchai Kaiser Kluin−Nelemans Martelli Martelli 2003 Milpied Rodriguez 2003 Santini Santini−2 Verdonck Vitolo Events 14 46 119 10 110 67 3 57 74 39 46 80 25 35 Total 1072 28 48 189 23 158 98 22 75 98 55 63 117 38 60 Experimental Events 10 35 116 9 97 56 4 51 56 30 34 71 26 46 Total 1054 26 50 181 25 154 96 27 75 99 53 61 106 35 66 Control 0.1 0.5 1 2 10 Odds Ratio OR 1.35 1.60 9.86 0.95 1.37 1.35 1.54 0.91 1.49 2.37 1.87 2.15 1.07 0.67 0.61 95%−CI [1.12; 1.61] [0.54; 4.73] [2.11; 45.96] [0.62; 1.45] [0.43; 4.36] [0.84; 2.16] [0.86; 2.78] [0.18; 4.57] [0.73; 3.06] [1.29; 4.35] [0.84; 4.14] [1.01; 4.56] [0.61; 1.87] [0.24; 1.83] [0.29; 1.27] W(fixed) 100% 2.6% 0.7% 21.8% 2.4% 14.8% 8.9% 1.5% 6.1% 6.8% 4.4% 4.6% 11.7% 4.6% 9.1%

Summary

Meta-analysis

I Central to evidence based medicine (see, e.g. website of Cochrane

Collaboration http://www.cochrane.org/).

I Only as good as the evidence that it relies on.

I Some important issues are:

I Heterogeneity of study results. The more you can explain, the better →subgroup-analysis / meta-regression.

I Bias in contributing studies / non-representativeness of studies (e.g. due to publication bias).

I Binary data raises some special issues, especially if event rates are low. R

I Modern statistical packages for data analysis, management & graphics

I Use of additional software easily possible, e.g. for meta-analysis

I Use of command line necessary

I Extended documentation available (Online, Use-R! books, ...)

Introduction R in Action Meta-Analysis Binary Data Summary References References

Carpenter, J., R¨ucker, G., and Schwarzer, G. (2009). copas: An R package for fitting the Copas selection model. The R Journal, 1(2):31–36.

Egger, M. and Smith, G. D. (1997). Meta-analysis: Potentials and promise. British Medical Journal, 315:1371–1374.

Fleiss, J. L. (1993). The statistical basis of meta-analysis. Statistical Methods in Medical Research, 2:121–145.

Gart, J. J. and Zweifel, J. R. (1967). On the bias of various estimators of the logit and its variance with application to quantal bioassay. Biometrika, 54:181–187.

Gasparrini, A. (2014). mvmeta: Multivariate and univariate meta-analysis and meta-regression. R package version 0.4.3.

Greb, A., Bohlius, J., Schiefer, D., Schwarzer, G., Schulz, H., and Engert, A. (2008). High-dose chemotherapy with autologous stem cell transplantation in the first line treatment of aggressive non-hodgkin lymphoma (nhl) in adults. Cochrane Database Syst Rev, 1:CD004024. DOI: 10.1002/14651858.CD004024.pub2.

Lumley, T. (2012). rmeta: Meta-analysis. R package version 2.16.

Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22(4):719–748.

Pettigrew, H. M., Gart, J. J., and Thomas, D. G. (1986). The bias and higher cumulants of the logarithm of a binomial variate.Biometrika, 73:425–435.

R Core Team (2014). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.

R¨ucker, G., Schwarzer, G., Krahn, U., and K¨onig, J. (2014). netmeta: Network meta-Analysis with R. R package version 0.5-0.

Schwarzer, G. (2007). meta: An R package for meta-analysis.R News, 7(3):40–45. Schwarzer, G., Carpenter, J., and R¨ucker, G. (2014).metasens: Advanced statistical

methods to model and adjust for bias in meta-analysis. R package version 0.1-0. Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package.

Journal of Statistical Software, 36(3):1–48.