## Introduction

We can analyze different scientific studies that address the same question by applying a meta-analysis. The assumption is that every individual study contains some degree of error. For example, a study could be the mortality rate of two treatments for a specific disease. The **goal **is to obtain pooled summary estimates from individual studies by taking into consideration the heterogeneity among individual studies. The aggregated data from the individual studies leads to higher statistical power.

## Procedure

- Define the research question
- Define inclusion/exclusion criteria for individual studies screened
- Search literature
- Select eligible studies
- Collect data
- Aggregate findings across studies and obtain pooled estimates of effect size
- Evaluate heterogeneity of included studies
- Conduct sensitivity and subgroup analyses

## Statistical Models

- Pool effect sizes (of individual studies) into one overall effect
- Two types of statistical models to combine data, Fixed Effect Model and Random Effects Model
- Frequentist vs Bayesian approach

## Example of Meta Analysis in R

We will provide an example of Meta Analysis in R using the meta library. Let’s start.

library(meta) data("Fleiss1993cont") head(Fleiss1993cont)

We will work with the **Fleiss1993cont** dataset where:

study | study label |

year | year of publication |

n.psyc | number of observations in psychotherapy group |

mean.psyc | estimated mean in psychotherapy group |

sd.psyc | standard deviation in psychotherapy group |

n.cont | number of observations in control group |

mean.cont | estimated mean in control group |

sd.cont | standard deviation in control group |

# meta-analysis with continuout outcome # comb.fixed/comb.random: indicator whether a fix/random effect mata-analysis to be conducted. # sm: Three different types of summary measures to choose,standardized mean difference (SMD),mean difference (MD), ratio of means (ROM) res.flesiss = metacont(n.psyc, mean.psyc, sd.psyc, n.cont, mean.cont, sd.cont, comb.fixed = T, comb.random = T, studlab = study, data = Fleiss1993cont, sm = "SMD") res.flesiss

**Output**

```
SMD 95%-CI %W(fixed) %W(random)
Davis -0.3399 [-1.1152; 0.4354] 11.5 11.5
Florell -0.5659 [-1.0274; -0.1044] 32.6 32.6
Gruen -0.2999 [-0.7712; 0.1714] 31.2 31.2
Hart 0.1250 [-0.4954; 0.7455] 18.0 18.0
Wilson -0.7346 [-1.7575; 0.2883] 6.6 6.6
Number of studies combined: k = 5
Number of observations: o = 232
SMD 95%-CI z p-value
Fixed effect model -0.3434 [-0.6068; -0.0800] -2.56 0.0106
Random effects model -0.3434 [-0.6068; -0.0800] -2.56 0.0106
Quantifying heterogeneity:
tau^2 = 0 [0.0000; 0.6936]; tau = 0 [0.0000; 0.8328]
I^2 = 0.0% [0.0%; 79.2%]; H = 1.00 [1.00; 2.19]
Test of heterogeneity:
Q d.f. p-value
3.68 4 0.4515
Details on meta-analytical method:
- Inverse variance method
- DerSimonian-Laird estimator for tau^2
- Jackson method for confidence interval of tau^2 and tau
- Hedges' g (bias corrected standardised mean difference)
```

**Forest**

forest(res.flesiss, leftcols = c('studlab'))

- According to the pooled results of meta-analysis, both fixed and random effects models yield a significant benefit of the intervention group against the control group (for the days of hospital stay, the lower, the better).
- The p-value =0.45 for the Cochran’s Q test, indicating no heterogeneity.

**Funnel Plot**

funnel(res.flesiss)

*metabias*: Test for funnel plot asymmetry, based on rank correlation or linear regression method.- Use Egger’s test to check publication bias, can take string ‘Egger’ or ‘linreg’.

metabias(res.flesiss, method.bias = 'linreg', k.min = 5, plotit = T)

**Output**

```
Linear regression test of funnel plot asymmetry
Test result: t = -0.04, df = 3, p-value = 0.9730
Sample estimates:
bias se.bias intercept se.intercept
-0.0668 1.8154 -0.3241 0.5455
Details:
- multiplicative residual heterogeneity variance (tau^2 = 1.2251)
- predictor: standard error
- weight: inverse variance
- reference: Egger et al. (1997), BMJ
```

The p-value is 0.973 which implies no publication bias. However, this meta-analysis contains k=5 studies. Egger’s test may lack the statistical power to detect bias when the number of studies is small (i.e., k<10).

## Binary case

At this point, we will provide an example by taking into consideration a binary case.

load("binarydata.RData") binarydata

- Author: This signifies the column for the study label (i.e., the first author)
- Ee: Number of events in the experimental treatment arm
- Ne: Number of participants in the experimental treatment arm
- Ec: Number of events in the control arm
- Nc: Number of participants in the control arm

The Analysis:

- Use
*metabin*to do the calculation. - As we want to have a pooled effect for binary data, we have to choose another summary measure now. We can choose from “OR” (Odds Ratio), “RR” (Risk Ratio), or RD (Risk Difference), among other things.
*method*: indicating which method is to be used for pooling of studies.

m.bin <- metabin(Ee,Ne,Ec,Nc, data = binarydata, studlab = paste(Author), comb.fixed = T,comb.random = T, method = 'MH',sm = "RR") # Mantel Haenszel weighting

**Output**

```
RR 95%-CI %W(fixed) %W(random)
Alcorta-Fleischmann 0.5018 [0.0462; 5.4551] 1.4 1.4
Craemer 1.0705 [0.5542; 2.0676] 15.2 18.9
Eriksson 1.1961 [0.3657; 3.9124] 4.5 5.8
Jones 0.5286 [0.1334; 2.0945] 5.3 4.3
Knauer 0.3278 [0.0134; 8.0140] 1.4 0.8
Kracauer 0.9076 [0.3512; 2.3453] 8.0 9.1
La Sala 0.9394 [0.4233; 2.0847] 11.2 12.9
Maheux 0.0998 [0.0128; 0.7768] 9.0 1.9
Schmidthauer 0.7241 [0.2674; 1.9609] 7.9 8.3
van der Zee 0.8434 [0.4543; 1.5656] 18.5 21.4
Wang 0.5519 [0.2641; 1.1534] 17.5 15.1
Number of studies combined: k = 11
Number of observations: o = 17604
Number of events: e = 194
RR 95%-CI z p-value
Fixed effect model 0.7536 [0.5696; 0.9972] -1.98 0.0478
Random effects model 0.7885 [0.5922; 1.0499] -1.63 0.1038
Quantifying heterogeneity:
tau^2 = 0; tau = 0; I^2 = 0.0% [0.0%; 60.2%]; H = 1.00 [1.00; 1.59]
Test of heterogeneity:
Q d.f. p-value
7.29 10 0.6976
Details on meta-analytical method:
- Mantel-Haenszel method
- DerSimonian-Laird estimator for tau^2
- Mantel-Haenszel estimator used in calculation of Q and tau^2 (like RevMan 5)
- Continuity correction of 0.5 in studies with zero cell frequencies
```

**Forest**

forest(m.bin, leftcols = c('studlab'))

funnel(m.bin) metabias(m.bin, method.bias = 'linreg', plotit = T)

```
Linear regression test of funnel plot asymmetry
Test result: t = -2.05, df = 9, p-value = 0.0701
Sample estimates:
bias se.bias intercept se.intercept
-1.1286 0.5493 0.2623 0.2661
Details:
- multiplicative residual heterogeneity variance (tau^2 = 0.5443)
- predictor: standard error
- weight: inverse variance
- reference: Egger et al. (1997), BMJ
```

## Limitations

The limitation of Meta Analysis is that it can only conduct pariwise comparisions, and cannot include multi-arm trials.

## References

[1] Dr. Na Zhao