# metareg: Fixed and random effects model for meta-analysis In gap: Genetic Analysis Package

## Description

Given k=n studies with b_1, ..., b_N being β's and se_1, ..., se_N standard errors from regression, the fixed effects model uses inverse variance weighting such that w_1=1/se_1^2, ..., w_N=1/se_N^2 and the combined β as the weighted average, β_f=(b_1*w_1+...+b_N*w_N)/w, with w=w_1+...+w_N being the total weight, the se for this estimate is se_f=√{1/w}. A normal z-statistic is obtained as z_f=β_f/se_f, and the corresponding p value p_f=2*pnorm(-abs(z_f)). For the random effects model, denote q_w=w_1*(b_1-β_f)^2+...+w_N*(b_N-β_f)^2 and dl=max(0,(q_w-(k-1))/(w-(w_1^2+...+w_N^2)/w)), corrected weights are obtained such that {w_1}_c=1/(1/w_1+dl), ..., {w_N}_c=1/(1/w_N+dl), totaling w_c={w_1}_c+...+{w_N}_c. The combined β and se are then β_r=(b_1*{w_1}_c+...+b_N*{w_N}_c)/w_c and se_r=sqrt(1/wc), leading to a z-statistic z_r=β_r/se_r and a p-value p_r=2*pnorm(-abs(z_r)). Moreover, a p-value testing for heterogeneity is p_{heter}=pchisq(q_w,k-1,lower.tail=FALSE).

## Usage

 `1` ```metareg(data, N, verbose="Y", prefixb="b", prefixse="se") ```

## Arguments

 `data` Data frame to be used `N` Number of studies `verbose` A control for screen output `prefixb` Prefix of estimate; default value is "b" `prefixse` Prefix of standard error; default value is "se"

The function accepts a wide format data with estimates as b1,...,bN and standard errors as se1,...,seN. More generally, they can be specified by prefixes in the function argument.

## Value

The returned value is a data frame with the following variables:

 `p_f` P value (fixed effects model) `p_r` P value (random effects model) `beta_f` regression coefficient `beta_r` regression coefficient `se_f` standard error `se_r` standard error `z_f` z value `z_r` z value `p_heter` heterogeneity test p value `i2` I^2 statistic `k` No of tests used `eps` smallest double-precision number

## References

JPT Higgins, SG Thompson, JJ Deeks, DG Altman. Measuring inconsistency in meta-analyses. BMJ 327:557-60

## Note

Adapted from a SAS macro

## Author(s)

Shengxu Li, Jing Hua Zhao

## Examples

 ```1 2 3 4 5 6 7 8``` ```## Not run: abc <- data.frame(chromosome=1,rsn='abcd',startpos=1234, b1=1,se1=2,p1=0.1,b2=2,se2=6,p2=0,b3=3,se3=8,p3=0.5) metareg(abc,3) abc2 <- data.frame(b1=c(1,2),se1=c(2,4),b2=c(2,3),se2=c(4,6),b3=c(3,4),se3=c(6,8)) print(metareg(abc2,3)) ## End(Not run) ```

### Example output

```gap version 1.1-17

Meta-analysis of 3 studies:

p_f= 0.5152782
p_r= 0.5152782
beta_f= 1.201183
beta_r= 1.201183
se_f= 1.846154
se_r= 1.846154
z_f= 0.650641
z_r= 0.650641
p_heter= 0.9615572
i2= 0
k= 3
eps= 2.220446e-16

where

p_f=P value (fixed effects model)
p_r=P value (random effects model)
beta_f=regression coefficient
beta_r=regression coefficient
se_f=standard error
se_r=standard error
z_f=z value
z_r=z value
p_heter=heterogeneity test p value
i2=I^2
k=No of tests used
eps=smallest double-precision number

Meta-analysis of 3 studies:

p_f= 0.4320349
p_r= 0.4320349
beta_f= 1.346939
beta_r= 1.346939
se_f= 1.714286
se_r= 1.714286
z_f= 0.7857143
z_r= 0.7857143
p_heter= 0.9358252
i2= 0
k= 3
eps= 2.220446e-16

Meta-analysis of 3 studies:

p_f= 0.4052736
p_r= 0.4052736
beta_f= 2.557377
beta_r= 2.557377
se_f= 3.072885
se_r= 3.072885
z_f= 0.8322397
z_r= 0.8322397
p_heter= 0.9717191
i2= 0
k= 3
eps= 2.220446e-16

where

p_f=P value (fixed effects model)
p_r=P value (random effects model)
beta_f=regression coefficient
beta_r=regression coefficient
se_f=standard error
se_r=standard error
z_f=z value
z_r=z value
p_heter=heterogeneity test p value
i2=I^2
k=No of tests used
eps=smallest double-precision number
beta_f     se_f       z_f   beta_r     se_r   p_heter i2 k          eps
1 1.346939 1.714286 0.7857143 1.346939 1.714286 0.9358252  0 3 2.220446e-16
2 2.557377 3.072885 0.8322397 2.557377 3.072885 0.9717191  0 3 2.220446e-16
```

gap documentation built on May 29, 2017, 9:09 p.m.