R practice: Meta-analysis (1)Minato Nakazawa
17 January 2011
1 Notes before starting• The meta-analysis is very difficult. It requires very sophisticated manner of statistical thinking. You
must pay substantial effort to learn.
• This practice is based on the textbook “Introduction to meta-analysis: the statistical technique to integratevarious evidences (in Japanese: meta anarishisu nyumon: ebidensu no tougou wo mezasu toukei shuho.)”,written by Dr. Toshiro Tango, Asakura-Shoten Pub., 2002. I will mostly pick up the contents of thispractice from this textbook.

2 Definition
A statistical analysis to integrate (including the possibility to do so) the various previous studies.
Egger et al. (1997) states “something occurring later, more comprehensive, and is often used to name a new
but related discipline designated to deal critically with the original one.” as “meta”.

3 History
The trials to integrate or summarize the previously conducted studies are not new. Sir Wright (1896) developed
a new vaccine against typhoid fever and tested the effectiveness of the same vaccine in several different groups.
Karl Pearson (1904) re-evaluated the effectiveness of that vaccine ever used.
Table 1.1 shows the 6 studies that Pearson evaluated. Pearson calculated a tetrachoric correlation*1 for eachresult.

*1 This is different from a simple correlation coefficient. Assuming the latent continuous variables behind the two binary variables

which obeys bivariate normal distribution, then estimate the correlation coefficient for those continuous variables.

1

6. Hospital Staffs in South Africa
Recovered
DiedTotals
7. Garrison of Ladysmith
Recovered
DiedTotals
8. Special Regimens in South AfricaRecovered
Died
Totals
9. Special Hospitals in South Africa
Recovered
DiedTotals
10. Various Military Hospitals in South Africa
Recovered
DiedTotals
11. Army in India, 1900-1.
Recovered
Died
Totals

Inoculated

Non-inoculated

Totals

30
2
32

63
12
75

93
14
107

27
8
35

1160
329
1489

1187
337
1524

63
972

61
21
82

124
30
154

1088
86
1174

4453
538
4991

5541
624
6165

701
63
764

2864
510
3374

3565
573
4138

73
11
84

1052
423
1475

1125
434
1559

Pearson’s tetrachoric correlation can be calculated by polychor() function included in polycor package (developed by Dr. John Fox, available from CRAN).
For instance, calculating that of Hospital Staffs in South Africa of table 1.1, type as below.
¶³library(polycor)
polychor(matrix(c(30,2,63,12),2,2),std.err=TRUE)µ´

Pearson calculated the 6 tetrachoric correlations (with standard errors) and those mean as follows*2 .
¶³

6. 0.307 (0.128)
7. -0.010 (0.081)
8. 0.300 (0.093)
9. 0.119 (0.023)
10. 0.194 (0.022)
11. 0.248 (0.050)
mean value 0.193µ´The major motivation of this summarizing was that they cannot provide the clear evidences by themselves due
to small sample size. This motivation is the same as most current meta-analyses.
Pearson concluded the 0.193 is too small to recommend the concurrent vaccine, although calculating mean is
not appropriate now. The appropriate method will be shown later. Anyway, this was the first meta-analysis.
(Continue to 31 Jan, 2011)*2 The point estimates of these tetrachoric correlations were the same as the results by polychor function, but the standard errors

are different. I cannot find the reason until now.

2