Meta analysis of continuous data
Kushan De Silva
August 24, 2017
The results of some of the meta-analyses presented in the paper titled “Meta-analysis of laparoscopic vs. open resection of gastric gastrointestinal stromal tumors” by L. Ye et al (2017) are re-analyzed and reported here. In addition, manual calculations in R are also presented, besides using the functions in R packages.
# Type the data
Matthews = c(169, 58.8,21,160, 48.1,12)
Mochizuki = c(111.3, 33.8,12, 124.5,41.6, 10)
Ishikawa = c(115, 56, 21,134 ,33, 25)
Pitsinis = c( 118,55, 14, 165,108, 7)
Catena = c( 129.6, 36.1,15, 110.8,38.1 , 13)
Goh = c(107.5,8.8,6, 157.5 ,53.6, 7)
Wu = c(130, 17.5,18,105, 32.5,30)
Karakousis = c(96, 38,40, 89,54.8, 40)
Dal = c(135, 45, 17,157 ,60, 29)
Pucci = c( 90, 45,68, 125,60 , 88)
Wan = c(145,26.1,14, 95 ,42.5, 39)
Kim = c(160, 76.3,57,191, 85.3,47)
Melstrom = c(119.8, 62.2,24, 154.3,53.5, 14)
Vogelaere = c( 116.6,26.1, 30, 119.6,48.5, 32)
Shu = c( 147.8, 59.3,15, 139.2,62.1 , 21)
Lee = c(48.5,16,37, 155 ,48.1, 16)
Lin = c(124.1, 50.3,23,196.5, 65.8,23)
Kasetsemwiriya = c(152.3, 64.8,23, 198.5,75.1, 10)
Takahashi = c(123.8, 44.8, 12,119.5 ,44.2, 15)
Sista = c( 130,70, 282, 150,79, 384)
Cal = c(106.6, 40.1,90, 119.9,59.9 , 66)
deAngelis = c(146.6,50.2,18, 113.3 ,42.9, 21)
Hsiao = c(129.3, 28.2,25,136.8, 27.5,25)
Piessen = c(245, 40,30, 228,52, 33)
Yan = c(138.8, 69.2, 158,173.3 ,74.5, 68)
# Make a data frame
dat = as.data.frame(rbind(Matthews,Mochizuki,Ishikawa, Pitsinis,Catena, Goh,Wu, Karakousis, Dal,
Pucci,Wan,Kim, Melstrom,Vogelaere,Shu,Lee,Lin,Kasetsemwiriya,Takahashi,
Sista,Cal,deAngelis,Hsiao,Piessen,Yan))
colnames(dat) = c("m.t","sd.t","n.t","m.c","sd.c","n.c")
# Print the data
dat## m.t sd.t n.t m.c sd.c n.c
## Matthews 169.0 58.8 21 160.0 48.1 12
## Mochizuki 111.3 33.8 12 124.5 41.6 10
## Ishikawa 115.0 56.0 21 134.0 33.0 25
## Pitsinis 118.0 55.0 14 165.0 108.0 7
## Catena 129.6 36.1 15 110.8 38.1 13
## Goh 107.5 8.8 6 157.5 53.6 7
## Wu 130.0 17.5 18 105.0 32.5 30
## Karakousis 96.0 38.0 40 89.0 54.8 40
## Dal 135.0 45.0 17 157.0 60.0 29
## Pucci 90.0 45.0 68 125.0 60.0 88
## Wan 145.0 26.1 14 95.0 42.5 39
## Kim 160.0 76.3 57 191.0 85.3 47
## Melstrom 119.8 62.2 24 154.3 53.5 14
## Vogelaere 116.6 26.1 30 119.6 48.5 32
## Shu 147.8 59.3 15 139.2 62.1 21
## Lee 48.5 16.0 37 155.0 48.1 16
## Lin 124.1 50.3 23 196.5 65.8 23
## Kasetsemwiriya 152.3 64.8 23 198.5 75.1 10
## Takahashi 123.8 44.8 12 119.5 44.2 15
## Sista 130.0 70.0 282 150.0 79.0 384
## Cal 106.6 40.1 90 119.9 59.9 66
## deAngelis 146.6 50.2 18 113.3 42.9 21
## Hsiao 129.3 28.2 25 136.8 27.5 25
## Piessen 245.0 40.0 30 228.0 52.0 33
## Yan 138.8 69.2 158 173.3 74.5 68library(meta)## Loading 'meta' package (version 4.8-4).
## Type 'help(meta)' for a brief overview.# call the metacont
mod = metacont(n.t,m.t,sd.t,n.c,m.c,sd.c,
data=dat,studlab=rownames(dat),sm="SMD")
# print the meta-analysis
mod## SMD 95%-CI %W(fixed) %W(random)
## Matthews 0.1589 [-0.5515; 0.8694] 1.5 3.7
## Mochizuki -0.3385 [-1.1850; 0.5079] 1.1 3.2
## Ishikawa -0.4156 [-1.0025; 0.1714] 2.3 4.1
## Pitsinis -0.5949 [-1.5238; 0.3341] 0.9 2.9
## Catena 0.4928 [-0.2628; 1.2485] 1.4 3.5
## Goh -1.1620 [-2.3766; 0.0526] 0.5 2.2
## Wu 0.8810 [ 0.2684; 1.4936] 2.1 4.0
## Karakousis 0.1470 [-0.2919; 0.5859] 4.1 4.6
## Dal -0.3930 [-0.9975; 0.2116] 2.1 4.0
## Pucci -0.6451 [-0.9698; -0.3205] 7.4 5.0
## Wan 1.2637 [ 0.6039; 1.9236] 1.8 3.8
## Kim -0.3823 [-0.7721; 0.0075] 5.1 4.8
## Melstrom -0.5705 [-1.2434; 0.1024] 1.7 3.8
## Vogelaere -0.0754 [-0.5737; 0.4229] 3.1 4.4
## Shu 0.1379 [-0.5255; 0.8014] 1.8 3.8
## Lee -3.5755 [-4.4944; -2.6566] 0.9 3.0
## Lin -1.2150 [-1.8486; -0.5814] 1.9 3.9
## Kasetsemwiriya -0.6633 [-1.4251; 0.0984] 1.3 3.5
## Takahashi 0.0938 [-0.6658; 0.8533] 1.4 3.5
## Sista -0.2652 [-0.4196; -0.1109] 32.8 5.4
## Cal -0.2677 [-0.5868; 0.0513] 7.7 5.0
## deAngelis 0.7031 [ 0.0524; 1.3538] 1.8 3.9
## Hsiao -0.2650 [-0.8220; 0.2919] 2.5 4.2
## Piessen 0.3597 [-0.1390; 0.8583] 3.1 4.4
## Yan -0.4855 [-0.7733; -0.1976] 9.4 5.1
##
## Number of studies combined: k = 25
##
## SMD 95%-CI z p-value
## Fixed effect model -0.2543 [-0.3427; -0.1659] -5.64 < 0.0001
## Random effects model -0.2384 [-0.4710; -0.0058] -2.01 0.0445
##
## Quantifying heterogeneity:
## tau^2 = 0.2534; H = 2.34 [1.96; 2.79]; I^2 = 81.7% [73.9%; 87.2%]
##
## Test of heterogeneity:
## Q d.f. p-value
## 131.22 24 < 0.0001
##
## Details on meta-analytical method:
## - Inverse variance method
## - DerSimonian-Laird estimator for tau^2
## - Hedges' g (bias corrected standardised mean difference)# call the metacont
mod2 = metacont(n.t,m.t,sd.t,n.c,m.c,sd.c,
data=dat,studlab=rownames(dat),sm="MD")
# print the meta-analysis
mod2## MD 95%-CI %W(fixed) %W(random)
## Matthews 9.0000 [ -28.0553; 46.0553] 1.5 3.6
## Mochizuki -13.2000 [ -45.3015; 18.9015] 2.0 3.9
## Ishikawa -19.0000 [ -46.2212; 8.2212] 2.7 4.1
## Pitsinis -47.0000 [-132.0353; 38.0353] 0.3 1.6
## Catena 18.8000 [ -8.8169; 46.4169] 2.6 4.1
## Goh -50.0000 [ -90.3262; -9.6738] 1.2 3.4
## Wu 25.0000 [ 10.8363; 39.1637] 10.0 4.7
## Karakousis 7.0000 [ -13.6659; 27.6659] 4.7 4.5
## Dal -22.0000 [ -52.5689; 8.5689] 2.2 3.9
## Pucci -35.0000 [ -51.4787; -18.5213] 7.4 4.6
## Wan 50.0000 [ 30.8995; 69.1005] 5.5 4.5
## Kim -31.0000 [ -62.4173; 0.4173] 2.0 3.9
## Melstrom -34.5000 [ -71.9783; 2.9783] 1.4 3.6
## Vogelaere -3.0000 [ -22.2251; 16.2251] 5.4 4.5
## Shu 8.6000 [ -31.4750; 48.6750] 1.3 3.4
## Lee -106.5000 [-130.6258; -82.3742] 3.5 4.3
## Lin -72.4000 [-106.2484; -38.5516] 1.8 3.8
## Kasetsemwiriya -46.2000 [ -99.7529; 7.3529] 0.7 2.8
## Takahashi 4.3000 [ -29.5056; 38.1056] 1.8 3.8
## Sista -20.0000 [ -31.3658; -8.6342] 15.6 4.8
## Cal -13.3000 [ -29.9575; 3.3575] 7.2 4.6
## deAngelis 33.3000 [ 3.7285; 62.8715] 2.3 4.0
## Hsiao -7.5000 [ -22.9402; 7.9402] 8.4 4.7
## Piessen 17.0000 [ -5.7957; 39.7957] 3.9 4.4
## Yan -34.5000 [ -55.2357; -13.7643] 4.7 4.5
##
## Number of studies combined: k = 25
##
## MD 95%-CI z p-value
## Fixed effect model -9.6466 [-14.1307; -5.1625] -4.22 < 0.0001
## Random effects model -13.4975 [-26.7784; -0.2166] -1.99 0.0464
##
## Quantifying heterogeneity:
## tau^2 = 919.4266; H = 2.79 [2.38; 3.28]; I^2 = 87.2% [82.3%; 90.7%]
##
## Test of heterogeneity:
## Q d.f. p-value
## 187.39 24 < 0.0001
##
## Details on meta-analytical method:
## - Inverse variance method
## - DerSimonian-Laird estimator for tau^2forest(mod)
forest(mod2)
#Step-by-Step Calculations in R
# First get the pooled sd to calculate the SMD
pooled.sd = sqrt(((dat$n.t-1)*dat$sd.t^2+(dat$n.c-1)*dat$sd.c^2)/(dat$n.t+dat$n.c-2))
# Print the SD
pooled.sd## [1] 55.24097 37.51125 44.93834 75.84923 37.03650 40.02835 27.91193
## [8] 47.15422 55.02066 53.98864 38.98045 80.48351 59.20599 39.30115
## [15] 60.96264 29.34586 58.56505 67.95135 44.46500 75.32263 49.43404
## [22] 46.39690 27.85220 46.68134 70.82685# The standardized mean difference(SMD)
g = (dat$m.t-dat$m.c)/pooled.sd
# Print the SMD
g## [1] 0.16292255 -0.35189442 -0.42280154 -0.61965035 0.50760736
## [6] -1.24911458 0.89567421 0.14844908 -0.39984982 -0.64828459
## [11] 1.28269444 -0.38517205 -0.58271132 -0.07633364 0.14107002
## [16] -3.62913220 -1.23623220 -0.67989818 0.09670528 -0.26552445
## [21] -0.26904538 0.71772035 -0.26927856 0.36417121 -0.48710339# Hedges correction factor
N = dat$n.t+dat$n.c
J = 1- 3/(4*N-9)
# Print the correction factor J
J## [1] 0.9756098 0.9620253 0.9828571 0.9600000 0.9708738 0.9302326 0.9836066
## [8] 0.9903537 0.9828571 0.9951220 0.9852217 0.9926290 0.9790210 0.9874477
## [15] 0.9777778 0.9852217 0.9828571 0.9756098 0.9696970 0.9988701 0.9951220
## [22] 0.9795918 0.9842932 0.9876543 0.9966480# now the Hedges g*
gstar = J*g
# Print it
gstar## [1] 0.15894883 -0.33853134 -0.41555351 -0.59486433 0.49282268
## [6] -1.16196705 0.88099103 0.14701710 -0.39299525 -0.64512222
## [11] 1.26373836 -0.38233295 -0.57048660 -0.07537547 0.13793513
## [16] -3.57549970 -1.21503965 -0.66331530 0.09377482 -0.26522443
## [21] -0.26773296 0.70307299 -0.26504905 0.35967526 -0.48547064# Variance of SMD
var.gstar = (dat$n.t+dat$n.c)/(dat$n.t*dat$n.c)
+ gstar^2/(2*(dat$n.t+dat$n.c-3.94))## [1] 4.346994e-04 3.172853e-03 2.052838e-03 1.037115e-02 5.047261e-03
## [6] 7.451255e-02 8.807821e-03 1.420854e-04 1.836011e-03 1.368482e-03
## [11] 1.627634e-02 7.304541e-04 4.777671e-03 4.892751e-05 2.967264e-04
## [16] 1.302915e-01 1.755018e-02 7.570323e-03 1.906703e-04 5.312509e-05
## [21] 2.356995e-04 7.049510e-03 7.626031e-04 1.095211e-03 5.306713e-04# Print it
var.gstar## [1] 0.130952381 0.183333333 0.087619048 0.214285714 0.143589744
## [6] 0.309523810 0.088888889 0.050000000 0.093306288 0.026069519
## [11] 0.097069597 0.038820455 0.113095238 0.064583333 0.114285714
## [16] 0.089527027 0.086956522 0.143478261 0.150000000 0.006150266
## [21] 0.026262626 0.103174603 0.080000000 0.063636364 0.021034996lowCI.gstar = gstar-1.96*sqrt(var.gstar)
upCI.gstar = gstar+1.96*sqrt(var.gstar)
# Print the CIs and the SMD
cbind(lowCI.gstar, gstar, upCI.gstar)## lowCI.gstar gstar upCI.gstar
## [1,] -0.55032309 0.15894883 0.868220760
## [2,] -1.17775320 -0.33853134 0.500690525
## [3,] -0.99572360 -0.41555351 0.164616577
## [4,] -1.50216803 -0.59486433 0.312439364
## [5,] -0.24988477 0.49282268 1.235530136
## [6,] -2.25241038 -1.16196705 -0.071523714
## [7,] 0.29663193 0.88099103 1.465350129
## [8,] -0.29125223 0.14701710 0.585286422
## [9,] -0.99169838 -0.39299525 0.205707875
## [10,] -0.96158496 -0.64512222 -0.328659486
## [11,] 0.65308088 1.26373836 1.874395846
## [12,] -0.76850993 -0.38233295 0.003844035
## [13,] -1.22962746 -0.57048660 0.088654251
## [14,] -0.57347520 -0.07537547 0.422724248
## [15,] -0.52466581 0.13793513 0.800536063
## [16,] -4.16195262 -3.57549970 -2.989046783
## [17,] -1.79301212 -1.21503965 -0.637067184
## [18,] -1.40573438 -0.66331530 0.079103780
## [19,] -0.66532992 0.09377482 0.852879552
## [20,] -0.41893474 -0.26522443 -0.111514108
## [21,] -0.58536562 -0.26773296 0.049899694
## [22,] 0.07350524 0.70307299 1.332640747
## [23,] -0.81942077 -0.26504905 0.289322666
## [24,] -0.13475921 0.35967526 0.854109744
## [25,] -0.76973820 -0.48547064 -0.201203091# The individual weight
w = 1/var.gstar
w## [1] 7.636364 5.454545 11.413043 4.666667 6.964286 3.230769
## [7] 11.250000 20.000000 10.717391 38.358974 10.301887 25.759615
## [13] 8.842105 15.483871 8.750000 11.169811 11.500000 6.969697
## [19] 6.666667 162.594595 38.076923 9.692308 12.500000 15.714286
## [25] 47.539823# The total weight
tot.w = sum(w)
tot.w## [1] 511.2536# And the relative weight
rel.w = w/tot.w
rel.w## [1] 0.014936547 0.010668962 0.022323643 0.009127890 0.013621978
## [6] 0.006319308 0.022004734 0.039119527 0.020962964 0.075029246
## [11] 0.020150247 0.050385198 0.017294949 0.030286085 0.017114793
## [16] 0.021847887 0.022493728 0.013632562 0.013039842 0.318031180
## [21] 0.074477561 0.018957925 0.024449704 0.030736771 0.092986769# Meta-estimate
M = sum(rel.w*gstar)
M## [1] -0.296462# The variance of M
var.M = 1/tot.w
var.M## [1] 0.001955976# The SE
se.M = sqrt(var.M)
se.M## [1] 0.04422642# The lower 95% CI bound
lowCI.M = M-1.96*se.M
lowCI.M## [1] -0.3831458# The upper 95% CI bound
upCI.M = M+1.96*se.M
upCI.M## [1] -0.2097782# Compute z
z = M/se.M
z## [1] -6.703278# Compute p-value
pval = 2*(1-pnorm(abs(z)))
pval## [1] 2.037948e-11# The Q statistic
Q = sum(w*gstar^2)-(sum(w*gstar))^2/tot.w
Q## [1] 211.6667# The degrees of freedom from 6 studies
df= 6-1
# C quantity
C = tot.w - sum(w^2)/tot.w
C## [1] 443.6353# The tau-square estimate
tau2 = (Q-df)/C
tau2## [1] 0.4658483# Now compute the weights incorporating heterogeneity
wR = 1/(var.gstar+tau2)
wR## [1] 1.675601 1.540401 1.806791 1.470299 1.640856 1.289703 1.802656
## [8] 1.938555 1.788414 2.032860 1.776458 1.981498 1.727284 1.885257
## [15] 1.723740 1.800584 1.808957 1.641156 1.623777 2.118651 2.032062
## [22] 1.757399 1.832011 1.888629 2.053880# The total weight
tot.wR = sum(wR)
tot.wR## [1] 44.63748# The relative weight
rel.wR = wR/tot.wR
rel.wR## [1] 0.03753799 0.03450914 0.04047700 0.03293866 0.03675960 0.02889284
## [7] 0.04038435 0.04342885 0.04006531 0.04554155 0.03979746 0.04439090
## [13] 0.03869583 0.04223485 0.03861642 0.04033795 0.04052552 0.03676633
## [19] 0.03637698 0.04746349 0.04552368 0.03937047 0.04104199 0.04231039
## [25] 0.04601246# The meta-estimate
MR = sum(rel.wR*gstar)
MR## [1] -0.276899# The variance of MR
var.MR = 1/tot.wR
var.MR## [1] 0.0224027# The SE of MR
se.MR = sqrt(var.MR)
se.MR## [1] 0.1496753# The lower bound of 95% CI
lowCI.MR = MR - 1.96*se.MR
lowCI.MR## [1] -0.5702626# The upper 95% CI
upCI.MR = MR + 1.96*se.MR
upCI.MR## [1] 0.01646461# The z value
zR = MR/se.MR
zR## [1] -1.849998# The p-value
pval.R = 2*(1-pnorm(abs(zR)))
pval.R## [1] 0.06431386# The summary table
sumTab = data.frame(
SMD = round(gstar,4),
lowCI = round(lowCI.gstar,4),
upperCI = round(upCI.gstar,4),
pctW.fixed = round(rel.w*100,2),
pctW.random = round(rel.wR*100,2))
rownames(sumTab) = rownames(dat)
# Print it
sumTab## SMD lowCI upperCI pctW.fixed pctW.random
## Matthews 0.1589 -0.5503 0.8682 1.49 3.75
## Mochizuki -0.3385 -1.1778 0.5007 1.07 3.45
## Ishikawa -0.4156 -0.9957 0.1646 2.23 4.05
## Pitsinis -0.5949 -1.5022 0.3124 0.91 3.29
## Catena 0.4928 -0.2499 1.2355 1.36 3.68
## Goh -1.1620 -2.2524 -0.0715 0.63 2.89
## Wu 0.8810 0.2966 1.4654 2.20 4.04
## Karakousis 0.1470 -0.2913 0.5853 3.91 4.34
## Dal -0.3930 -0.9917 0.2057 2.10 4.01
## Pucci -0.6451 -0.9616 -0.3287 7.50 4.55
## Wan 1.2637 0.6531 1.8744 2.02 3.98
## Kim -0.3823 -0.7685 0.0038 5.04 4.44
## Melstrom -0.5705 -1.2296 0.0887 1.73 3.87
## Vogelaere -0.0754 -0.5735 0.4227 3.03 4.22
## Shu 0.1379 -0.5247 0.8005 1.71 3.86
## Lee -3.5755 -4.1620 -2.9890 2.18 4.03
## Lin -1.2150 -1.7930 -0.6371 2.25 4.05
## Kasetsemwiriya -0.6633 -1.4057 0.0791 1.36 3.68
## Takahashi 0.0938 -0.6653 0.8529 1.30 3.64
## Sista -0.2652 -0.4189 -0.1115 31.80 4.75
## Cal -0.2677 -0.5854 0.0499 7.45 4.55
## deAngelis 0.7031 0.0735 1.3326 1.90 3.94
## Hsiao -0.2650 -0.8194 0.2893 2.44 4.10
## Piessen 0.3597 -0.1348 0.8541 3.07 4.23
## Yan -0.4855 -0.7697 -0.2012 9.30 4.60