Mycorrhizal responsiveness trends in annual crop plants and their wild relatives- A meta-analysis on studies from 1981 to 2010
Plant and Soil
Anika Lehmann1, E. Kathryn Barto1, Jeff R. Powell2 and Matthias C. Rillig1, *
1
Freie Universität Berlin, Institut für Biologie, Dahlem Center of Plant Sciences, Altensteinstr. 6, D-14195 Berlin, Germany
2
Hawkesbury Institute for the Environment, University of Western Sydney, Hawkesbury Campus, Bourke St, Richmond 2753, NSW, Australia
* Author for correspondence
Matthias C. Rillig
Institut für Biologie, Dahlem Center of Plant Sciences
Ökologie der Pflanzen
Altensteinstr. 6
D- 14195 Berlin
matthias.rillig@fu-berlin.de
+49 (0)30 838-53165; fax -53886
Online Resource 3: “Error” bootstrap code for R
## function for implementing bootstrap methods in meta-analysis ##
## based on Van den Noortgate and Onghena 2005 Behavior Research Methods 37, 11-22 ##
## coded by Jeff Powell (jeffpowell2@gmail.com) ##
## data from Van den Noortgate and Onghena 2005
study<-factor(1:10)
grade<-c(6,5,3,3,2,4,8,1,3,5)
n1<-n2<-c(90,40,36,20,22,10,10,10,39,50)
N<-n1+n2
d<-c(-0.583,0.535,0.779,1.052,0.563,0.308,0.081,0.598,-0.178,-0.234)
err<-(N/(n1*n2))+((d^2)/(2*N))
#mima(yi=d,vi=err,mods=grade)
library(metafor)
z<-rma.uni(yi=d,vi=err) #init w/o mods
zz<-rma.uni(yi=d,vi=err,mods=grade) #init w/ mods
# error here is estimated as described in Van den Noortgate and Onghena 2005, which may be different from how you want to estimate error
boot.rma<-function(init.mod,type=c('effect size','error'),n=list(),boot.reps=10,control=list(maxit=100)){
if(type!='effect size'&type!='error') stop('indicate correct type of analysis')
boot.out<-matrix(NA,nrow=boot.reps,ncol=ncol(init.mod[['X']]),dimnames=list(NULL,colnames(init.mod[['X']])))
n1<-n[[1]]
n2<-n[[2]]
N<-n1+n2
boot.samp<-function(){
y<-init.mod
if(type=='effect size'){ #effect size bootstrap (parametric)
u.boot<-rnorm(length(resid(y)),0,sqrt(y[['tau2']])) # step 1 (u.boot = u*)
del.boot<-fitted(y)+u.boot # step 2 (del.boot = delta*)
err.boot<-(N/(n1*n2))+((del.boot^2)/(2*N)) # step 3 (err.boot = vi*) --> is this how error is estimated?
e.boot<-rnorm(length(del.boot),0,sqrt(err.boot)) # step 3 (e.boot = e*)
d.boot<-del.boot+e.boot # step 4 (d.boot = d*)
}
if(type=='error'){ #error bootstrap (nonparametric)
u<-(y[['tau2']]/(y[['tau2']]+y$vi))*(y$yi-fitted(y)) # step 1 using equation 8 (u = level 2 residuals from y)
#r<-(y$yi-u)/sqrt(y$vi) # step 3 using equation 8 (r = level 1 residuals from y) <-- this is wrong, don't use this (see message at end)
r<-(y$yi-fitted(y)-u)/sqrt(y$vi) # step 3 using equation 3 (r = level 1 residuals from y)
u.infl<-function(p) (y[['tau2']]-var(p*u))^2 # function for reflation of residuals (from paragraph containing equation 8)
u<-u*optimize(u.infl,c(0,10))$minimum # reflation of residuals
u.boot<-sample(u,replace=T) # step 1 (u.boot = u*)
del.boot<-fitted(y)+u.boot # step 2 using equation 5 (del.boot = delta*)
r.infl<-function(p) (1-var(p*r))^2 # function for reflation of residuals (from paragraph containing equation 8)
r<-r*optimize(r.infl,c(0,10))$minimum # reflation of residuals
r.boot<-sample(r,replace=T) # step 3 (r.boot = r*)
err.boot<-(N/(n1*n2))+((del.boot^2)/(2*N)) # step 3 (err.boot = vi*) --> is this how error is estimated?
e.boot<-r.boot*sqrt(err.boot) # step 3 from equations 2+3 (e.boot = e*)
d.boot<-del.boot+e.boot # step 4 (d.boot = d*)
}
res<-list(d.boot,err.boot)
names(res)<-c('d.boot','err.boot')
res
}
for(i in 1:boot.reps){
go<-T
yy<-NULL
while(go){ # for dealing with convergence errors during estimation on bootstrap replicates, resamples bootstrap replicate on error
bootsamp<-boot.samp()
if(ncol(init.mod[['X']])==1) try(yy<-rma.uni(yi=bootsamp[['d.boot']],vi=bootsamp[['err.boot']],control=control),silent=T) else try(yy<rma.uni(yi=bootsamp[['d.boot']],vi=bootsamp[['err.boot']],mods=init.mod[['X']][,2:ncol(init.mod[['X']])],control=control),silent=T)
if(!is.null(yy)) ifelse(any(attr(yy,'class')=='try-error'),go<-T,go<-F)
}
boot.out[i,]<-yy[['b']][,1]
}
boot.out
}
# initial testing
para.boot<-boot.rma(z,type='effect size',n=list(n1,n2),boot.reps=10)
summary(para.boot)
mean(para.boot)
sd(para.boot)
nonpara.boot<-boot.rma(z,type='error',n=list(n1,n2),boot.reps=10)
summary(nonpara.boot)
mean(nonpara.boot)
sd(nonpara.boot)
para.boot<-boot.rma(zz,type='effect size',n=list(n1,n2),boot.reps=10)
summary(para.boot)
mean(para.boot)
sd(para.boot)
1-mean(para.boot[,'mods']<0)
nonpara.boot<-boot.rma(zz,type='error',n=list(n1,n2),boot.reps=10)
summary(nonpara.boot)
mean(nonpara.boot)
sd(nonpara.boot)
1-mean(nonpara.boot[,'mods']<0)
# with more than one moderator
fake<-sample(15,length(grade))
zzz<-rma.uni(yi=d,vi=err,mods=cbind(grade,fake)) #init w/ mods
para.boot<-boot.rma(zzz,type='effect size',n=list(n1,n2),boot.reps=10)
summary(para.boot)
mean(para.boot)
sd(para.boot)
1-mean(para.boot[,'grade']<0)
1-mean(para.boot[,'fake']<0)
nonpara.boot<-boot.rma(zzz,type='error',n=list(n1,n2),boot.reps=10)
summary(nonpara.boot)
mean(nonpara.boot)
sd(nonpara.boot)
1-mean(nonpara.boot[,'grade']<0)
1-mean(nonpara.boot[,'fake']<0)
# try forcing convergence error
boot.rma(zz,type='effect size',n=list(n1,n2),boot.reps=1,control=list(maxit=1))
boot.rma(zz,type='error',n=list(n1,n2),boot.reps=1,control=list(maxit=1))
### testing -- bias observed in the intercept estimate for 'error' bootstrap, moderator estimate appears fine
boot.rma(z,type='effect size',n=list(n1,n2))
boot.rma(z,type='error',n=list(n1,n2))
boot.rma(zz,type='effect size',n=list(n1,n2))
boot.rma(zz,type='error',n=list(n1,n2))
boot.rma(zzz,type='effect size',n=list(n1,n2))
boot.rma(zzz,type='error',n=list(n1,n2))
summary(boot.rma(z,type='effect size',n=list(n1,n2),boot.reps=10000))
summary(boot.rma(z,type='error',n=list(n1,n2),boot.reps=10000))
#> summary(boot.rma(z,type='effect size',n=list(n1,n2),boot.reps=10000))
# intrcpt
# Min. :-0.5280
# 1st Qu.: 0.1339
# Median : 0.2514
# Mean : 0.2522
# 3rd Qu.: 0.3734
# Max. : 0.9184
#> summary(boot.rma(z,type='error',n=list(n1,n2),boot.reps=10000))
# intrcpt
# Min. :-0.3551
# 1st Qu.: 0.1893
# Median : 0.3089
# Mean : 0.3055
# 3rd Qu.: 0.4243
# Max. : 0.8530
#>
#> z
#
#Random-Effects Model (k = 10; tau^2 estimator: REML)
#
#tau^2 (estimate of total amount of heterogeneity): 0.2301 (SE = 0.1497)
#tau (sqrt of the estimate of total heterogeneity): 0.4797
#I^2 (% of total variability due to heterogeneity): 77.35%
#H^2 (total variability / within-study variance): 4.42
#
#Test for Heterogeneity:
#Q(df = 9) = 47.1106, p-val < .0001
#
#Model Results:
#
#estimate
se zval pval ci.lb ci.ub
# 0.2521 0.1793 1.4062 0.1597 -0.0993 0.6036
#
#--#Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
#>
#
summary(boot.rma(zz,type='effect size',n=list(n1,n2),boot.reps=10000))
summary(boot.rma(zz,type='error',n=list(n1,n2),boot.reps=10000))
#> summary(boot.rma(zz,type='effect size',n=list(n1,n2),boot.reps=10000))
# intrcpt
mods
# Min. :-0.5564 Min. :-0.4991
# 1st Qu.: 0.6213 1st Qu.:-0.2177
# Median : 0.8822 Median :-0.1575
# Mean : 0.8806 Mean :-0.1582
# 3rd Qu.: 1.1433 3rd Qu.:-0.0992
# Max. : 2.4020 Max. : 0.1714
#> summary(boot.rma(zz,type='error',n=list(n1,n2),boot.reps=10000))
# intrcpt
mods
# Min. :-0.4631 Min. :-0.4743
# 1st Qu.: 0.6797 1st Qu.:-0.2163
# Median : 0.9331 Median :-0.1577
# Mean : 0.9330 Mean :-0.1591
# 3rd Qu.: 1.1884 3rd Qu.:-0.1010
# Max. : 2.4071 Max. : 0.1704
#>
#> zz
#
#Mixed-Effects Model (k = 10; tau^2 estimator: REML)
#
#tau^2 (estimate of residual amount of heterogeneity): 0.1627 (SE = 0.1226)
#tau (sqrt of the estimate of residual heterogeneity): 0.4034
#
#Test for Residual Heterogeneity:
#QE(df = 8) = 27.7876, p-val = 0.0005
#
#Test of Moderators (coefficient(s) 2):
#QM(df = 1) = 3.2550, p-val = 0.0712
#
#Model Results:
#
#
estimate
se zval pval ci.lb ci.ub
#intrcpt 0.8896 0.3936 2.2604 0.0238 0.1183 1.6610 *
#mods
-0.1597 0.0885 -1.8042 0.0712 -0.3333 0.0138 .
#
#--#Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
#>
#
summary(boot.rma(zzz,type='effect size',n=list(n1,n2),boot.reps=10000))
summary(boot.rma(zzz,type='error',n=list(n1,n2),boot.reps=10000))
#> summary(boot.rma(zzz,type='effect size',n=list(n1,n2),boot.reps=10000))
# intrcpt
grade
fake
# Min. :-1.0751 Min. :-0.59458 Min. :-0.124792
# 1st Qu.: 0.5880 1st Qu.:-0.24120 1st Qu.:-0.018217
# Median : 0.8611 Median :-0.16591 Median : 0.007636
# Mean : 0.8635 Mean :-0.16620 Mean : 0.007355
# 3rd Qu.: 1.1459 3rd Qu.:-0.09213 3rd Qu.: 0.032733
# Max. : 3.0354 Max. : 0.34232 Max. : 0.161243
#> summary(boot.rma(zzz,type='error',n=list(n1,n2),boot.reps=10000))
# intrcpt
grade
fake
# Min. :-0.5467 Min. :-0.54926 Min. :-0.105141
# 1st Qu.: 0.6423 1st Qu.:-0.24152 1st Qu.:-0.018047
# Median : 0.9237 Median :-0.16796 Median : 0.006882
# Mean : 0.9196 Mean :-0.16931 Mean : 0.007162
# 3rd Qu.: 1.1965 3rd Qu.:-0.09547 3rd Qu.: 0.032170
# Max. : 2.3813 Max. : 0.18150 Max. : 0.142709
#>
#> zzz
#
#Mixed-Effects Model (k = 10; tau^2 estimator: REML)
#
#tau^2 (estimate of residual amount of heterogeneity): 0.1927 (SE = 0.1500)
#tau (sqrt of the estimate of residual heterogeneity): 0.4390
#
#Test for Residual Heterogeneity:
#QE(df = 7) = 27.5003, p-val = 0.0003
#
#Test of Moderators (coefficient(s) 2,3):
#QM(df = 2) = 2.8732, p-val = 0.2377
#
#Model Results:
#
#
estimate
se zval pval ci.lb ci.ub
#intrcpt 0.8751 0.4162 2.1027 0.0355 0.0594 1.6907 *
#grade -0.1682 0.1108 -1.5182 0.1290 -0.3854 0.0490
#fake
0.0071 0.0377 0.1876 0.8512 -0.0667 0.0809
#
#--#Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
#>
#
### message regarding nonparametric bootstrap estimation of r_j ###
#From: Wim Van Den Noortgate <Wim.VandenNoortgate@kuleuven-kortrijk.be>
#Date: March 4, 2011 8:58:41 PM GMT+01:00
#To: Jeff Powell <jeff.powell@fu-berlin.de>
#Subject: RE: nonparametric bootstrap for meta-analysis
#
#Dear Jeff,
#
#You are right, the formula is not complete... It should be the way you describe.
#I am sorry for the confusion!
#
#Kind regards and good luck,
#
#Wim
#
#-----Oorspronkelijk bericht----#Van: Jeff Powell [mailto:jeff.powell@fu-berlin.de]
#Verzonden: vrijdag 4 maart 2011 17:19
#Aan: wim.vandennoortgate@ped.kuleuven.ac.be
#Onderwerp: nonparametric bootstrap for meta-analysis
#
#Dear Dr. Van den Noortgate,
#
#I'm trying to implement the 'error bootstrap' from your 2005 Behaviour Â
#Research Methods paper in an R script for a colleague. I've gotten to Â
#step 3 but I'm a little confused by the estimate of r_j in equation 8. Â
#It seems to me that based on equation 3 there should be a term for the Â
#initial estimates of the fixed parameters in this equation as well, Â
#i.e.:
#
#rhat_j = ( d_j - gammahat_0 - sum( gammahat_s x W_sj ) - uhat_j ) / Â
#sigmahat_j
#
#instead of:
#
#Â rhat_j = ( d_j - uhat_j ) / sigmahat_j
#
#If I am wrong about this could you please tell me why or point me to Â
#some literature that will help me understand?
#
#Thanks and best regards,
#Jeff Powell
#
## plotting code for troubleshooting:
#plot(y$yi,pch=16,ylim=c(-1,2)) # observed effect size (d_j)
#points(fitted(y),pch=1) # model prediction
#points(fitted(y)+u,pch='u') # delta_j
#points(fitted(y)+u+r*sqrt(y$vi),pch='E') # d_j
#points(u,col='red') # level 2 residuals
#points(r*sqrt(y$vi),col='blue') # level 1 residuals
#