[R-meta] Nonlinear meta-regression with factorial moderator

[Cesar Terrer Moreno]

Hi Wolfgang,

Thanks for your response.

I have followed your code and adapted for a factor with 5 levels (data attached)

df <- read.csv("~/OneDrive/OneDrive - Universitat Autònoma de Barcelona/FACEreview/data/AGB_effects.csv",na.strings=c("",NA))
df <- df[complete.cases(df$id),]
df2 <- filter(df, Site.Name != "Nevada FACE",!is.na(CNr)) %>% 
  rename(es=yi, var=vi)
df2$var <- ifelse(is.na(df2$var),max(df2$var,na.rm=T), df2$var)


am.df <- filter(df2,Myc=="AM") %>% mutate(Biome = relevel(Biome, ref= "Grassland")) %>% droplevels()

# Biome has 5 levels. Then create 4 dummy variables (so level 1 is the reference level). Let's call these dummies B2, B3, B4, B5
library(dummies)
am.df <- dummy.data.frame("Biome",data=am.df, sep = "")

nlfun <- function(x, B2, B3, B4, B5, p1, p2, p3, p4, p5, p6)
  (p1 + B2*p2 + B3*p3 + B4*p4 + B5*p5) * exp(-p6*x)

# optimization function
llfun <- function(par, yi, vi, x, B2, B3, B4, B5, random=TRUE) {
  p1 <- par[1]
  p2 <- par[2]
  p3 <- par[3]
  p4 <- par[4]
  p5 <- par[5]
  p6 <- par[6]
  if (random) {
    tau2 <- exp(par[7])
  } else {
    tau2 <- 0
  }
  mu <- nlfun(x, B2, B3, B4, B5, p1, p2, p3, p4, p5, p6)
  -sum(dnorm(yi, mean=mu, sd=sqrt(vi + tau2), log=TRUE))
}

# optimize
res <- optim(par=c(8,0,0,0,0,0.4,log(.01)), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, 
            B2=am.df$BiomeCropland, B3=am.df$BiomeShrubland, 
            B4=am.df$BiomeTemperate_Forest, B5=am.df$BiomeTropical_Forest, 
             hessian=TRUE)
# back-transform log(tau2) to tau2
res$par[7] <- exp(res$par[7])
tau2_ME <- res$par[7]

# fit model with tau2=0
res0 <- optim(par=c(8,0,0,0,0,0.4,log(.01)), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, 
              B2=am.df$BiomeCropland, B3=am.df$BiomeShrubland, 
              B4=am.df$BiomeTemperate_Forest, B5=am.df$BiomeTropical_Forest,
              random=FALSE, hessian=TRUE)

# LRT of H0: tau2=0
x2 <- -2 * (res0$value - res$value)
x2 ### test statistic
pchisq(x2, df=1, lower.tail=FALSE) ### p-value

# compute standard errors
vb <- chol2inv(chol(res$hessian))

However, in this last step, I get the following error:
Error in chol.default(res$hessian) : 
  the leading minor of order 6 is not positive definite

Google says this may be due to overfitting. What can I do?
Thanks
César

> On 24 Mar 2018, at 16:39, Viechtbauer Wolfgang (SP) <wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
> 
> Let's say F has three levels. Then create two dummy variables for levels 2 and 3 (so level 1 is the reference level). Let's call these dummies F2 and F2. Then I think this should be it (I haven't tested this):
> 
> nlfun <- function(x, F2, F3, p1, p2, p3, p4)
>  (p1 + F2*p2 + F3*p3) * exp(-p4*x)
> 
> # optimization function
> llfun <- function(par, yi, vi, x, F2, F3, random=TRUE) {
>  p1 <- par[1]
>  p2 <- par[2]
>  p3 <- par[3]
>  p4 <- par[4]
>  if (random) {
>    tau2 <- exp(par[5])
>  } else {
>    tau2 <- 0
>  }
>  mu <- nlfun(x, F2, F3, p1, p2, p3, p4)
>  -sum(dnorm(yi, mean=mu, sd=sqrt(vi + tau2), log=TRUE))
> }
> 
> # optimize
> res <- optim(par=c(8,0,0,0.4,log(.01)), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, F2=am.df$F2, F3=am.df$F3, hessian=TRUE)
> # back-transform log(tau2) to tau2
> res$par[5] <- exp(res$par[5])
> tau2_ME <- res$par[5]
> 
> # fit model with tau2=0
> res0 <- optim(par=c(8,0,0,0.4), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, F2=am.df$F2, F3=am.df$F3, random=FALSE, hessian=TRUE)
> 
> # LRT of H0: tau2=0
> x2 <- -2 * (res0$value - res$value)
> x2 ### test statistic
> pchisq(x2, df=1, lower.tail=FALSE) ### p-value
> 
> # compute standard errors
> vb <- chol2inv(chol(res$hessian))
> se <- sqrt(diag(vb))
> 
> # ignore SE for tau2
> se[5] <- NA
> 
> # estimates, standard errors, z-values, and p-values
> data.frame(estimate=round(res$par,4), se=round(se,4), zval=round(res$par/se,3), pval=round(2*pnorm(abs(res$par/se), lower.tail=FALSE),5), row.names=c("p1","p2","p3","p4","tau2"))
> 
> # Function to predict the nonlinear effect size (ES)
> ESpred <- function(x, F2, F3) {
>  p1 <- res$par[1]
>  p2 <- res$par[2]
>  p3 <- res$par[3]
>  p4 <- res$par[4]
>  (p1 + F2*p2 + F3*p3)*exp(-p4*x)
> }
> 
> # Function to predict the nonlinear standard error (SE)
> SEpred <- function(x, F2, F3) {
>  p1 <- res$par[1]
>  p2 <- res$par[2]
>  p3 <- res$par[3]
>  p4 <- res$par[4]
>  g <- matrix(c((1 + F2*p2 + F3*p3) * exp(-p4*x), (p1 + F2 + F3*p3) * exp(-p4*x), (p1 + F2*p2 + F3) * exp(-p4*x), -(p1 + F2*p2 + F3*p3)*exp(-p4*x)*x), ncol=1, nrow=2)
>  vb <- vb[1:4,1:4]
>  c(sqrt(t(g) %*% vb %*% g))
> }
> 
> Best,
> Wolfgang
> 
> -----Original Message-----
> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-project.org] On Behalf Of Cesar Terrer Moreno
> Sent: Monday, 19 March, 2018 15:57
> To: r-sig-meta-analysis at r-project.org
> Subject: [R-meta] Nonlinear meta-regression with factorial moderator
> 
> Dear all,
> 
> I have an effect size that is driven by two moderators: a continuous “C” and a factorial “F”. Let’s say, in this case, that C is temperature, while F is ecosystem type (e.g. grassland, forest, cropland):
> 
> rma(ES, VAR, data= dat, mods= ~ 1 + C + F)
> 
> The relationship between ES and C is nonlinear, with high ES when C is low, but ES quickly approaching 0 when C becomes higher towards infinite, but never becoming negative. 
> 
> Previously, you guys helped me define a nonlinear meta-regression ES ~ C like: ES ~ p1 × exp(-p2*C), including a function to predict the nonlinear ES and SE.
> 
> nlfun <- function(x, p1, p2)
>  p1 * exp(-p2*x)
> 
> # optimization function
> llfun <- function(par, yi, vi, x, random=TRUE) {
>  p1 <- par[1]
>  p2 <- par[2]
>  if (random) {
>    tau2 <- exp(par[3])
>  } else {
>    tau2 <- 0
>  }
>  mu <- nlfun(x, p1, p2)
>  -sum(dnorm(yi, mean=mu, sd=sqrt(vi + tau2), log=TRUE))
> }
> 
> # optimize
> res <- optim(par=c(8,0.4,log(.01)), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, hessian=TRUE)
> # back-transform log(tau2) to tau2
> res$par[3] <- exp(res$par[3])
> tau2_ME <- res$par[3]
> 
> # fit model with tau2=0
> res0 <- optim(par=c(8,0.4), llfun, yi=am.df$es, vi=am.df$var, x=am.df$CNr, random=FALSE, hessian=TRUE)
> 
> # LRT of H0: tau2=0
> x2 <- -2 * (res0$value - res$value)
> x2 ### test statistic
> pchisq(x2, df=1, lower.tail=FALSE) ### p-value
> 
> # compute standard errors
> vb <- chol2inv(chol(res$hessian))
> se <- sqrt(diag(vb))
> 
> # ignore SE for tau2
> se[3] <- NA
> 
> # estimates, standard errors, z-values, and p-values
> data.frame(estimate=round(res$par,4), se=round(se,4), zval=round(res$par/se,3), pval=round(2*pnorm(abs(res$par/se), lower.tail=FALSE),5), row.names=c("p1","p2","tau2"))
> 
> # Function to predict the nonlinear effect size (ES)
> ESpred <- function(x) {
>  p1 <- res$par[1]
>  p2 <- res$par[2]
>  p1*exp(-p2*x)
> }
> 
> # Function to predict the nonlinear standard error (SE)
> SEpred <- function(x) {
>  p1 <- res$par[1]
>  p2 <- res$par[2]
>  g <- matrix(c(exp(-p2*x), -p1*exp(-p2*x)*x), ncol=1, nrow=2)
>  vb <- vb[1:2,1:2]
>  c(sqrt(t(g) %*% vb %*% g))
> }
> 
> I wouldn’t be able to figure out how to model the nonlinear relationship ES ~ C in my entire life, so I really appreciate the help.
> 
> My question is: how can I incorporate a factorial moderator F in these functions to model ES ~ C + F.?
> 
> Many thanks in advance
> César 

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