R probit regression marginal effects

Question

I am using R to replicate a study and obtain mostly the same results the author reported. At one point, however, I calculate marginal effects that seem to be unrealistically small. I would greatly appreciate if you could have a look at my reasoning and the code below and see if I am mistaken at one point or another.

My sample contains 24535 observations, the dependent variable "x028bin" is a binary variable taking on the values 0 and 1, and there are furthermore 10 explaining variables. Nine of those independent variables have numeric levels, the independent variable "f025grouped" is a factor consisting of different religious denominations.

I would like to run a probit regression including dummies for religious denomination and then compute marginal effects. In order to do so, I first eliminate missing values and use cross-tabs between the dependent and independent variables to verify that there are no small or 0 cells. Then I run the probit model which works fine and I also obtain reasonable results:

probit4AKIE <- glm(x028bin ~ x003 + x003squ + x025secv2 + x025terv2 + x007bin + x04chief + x011rec + a009bin + x045mod + c001bin + f025grouped, family=binomial(link="probit"), data=wvshm5red2delna, na.action=na.pass)

summary(probit4AKIE)

However, when calculating marginal effects with all variables at their means from the probit coefficients and a scale factor, the marginal effects I obtain are much too small (e.g. 2.6042e-78). The code looks like this:

ttt <- cbind(wvshm5red2delna$x003,
wvshm5red2delna$x003squ,
wvshm5red2delna$x025secv2,
wvshm5red2delna$x025terv2,
wvshm5red2delna$x007bin,
wvshm5red2delna$x04chief,
wvshm5red2delna$x011rec,
wvshm5red2delna$a009bin,
wvshm5red2delna$x045mod,
wvshm5red2delna$c001bin,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped,
wvshm5red2delna$f025grouped) #I put variable "f025grouped" 9 times because this variable consists of 9 levels

ttt <- as.data.frame(ttt)

xbar <- as.matrix(mean(cbind(1,ttt[1:19]))) #1:19 position of variables in dataframe ttt

betaprobit4AKIE <- probit4AKIE$coefficients

zxbar <- t(xbar) %*% betaprobit4AKIE

scalefactor <- dnorm(zxbar)

marginprobit4AKIE <- scalefactor * betaprobit4AKIE[2:20] #2:20 are the positions of variables in the output of the probit model 'probit4AKIE' (variables need to be in the same ordering as in data.frame ttt), the constant in the model occupies the first position

marginprobit4AKIE #in this step I obtain values that are much too small

I apologize that I can not provide you with a working example as my dataset is much too large. Any comment would be greatly appreciated. Thanks a lot.

Best,

Tobias

Answer 1

@Gavin is right and it's better to ask at the sister site.

In any case, here's my trick to interpret probit coefficients.

The probit regression coefficients are the same as the logit coefficients, up to a scale (1.6). So, if the fit of a probit model is Pr(y=1) = fi(.5 - .3*x), this is equivalent to the logistic model Pr(y=1) = invlogit(1.6(.5 - .3*x)).

And I use this to make a graphic, using the function invlogit of package arm. Another possibility is just to multiply all coefficients (including the intercept) by 1.6, and then applying the 'divide by 4 rule' (see the book by Gelman and Hill), i.e, divide the new coefficients by 4, and you will find out an upper bound of the predictive difference corresponding to a unit difference in x.

Here's an example.

x1 = rbinom(100,1,.5)
x2 = rbinom(100,1,.3)
x3 = rbinom(100,1,.9)
ystar = -.5  + x1 + x2 - x3 + rnorm(100)
y = ifelse(ystar>0,1,0)
probit = glm(y~x1 + x2 + x3, family=binomial(link='probit'))
xbar <- as.matrix(mean(cbind(1,ttt[1:3])))

# now the graphic, i.e., the marginal effect of x1, x2 and x3
library(arm)
curve(invlogit(1.6*(probit$coef[1] + probit$coef[2]*x + probit$coef[3]*xbar[3] + probit$coef[4]*xbar[4]))) #x1
curve(invlogit(1.6*(probit$coef[1] + probit$coef[2]*xbar[2] + probit$coef[3]*x + probit$coef[4]*xbar[4]))) #x2
curve(invlogit(1.6*(probit$coef[1] + probit$coef[2]*xbar[2] + probit$coef[3]*xbar[3] + probit$coef[4]*x))) #x3

Answer 2

This will do the trick for probit or logit:

mfxboot <- function(modform,dist,data,boot=1000,digits=3){
  x <- glm(modform, family=binomial(link=dist),data)
  # get marginal effects
  pdf <- ifelse(dist=="probit",
                mean(dnorm(predict(x, type = "link"))),
                mean(dlogis(predict(x, type = "link"))))
  marginal.effects <- pdf*coef(x)
  # start bootstrap
  bootvals <- matrix(rep(NA,boot*length(coef(x))), nrow=boot)
  set.seed(1111)
  for(i in 1:boot){
    samp1 <- data[sample(1:dim(data)[1],replace=T,dim(data)[1]),]
    x1 <- glm(modform, family=binomial(link=dist),samp1)
    pdf1 <- ifelse(dist=="probit",
                   mean(dnorm(predict(x, type = "link"))),
                   mean(dlogis(predict(x, type = "link"))))
    bootvals[i,] <- pdf1*coef(x1)
  }
  res <- cbind(marginal.effects,apply(bootvals,2,sd),marginal.effects/apply(bootvals,2,sd))
  if(names(x$coefficients[1])=="(Intercept)"){
    res1 <- res[2:nrow(res),]
    res2 <- matrix(as.numeric(sprintf(paste("%.",paste(digits,"f",sep=""),sep=""),res1)),nrow=dim(res1)[1])
    rownames(res2) <- rownames(res1)
  } else {
    res2 <- matrix(as.numeric(sprintf(paste("%.",paste(digits,"f",sep=""),sep="")),nrow=dim(res)[1]))
    rownames(res2) <- rownames(res)
  }
  colnames(res2) <- c("marginal.effect","standard.error","z.ratio")
  return(res2)
}

Source: http://www.r-bloggers.com/probitlogit-marginal-effects-in-r/