How to get average marginal effects (AMEs) with standard errors of a multinomial logit model?

问题

I want to get the average marginal effects (AME) of a multinomial logit model with standard errors. For this I've tried different methods, but they haven't led to the goal so far.

Best attempt

My best attempt was to get the AMEs by hand using mlogit which I show below.

library(mlogit)
ml.d <- mlogit.data(df1, choice="Y", shape="wide")  # shape data for `mlogit()`
ml.fit <- mlogit(Y ~ 1 | D + x1 + x2, reflevel="1", data=ml.d)  # fit the model

# coefficient names
c.names <- names(ml.fit$model)[- c(1, 5:6)]

# get marginal effects
ME.mnl <- sapply(c.names, function(x) 
  stats::effects(ml.fit, covariate=x, data=ml.d), 
  simplify=FALSE) 

# get AMEs
(AME.mnl <- t(sapply(ME.mnl, colMeans)))
#              1            2            3           4          5
# D  -0.03027080 -0.008806072 0.0015410569 0.017186531 0.02034928
# x1 -0.02913234 -0.015749598 0.0130577842 0.013240212 0.01858394
# x2 -0.02724650 -0.005482753 0.0008575982 0.005331181 0.02654047

I know these values are the correct ones. However, I could not get the correct standard errors by simply doing the columns' standard deviations:

# standard errors - WRONG!
(AME.mnl.se <- t(sapply(E.mnl, colSdColMeans)))

(Note: colSdColMeans() for columns' SD is provided here.)

Accordingly this also led me to the wrong t-values:

# t values - WRONG!
AME.mnl / AME.mnl.se
#             1          2          3         4         5
# D  -0.7110537 -0.1615635 0.04013228 0.4190057 0.8951484
# x1 -0.7170813 -0.2765212 0.33325968 0.3656893 0.8907836
# x2 -0.7084573 -0.1155825 0.02600653 0.1281190 0.8559794

Whereas I know the correct t-values for this case are these:

# D  -9.26 -1.84  0.31 4.29 8.05   
# x1 -6.66 -2.48  1.60 1.50 3.22  
# x2 -2.95 -0.39  0.06 0.42 3.21

I learned that there should be a "delta method", but I only found some code for a very special case with interactions at Cross Validated.

Failed attempts

1.) Package margins doesn't seem to be able to handle "mlogit" objects:

library(margins)
summary(margins(ml.fit))

2.) There's another package for mlogits, nnet,

library(nnet) 
ml.fit2 <- multinom(Y ~ D + x1 + x2, data=df1)
summary(ml.fit2)

but margins can't handle this correctly either:

> summary(margins(ml.fit2))
 factor     AME SE  z  p lower upper
      D -0.0303 NA NA NA    NA    NA
     x1 -0.0291 NA NA NA    NA    NA
     x2 -0.0272 NA NA NA    NA    NA

3.) There's also a package around that claims to calculate "Average Effects for Multinomial Logistic Regression Models",

library(DAMisc)
mnlChange2(ml.fit2, varnames="D", data=df1)

but I couldn't get a drop of milk out of it, since the function yields just nothing (even not with the function's example).

How now can we get AMEs with standard errors / t-statistics of a multinomial logit model with R?

Data

df1 <- structure(list(Y = c(3, 4, 1, 2, 3, 4, 1, 5, 2, 3, 4, 2, 1, 4, 
1, 5, 3, 3, 3, 5, 5, 4, 3, 5, 4, 2, 5, 4, 3, 2, 5, 3, 2, 5, 5, 
4, 5, 1, 2, 4, 3, 1, 2, 3, 1, 1, 3, 2, 4, 2, 2, 4, 1, 5, 3, 1, 
5, 2, 3, 4, 2, 4, 5, 2, 4, 1, 4, 2, 1, 5, 3, 2, 1, 4, 4, 1, 5, 
1, 1, 1, 4, 5, 5, 3, 2, 3, 3, 2, 4, 4, 5, 3, 5, 1, 2, 5, 5, 1, 
2, 3), D = c(12, 8, 6, 11, 5, 14, 0, 22, 15, 13, 18, 3, 5, 9, 
10, 28, 9, 16, 17, 14, 26, 18, 18, 23, 23, 12, 28, 14, 10, 15, 
26, 9, 2, 30, 18, 24, 27, 7, 6, 25, 13, 8, 4, 16, 1, 4, 5, 18, 
21, 1, 2, 19, 4, 2, 16, 17, 23, 15, 13, 21, 24, 14, 27, 6, 20, 
6, 19, 8, 7, 23, 11, 11, 1, 22, 21, 4, 27, 6, 2, 9, 18, 30, 26, 
22, 10, 1, 4, 7, 26, 15, 26, 18, 30, 1, 11, 29, 25, 3, 19, 15
), x1 = c(13, 12, 4, 3, 16, 16, 15, 13, 1, 15, 10, 16, 1, 17, 
7, 13, 12, 6, 8, 16, 16, 11, 7, 16, 5, 13, 12, 16, 17, 6, 16, 
9, 14, 16, 15, 5, 7, 2, 8, 2, 9, 9, 15, 13, 9, 4, 16, 2, 11, 
13, 11, 6, 4, 3, 7, 4, 12, 2, 16, 14, 3, 13, 10, 11, 10, 4, 11, 
16, 8, 12, 14, 9, 4, 16, 16, 12, 9, 10, 6, 1, 3, 8, 7, 7, 5, 
16, 17, 10, 4, 15, 10, 8, 3, 13, 9, 16, 12, 7, 4, 11), x2 = c(12, 
19, 18, 19, 15, 12, 15, 16, 15, 11, 12, 16, 17, 14, 12, 17, 17, 
16, 12, 20, 11, 11, 15, 14, 18, 10, 14, 13, 10, 14, 18, 18, 18, 
17, 18, 14, 16, 19, 18, 16, 18, 14, 17, 10, 16, 12, 16, 15, 11, 
18, 19, 15, 19, 11, 16, 10, 20, 14, 10, 12, 10, 15, 13, 15, 11, 
20, 11, 12, 16, 16, 11, 15, 11, 11, 10, 10, 16, 11, 20, 17, 20, 
17, 16, 11, 18, 19, 18, 14, 17, 11, 16, 11, 18, 14, 15, 16, 11, 
14, 11, 13)), class = "data.frame", row.names = c(NA, -100L))

回答1:

We can do something very similar to what is done in your linked answer. In particular, first we want a function that would compute AMEs at a given vector of coefficients. For that we can define

AME.fun <- function(betas) {
  tmp <- ml.fit
  tmp$coefficients <- betas
  ME.mnl <- sapply(c.names, function(x) 
    effects(tmp, covariate = x, data = ml.d), simplify = FALSE)
  c(sapply(ME.mnl, colMeans))
}

where the second half is yours, while in the first one I use a trick to take the same ml.fit object and to change its coefficients. Next we find the jacobian with

require(numDeriv)
grad <- jacobian(AME.fun, ml.fit$coef)

and apply the delta method. Square roots of the diagonal of grad %*% vcov(ml.fit) %*% t(grad) is what we want. Hence,

(AME.mnl.se <- matrix(sqrt(diag(grad %*% vcov(ml.fit) %*% t(grad))), nrow = 3, byrow = TRUE))
#             [,1]        [,2]        [,3]        [,4]        [,5]
# [1,] 0.003269320 0.004788536 0.004995723 0.004009762 0.002527462
# [2,] 0.004375795 0.006348496 0.008168883 0.008844684 0.005763966
# [3,] 0.009233616 0.014048212 0.014713090 0.012702188 0.008261734
AME.mnl / AME.mnl.se
#            1          2          3         4        5
# D  -9.259050 -1.8389907 0.30847523 4.2861720 8.051269
# x1 -6.657611 -2.4808393 1.59847852 1.4969683 3.224159
# x2 -2.950794 -0.3902812 0.05828811 0.4197057 3.212458

which coincides with Stata's results.

来源：https://stackoverflow.com/questions/54079553/how-to-get-average-marginal-effects-ames-with-standard-errors-of-a-multinomial

标签

multinomial

mlogit

marginal-effects