r - How to get marginal effects for categorical variables in mlogit?

Question

I want to compute marginal effects for a "mlogit" object where explanatory variables is categorical (factors). While with numerical data effects() throws something, with categorical data it won't.

For simplicity I show a bivariate example below.

numeric variables

# with mlogit
library(mlogit)
ml.dat <- mlogit.data(df3, choice="y", shape="wide")
fit.mnl <- mlogit(y ~ 1 | x, data=ml.dat)

head(effects(fit.mnl, covariate="x", data=ml.dat))
#         FALSE       TRUE
# 1 -0.01534581 0.01534581
# 2 -0.01534581 0.01534581
# 3 -0.20629452 0.20629452
# 4 -0.06903946 0.06903946
# 5 -0.24174312 0.24174312
# 6 -0.39306240 0.39306240

# with glm
fit.glm <- glm(y ~ x, df3, family = binomial)

head(effects(fit.glm))
# (Intercept)           x                                                 
#  -0.2992979  -4.8449254   2.3394989   0.2020127   0.4616640   1.0499595

factor variables

# transform to factor
df3F <- within(df3, x <- factor(x))
class(df3F$x) == "factor"
# [1] TRUE

While glm() still throws something,

# with glm
fit.glmF <- glm(y ~ x, df3F, family = binomial)

head(effects(fit.glmF))
# (Intercept)           x2           x3           x4           x5           x6 
# 0.115076511 -0.002568206 -0.002568206 -0.003145397 -0.003631992 -0.006290794

the mlogit() approach

# with mlogit
ml.datF <- mlogit.data(df3F, choice="y", shape="wide")
fit.mnlF <- mlogit(y ~ 1 | x, data=ml.datF)

head(effects(fit.mnlF, covariate="x", data=ml.datF))

throws this error:

Error in `contrasts<-`(`*tmp*`, value = contr.funs[1 + isOF[nn]]) : 
  contrasts can be applied only to factors with 2 or more levels
In addition: Warning message:
In Ops.factor(data[, covariate], eps) :

 Error in `contrasts<-`(`*tmp*`, value = contr.funs[1 + isOF[nn]]) : 
  contrasts can be applied only to factors with 2 or more levels

How could I solve this?

I already tried to manipulate effects.mlogit() with this answer but it didn't help to solve my problem.

Note: This question is related to this solution, which I want to apply to categorical explanatory variables.

edit

(To demonstrate the issue when applying the given solution to an underlying problem related to a question linked above. See comments.)

# new example ----
library(mlogit)
ml.d <- mlogit.data(df1, choice="y", shape="wide")
ml.fit <- mlogit(y ~ 1 | factor(x), reflevel="1", data=ml.d)

AME.fun2 <- function(betas) {
  aux <- model.matrix(y ~ x, df1)[, -1]
  ml.datF <- mlogit.data(data.frame(y=df1$y, aux), 
                         choice="y", shape="wide")
  frml <- mFormula(formula(paste("y ~ 1 |", paste(colnames(aux), 
                                                  collapse=" + "))))
  fit.mnlF <- mlogit(frml, data=ml.datF)
  fit.mnlF$coefficients <- betas  # probably?
  colMeans(effects(fit.mnlF, covariate="x2", data=ml.datF))  # first co-factor?
}

(AME.mnl <- AME.fun2(ml.fit$coefficients))

require(numDeriv)
grad <- jacobian(AME.fun2, ml.fit$coef)
(AME.mnl.se <- matrix(sqrt(diag(grad %*% vcov(ml.fit) %*% t(grad))), 
                      nrow=3, byrow=TRUE))
AME.mnl / AME.mnl.se
#  doesn't work yet though...

# probably "true" values, obtained from Stata:
# # ame
#         1      2      3      4      5
# 1.     NA     NA     NA     NA     NA   
# 2. -0.400  0.121 0.0971  0.113 0.0686   
# 3. -0.500 -0.179 0.0390  0.166 0.474 
#
# # z-values
#        1     2     3     4     5
# 1.    NA    NA    NA    NA    NA
# 2. -3.86  1.25  1.08  1.36  0.99
# 3. -5.29 -2.47  0.37  1.49  4.06

data

df3 <- structure(list(x = c(11, 11, 7, 10, 9, 8, 9, 6, 9, 9, 8, 9, 11, 
7, 8, 11, 12, 5, 8, 8, 11, 6, 13, 12, 5, 8, 7, 11, 8, 10, 9, 
10, 7, 9, 2, 10, 3, 6, 11, 9, 7, 8, 4, 12, 8, 12, 11, 9, 12, 
9, 7, 7, 7, 10, 4, 10, 9, 6, 7, 8, 9, 13, 10, 8, 10, 6, 7, 10, 
9, 6, 4, 6, 6, 8, 6, 9, 3, 7, 8, 2, 8, 6, 7, 9, 10, 8, 6, 5, 
5, 7, 9, 1, 6, 11, 11, 9, 7, 8, 9, 9), y = c(TRUE, TRUE, TRUE, 
TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, TRUE, FALSE, TRUE, TRUE, 
TRUE, FALSE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, FALSE, TRUE, 
TRUE, FALSE, TRUE, FALSE, TRUE, FALSE, TRUE, TRUE, TRUE, FALSE, 
TRUE, FALSE, TRUE, FALSE, FALSE, TRUE, TRUE, TRUE, FALSE, FALSE, 
TRUE, FALSE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, FALSE, FALSE, 
TRUE, FALSE, TRUE, TRUE, FALSE, FALSE, TRUE, TRUE, TRUE, TRUE, 
FALSE, TRUE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, 
FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, TRUE, 
FALSE, FALSE, TRUE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, 
TRUE, FALSE, FALSE, TRUE, TRUE, TRUE, FALSE, FALSE, TRUE, FALSE
)), class = "data.frame", row.names = c(NA, -100L))

> summary(df3)
       x             y          
 Min.   : 1.00   Mode :logical  
 1st Qu.: 7.00   FALSE:48       
 Median : 8.00   TRUE :52       
 Mean   : 8.08                  
 3rd Qu.:10.00                  
 Max.   :13.00  

df1 <- structure(list(y = c(5, 4, 2, 2, 2, 3, 5, 4, 1, 1, 2, 4, 1, 4, 
5, 5, 2, 3, 3, 5, 5, 3, 2, 4, 5, 1, 3, 3, 4, 3, 5, 2, 4, 4, 5, 
5, 5, 2, 1, 5, 1, 3, 1, 4, 1, 2, 2, 4, 3, 1, 4, 3, 1, 1, 5, 2, 
5, 4, 2, 2, 4, 2, 3, 5, 4, 1, 2, 2, 3, 5, 2, 5, 3, 3, 3, 1, 3, 
1, 1, 4, 3, 4, 5, 2, 1, 1, 3, 1, 5, 4, 4, 2, 5, 3, 4, 4, 3, 1, 
5, 2), x = structure(c(2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 2L, 2L, 
2L, 1L, 1L, 2L, 2L, 3L, 2L, 2L, 2L, 2L, 3L, 2L, 2L, 3L, 3L, 2L, 
3L, 2L, 2L, 2L, 3L, 2L, 1L, 3L, 2L, 3L, 3L, 1L, 1L, 3L, 2L, 2L, 
1L, 2L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 3L, 2L, 
2L, 2L, 3L, 2L, 3L, 1L, 2L, 1L, 2L, 2L, 1L, 3L, 2L, 2L, 1L, 2L, 
2L, 1L, 3L, 1L, 1L, 2L, 2L, 3L, 3L, 2L, 2L, 1L, 1L, 1L, 3L, 2L, 
3L, 2L, 3L, 1L, 2L, 3L, 3L, 1L, 2L, 2L), .Label = c("1", "2", 
"3"), class = "factor")), row.names = c(NA, -100L), class = "data.frame")

score 1 · Accepted Answer

预计它effects不适用于因子，因为否则输出将包含另一个维度，这会使结果有些复杂，并且就像在我下面的解决方案中一样，人们可能只希望对某个因子产生影响，这是非常合理的级别，而不是所有级别。此外，正如我在下面解释的那样，分类变量的边际效应不是唯一定义的，因此这将是effects.

一个自然的解决方法是手动将因子变量转换为一系列虚拟变量，如

aux <- model.matrix(y ~ x, df3F)[, -1]
head(aux)
#   x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
# 1  0  0  0  0  0  0  0  0   0   1   0   0
# 2  0  0  0  0  0  0  0  0   0   1   0   0
# 3  0  0  0  0  0  1  0  0   0   0   0   0
# 4  0  0  0  0  0  0  0  0   1   0   0   0
# 5  0  0  0  0  0  0  0  1   0   0   0   0
# 6  0  0  0  0  0  0  1  0   0   0   0   0

这样数据就是

ml.datF <- mlogit.data(data.frame(y = df3F$y, aux), choice = "y", shape = "wide")

我们还需要手动构造公式

frml <- mFormula(formula(paste("y ~ 1 |", paste(colnames(aux), collapse = " + "))))

到目前为止，一切都很好。现在如果我们运行

fit.mnlF <- mlogit(frml, data = ml.datF)
head(effects(fit.mnlF, covariate = "x2", data = ml.datF))
#           FALSE         TRUE
# 1 -1.618544e-15 0.000000e+00
# 2 -1.618544e-15 0.000000e+00
# 3 -7.220891e-08 7.221446e-08
# 4 -1.618544e-15 0.000000e+00
# 5 -5.881129e-08 5.880851e-08
# 6 -8.293366e-08 8.293366e-08

那么结果是不正确的。这里effects所做的是将其x2视为一个连续变量并计算这些情况下通常的边际效应。也就是说，如果对应的系数x2是 b2 并且我们的模型是 f(x,b2)，effects则计算 f 对 b2 的导数，并在每个观察到的向量 x _i处进行评估。这是错误的，因为x2只取值 0 和 1，而不是 0 或 1 左右的值，这是取导数所假设的（极限的概念）！例如，考虑您的其他数据集df1。在这种情况下，我们错误地得到

colMeans(effects(fit.mnlF, covariate = "x2", data = ml.datF))
#           1           2           3           4           5 
# -0.25258378  0.07364406  0.05336283  0.07893391  0.04664298

这是另一种方法（使用导数近似）来得到这个不正确的结果：

temp <- ml.datF
temp$x2 <- temp$x2 + 0.0001
colMeans(predict(fit.mnlF, newdata = temp, type = "probabilities") - 
             predict(fit.mnlF, newdata = ml.datF, type = "probabilities")) / 0.0001
#           1           2           3           4           5 
# -0.25257597  0.07364089  0.05336032  0.07893273  0.04664202

我没有使用effects，而是通过使用两次手动计算了错误的边际效应predict：结果是平均值（{x2new 的拟合概率 = x2old + 0.0001} - {x2new = x2old 的拟合概率}）/0.0001。也就是说，我们通过x2向上移动 0.0001 来查看预测概率的变化，即从 0 到 0.0001 或从 1 到 0.0001。这两个都没有意义。当然，我们不应该期望其他任何东西，effects因为x2数据是数字的。

那么问题是如何计算正确的（平均）边际效应。正如我所说，分类变量的边际效应不是唯一定义的。假设 x_i 是个人 i 是否有工作，y_i 是他们是否有车。所以，那么至少有以下六点需要考虑。

从 x_i=0 到 x_i=1 时对 y_i = 1 概率的影响。
从 x_i=0 到 x_i（观察值）时。
从 x_i 到 1。

现在，当我们对平均边际效应感兴趣时，我们可能只想对那些 1-3 的变化产生影响的个体进行平均。那是，

如果观测值不是 1，则从 x_i=0 到 x_i=1。
如果观测值不为 0，则从 x_i=0 到 x_i。
如果观测值不是 1，则从 x_i 到 1。

根据您的结果，Stata 使用选项 5，因此我将重现相同的结果，但实现任何其他选项都很简单，我建议您考虑哪些选项在您的特定应用程序中很有趣。

AME.fun2 <- function(betas) {
  aux <- model.matrix(y ~ x, df1)[, -1]
  ml.datF <- mlogit.data(data.frame(y = df1$y, aux), choice="y", shape="wide")
  frml <- mFormula(formula(paste("y ~ 1 |", paste(colnames(aux), collapse=" + "))))
  fit.mnlF <- mlogit(frml, data = ml.datF)
  fit.mnlF$coefficients <- betas
  aux <- ml.datF # Auxiliary dataset
  aux$x3 <- 0 # Going from 0 to the observed x_i
  idx <- unique(aux[aux$x3 != ml.datF$x3, "chid"]) # Where does it make a change?
  actual <- predict(fit.mnlF, newdata = ml.datF)
  counterfactual <- predict(fit.mnlF, newdata = aux)
  colMeans(actual[idx, ] - counterfactual[idx, ])
}
(AME.mnl <- AME.fun2(ml.fit$coefficients))
#           1           2           3           4           5 
# -0.50000000 -0.17857142  0.03896104  0.16558441  0.47402597 

require(numDeriv)
grad <- jacobian(AME.fun2, ml.fit$coef)
AME.mnl.se <- matrix(sqrt(diag(grad %*% vcov(ml.fit) %*% t(grad))), nrow = 1, byrow = TRUE)
AME.mnl / AME.mnl.se
#           [,1]      [,2]    [,3]     [,4]     [,5]
# [1,] -5.291503 -2.467176 0.36922 1.485058 4.058994

r - How to get marginal effects for categorical variables in mlogit?

numeric variables

factor variables

edit

data

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