# Predicted probabilities and marginal effects relationship (R, margins package)

Working on a logit model, i get the following results:

Predicted Probs.:

#(GlmModel, x2[in this case, years])
> logit2prob(m3, 0.5)
0.3178728
> logit2prob(m3, 1)
0.2200801
> logit2prob(m3, 2)
0.0937683
> logit2prob(m3, 3)
0.03655361
> logit2prob(m3, 4)
0.01372108
> logit2prob(m3, 5)
0.005075335
> logit2prob(m3, 6)
0.001867019


Marginal Effects:

> head(marginal_effects(m3))
dydx_x2
1 -0.20786769
2 -0.21936998
3 -0.20786769
4 -0.20330973
5 -0.11609512
6 -0.04823988


And graphs for both using cplot(m3, "x2", what = "predict") and cplot(m3, "x2", what = "effect"):

The numbers i get from marginal_effects doesn't seems to match "effect" clplot. And both instantaneous marginal effects (table and graph) doesn't seems to match predicted values rate of change. What am i missing here?

• Is it possible that the marginal effects in your second plot are expressed on the log odds scale? Recall that log odds are computed as log(p/(1-p)), where p is the probability of interest. – Isabella Ghement May 18 '18 at 14:23
• So, supposing that at point 2 yaxis = -0.10, you mean log(P/(1-P))=-0.10? Then if i do exp(log(P/(1-P))=P/(1-P) then [P/(1-P)]*(1-P)=P that P would be the marginal fx? – Alberson Miranda May 18 '18 at 15:27
• And what about the results from marginal_effects(m3)? What's the correct interpretation? They doesn't seems to be percentual points to actual predicted probs. Can it be dx/dy to odds or log odds maybe? – Alberson Miranda May 18 '18 at 15:31
• Let me look into this margins package first, Alberson, as I am not familiar with it. I'll come back here once I understand how it works. – Isabella Ghement May 18 '18 at 15:31

First, I must confess that I don't understand your use of the logit2prob function. I suspect this function comes from the rcfss package. If so, the example below shows how it can be used to compute predicted probabilities from a binary logistic regression model. The data for this example come from https://stats.idre.ucla.edu/r/dae/logit-regression/.

# install rcfss package
library(devtools)
install_github("uc-cfss/rcfss")

require(rcfss)

# read data for use in binary logistic regression model example

# convert rank variable to factor (i.e., categorical variable)

mydata$rank <- factor(mydata$rank)

# fit binary logistic regression model to data which will predict
# probability of admission to college for a student based
# on their gre, gpa and current school rank/prestige
# (ranks range from 1 = highest prestige to 4 = lowest prestige)

mymodel <- glm(admit ~ gre + gpa + rank,
data = mydata,

# display model summary

summary(mymodel)

# compute the log odds of admission for each student represented in the data

predict(mymodel)

# convert the log odds computed above to predicted probabilities

logit2prob(predict(mymodel))

# calculate the predicted probabilities of admission directly using
# the predict() function

predict(mymodel, type="response")

# compare the two methods for computing predicted probabilities
# to ensure they yield the same results

cbind(logit2prob(predict(mymodel)), predict(mymodel, type="response"))


As for the conditional plot produced by cplot, I can't figure out how exactly it is computed via the command:

## conditional effect plot
cplot(mymodel, "gre", what="predict", draw=FALSE)


If I use a different way of computing conditional probabilities (as per the effects package), I can replicate the reasoning involved:

require(effects)

allEffects(mymodel)

model: admit ~ gre + gpa + rank

gre effect
gre
200       400       500       700       800
0.1503641 0.2177438 0.2587605 0.3544499 0.4077938

gpa effect
gpa
2.3       2.7       3.1       3.6         4
0.1505682 0.1964650 0.2521985 0.3351680 0.4101640

rank effect
rank
1         2         3         4
0.5166016 0.3522846 0.2186120 0.1846684


Indeed:

# extract model coefficients
b0 <- coef(mymodel)[1]
b1 <- coef(mymodel)[2]
b2 <- coef(mymodel)[3]
b3 <- coef(mymodel)[4]
b4 <- coef(mymodel)[5]
b5 <- coef(mymodel)[6]

# compute proportion of data values represented in each
# category of rank
table(mydata$rank)/sum(table(mydata$rank))

log.odds <-  b0 + b1*(200) + b2*mean(mydata$gpa)+ b3 *(0.3775) + b4*(0.3025) + b5*(0.1675) exp(log.odds)/(1+exp(log.odds))  In the above log odds formula, I plugged in 200 for gre (which is the same value used by the effects package), I replaced gpa with its observed mean value and replaced the dummy variables used to encode the effect of rank with their mean value (i.e., with the proportion of values falling into the categories they represent). Exponentiating the log odds enabled me to obtain the first predicted probability obtained by the effects package (i.e., 0.1503641) when gre is set to 200, gpa is set to its observed mean value and the dummy variables rank2, rank3 and rank4 are set to their observed mean values. It seems that the effects package computes what the documentation of the margins package refers to as "Fitted values at the mean of X" (i.e., predicted probabilities at the mean values of the non-focal predictor variables, evaluated over a range of values for the focal variable gre). I would have thought that cplot uses a similar argument to obtain conditional probabilities, but it doesn't. What I think it does is compute what the documentation of the margins package refers to as "Average" fitted values (i.e., average predicted probabilities). By documentation, I mean the vignette available at https://cran.r-project.org/web/packages/margins/vignettes/TechnicalDetails.pdf. In other words, for each gre value in a grid, gpa and rank are set in turns to all the combinations of values observed in the data. The predicted probability of admission is computed for each given gra value across all of these combinations of gpa and rank values and then averaged across combinations. In the above-mentioned vignette, the author of the margins package clarifies that, for binary logistic regression models, the margins function computes marginal effects as changes in the predicted probability of the outcome corresponding to changes in the values of a focal predictor. If the focal predictor is categorical (e.g., rank), changes are expressed moving from the base category to a particular category (e.g., from rank = 1 to rank = 2). If the focal predictor is numeric (e.g., gpa), changes correspond (by default) to an infinitesimal increment in the value of that predictor. However, a discrete change in the values of a numeric predictor can be requested via the change argument to the workhorse dydx() function, which allows for expression of changes quantified by the following: observed minimum to observed maximum values of the focal predictor, first quartile to the third quartile of the focal predictor, mean +/- a standard deviation of the focal predictor, or any arbitrary change. See http://www.brodrigues.co/blog/2017-10-26-margins_r/ for an example involving the use of the dydx() function. By default, margins calculates the average marginal effects of every variable included in the glm model. • That was awesome, thank you! I did forget to explain the function. It's just a shortcut to transform back logit to predicted probs when working a 1 variable model: logit2prob <- function(model, x){ logit<-model$coefficients[1]+model\$coefficients[2]*x odds <- exp(logit) prob <- odds / (1 + odds) return(prob) } {effects} did matched the results i get using the function above (while {margins} did not), which is what i'm looking for, so it's a go. – Alberson Miranda May 22 '18 at 12:55
• On the other hand, with {effects} i'm getting issues plotting graphics. The y axis scale is altered to force the line to be straight, even if i set rescale.axis=FALSE. Are you experiencing it too? – Alberson Miranda May 22 '18 at 13:57