Multinomial Logit as an Iterated Logit Regression

For the second section of the course at ENSAE, yesterday, we’ve seen how to run a multinomial logistic regression model. It is simply an extension of the binomial logistic regression. But actually, it is also possible to consider iterative binomial regressions.

Consider here a response variable Y with a multinomial distribution (3 factors to have something more general than the binomial), taking values \{A,B,C\}, with respective probabilities \mathbf{p}=(p_A,p_B,p_C). Here is a code to generate some multinomial variables

msample=function(A,B,C){
Y=rep(NA,B)
for(i in 1:B){Y[i]=sample(A,size=1,prob=C[i,])}
return(Y)
}

and here is a code to generate a dataset with n rows,

generate3=function(n,x,pb=c(-2,0)){
set.seed(x)
X1=runif(n)
X2=runif(n)
X3=runif(n)
s1=pb[1]+X1+X2
s2=pb[2]-X1+X2
P1=exp(s1)/(1+exp(s1)+exp(s2))
P2=exp(s2)/(1+exp(s1)+exp(s2))
Y=msample(0:2,n,cbind(1-P1-P2,P1,P2))
df=data.frame(Y=Y,X1=X1,X2=X2,X3=X3)
return(df)
}

Let us generate a training dataset and a validation one

pb=c(.31,.42)
DF1=generate3(1000,1,pb=pb)
DF2=generate3(500,2,pb=pb)

With a multivariate logistic regression
\mathbb{P}[Y=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}
\mathbb{P}[Y=B|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}
\mathbb{P}[Y=B|\mathbf{x}]=\frac{1}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}

For convenience, consider the most popular factor in our training dataset

modalite=names(sort(table(DF1$Y),decreasing = TRUE))

Consider a regression model on the simulated dataset (with several covariates), let us estimate it, and let us get predictions.

library(nnet)
reg=multinom(as.factor(Y) ~ ., data = DF1)
mp1=predict (reg, DF1, "probs")
mp2=predict (reg, DF2, "probs")

An alternative can be the following.
consider a first regression model on the Bernoulli variable Y_A=\mathbf{1}(Y=A). Actually, we will consider the most important factor, but for convenience, assume that it is A.
\mathbb{P}[Y_A=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}
On our dataset, estimate that model, and get predictions. In the case where Y\neq A, define another Bernoulli variable Y_B=\mathbf{1}(Y=B|Y\neq A). We can estimate that model and derive two probabilities, \mathbb{P}(Y=B|Y\neq A) and \mathbb{P}(Y=C|Y\neq A) (the sum of the two being equal to 1). Based on those two models, it is possible to compute the three probabilities we are looking for. \mathbb{P}[Y=A] is obtained from the first model, and we can derive the other two from \mathbb{P}[Y=B|Y\neq A]\cdot\mathbb{P}[Y\neq A] and \mathbb{P}[Y=C|Y\neq A]\cdot\mathbb{P}[Y\neq A].

reg1=glm((Y==modalite[1])~.,data=DF1,family=binomial)
reg2=glm((Y==modalite[2])~.,data=DF1[-which(DF1$Y==modalite[1]),],family=binomial)
p11=predict (reg1, newdata=DF1, type="response")
p12=predict (reg2, newdata=DF1, type="response")
p21=predict (reg1, newdata=DF2, type="response")
p22=predict (reg2, newdata=DF2, type="response")
mmp1=cbind(p11,(1-p11)*p12,(1-p11)*(1-p12))
mmp2=cbind(p21,(1-p21)*p22,(1-p21)*(1-p22))
colnames(mmp1)=colnames(mmp2)=modalite

Let us compare the predicted probabilites, on the same dataset (here the training dataset)

> mmp1[1:9,c("0","1","2")]
0 1 2
1 0.19728737 0.4991805 0.3035321
2 0.17244580 0.5648537 0.2627005
3 0.19291753 0.5971058 0.2099767
4 0.09087176 0.7787304 0.1303978
5 0.23400225 0.4083022 0.3576955
6 0.18063647 0.6637352 0.1556283
7 0.13188881 0.7402710 0.1278401
8 0.13776970 0.6524959 0.2097344
9 0.12325864 0.6790336 0.1977078
> mp1[1:9,c("0","1","2")]
0 1 2
1 0.19691036 0.5022692 0.3008205
2 0.17123189 0.5680647 0.2607034
3 0.19293066 0.5984402 0.2086291
4 0.08821851 0.7813318 0.1304497
5 0.23470739 0.4109990 0.3542936
6 0.18249687 0.6602168 0.1572863
7 0.13128711 0.7400898 0.1286231
8 0.13525341 0.6553618 0.2093848
9 0.12090016 0.6815915 0.1975084

The two are very close. So yes, it is possible to see the multinomial regression as some sequential binomial regressions.


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