Consider a random pair of binary responses, i.e. with taking values 1 or 2. Assume that probability can be function of some covariates.

- The Gaussian vector latent structure

A standard model is based a latent Gaussian structure, i.e. there exists some random vector such that if is lower than a given threshold, and 1 otherwise.

As in standard probit models, assume that

where we can assume that is a Gaussian random vector. This assumption can be used to derive the likelihood of a sample .

> logV=function(parameter){
+ CORRELATION=parameter[1]
+ BETA=matrix(parameter[2:length(parameter)],ncol(Y),ncol(X))
+ z=cbind(X%*%(BETA[1,]),X%*%(BETA[2,]))
+ sigma=matrix(c(1,CORRELATION,CORRELATION,1),2,2)
+ a11=pmnorm(z[1,],rep(0,ncol(Y)),varcov=sigma)
+ for(i in 2:nrow(z)){a11=c(a11,pmnorm(z[i,],rep(0,ncol(Y)),varcov=sigma))}
+ a10=pnorm(z[1,1],sd=sqrt(sigma[1,1]))-pmnorm(z[1,],varcov=sigma)
+ for(i in
+ 2:nrow(z)){a10=c(a10,pnorm(z[i,1],sd=sqrt(sigma[1,1]))-pmnorm(z[i,],varcov=sigma))}
+ a01=pnorm(z[1,2],sd=sqrt(sigma[2,2]))-pmnorm(z[1,],varcov=sigma)
+ for(i in
+ 2:nrow(z)){a01=c(a01,pnorm(z[i,2],sd=sqrt(sigma[2,2]))-pmnorm(z[i,],varcov=sigma))}
+ a00=1-a10-a01-a11
+ -sum(((Y[,1]==1)&(Y[,2]==1))*log(a11) +
+ ^{1}*log(a01) +
+ ^{2}*log(a10) +
+ ^{3}*log(a00) )
+ }
> OPT=optim(fn=logV,par=c(0,1,1,1,1,1,1),method="BFGS")$par

(the code is a bit long since I had trouble working properly with matrices – or more precisely to vectorize my functions – so I used loops… I am sure it is possible to write a better code).

It is possible to generate samples (based on that specific model) to check that we can actually derive proper maximum likelihood estimators,

> library(mnormt)
> set.seed(1)
> n=1000
> r=0.5
> X1=runif(n)
> X2=rnorm(n)
> Y1S=1+5*X1
> Y2S=8-5*X1
> RES=rmnorm(n,mean=c(0,0),varcov=matrix(c(1,r,r,1),2,2))
> YS=cbind(Y1S,Y2S)+RES
> Y1=(YS[,1]>quantile(YS[,1],.5))*1
> Y2=(YS[,2]>quantile(YS[,2],.5))*1
> base=data.frame(i,Y1,Y2,X1,X2,YS)
> head(base)
i Y1 Y2 X1 X2 Y1S Y2S
1 1 0 0 0.2655087 0.07730312 3.177587 5.533884
2 2 0 0 0.3721239 -0.29686864 1.935307 5.089524
3 3 1 0 0.5728534 -1.18324224 4.757848 5.172584
4 4 1 0 0.9082078 0.01129269 4.600029 3.878225
5 5 0 1 0.2016819 0.99160104 2.547362 6.743714
6 6 1 0 0.8983897 1.59396745 5.309974 4.421523

(the two columns on the right are latent observations, that cannot be used since theoretically they are unobservable). Note that it is a *simple* regression, one of the component is here only to bring some noise. First of all, let us look at *marginal* probit regression

> reg1=glm(Y1~X1+X2,data=base,family=binomial)
> reg2=glm(Y2~X1+X2,data=base,family=binomial)
> summary(reg1)
Call:
glm(formula = Y1 ~ X1 + X2, family = binomial, data = base)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.90570 -0.50126 -0.00266 0.49162 2.78256
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.291725 0.267149 -16.065 <2e-16
X1 8.656836 0.510153 16.969 <2e-16 ***
X2 0.007375 0.090530 0.081 0.935
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1386.29 on 999 degrees of freedom
Residual deviance: 726.48 on 997 degrees of freedom
AIC: 732.48
Number of Fisher Scoring iterations: 5
> summary(reg2)
Call:
glm(formula = Y2 ~ X1 + X2, family = binomial, data = base)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.74682 -0.51814 -0.00001 0.57969 2.58565
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.91709 0.24399 16.054 <2e-16 ***
X1 -7.89703 0.46277 -17.065 <2e-16 ***
X2 0.18360 0.08758 2.096 0.036 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1386.29 on 999 degrees of freedom
Residual deviance: 777.61 on 997 degrees of freedom
AIC: 783.61
Number of Fisher Scoring iterations: 5

Here, the optimization yields,

> OPT=optim(fn=logV,par=c(0,1,1,1,1,1,1),method="BFGS")$par
> OPT[1]
[1] 0.5261382
> matrix(OPT[2:7],2,3)
[,1] [,2] [,3]
[1,] -2.451721 4.908633 0.01600769
[2,] 2.241962 -4.539946 0.10614807

Note that the coefficients we have obtained are almost identical to the ones obtained with R standard procedure,

> library(Zelig)
> REG= zelig(list(mu1=Y1~X1+X2,
+ mu2=Y2~X1+X2,
+ rho=~1),
+ model="bprobit",data=base)
> summary(REG)
Call:
zelig(formula = list(mu1 = Y1 ~ X1 + X2, mu2 = Y2 ~ X1 + X2,
rho = ~1), model = "bprobit", data = base)
Pearson Residuals:
Min 1Q Median 3Q Max
probit(mu1) -10.5442 -0.377243 0.0041803 0.36709 8.60398
probit(mu2) -7.8547 -0.376888 0.0083715 0.42923 5.88264
rhobit(rho) -13.8322 -0.091502 -0.0080544 0.37218 0.85101
Coefficients:
Value Std. Error t value
(Intercept):1 -2.451699 0.135369 -18.11116
(Intercept):2 2.241964 0.125072 17.92536
(Intercept):3 1.169461 0.189771 6.16249
X1:1 4.908617 0.252683 19.42602
X1:2 -4.539951 0.233632 -19.43203
X2:1 0.015992 0.050443 0.31703
X2:2 0.106154 0.049092 2.16235
Number of linear predictors: 3
Names of linear predictors: probit(mu1), probit(mu2), rhobit(rho)
&n
bsp;
Dispersion Parameter for binom2.rho family: 1
Residual Deviance: 1460.355 on 2993 degrees of freedom
Log-likelihood: -730.1774 on 2993 degrees of freedom
Number of Iterations: 3
> matrix(coefficients(REG)[c(1:2,4:7)],2,3)
[,1] [,2] [,3]
[1,] -2.451699 4.908617 0.01599183
[2,] 2.241964 -4.539951 0.10615443

The correlation here is also the same

That procedure works well an can be extended to ordinal responses (not only binary ones, or to three dimensional problems,

logV=function(beta){
BETA=matrix(beta[4:(3+ncol(Y)*ncol(X))],ncol(Y),ncol(X))
z=cbind(X%*%(BETA[1,]),X%*%(BETA[2,]),X%*%(BETA[3,]))
r12=beta[1]
r23=beta[2]
r31=beta[3]
s1=s2=s3=1
sigma=matrix(c(s1^2,r12*s1*s2,r31*s1*s3,
r12*s1*s2,s2^2,r23*s2*s3,
r31*s1*s3,r23*s2*s3,s3^2),3,3)
sigma1=matrix(c(s2^2,r23*s2*s3,
r23*s2*s3,s3^2),2,2)
sigma2=matrix(c(s1^2,r31*s1*s3,
r31*s1*s3,s3^2),2,2)
sigma3=matrix(c(s1^2,r12*s1*s2,
r12*s1*s2,s2^2),2,2)
a111=pmnorm(z[1,],rep(0,ncol(Y)),varcov=sigma)
for(i in 2:nrow(z)){a111=c(a111,pmnorm(z[i,],rep(0,ncol(Y)),varcov=sigma))}
a011=pmnorm(z[1,2:3],varcov=sigma1)-pmnorm(z[1,],varcov=sigma)
for(i in 2:nrow(z)){a011=c(a011,pmnorm(z[i,2:3],varcov=sigma1)-pmnorm(z[i,],varcov=sigma))}
a101=pmnorm(z[1,c(1,3)],varcov=sigma2)-pmnorm(z[1,],varcov=sigma)
for(i in 2:nrow(z)){a101=c(a101,pmnorm(z[i,c(1,3)],varcov=sigma2)-pmnorm(z[i,],varcov=sigma))}
a110=pmnorm(z[1,1:2],varcov=sigma3)-pmnorm(z[1,],varcov=sigma)
for(i in 2:nrow(z)){a110=c(a110,pmnorm(z[i,1:2],varcov=sigma3)-pmnorm(z[i,],varcov=sigma))}
a100=pnorm(z[1,1],sd=s1)-pmnorm(z[1,c(1,2)],varcov=sigma3)-pmnorm(z[1,c(1,3)],varcov=sigma2)+pmnorm(z[1,],rep(0,ncol(Y)),varcov=sigma)
for(i in 2:nrow(z)){a100=c(a100,pnorm(z[i,1],sd=s1)-pmnorm(z[i,c(1,2)],varcov=sigma3)-pmnorm(z[i,c(1,3)],varcov=sigma2)+pmnorm(z[i,],rep(0,ncol(Y)),varcov=sigma))}
a010=pnorm(z[1,2],sd=s2)-pmnorm(z[1,c(1,2)],varcov=sigma3)-pmnorm(z[1,c(2,3)],varcov=sigma1)+pmnorm(z[1,],rep(0,ncol(Y)),varcov=sigma)
for(i in 2:nrow(z)){a010=c(a010,pnorm(z[i,2],sd=s2)-pmnorm(z[i,c(1,2)],varcov=sigma3)-pmnorm(z[i,c(2,3)],varcov=sigma1)+pmnorm(z[i,],rep(0,ncol(Y)),varcov=sigma))}
a001=pnorm(z[1,3],sd=s3)-pmnorm(z[1,c(2,3)],varcov=sigma1)-pmnorm(z[1,c(1,3)],varcov=sigma2)+pmnorm(z[1,],rep(0~~,~~ncol(Y)),varcov=sigma)
for(i in 2:nrow(z)){a001=c(a001,pnorm(z[i,3],sd=s3)-pmnorm(z[i,c(2,3)],varcov=sigma1)-pmnorm(z[i,c(1,3)],varcov=sigma2)+pmnorm(z[i,],rep(0,ncol(Y)),varcov=sigma))}
a000=1-a111-a011-a101-a110-a001-a010-a100
a111[a111<=0]=1e-50
a110[a110<=0]=1e-50
a101[a101<=0]=1e-50
a011[a011<=0]=1e-50
a100[a100<=0]=1e-50
a010[a010<=0]=1e-50
a001[a001<=0]=1e-50
a000[a000<=0]=1e-50
-sum(((Y[,1]==0)&(Y[,2]==0)&(Y[,3]==0))*log(a111) +
^{4}*log(a011) +
^{5}*log(a101) +
^{6}*log(a110) +
^{7}*log(a001) +
^{8}*log(a010) +
^{9}*log(a100) +
^{10}*log(a000) )
}

A *strong* assumption in that bivariate model is that residuals have a Gaussian structure. It is possible to change that assumption

- marginally: for instance if we use a logistic cumulative distribution function, then we will have a bivariate
*logit* regression
- in terms of dependence structure: it is possible to consider another copula than the
*gaussian *one, e.g. Gumbel’s copula (also called the *bivariate logistic* copula), or Clayton’s

Here, the following code can be used to extend the model to non Gaussian structures,

> F=function(x,r){pmnorm(x,rep(0,length(x)),
+ varcov=matrix(c(1,r,r,1),2,2))}
> Fx=function(x1){F(c(x1,1e40),0)}
> Fy=function(x2){Fx(x2)}
>
> logVgen=function(parameter){
+ CORRELATION=parameter[1]
+ BETA=matrix(parameter[2:length(parameter)],ncol(Y),ncol(X))
+ z=cbind(X%*%(BETA[1,]),X%*%(BETA[2,]))
+ a11=F(z[1,],r=CORRELATION)
+ for(i in 2:nrow(z)){a11=c(a11,F(z[i,],r=CORRELATION))}
+ a10=Fx(z[1,1])-F(z[1,],r=CORRELATION)
+ for(i in 2:nrow(z)){a10=c(a10,Fx(z[i,1])-F(z[i,],r=CORRELATION))}
+ a01=Fy(z[1,2])-F(z[1,],r=CORRELATION)
+ for(i in 2:nrow(z)){a01=c(a01,Fy(z[i,2])-F(z[i,],r=CORRELATION))}
+ a00=1-a10-a01-a11
+ -sum(((Y[,1]==1)&(Y[,2]==1))*log(a11) +
+ ^{11}*log(a01) +
+ ^{12}*log(a10) +
+ ^{13}*log(a00) )
+ }
>
> beta0=c(0,1,1,1,1,1,1)
> (OPT=optim(fn=logVgen,par=beta0,method="BFGS")$par)
[1] 0.52613820 -2.45172059 2.24196154 4.90863292 -4.53994592 0.01600769
[7] 0.10614807
There were 23 warnings (use warnings() to see them)

E.g.

- An application to school tests

Consider the following dataset,

hsb2=read.table("http://freakonometrics.free.fr/hsb2.csv",
header=TRUE, sep=",")
math_male=hsb2$math[female==0]
write_male=hsb2$write[female==0]
math_female=hsb2$math[female==1]
write_female=hsb2$write[female==1]
plot(math_female, write_female, type="p",
pch=19,col="red",xlab="maths",ylab="writing",cex=.8)
points(math_male, write_male, cex=1.2, col="blue")

with here maths versus writing, with girls in red and boys in blue, where variables here are

** female :
0: male
1: female
race :
1: hispanic
2: asian
3: african-amer
4: white
ses :
1: low
2: middle
3: high
schtyp : type of school
1: public
2: private
prog : type of program
1: general
2: academic
3: vocation
read : reading score
write : writing score
math : math score
science : science score
socst : social studies score
**

We can try to understand correlation between math and writing skills. Covariates can be the sex of the child, and his *reading* skills. The question will then be: are good students in maths and writing simply students that can read well ?

Here the code is simply

> W=hsb2$write>=50
> M=hsb2$math>=50
> base=data.frame(Y1=W,Y2=M,
+ X1=hsb2$female,X2=hsb2$read)
>
> library(Zelig)
> REG= zelig(list(mu1=Y1~X1+X2,
+ mu2=Y2~X1+X2,
+ rho=~1),
+ model="bprobit",data=base)
> summary(REG)
Call:
zelig(formula = list(mu1 = Y1 ~ X1 + X2, mu2 = Y2 ~ X1 + X2,
rho = ~1), model = "bprobit", data = base)
Pearson Residuals:
Min 1Q Median 3Q Max
probit(mu1) -4.7518 -0.502594 0.15038 0.53038 1.8592
probit(mu2) -3.4243 -0.653537 0.23673 0.67011 2.6072
rhobit(rho) -4.9821 0.010481 0.13500 0.40776 2.9171
Coefficients:
Value Std. Error t value
(Intercept):1 -5.484711 0.787101 -6.96825
(Intercept):2 -4.061384 0.633781 -6.40818
(Intercept):3 1.332187 0.322175 4.13497
X1:1 1.125924 0.233550 4.82092
X1:2 0.167258 0.202498 0.82598
X2:1 0.103997 0.014662 7.09286
X2:2 0.082739 0.012026 6.88017
Number of linear predictors: 3
Names of linear predictors: probit(mu1), probit(mu2), rhobit(rho)
Dispersion Parameter for binom2.rho family: 1
Residual Deviance: 364.51 on 593 degrees of freedom
Log-likelihood: -182.255 on 593 degrees of freedom
Number of Iterations: 3
> (exp(summary(REG)@coef3[3])-1)/(exp(
summary(REG)@coef3[3])+1)
[1] 0.5824045

with a remaining correlation among residuals of 0.58. So with only the sex of the student, and his or her reading skill, we cannot explain the correlation between maths and writing skills. With our previous code, we have here

> beta0=c((exp(summary(REG)@coef3[3])-1)/(exp(summary(REG)@coef3[3])+1),
+ summary(REG)@coef3[c(1:2,4:7),1])
> beta0
(Intercept):1 (Intercept):2 X1:1 X1:2
0.58240446 -5.48471133 -4.06138412 1.12592427 0.16725842
X2:1 X2:2
0.10399668 0.08273879
> (OPT=optim(fn=logV,par=beta0,method="BFGS")$par)
(Intercept):1 (Intercept):2 X1:1 X1:2
0.5824045 -5.4847113 -4.0613841 1.1259243 0.1672584
X2:1 X2:2
0.1039967 0.0827388

i.e. we obtain (almost) exactly the same estimators. But here I have used as starting values for the optimization procedure the estimators given by R. If we change them, hopefully we have a robust maximum likelihood estimator,

> (OPT=optim(fn=logV,par=beta0/2,method="BFGS")$par)
(Intercept):1 (Intercept):2 X1:1 X1:2
0.58233360 -5.49428984 -4.06839571 1.12696594 0.16760347
X2:1 X2:2
0.10417767 0.08287409
There were 12 warnings (use warnings() to see them)

So once again, it is possible to optimize numerically a likelihood function, and it works.