In my previous post on GLMs, I discussed power link functions. But there are much more links that can be used :

**The square root link (for the Poisson model)**

Consider some random variable Y with mean \mu and variance \sigma^2. Using Taylor’s expansion,g(Y)\sim g(\mu)+(Y-\mu)g'(\mu)+\frac{1}{2}(Y-\mu)^2g''(\mu)we can write\mathbb{E}[g(Y)]\sim g(\mu)+\frac{\sigma^2}{2}g''(\mu) \text{Var}[g(Y)]\sim [g'(\mu)]^2\sigma^2

Assume that Y\sim\mathcal{P}(\lambda), a consider a square root transformation, g(y)=\sqrt{y}, then the second equality becomes \text{Var}[\sqrt{Y}]\sim \left[\frac{1}{2\sqrt{\mathbb{E}[Y]}}\right]^2\text{Var}[Y]=\frac{1}{4}

So, somehow, with a square-root transformation, we have variance stability, which might be interpreted as some homoscedasticity.

**The complementary log-log function for the Bernoulli model**

Assume that the true variable of interest is a Poisson one, N|\mathbf{X}=\mathbf{x}\sim\mathcal{P}(\lambda_{\mathbf{x}}) where \lambda_{\mathbf{x}}=\exp[\mathbf{x}^T\mathbf{\beta}]Thus,\mathbb{P}[N=0|\mathbf{X}=\mathbf{x}]=\exp[-\lambda_{\mathbf{x}}]=\exp[-(\exp[\mathbf{x}^T\mathbf{\beta}])]while\mathbb{P}[N>0|\mathbf{X}=\mathbf{x}]=1-\exp[-(\exp[\mathbf{x}^T\mathbf{\beta}])]=H(\mathbf{x}^T\mathbf{\beta})where H(s)=1-\exp[-\exp(s)]. Let Y=\mathbf{1}(N>0). The previous model seems like a Bernoulli regression with H as link function,\mathbb{P}[Y=1|\mathbf{X}=\mathbf{x}]=H(\mathbf{x}^T\mathbf{\beta})

So, assume now that instead of observing N we observe Y=\boldsymbol{1}(N>0). In that case, running a Bernoulli regression with a complementary log-log link function would be the same (?) as running first a Poisson regression on the original data, and then use it on our binary variable, zero vs. non-zero. Let us generate some data, and see what’s going on. Let us compare e^{\lambda_{\mathbf{x}}} and p_{\mathbf{x}} obtained from a standard logistic regression

n=563 set.seed(1) base=data.frame(X1=rnorm(n),X2=rnorm(n)) lambda=base$X1+base$X2 base$Y=rpois(n,exp(lambda)) regPois = glm(Y~.,data=base,family=poisson(link="log")) lambda = predict(regPois,type="response") regBinom = glm((Y==0)~.,data=base,family=binomial(link="probit")) prob = predict(regBinom, type="response") plot(prob,exp(-lambda),xlim=0:1,ylim=0:1) abline(a=0,b=1,lty=2,col="red") |

What if p_{\mathbf{x}} was obtained from a Bernoulli regression, with a cloglog link function ?

regBinom = glm((Y>0)~.,data=base,family=binomial(link="cloglog")) prob = predict(regBinom, type="response") plot(prob,1-exp(-lambda),xlim=0:1,ylim=0:1) abline(a=0,b=1,lty=2,col="red") |

It looks like the fit is very good here ! Now, what if we have real data, like the dataset from *A Theory of Extramarital Affairs*, by Ray Fair, published in 1978 in the *Journal of Political Economy* (with 563 observations, and nine variables)

base = read.table("http://freakonometrics.free.fr/baseaffairs.txt",header=TRUE) str(base) x=base$SEX base$SEX="M" base$SEX[x=="0"]="F" x=base$CHILDREN base$CHILDREN="YES" base$CHILDREN[x==0]="NO" regPois = glm(Y~.,data=base,family=poisson(link="log")) lambda = predict(regPois,type="response") regBinom = glm((Y==0)~.,data=base,family=binomial(link="probit")) prob = predict(regBinom, type="response") plot(prob,exp(-lambda),xlim=0:1,ylim=0:1) abline(a=0,b=1,lty=2,col="red") |

In that case the two models are very different. But actually, so is the second one

regBinom = glm((Y>0)~.,data=base,family=binomial(link="cloglog")) prob = predict(regBinom, type="response") plot(prob,1-exp(-lambda),xlim=0:1,ylim=0:1) abline(a=0,b=1,lty=2,col="red") |

How can we interpret that ? Could it be because the Poisson model is not good ? Actually, if we run a zero-inflated model here,

library(pscl) regZIP = zeroinfl(Y ~ . | ., data = base) summary(regZIP) Count model coefficients (poisson with log link): Estimate Std. Error z value Pr(>|z|) (Intercept) -0.002274 0.048413 -0.047 0.963 X1 1.019814 0.026186 38.945 <2e-16 *** X2 1.004814 0.024172 41.570 <2e-16 *** Zero-inflation model coefficients (binomial with logit link): Estimate Std. Error z value Pr(>|z|) (Intercept) -4.90190 2.07846 -2.358 0.0184 * X1 -2.00227 0.86897 -2.304 0.0212 * X2 -0.01545 0.96121 -0.016 0.9872 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 |

Hence, we reject here the Poisson distribution assumption, because of the inflation of zeros… It looks like the cloglog link can be used to check if the Poisson distribution is a good model, or not…