Intuition Behind GLM

This is a question regarding using logistic regression, and relating it to gaussian distribution or a binomial distribution.

model<-glm(target~ x1, data=data, type='response', family='binomial')
model<-glm(target~ x1, data=data, type='response')  #defaults to gaussian

My understanding of binomial is that it is

theta=chance of success
z=trails ending in success
k=trials ending in failure
(theta^z)*(1-theta)^k

And something Gaussian is

theta = standard deviation
x = success
u = mean
Y = [ 1/σ * sqrt(2π) ] * e -(x - μ)2/2σ2

So I understand how to do GLM with R, I kind of understand what binomial and gaussian means, but I have no understanding of how you relate binomial or gaussian to logistic regression, and how binomial and gaussian are different in this context.

Question 1- Can someone explain the intuition behind how "family='binomial'" is used when building a model with GLM?

Question 2- Given that the shapes of a binomial distribution and a gaussian distribution look very much the same (they both peak in the middle and gradually go down towards the ends), how does choosing either binomial or guassian lead to different models built from GLM?

thanks!!

• If $np \geq 10$ and $n(1-p) \geq 10$ then the normal distribution is a good approximation to the binomial. – PEV Jul 24 '13 at 19:34

Lets say your response variable is $Y$. In regression we want to model our response variable as a linear combination of our predictor variables ($X$) e.g. $Y=\beta_0 + \beta_1X + \epsilon$ or $E[Y]=\beta_0 + \beta_1X$. But what happens when our response variable is only in $[0,1]$ (i.e. it is a probability, proportion or strictly only 0 or 1). Notice that $\beta_0 + \beta_1X$ may take any value on the real line! It could be 0 or 1 or 100 or even negative! If our response variable is strictly in $[0,1]$ it makes no sense to try to use a model that can take values outside of that range.

Therefore, when we want to model a probability or a proportion, we instead model a function of $Y$. For example $g(E[Y])=\beta_0 + \beta_1X$. This function is called the link function.

$g(E[Y]) = E[Y]$ Identity link. Used in Linear regression

$g(P(Y=1)) = \log{\dfrac{P(Y=1)}{1-P(Y=1)}}$ Logit link, Used in logistic regression. Notice here we are modeling the probability $Y=1$ which is also the expected value. Then we can solve for what we want: $P(Y=1) = \dfrac{e^{\beta_0+\beta_1X}}{1+e^{\beta_0+\beta_1X}}$

$g(E[Y]) = \log{E[Y]}$ log link, used in poisson regression

Question 1: In the GLM R function, the family parameter allows you to specify the link function.

Question 2: First of all, it is not true that binomial always looks like normal. If $X\sim Binomial(n, p=0.1)$ it is a skewed distribution, which does not look like a bell shaped distribution.

• I think the statement "we are modeling the probability Y=1 instead of the expected value" is wrong because E(Y) is P(Y=1). – HelloWorld Sep 16 '14 at 6:47
• Benjamin; I approved the edit that came through because I think it's correct - but usually if I think there's a correction to be made, normally I myself prefer to comment and let the original author decide to change it (or not), unless they're unlikely to respond. You should feel free to clarify your answer, to edit to improve further, or to roll back if you think it's necessary. – Glen_b -Reinstate Monica Sep 16 '14 at 7:19

You use logistic regression when your response variable is binary (0/1) or a proportion (10/30) so you can't relate it to a gaussian distribution which is continuos and has no boundaries. That's why you specify "family="binomial" to perform logistic regression in R and family="gaussian" to perform linear regression.