What to do ordinal response variable?

[FallenApple]

So what if my response variable, y, is say a scale. For example, ranking, something like 1-10. How would I transform the response to make linear regression work?

 

[andrewkirk]

Do a glm where the response variable is uniformly distributed on [0,1] and the link function is the cdf of the standard normal.
Code response variable values 1,...,10 as 0.05, 0.15,...,0.95.

 

[FallenApple]

Do a glm where the response variable is uniformly distributed on [0,1] and the link function is the cdf of the standard normal.
Code response variable values 1,...,10 as 0.05, 0.15,...,0.95.

Oh ok got it. But why the mid point, is it because the rankings are evenly spaced? so 1 becomes 0.05.

But what if its something else. Like grades? A,B,C,D,F then would I split it into five? So 100/5=20, then take the mid point so A=10, B=30, C=50, D=70, F=90?

So Prob( Z < transformed(y))= sum of regressors?

so in R it would be qnorm(new_y, 1,0)~x1+x2+...+xp ?

 

[andrewkirk]

I think the code would be something like

glm(y~x1+x2+ ... + xn, family=quasi(link = "probit", variance = "constant"))

But I am not completely sure about the variance argument. Unfortunately, the R documentation on the 'quasi' family is almost non-existent. Best to try it and see what happens.

So Prob( Z < transformed(y))= sum of regressors?

We need to apply $\Phi$ to the sum of regressors.

If you have an ordered factor variable fac and the vector of corresponding integer values is fac.int then the transformation would be

n<-length(levels(fac))
y.transformed<-  (2 * fac.int - 1) / (2 * n)

You could also try searching 'Ordinal regression', which is the term for what you are trying to do.

 

[FallenApple]

I think the code would be something like

glm(y~x1+x2+ ... + xn, family=quasi(link = "probit", variance = "constant"))

But I am not completely sure about the variance argument. Unfortunately, the R documentation on the 'quasi' family is almost non-existent. Best to try it and see what happens.We need to apply $\Phi$ to the sum of regressors.

If you have an ordered factor variable fac and the vector of corresponding integer values is fac.int then the transformation would be

n<-length(levels(fac))
y.transformed<-  (2 * fac.int - 1) / (2 * n)

You could also try searching 'Ordinal regression', which is the term for what you are trying to do.

Oh ok. I'll research into the probit glm.What about dichotomizing it? Would that hurt?

For grades, I could do C-A as 1 for pass and D-F as 0 for fail. So it depends on context if I can do this?

 

[andrewkirk]

So it depends on context if I can do this?

Yes, it depends on what you are trying to achieve.

 

[FallenApple]

Yes, it depends on what you are trying to achieve.

Well, mostly its just to make it simpler. But what is the tradeoff? If I dichotomize say grades, to pass fail? Would I lose power? Presumably, if there is an effect of say x on grades from level to level of grades, then there would be a cooresponding effect of grades from fail to pass and vice versa.