# A What to do ordinal response variable?

1. Jun 23, 2017

### 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?

2. Jun 23, 2017

### 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.

3. Jun 23, 2017

### FallenApple

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 ?

4. Jun 23, 2017

### andrewkirk

I think the code would be something like

Code (Text):
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

Code (Text):

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.

Last edited: Jun 23, 2017
5. Jun 23, 2017

### FallenApple

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?

6. Jun 23, 2017

### andrewkirk

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

7. Jun 24, 2017

### FallenApple

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.