Finding predictors of upper level of a variable

Question

I am analyzing data on a health variable and its relation to age, gender, height etc. I am more interested in 90th percentile of the health variable, which can be called upper limit of 'normal'. How will this analysis be different from regular analysis where one is trying to find out factors predicting the value of health variable?

Answer 1

It sounds like you're seeking quantile regression.

In practice this works rather like ordinary regression except that you fit one or more conditional quantiles (which are assumed linear in predictors) rather than the conditional mean.

Parameter estimation involves finding the $\hat{\beta}_\tau$ minimizing $\sum_{i=1}^{n}\rho_{\tau}(Y_{i}-X\beta)$, where $\rho_{\tau}(y)=|y(\tau-\mathbf{I}_{(y<0)})|$ and $\tau$ is the desired quantile.

http://en.wikipedia.org/wiki/Quantile_regression

This minimization problem can be cast as a linear programming problem, for which a number of algorithms exist.

There are numerous questions and answers here relating to quantile regression.

Here's a small example (with a single quantile, the 90th percentile), done in R:

 library(quantreg)
 carsfit<- rq(dist~speed,tau=.9,data=cars)
 plot(dist~speed,cars)
 lines(carsfit$fitted~cars$speed,col=2)

and as with ordinary regression you can get summary output as well, but for coefficients you get an interval rather than a standard error:

> summary(carsfit)

Call: rq(formula = dist ~ speed, tau = 0.9, data = cars)

tau: [1] 0.9

Coefficients:
            coefficients lower bd  upper bd 
(Intercept)  -8.85714    -19.13570  40.37029
speed         4.71429      3.46632   5.83520
Warning message:
In rq.fit.br(x, y, tau = tau, ci = TRUE, ...) : Solution may be nonunique

(Normally you'd want a lot more data than this to try to estimate a high quantile.)

I'm barely touching on the details and capabilities of quantile regression here; you might find this vignette of use, even if you don't use R to perform quantile regression.