[R] Pointwise Confidence Bounds on Logistic Regression

[Gavin Simpson]

On Thu, 2008-06-19 at 10:42 -0400, Bryan Hanson wrote:
> [I've ommitted some of the conversation so far...]
> 
> > E.g. in a logistic model, with (say) eta = beta_0 + beta_1*x one may
> > find, on the
> > linear predictor scale, A and B (say) such that P(A <= eta <= B) = 0.95.
> > 
> > Then P(expit(A) <= expit(eta) <= expit(B)) = 0.95, which is exactly
> > what is wanted.
> 
> I think I follow the above conceptually, but I don't know how to implement
> it, though I fooled around (unsuccessfully) with some of the variations on
> predict().
> 
> I'm trying to learn this in response to a biology colleague who did
> something similar in SigmaPlot. I can already tell that SigmaPlot did a lot
> of stuff for him in the background.  The responses are 0/1 of a particular
> observation by date.  The following code simulates what's going on (note
> that I didn't use 0/1 since this leads to a recognized condition/warning of
> fitting 1's and 0's. I've requested Brian's Pattern Recognition book so I
> know what the problem is and how to solve it).  My colleague is looking at
> two populations in which the "LD50" would differ.  I'd like to be able to
> put the "pointwise confidence bounds" on each curve so he can tell if the
> populations are really different.
> 
> By the way, this code does give a (minor?) error from glm (which you will
> see).
> 
> Can you make a suggestion about how to get those "confidence bounds" on
> there?
> 
> Also, is a probit link more appropriate here?
> 
> Thanks,  Bryan
> 
> x <- c(1:40)
> y1 <- c(rep(0.1,10), rep(NA, 10), rep(0.9,20))
> y2 <- c(rep(0.1,15), rep(NA, 10), rep(0.9,15))
> data <- as.data.frame(cbind(x,y1,y2))
> plot(x, y1, pch = 1, ylim = c(0,1), col = "red")
> points(x, y2, pch = 3, col = "blue")
> abline(h = 0.5, col = "gray")
> fit1 <- glm(y1~x, family = binomial(link = "logit"), data, na.action =
> na.omit)
> fit2 <- glm(y2~x, family = binomial(link = "logit"), data, na.action =
> na.omit)
> lines(fit1$model$x, fit1$fitted.values, col = "red")
> lines(fit2$model$x, fit2$fitted.values, col = "blue")

The point is to get predictions on the scale of the link function,
generate 95% confidence bands in the normal way and then transform the
confidence bands onto the scale of the response using the inverse of the
link function used to fit the model.

[note, am doing this from memory, so best to check this is right -- I'm
sure someone will tell me very quickly if I have gone wrong anywhere!]

## predictions on scale of link function
pred1 <- predict(fit1, se.fit = TRUE)
pred2 <- predict(fit2, se.fit = TRUE)

## constant for 95% confidence bands
## getting two t values is redundant here as fit1 and fit2
## have same residual df, but the real world may be different
res.df <- c(fit1$df.residual, fit2$df.residual)
## 0.975 because we want 2.5% in upper and lower tail
const <- qt(0.975, df = res.df)

## confidence bands on scale of link function
upper1 <- pred1$fit + (const[1] * pred1$se.fit)
lower1 <- pred1$fit - (const[1] * pred1$se.fit)
upper2 <- pred2$fit + (const[2] * pred2$se.fit)
lower2 <- pred2$fit - (const[2] * pred2$se.fit)

## bind together into a data frame
bands <- data.frame(upper1, lower1, upper2, lower2)

## transform on to scale of response
bands <- data.frame(lapply(bands, binomial(link = "logit")$linkinv))

## plot confidence bands
lines(fit1$model$x, bands$upper1, col = "red", lty = "dotted")
lines(fit1$model$x, bands$lower1, col = "red", lty = "dotted")
lines(fit2$model$x, bands$upper2, col = "blue", lty = "dotted")
lines(fit2$model$x, bands$lower2, col = "blue", lty = "dotted")

HTH

G

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