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I am using logistic regression with white cell count and temperature as predictors and hospital admission>3 days as the outcome of interest. I'm using the rms package in R to assess calibration (curve generated by val.prob) and having some difficulty interpreting the output. My specific questions are, how is slope calculated and how is the intercept calculated? The smooth line logistic calibration I believe is generated by locally estimated scatterplot smoothing (loess). And qualitatively I believe this is a poorly calibrated model. However, I don't understand how to interpret the slope or intercept given that the slope differs along the curve. Any help would be appreciated. enter image description here

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It transforms your predicted probabilities to log odds ratios (or logit) and then uses that as a dependent variable to fit a logistic regression. If your prediction can clearly separate the labels, you would get an intercept of 0 and slope 1..

If we check the vignette of the function:

Given a set of predicted probabilities p or predicted log odds logit, and a vector of binary outcomes y that were not used in developing the predictions p or logit, val.prob computes the following indexes and ... chi- square with 2 d.f. for testing unreliability (H0: intercept=0, slope=1), its P-value, ..., Intercept, and Slope

Of course, bear in mind this is one of many test, and we can use the example from the vignette below, and the slope and intercept values are great because it's simulated:

library(rms)

set.seed(1)
n <- 200
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
logit <- 2*(x1-.5)
P <- 1/(1+exp(-logit))
y <- ifelse(runif(n)<=P, 1, 0)
d <- data.frame(x1,x2,x3,y)
f <- lrm(y ~ x1 + x2 + x3, subset=1:100)
pred.logit <- predict(f, d[101:200,])
phat <- 1/(1+exp(-pred.logit))

res = val.prob(phat, y[101:200], m=20, cex=.5)

res[c("Intercept","Slope")]
 Intercept      Slope 
0.05228721 0.95651781 

It's the same as doing:

glm(y[101:200] ~ log(phat/(1-phat)),family="binomial")

Call:  glm(formula = y[101:200] ~ log(phat/(1 - phat)), family = "binomial")

Coefficients:
         (Intercept)  log(phat/(1 - phat))  
             0.05229               0.95652  
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