clear all
set obs 101
generate x = (_n-51)/10
generate pr = invlogit(x)
generate sd = sqrt(p*(1-p))

line pr x, name(prob) title("Cumulative Probability")
line sd x, name(stddev) title("Standard Deviations")
graph combine prob stddev, col(1) title("Logits") xcommon ycommon

// estimated as a blocked logit
generate pos = round(pr*100) // truncates exact values
generate pop = 100
glm pos x, family(binomial pop) link(logit)

// this should be ~ 0.5
di exp(_se[_cons])/(1+exp(_se[_cons]))
// this should be a little smaller
di exp(_se[x])/(1+exp(_se[x]))

glm, eform
// although the results show odds, the
//  e(b) vector is in log-odds
// Probability at x=1
// should be ~ exp(1)/(1+exp(1)), about 0.73106
display exp(_b[x])/(1+exp(_b[x]))
margins, at(x=(0))

// Now if our data are all in a range to the right,
// we are still estimating the same logit function
clear all
set obs 101
generate x = (_n-1)/10
generate pr = invlogit(x)
generate sd = sqrt(p*(1-p))

line pr x, name(prob) title("Cumulative Probability")
line sd x, name(stddev) title("Standard Deviations")
graph combine prob stddev, col(1) title("Logits") xcommon ycommon

// estimated as a blocked logit
generate pos = round(pr*100) // truncates exact values
generate pop = 100
glm pos x, family(binomial pop) link(logit)

// the first should be ~ 0.5
di exp(_se[_cons])/(1+exp(_se[_cons]))
// this should be a little smaller
di exp(_se[x])/(1+exp(_se[x]))

// Probability at x=1
// should be ~ exp(1)/(1+exp(1)), about 0.73106
display exp(_b[x])/(1+exp(_b[x]))