Getting a data frame of logit probabilities and their confidence intervals

Question

I have the following model and have used the effects package to plot the predicted probabilities and the confidence interval lines. However, I was wondering how I'd go about spitting out a data frame in R which has the response value, the low and high ci values, and the predicted values. Kind of like the following

mod1 = glm(won_ping ~ our_bid, data=ndat, family=binomial(link="probit"))
summary(mod1)

library(effects)
plot(effect("our_bid", mod1), rescale.axis=FALSE, multiline=TRUE, xlim=c(0,2000), main="129- AH - Bid model")

Desired output with our response variable and the confidence intervals for the predicted probabilities:

our bid    low_ci     hi_ci      prob
1           0.15       0.21       0.17
2           0.18       0.23       0.20 
3           0.20       0.30       0.25   
4           0.21       0.25       0.23  
...

How would I go about getting the following result in R.

I tried the following but it doesn't work as I want the probabilities and not the log-odds values.

> as.data.frame(effect("our_bid", mod1))
   our_bid        fit         se       lower      upper
1       25 -2.3549908 0.04598536 -2.44512045 -2.2648612
2      238 -2.0297771 0.03794491 -2.10414781 -1.9554065
3      451 -1.7045635 0.03724233 -1.77755712 -1.6315699
4      664 -1.3793498 0.04422870 -1.46603649 -1.2926632
5      877 -1.0541362 0.05610148 -1.16409305 -0.9441793
6     1090 -0.7289225 0.07043143 -0.86696557 -0.5908795
7     1303 -0.4037089 0.08599888 -0.57226356 -0.2351541
8     1516 -0.0784952 0.10224011 -0.27888213  0.1218917
9     1729  0.2467185 0.11887929  0.01371934  0.4797176
10    1942  0.5719321 0.13577018  0.30582746  0.8380368

Answer 1

if $\hat{\eta}_i$ is the $i^\rm{th}$ fitted log-odds then $$ \hat{p}_i=\frac{\exp(\hat{\eta}_i)}{1+\exp(\hat{\eta}_i)} = \frac{1}{1+\exp(-\hat{\eta}_i)} $$

You can transform the limits of confidence intervals in similar fashion.

Transforming s.e.'s is trickier, but you can do it approximately via Taylor expansion.

You can get R to produce these directly, you don't have to calculate them yourself. See the help relating to the type argument of the predict.glm function.