Method to identify the point in which the slope of a predicted probability becomes significant I'm running a logistic regression in which I'm predicted a binary response from a continuous predictor... I'm interested in determining the exact point in which the predicted probability (exponentiated y-hat / (1 + exponentiated y-hat) becomes significantly different than the average y-hat (or average predicted probability of the discrete event...
I was thinking of some sort of Johnson Neyman technique, but am unsure how to begin this exercise...
Would like to do this in R at the end of the day, but not required for an answer.
 A: A logit model is a linearization of a nonlinear relationship between the X's and the dichotomous outcome. The slope is therefore constant everywhere in X. 
As far as the predicted probabilities, the average predicted probability is just the proportion of 1's in your data. So you want to identify for what value of X does an observation have a higher probability of being 1 than the overall proportion. You can do that a number of ways - one way would be to estimate the predicted probabilities for each level of X (keeping other variables in your model at their means), sort them and see at which point is it higher than the overall. (You can find this out by looking at the coefficients and doing some arithmetic, but this may be easier if you have trouble with converting log-odds to p).
You can also estimate a confidence interval around the marginal effect (i.e. average probability) at each level. I haven't done this in R though it probably isn't that complicated - in Stata, you run your logit and then do margins. For example, I simulated some data where the overall sample probability is 0.391, and one standard normal predictor x, where the true logit coefficient was 2. In Stata, here is my output:
    logit y x

Logistic regression                               Number of obs   =       1000
                                                  LR chi2(1)      =     504.19
                                                  Prob > chi2     =     0.0000
Log likelihood = -417.09877                       Pseudo R2       =     0.3767

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   2.192613   .1399275    15.67   0.000      1.91836    2.466866
       _cons |  -.8138537   .0933571    -8.72   0.000    -.9968302   -.6308772

margins

             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |       .391   .0116073    33.69   0.000     .3682501    .4137499
------------------------------------------------------------------------------

margins, at(x==0.2)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .4072535    .020886    19.50   0.000     .3663176    .4481893
------------------------------------------------------------------------------

Look at the confidence intervals for the two margins commands and you will see that they overlap, so even though at x=0.2 you see a predicted probability higher than the average, it isn't statistically significant. 
EDIT: Per whuber's comments below, this approach doesn't account for the sampling variation of the mean from the data. It will tell you a difference from a particular value, which may happen in this instance to be the mean of your sample. But it may not address your specific question.
