A.     Explain the logit and probit transformations. A one sentence explanation is sufficient.

B.     Fill in the table below to compare the two transformations.

C.    In what range are the two transformations practically equivalent? In what ranges might the two transformations give different results? (The formula for the logit transformation is in the textbook. Excel gives the probit transformation as the inverse of the CDF of the standard normal distribution.)

 

 

 

 

 

Part C) I'm not seeing a clear range where the two transformations are practically equivalent. Is any else?

[NEAS: The two transformations are not exactly the same in any range. But they are reasonably close to 50%. For transforming a response variable that shows probabilities into a variable that can be modeled by a normal distribution, both transformations work equally well if the probabilities are near 50%.]

In Excel, I put P in cell A4 and used =ln(A4/(1-A4)) for logit and =NORMSINV(A4) for the probit. Did I do something wrong? I don't see another way to solve this.

I don't see a "range" either. But obviously when P = 0.5 they are equal. This is what I put down for answer, but I doubt if it's correct.

My understanding is when you apply the formula in the text above the problems to equate the probit with the logit, you can then compare that # to the original logit # you got.  Once you apply the formula it is fairly clear to see where the ranges are.

I did use the factor.

 

P	% error = 1 – (probit*pi/sqrt(3))/logit
0.001	19%
0.002	16%
0.01	8%
0.02	4%
0.1	-6%
0.2	-10%
0.4	-13%
0.5	n/a
0.6	-13%
0.8	-10%
0.9	-6%
0.98	4%
0.99	8%
0.998	16%
0.999	19%

P	Logit	Probit	probit * pi/sqrt(3)
0.001	-6.91	-3.09	-5.61
0.002	-6.21	-2.88	-5.22
0.010	-4.60	-2.33	-4.22
0.020	-3.89	-2.05	-3.73
0.100	-2.20	-1.28	-2.32
0.200	-1.39	-0.84	-1.53
0.400	-0.41	-0.25	-0.46
0.500	0.00	0.00	0.00
0.600	0.41	0.25	0.46
0.800	1.39	0.84	1.53
0.900	2.20	1.28	2.32
0.980	3.89	2.05	3.73
0.990	4.60	2.33	4.22
0.998	6.21	2.88	5.22
0.999	6.91	3.09	5.61

It's hard to pick a range where they are equal, but I think you can say 'near' p=1/2.

[NEAS: Correct; the two transformations are not exactly the same, but they give the same type of transformation for probabilities near p = 50%. A logit GLM and a probit GLM give nearly the same predictions for probabilities between 25% and 75%. Because of the stochasticity of observed values, we can't choose between the logit and probit GLMS in most analyses. We use the logit GLM because it is simpler.]