Providing 95% confidence intervals for VTT calculation using logistic regression coefficients

Question

I appreciate that similar questions have been asked and answered to this, but I think that my case is substantively different as the specific interpretation of the coefficient is not relevant.

I have a set of choice experiment results that I am using to calculate the 'Valuation of Travel Time' for the participants. The standard process for doing this is to fit a logistic regression model to the result of whether someone accepts a specific level of compensation for a specific travel time. You can then divide the time coefficient by the compensation one to generate the valuation of travel time (don't worry if doesn't make sense). Hence for the below model:

term	estimate	std.error	z value	p.value
(Intercept)	1.256319853	0.1096819372	11.45420919	2.24E-30
time	0.1015354957	0.003527728861	-28.78211442	3.59E-182
comp	0.1532430136	0.005337402709	28.7111582	2.77E-181

The 'VTT' is (approximately) 0.102/0.153 - which, is £0.67 per minute, or £40 per hour.

I want to provide a 95% confidence interval for this valuation and I've been searching for the correct method. My instinct is that it's just:

upper limit: time_coef + 1.96time_SE / comp_coef - 1.96comp_SE

lower limit: time_coef - 1.96time_SE / comp_coef + 1.96comp_SE

Without a detailed understanding of VTT methodology, does this look correct?

Edit - following useful feedback, I can include the covariance table if anyone else is able to use this data to help me validate that I am using the correct method.

	(Intercept)	time	comp
(Intercept	1.203013e-02	-1.265604e-04	9.785909e-06
time	-1.265804e	1.244487e-05	-1.437472e-05
comp	9.785909e-06	-1.437472e-05	2.848787e-05

Answer 1

Expanding over Bjorn's answer, you are trying to perform statistical inference on the ratio of two regression coefficients and since this is a nonlinear function you have to use additional tools. The most accessible tool is the delta method; other possible procedures may be higher-order asymptotics (e.g. Brazzale et al. Applied Asymptotics: Case Studies in Small-Sample Statistics, Cambridge, 2007).

R implementations for the delta method and higher-order asymptotics can be found in the car and likelihoodAsy (see also hoa) packages respectively.

Here is an example of the delta method using the car package. Note that in order to use the delta method you'll need the full estimated covariance matrix of the estimator of your regression coefficients. The function deltaMethod automatically takes care of this.

library(car)

# create a fictitius binary variable
binary_time <- ifelse(Transact$time <= 5000, 1, 0)

# run binary logistic regression
m1 <- glm(binary_time ~ t1 + t2, data = Transact, 
          family = "binomial") 

# get the CI by the delta method
deltaMethod(m1, "b1/b2", parameterNames= paste("b", 0:2, sep="")) 

Answer 2

No, that's not correct. You don't have the necessary information in the output, you would additionally need the information on the covariance matrix of the regression coefficients (with the standard errors, you have the square root of the diagonal entries of that matrix). Luckily, if you have that, you can use something like the delta method, which there are existing packages that will handle the details for you, e.g. in R the deltaMethod function from the car package. That's a useful search term to look for/the background information given there would be a good starting point.