I made a basic mistake, one that I've made throughout my depressingly long career [a], in my comment on DMT (2015). In that post I said:
Probit models estimating the effect of the bid on the yes responses with the $10 bid removed are in Table 2. Only two of the five coefficients (B) on the bid variable are statistically significant at the p=.10 level.
This is incorrect because I'm using a two-tailed test when the proper test should be one-tailed. Theory suggests that the effect of cost on a vote in favor of a policy is negative so the appropriate null and alternative hypotheses are:
H0: Bc = 0
HA: Bc < 0
Where Bc is the coefficient on the cost variable in a regression analysis. This mistake usually doesn't make a difference but if the coefficient is marginally significant in a two-tailed test (e.g., .05 < p < .10) then it will be significant in a one-tailed test (i.e., p = .05 when t = 1.645). Any use of a two-tailed test to argue that statistical results are as statistically significant as they seem when a one-tailed test is more appropriate is overly harsh.
[a] Most everyone who does nonmarket valuation has made the same mistake and many others. In CVM a scope test should be one-tailed. In TCM the travel cost coefficient should be judged on a one-tailed test. Why do we do this? Do we not trust our theory?
Here is the corrected text:
Visual inspection of the data suggests that in the whole scenario there are three distinct sections to the bid curve. The only statistically significant difference in the yes responses at different cost amounts is between $80 and $125. In a linear probability model the slope of this portion of the bid function, B = -0.00513, is statistically significant at the p=.05 level in a one-tailed test (t=-1.70). The slopes on the upper (cost = $10 to $80) and lower (cost = $125 to $405) portions of the bid curve are B = 0.0004 (t=0.20) and B = -0.0000016 (t=0.003).
Visual inspection of the data suggests that there is a kink in the survival function at the $45 bid amount for each of the incremental parts versions. I estimate linear probability models of the cost variable on the yes responses with (a) the costs greater than $45 removed and (b) the $10 cost removed. All four of the coefficients on the slope between $10 and $45 are statistically significant at, at least, the p=.10 level using a one-tailed test (Table 1). Three of the four coefficients on the cost variables are statistically significant at, at least, the p=.10 level using a one-tailed test from the $45 to $405 bid range. The bid curve for the third increment for bids over $45.
Table 1. Linear probability model slopes
The fitted probabilities for the whole and third scenarios are plotted in Figures 1. The bid curves for the whole and third versions do not exhibit theoretical validity over 91% and 94% of the bid range, respectively. Conduct of any further validity tests (e.g., scope, adding-up) with these two sub-samples is of dubious value.
Figure 1. Fitted probabilities for the whole and third scenarios
