Stata command for S-shapes analogue to Utest command by Lind and Mehlum

[Franz Hopp]

Dear Stata comunity,

I was wondering whether there is any statistical test and command in Stata to test for S-shapes -- analogue to the Utest command which Lind and Mehlum developed for quadratic relationships.

I am currently working on a paper where we hypothesize an S-shaped relationship. The IV of interest is a certain firm_strategy, and the DV is firm_performance.

Of course I can just check whether the normal, the squared, and the cubic term are all significant (and whether the signs of the three variables are negative/positive/negative).
I.e., I would look at: firm_strategy, firm_strategy_squared, firm_strategy_cubed

Now I have been advised by a colleague that when one hypothesizes a U-shaped (quadratic) relationship, just looking at the term of firm_strategy and firm_strategy_squared is not enough. Apparently the Utest of Lind and Mehlum is more elaborate, as it combines a number of different tests in one test .For example, the Utest of Lind and Mehlum also tests whether the turning points are well within the data range, or whether the curves are steep enough around the turning points, etc.

http://fmwww.bc.edu/repec/bocode/u/utest.html

Now I have never heard about the existence of a similar test for S-shaped (cubic) relationships. And I could not find any such test when googling for it.

Yet I assume that potentially lots of the "additional things" that the Utest of Lind and Mehlum takes into consideration for U-shaped relationships could also be of importance when studying S-shaped cubic relationships. I.e., just looking at the significance and signs of firm_strategy, firm_strategy_squared, firm_strategy_cubed might not be enough.

Now I was wondering whether anyone in this community might indeed have heard of the existence of such a Stata command/ statistical test.

Thanks a lot.

Franz

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stat output, and sample data using dataex.

While I'm sure that the U test is more sophisticated than this, it looks to me that part of it is running linear regression on subsets of the data. You can certainly do this with your s-shape to see if the slopes on parts of the data look right.

[Joseph Coveney]

It sounds like what's called a sigmoid Emax model in pharmacology. You might want to Google that to see whether it sounds like what you're looking for. You can set it up with fewer or more parameters to estimate various shapes and locations, and fit it with Stata with a nonlinear regression command.

[Max Eber]

Franz Hopp Were you ever able to resolve this? I'd be interested in doing the same thing now if it were available but can't find anything

[Franz Hopp]

Max Eber Sadly not, I would still be most interested in learning about such a test/ command!

[Nick Cox]

I guess part of the problem is the lack of one-to-one correspondence between hypothesized shape and predictor choice. For example, an S shape to me implies a monotonic, convex down to convex up curve with an inflexion separating those sections. Hence if you have such a shape then with a cubic polynomial you would certainly expect to see a cubic term that is substantial. But cubic polynomials don't imply S-shapes uniquely and a good fit to a cubic is consistent with one or even two turning points and so failure even to be monotonic. (As always, what a curve does in the range of the data and what it does over an arbitrarily wider range are different questions, just as quadratics can capture some curvature without showing a turning point in the range of the data.)

In some contexts an S-shape means a logistic curve (whether the response is bounded. e.g. a proportion or percent that varies between 0 and 1 or100 -- or the response is unbounded in principle but expected to level off in practice).

Back in the 1920s and 1930s the then famous biostatistician Raymond Pearl published again and again on how a logistic curve could be a good fit to all sorts of graphs, especially for population growth, much to the wicked amusement of some more sceptical contemporaries. More recently Theodore Modis seems to be following the same trajectory.

[Franz Hopp]

That's very helpful, thanks for the reply Nick Cox!