Climate-smart agriculture in Zambia
The Bayesian Multinomial Logit
Three frightening words, taken one at a time. The third one was forced on me by a category with a single household in it.
Take the name apart and it stops being frightening.
Multinomial means the answer is not yes or no. Households do not adopt or fail to adopt. They end up holding a bundle, running from none at all through to several practices combined, so the outcome has seven possible values rather than two.
Logit means the model works in odds rather than in probabilities directly, which is mostly housekeeping to keep predictions between nought and one where they belong.
Bayesian was not chosen for elegance. It was chosen because the ordinary approach broke. One of the bundle categories contained a single household, and standard estimation cannot put an honest number on a category with one household in it. The maths runs towards infinity and the software hands back a coefficient with no measure of uncertainty attached. That is not a fault in the software. It is the data saying it does not know.
The Bayesian version starts from a deliberately vague statement of what is plausible before looking at the data, wide enough to admit any effect you would actually expect in farming, then reports how far the data moved it.
What it cannot do
There is a second limit, and it is the standard objection to any model of this shape. A multinomial logit assumes independence of irrelevant alternatives: that how a household chooses between two bundles does not depend on what else is on the menu. In farming that is questionable, because unobserved things like managerial skill or soil quality push on every option at once. The assumption is tested rather than assumed here, but a test that passes is not the same as a thing being true.
Where the data is rich, the data dominates and the starting point is irrelevant. Where the data is thin, the answer is largely that starting point handed back to you. The obligation is to say which is which, and in the two sparse categories I read the results as description only.
The equation
ln [ P(y = j) / P(y = 0) ] = xβj with βj ~ Normal(0, 2.5)
- P(y = j)
- the chance a household ends up holding bundle j
- P(y = 0)
- the chance it adopts nothing, the category everything is measured against
- x
- what we know about the household: land, schooling, credit, extension, and the rest
- βj ~ Normal(0, 2.5)
- the prior. Before looking, treat any effect between roughly 0.007 and 148 times the odds as possible
The prior is the whole reason this version exists. It is wide enough to allow anything you would actually expect in farming, and still finite, which is what keeps the one-household category from running off to infinity.
Further reading
- Gelman, A., Jakulin, A., Pittau, M. G. and Su, Y.-S. (2008). "A Weakly Informative Default Prior Distribution for Logistic and Other Regression Models", Annals of Applied Statistics. free Read it.The prior in this model comes from this paper, and it explains the separation problem that forced the choice.
- McFadden, D. (1974). "Conditional Logit Analysis of Qualitative Choice Behavior".The random utility foundation the whole model sits on.
From the Zambia climate-smart agriculture research, built on the Water and Soil Accelerator household survey, which was funded by USAID. The thesis is under examination and the three papers drawn from it are under anonymous peer review, so there is nothing to link to yet.