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The informative sampling process is accounted for by weighting the likelihood contributions of the observations by the inverses of the sampling probabilities.
In this appendix, we derive the likelihood contributions for time-grouped data, conditional on observed and unobserved variables.
To derive expressions for the likelihood contributions of all households, we introduce two times four couple-specific dummy variables d i r, 1 and d i r, 2 (r = 1,…,4) that indicate the (sub regime in which a couple operates, on the basis of the preference parameters of both spouses and their wage rates.
Hence, the likelihood function is given by the product of the likelihood contributions of the individual households.
To ease the calculations, the likelihood contributions from the screening and interval data have been taken as independent.
In the selective decoding model, the computed decoding weight of each neuron was then multiplied by that neuron's contribution to the likelihood function (Equation 4) before the likelihood contributions are summed across neurons.
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The likelihood contribution from worker i whose earnings fall on interval ]y 1i, y 2i ] is P r y 1i <Y i ≤y 2i ); in the case of the right-censored interval, it is P r(Y i >y Ri ), and for the left-censored interval, it is P r(Y i ≤y Li ).
The likelihood contribution function of a household h for an observation xi,h in each of consumption regime k (positive or null) is: L h, k x i, h = ∫ η : η ≤ r f η h d η h f e 2, h ∂ e 2, h ∂ s 2, h (14).
For the combined analysis unrelated subjects were assumed to have two missing parents and parameters are set to give the likelihood contribution for the family.
The likelihood contribution of the ith individual,, is weighted by n i to allow for differential call rates between samples.
Replacing this expected value in the likelihood contribution (equation A1), π cancels out and we are left with the original weights.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com