Computing optimal sampling designs for two-stage studies

Description

The meanscor command performs a weighted logistic regression using the mean score method. This function requires the
complete covariate(s) to be categorical, and the default output contains the regression coefficient estimates and their standard
errors in odds-ratio form.

An important area of application of this function is in the analysis of data from a two-stage study. In this type of study,
some variables are incomplete due to only a subset of the study subjects being sampled at the second stage (Reilly 1996).

Options

f irst Vaarlist) specifies the complete covariates.

second(aαrlist) specifies the incomplete covariates.

odd(#) specifies whether the odds-ratio (odd = 1) or regression coefficients (odd = 0) format is reported. Default value is 1.

Methods and Formulas

The mean score estimates will maximize the weighted likelihood

Z _nI(zi,yi) ∖

Σ (¹ + √⅛j) ⅛s^pβ(yih)

where          is the number of incomplete observations in each stratum defined by the different levels of response yι and

complete covariates Zi, and n^c^^i,yi^ is the number of complete observations in each stratum.

As the above equation indicates, the mean score method weights each complete observation according to the total number
of observations in the same stratum.

The asymptotic variance of the mean score estimate is given by

Var(3) = ɪ (Γ¹+Γ¹VΓ^v)

where n is total number of observations, and I is the usual information matrix. V is estimated by the matrix

n(y, Z)n¹^^y'^z'>

Σ  _nc(y,z)  V^ar(⅜*)

(У,*)

where Var{Sβ{y, z)) is the variance-covariance matrix of the score in each {y,z) stratum.

We can regard the second term of the variance expression as a penalty for the incompletely observed observations. Hence,
the mean score estimates will have larger variance than the estimates obtained if all observations were complete but smaller
variance than the estimates from an analysis of complete cases only.

Examples

We begin with a simulated dataset. We generated 1,000 observations of a predictor variable x from the standard normal
distribution. The response variable y was then generated as a Bernoulli random variable with p = ехр(ж)/{1 + ехр(ж)}. A
dichotomous surrogate variable for x. called z, was generated as one for positive x and zero otherwise.

A random subsample of 500 observations had their x value deleted (set to missing). The dataset, called sim_miss.dta is
provided with this insert as an illustration and can be analyzed using the mean score method by

. use sim_-miss

. meanscor у x,first(z) second(x)
meanscore estimates

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