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Model-Based Adjustment

A model based method of computing adjustments that would account for the changes in the NHSDA methodology was used for estimates of the use of the more prevalent drugs including lifetime, past year and past month use of alcohol, marijuana, cigarettes, any illicit drug as well as past month binge drinking and past month heavy drinking. The model that was used is based on a constrained exponential model originally proposed by Deville & Särndal (1992). Similar to the ratio adjustment, this method of adjusting previous estimates models the combined effect of all measurement error differences between the new and old methodologies. This model offers the primary advantages of allowing (1) a greater number of potentially significant explanatory variables in the adjustment and (2) bounding the resulting adjustment between predetermined thresholds. This apriori bounding eliminates extreme adjustments that might otherwise occur, particularly for small subpopulations. Additionally, the model fitting procedure used to compute the adjustment forces the adjusted 1994-A estimates to equal the 1994-B estimates within the subpopulations represented by the dummy variables in the vector of model predictors. Mathematically, this model can be expressed as follows:

R_i~=~{L(U-1) ~+~ U(1-L) e^{-AX_i beta} } over
{(U-1) ~+~ (1-L) e^{-AX_i beta} }
(1)

the ratio adjustment R I can be interpreted as:

R_i~={ FUNC {Probability~Of~Reporting~Use~With~The~New~Survey~Methodology}} over
{ FUNC {Probability~Of~Reporting~Use~With~The~Old~Survey~Methodology}}

In equation (1) the constant A is simply a scale factor set equal to
left[ U-L right]~ DIV ~left[ (1-L)(U-1) right]
,
beta
are the model coefficients, and
X_i

is a vector of explanatory variables. The explanatory variables considered in the models consisted of the categorical indicator variables for age group and race/ethnicity. The parameters L and U are the predetermined constants that force the estimated
R_i

to be

L ~<=~ R_i~ <=~ U ~~~~~~~~~ func {for ~\all~ i ~\and~\for~\any~\value~\of~X_i beta}

.

Notice that if the constant Lis set equal to zero and Uapproaches , then the constant A approaches 1, and equation (1) reduces to the familiar, unconstrained exponential model:

R_i~=~{e^{-X_i beta} }
.

The model parameter vector
beta

in (1) was estimated by solving the generalized raking equations:

Sum from {i in S_{1994-A}} w_i ~R_i~X_i^T~y_i ~~=~~
Sum from {i in S_{1994-B}} w_i ~X_i^T~y_i

subject to the constraints.

Notice from the above raking equations that the estimated adjustment
R_i

forces the 1994-A estimate to equal the 1994-B estimate for any subpopulation represented by an indicator variable in
X_i

. Therefore, for example, if an appropriate indicator for the age group=12-17 year-olds was included in
X_i

, then the model-based estimate of the
R_i
's would produce an adjusted prevalence estimate using the 1994-A sample that exactly equaled the prevalence estimate generated from the 1994-B sample for the 12-17 year-old age group.

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This page was last updated on June 16, 2008.