Investigating alternative estimators for the prevalence of serious mental illness based on a two-phase sample
Section 2. Some estimators

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2.1  Across all adults

Let $S$ denotes the relevant NSDUH respondent sample (adults 18 years or older) from 2008 through 2012, and $w_k$ the NSDUH analysis (first-phase) weight for an individual $k\in S.$ Let $S^{\prime }$ denotes the subsample of $S$ responding to a clinical evaluation of their SMI status. Let $y_k=1$ when $k$ is diagnosed to have serious mental illness, and $y_k=0$ when $k$ is diagnosed not to have serious mental illness. Let ${\omega }_k$ be the two-phase weight for an individual $k$ in $S^{\prime }\text{}.$  For convenience, we set ${\omega }_k$  to 0 for individuals in $S$ but not $S^{\prime }\text{}.$

In actual practice, both sets of weights have been adjusted to account for nonresponse and undercoverage and to increase their efficiency, but we will ignore that fact here for simplicity. Instead, we will assume $1/w_k$ is the probability of selection for a NSDUH respondent, $1/{\omega }_k$ the probability of selection for a MHSS subsample respondent, and thus $w_k/{\omega }_k$ the conditional selection probability of a subsample respondent given (s)he was a NSDUH respondent. A nearly unbiased estimator for the prevalence of SMI among adults between 2008 and 2012 based on the two-phase sample is ${\overline{y}}_U={\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{y}_k}/{\displaystyle {\sum }_{S^{\prime }}{\omega }_k}\text{},$ “nearly” because the denominator may contain some sampling error.

Suppose a ${\omega }_k-$ weighted logistic regression is run on the all-adult MHSS subsample respondents in $S^{\prime }$ with $y_k$ as the dependent variable using a reasonable vector of explanatory covariates, $x_k\text{},$ available for every respondent in the adult NSDUH sample. Exactly how the covariates have been chosen is beyond the scope of this investigation (for that, the reader is directed to Center for Behavioral Health Statistics and Quality, 2015; Chapter 4). Let the predictor for $y_k$ from this weighted-logistic regression be $p_k=p\left(x_k^{\prime }b\right)={\left[1+\exp \left(-x_k^{\prime }b\right)\right]}^{-1}\text{}.$

The use of weights in fitting the logistic-regression model protects against the possibility that the model residuals are correlated with the probabilities of selection. It is also consistent with how SMI prevalence was estimated; that estimate resulted from the weighted regression of $y_k$ on the constant 1 and no covariates.

Sorting the subsample by the $p_k$ values, one can find the cut point value $p_C$ such that

$${\displaystyle \sum _{\begin{array}{c}k\in S^{{}^{\prime }}\\ p_k\ge p_C\end{array}}{\omega }_k}={\displaystyle \sum _{k\in S^{{}^{\prime }}}{\omega }_k{y}_k}(2.1)$$

holds exactly or as nearly so as possible. That is to say, the estimated number of adults in the population having $p_k-$ values at or above the cut point approximately equals the estimated number of adults with SMI. Define an indicator random variable $c_k$ to be 1 when $p_k\ge p_C$ and 0 otherwise. A cut point determined using equation (2.1) also comes as close as possible to equalizing the weighted false positives $\left({\displaystyle {\sum }_{S^{{}^{\prime }}\text{}:c_k=1}{\omega }_k\left(1-y_k\right)}\right)$ and false negatives $\left({\displaystyle {\sum }_{S^{{}^{\prime }}\text{}:c_k=0}{\omega }_k{y}_k}\right)$ in $S^{\prime }\text{}.$

Two alternative estimators for SMI prevalence among adults are the model-driven cut point and probability estimators:

$${\overline{y}}_C=\frac{{\displaystyle {\sum }_S{w}_k{c}_k}}{{\displaystyle {\sum }_S{w}_k}},(2.2)$$

and

$${\overline{y}}_P=\frac{{\displaystyle {\sum }_S{w}_k{p}_k}}{{\displaystyle {\sum }_S{w}_k}},(2.3)$$

these estimators are computed using the entire NSDUH sample rather than the smaller MHSS subsample as is ${\overline{y}}_U.$

We assume now that one of the covariates in the logistic model is 1 or the equivalent $(x_k^{\prime }\gamma =1$ for some $\gamma ).$ Under this assumption, the probability estimator for SMI prevalence is exactly equal to a bias-corrected probability estimator given below:

$$\begin{array}{ll}{\overline{y}}_{P-\text{BC}}\hfill & =\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{y}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}+\left(\frac{{\displaystyle {\sum }_S{w}_k{p}_k}}{{\displaystyle {\sum }_S{w}_k}}-\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{p}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}\right)\hfill \\ \hfill & =\frac{{\displaystyle {\sum }_S{w}_k{p}_k}}{{\displaystyle {\sum }_S{w}_k}}+\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k(y_k-p_k)}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}.(2.4)\hfill \end{array}$$

The equality between ${\overline{y}}_P$ and ${\overline{y}}_{P-\text{BC}}$ results from the numerator of the bias-correction term in the second line of equation (2.4), ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-p_k\right)}/{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\text{},}$ equaling zero. Fitting a logistic regression forces ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-p_k\right)}{x}_k=0,$ and we have assumed $x_k$ contains 1 or the equivalent.

Since the expectation of the term in parentheses in the first line of equation (2.4) is nearly zero under mild conditions, ${\overline{y}}_P={\overline{y}}_{P-\text{BC}},$ like ${\overline{y}}_U,$ is nearly unbiased under survey-sampling theory. This is true whether or not the model used to determine the $p_k$ is correct so long as $b$ in $p_k=p\left(x_k^{\prime }b\right)={\left[1+\exp \left(-x_k^{\prime }b\right)\right]}^{-1}$ converges to something as the MHSS subsample and NSDUH sample sizes grow arbitrarily large.

The estimator ${\overline{y}}_{P-\text{BC}}$ is analogous to the popular GREG estimator. It follows Lehtonen and Veijanen (1998), and computes the $p_k$ with a logistic rather than the linear model of the GREG.

A bias-corrected cut point estimator is

$$\begin{array}{ll}{\overline{y}}_{C-\text{BC}}\hfill & =\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{y}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}+\left(\frac{{\displaystyle {\sum }_S{w}_k{c}_k}}{{\displaystyle {\sum }_S{w}_k}}-\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{c}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}\right)\hfill \\ \hfill & =\frac{{\displaystyle {\sum }_S{w}_k{c}_k}}{{\displaystyle {\sum }_S{w}_k}}+\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-c_k\right)}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}}.(2.5)\hfill \end{array}$$

Using the same logic as above, this estimator is also nearly unbiased under mild conditions. It is close to the model-driven cut point estimator since the bias-correction term, ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-c_k\right)}/{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k}\text{},$ is close to zero. The bias-correction term would be exactly zero if there were a cut point $p_C$ that satisfied equation (2.1) exactly.

2.2  Domain estimation

Let us now turn our attention to a subpopulation of all adults, such as males or all adults who have received treatment for mental illness (or all adults who live in a particular state). We call such a subpopulation a “domain” of interest. To estimate SMI prevalence in a domain, we can simply insert an indicator for domain membership, $d_k\text{},$ equaling 1 when $k$ is in the domain, 0 otherwise, into all our estimates:

$${\overline{y}}_{U\left(d\right)}=\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{y}_k{d}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}}(2.6)$$

$${\overline{y}}_{P\left(d\right)}=\frac{{\displaystyle {\sum }_S{w}_k{p}_k{d}_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}(2.7)$$

$${\overline{y}}_{C\left(d\right)}=\frac{{\displaystyle {\sum }_S{w}_k{c}_k{d}_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}(2.8)$$

$$\begin{array}{ll}{\overline{y}}_{P-\text{BC}\left(d\right)}\hfill & ={\overline{y}}_{U\left(d\right)}+\left(\frac{{\displaystyle {\sum }_S{w}_k{p}_k}{\displaystyle d_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}-\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{p}_k{d}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}}\right)\hfill \\ \hfill & =\frac{{\displaystyle {\sum }_S{w}_k{p}_k}{\displaystyle d_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}+\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-p_k\right)d_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}}(2.9)\hfill \end{array}$$

$$\begin{array}{ll}{\overline{y}}_{C-\text{BC}\left(d\right)}\hfill & ={\overline{y}}_{U\left(d\right)}+\left(\frac{{\displaystyle {\sum }_S{w}_k{c}_k}{\displaystyle d_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}-\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{c}_k{d}_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}}\right)\hfill \\ \hfill & =\frac{{\displaystyle {\sum }_S{w}_k{c}_k}{\displaystyle d_k}}{{\displaystyle {\sum }_S{w}_k{d}_k}}+\frac{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k\left(y_k-c_k\right)d_k}}{{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}}.(2.10)\hfill \end{array}$$

It is here where the bias-correction terms serve an important purpose. If the logistic model, which was fit on the subsample of all adults, holds within the domain, then ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k\left(y_k-p_k\right)}/{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}$ will be an estimate of zero, and the model-driven probability estimator, ${\overline{y}}_{P\left(d\right)}$ in equation (2.7), will be nearly unbiased. If the model does not hold in the domain (e.g., if males are more likely to have SMI than the model predicts), then the model-driven probability estimator can be significantly biased.

Adding the bias correction ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k\left(y_k-p_k\right)}/{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}$ to ${\overline{y}}_{P\left(d\right)}$ produces an estimator that is nearly unbiased under survey-sampling theory. When the model holds in the domain, however, applying the correction will almost certainly result in a decrease in accuracy. A similar argument can be made about the appropriateness of adding the ${\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k\left(y_k-c_k\right)}/{\displaystyle {\sum }_{S^{{}^{\prime }}}{\omega }_k{d}_k}$ term in equation (2.10) to the cut point estimator, ${\overline{y}}_{C\left(d\right)}\text{},$ in equation (2.8).

Equations (2.4) and (2.5) can be viewed as special cases of (2.9) and (2.10), respectively, with $d_k\equiv 1.$

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