An alternative jackknife variance estimator when calibrating weights to adjust for unit nonresponse in a complex survey
Section 5. The (bounded) logistic response model

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Up until this point, we have not specified a response function, $p(.)=1/f(.).$ Consider now a (bounded) logistic (or logit) response model having the form:

$$p({\gamma }^T{x}_k)=\frac{1+\exp ({\gamma }^T{x}_k)/U}{L+\exp ({\gamma }^T{x}_k)}.(5.1)$$

where $1\le L<U.$ When $L=1$ and $U$ is infinite, this is a standard logistic response model, where the response probability can range from 0 to 1, not including the endpoints. For finite values of $L$ and $U,$ the bounded probability of response falls between $1/U$ and $1/L.$ Consequently, the value of adjustment function ranges from $L$ to $U.$ In practice, $L$ is usually set to 1, while $U$ is frequently set as low as the sample allows for the calibration equation to hold.

The calibration procedures in the SUDAAN^® language (Research Triangle Institute, 2012), WTADJUST for when $x_k=z_k$ and WTADJX otherwise, fit an equivalent weight-adjustment function:

$$f(g^T{x}_k)=\frac{L+B\exp (A{g}^T{x}_k)}{1+B\exp (A{g}^T{x}_k)/U},(5.2)$$

where $A=\frac{U-L}{(C-L)(U-C)},$ $B=U\frac{C-L}{U-C},$ and $L<C<U.$

The choice of $C$ helps determine what $g$ satisfies the calibration equation but will not affect the value of the weight adjustment itself, $f_k=f(g^T{x}_k).$ Consequently, $C$ can be any value between $L$ and $U.$ When $L=1,$ $C=2,$ and $U$ is infinite, $\text{A}=\text{B}=1.$

A little calculus reveals with the weight-adjustment function in equation (5.2):

$$f_k^{\prime }=f^{\prime }(g^T{x}_k)=\frac{(U-f_k)(f_k-L)}{(U-C)(C-L)},$$

which is needed to compute equation (4.3) or (4.4). The general exponential model in the SUDAAN calibration procedures allow the $L,$ $C,$ and $U$ to vary from element to element, a flexibility hard to interpret in response modeling and not considered here.

What will be useful here, although not for modeling, is the possibility that $L$ in equation (5.2) is 0 and $U$ is infinite. When iteratively solving a calibration equation for $g$ with $f(g^T{x}_k)=\exp (g^{T}x)$ using Newton’s method, the SUDAAN calibration procedures first solve for $g_1$ in the calibration equation with $f(g_1^T{x}_k)=1+g_1^T{x}_k,$ which is a useful result when computing alternative jackknife weights. (The programs set the first iteration of the weight adjustment at $f_{k1}=\exp (g_1^T{x}_k)$ from which $1+g_1^T{x}_k$ is easily derived.)

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