An alternative way of estimating a cumulative logistic model with complex survey data
Section 3. Discussion

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When there is more than one explanatory variable in the cumulative logistic model then each one needs to be tested like $meds$ was in the previous section by adding an analogous class variable for each. A general F test can be used for testing whether every class variable is not significant (say at the 0.05 level). A better approach with complex survey data may be to follow Korn and Graubard (1990) and use the simple Bonferroni-adjusted $t$ -test. For significance at the 0.05 level, one would compute the $t$ -values for every tested component of each added class variable (there are three such in Table 2.3), then compare the $p$ -value of the smallest of these to 0.05/the number of components tested.

An advantage of the design-sensitive model-based approach to fitting a simple cumulative logistic model over the pseudo-maximum-likelihood approach is not apparent with our NSDUH data. When the parallel-lines assumption doesn’t hold, and an extended model is being fit, satisfying the first “equation” in (1.4) assures us that

${\displaystyle {\sum }_{k\in S}{w}_k{y}_{lk}={\displaystyle {\sum }_{k\in S}{w}_k\frac{\exp \left(a_l+x_{k}b\right)}{1+{\displaystyle {\sum }_{j=1}^{L-1}\exp \left(a_j+x_{k}b\right)}}}}$ for $l=1,\dots ,L-1.(3.1)$

When $x_k$ itself is a multi-level categorical variable (so that one and only one component of $x_k=\left(x_{k1},\dots ,x_{kQ}\right)$ is 1 while the other components are 0), equation (3.1) assures that the weighted mean of $y_{lk}$ for each $x$ -category (i.e., component of $x_k)$ and cumulative level $l$ equals its predicted value described by

$${\widehat{y}}_{lk}=\exp \left(a_l+x_k^{T}b\right)/\left[1+{\displaystyle \sum _{j=1}^{L-1}\exp \left(a_j+x_k^{T}b\right)}\right],$$

which is a reasonable property. Equation (1.4) is simply an extension of the property to more general $x_k.$

In our NSDUH example, although not generally, using the design-sensitive approach was slightly more efficient than using the pseudo-maximum-likelihood approach. This can be seen by comparing the $t$ -values of $meds$ (the inverses of their respective estimated coefficients of variation) in Tables 2.1 and 2.2. When we ignore the analysis weights, the strata, and the clustering (by setting the weights and strata to 1, and treating each respondent as a primary sampling unit), this result reverses as expected. The point here is that pseudo-maximum likelihood with complex survey data is indeed “pseudo” (in this case that is likely because of the impact of the weights on the estimates).

Finally, the data set we created dropped responding observations with missing values of the dependent and meds variables. When fitting the extended model, this is only valid (i.e., resulting estimates are asymptotically unbiased) when an in-scope respondent − an adolescent who had treatment for depression in the previous year − being dropped occurred completely at random. When fitting the standard model, the probability of being dropped can be a function only of whether an in-scope adolescent has taken medication for depression in the previous year but nothing else. This suggests it may have been prudent to add variables to the model that are never missing even when they are not significant. If we add class variables for age, sex, race/ethnicity, urbanicity, and family income (all of which have values imputed for them when missing in the NSDUH) to our simple cumulative logistic model, none are significant at the 0.05 level. The major results do not change meaningfully (the estimate for $\beta$ increases from roughly 0.45 to 0.50), although that the $t$ -value for meds using the design-sensitive approach $\left(b_{meds}=0.4948;t_{meds}=5.49\right)$ is slightly smaller than that from using the pseudo-maximum-likelihood approach $\left(b_{meds}=0.4987;t_{meds}=5.52\right).$

Appendix

/* PML is a data set of adolescents NSDUH respondents in the 2006 to 2010 survey years who reported having treatment for depression and whether they had taken drugs for depression. Variables include:

Y = 1 treatment was extremely helpful; Y = 2 treatment helped a lot; Y = 3 some; Y = 4 a little;

Y = 5 not at all

meds = 1 had taken drugs for depression, 0 otherwise

VESTR variance stratum

VEPSU variance primary sampling unit

IDNUM respondent identification number

ANALWT the analysis weight

This set is employed for pseudo-maximum-likelihood estimation of the simple cumulative logistic model and to create the DS_SIMPLE data set, which is used for design-sensitive estimation of the simple cumulative logistic model, and it is employed to create DS_GENERAL data set, which is used for design-sensitive estimation of the general cumulative logistic model. */

DATA DS_SIMPLE; SET PML; BY VESTR VEPSU IDNUM;
D = 0;
C = 1; IF Y < 2 THEN D = 1; OUTPUT;
C = 2; IF Y < 3 THEN D = 1; OUTPUT;
C = 3; IF Y < 4 THEN D = 1; OUTPUT;
C = 4; IF Y < 5 THEN D = 1; OUTPUT;

DATA DS_GENERAL; SET DS_SIMPLE;
M = 4;
IF C = 1 AND MEDS = 1 THEN M = 1;
IF C = 2 AND MEDS = 1 THEN M = 2;
IF C = 3 AND MEDS = 1 THEN M = 3;

/*The PROC below is used to produce Table 2.1*/

PROC SURVEYLOGISTIC DATA = PML; CLUSTER VEPSU;
MODEL Y = MEDS;
STRATA VESTR; WEIGHT ANALWT; RUN;

/*The PROC below is used to produce Table 2.2*/

PROC SURVEYLOGISTIC DATA = DS_SIMPLE; CLASS C;
CLUSTER VEPSU;
MODEL D(EVENT = '1') = C MEDS;
STRATA VESTR; WEIGHT ANALWT; RUN;

/*The PROC below is used to produce Tables 2.3 and 2.4*/

PROC SURVEYLOGISTIC DATA =DS_GENERAL; CLASS M C;
CLUSTER VEPSU;
MODEL D(EVENT = '1') = C MEDS M;
STRATA VESTR; WEIGHT ANALWT; RUN;

References

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