An alternative way of estimating a cumulative logistic model with complex survey data
Section 2. A simple example

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The National Survey on Drug Use and Health (NSDUH) is an annual survey of the civilian, noninstitutionalized population aged 12 or older living in the United States. Using NSDUH data from 2006 to 2010, we focus on a survey question given to adolescents (12-17) who received depression treatment in the past year:

During the past 12 months, how much has treatment or counseling helped you?

The viable responses were: Not at all (l); A little (2); Some (3); A lot (4); or Extremely (5).

We discarded missing and invalid responses both to this question and to the question of whether the respondent received depression treatment in the past year. We will return to this practice in the discussion section.

Using SAS, we estimated the following simple cumulative logistic model:

$E\left(y_{lk}|x_k\right)=\frac{\exp \left({\alpha }_l+med{s}_k\beta \right)}{1+\exp \left({\alpha }_l+med{s}_k\beta \right)}$ for $l=1,\dots ,L-1,(2.1)$

where $meds=1$ when respondent $k$ was taking medication for depression (0 otherwise), with both pseudo-maximum-likelihood and the design-sensitive technique. For pseudo-maximum-likelihood estimation, we reversed the order of the responses with $y_{1k}=1$ when $k$ responded that treatment (or counseling) helped extremely, $y_{2k}=1$ when $k$ responded that treatment helped extremely or a lot, $y_{3k}=1$ when $k$ responded that treatment helped more than a little, and $y_{4k}=1$ when $k$ responded that treatment helped at least a little. Finally, $y_{5k}=1-y_{4k}=1$ when $k$ responded that treatment did not help at all. In SAS, this meant dependent variable $\text{Y}$ was set equal to 1 when treatment helped extremely, to 2 when treatment helped a lot, $\dots ,$ and to 5 when treatment didn’t help at all.

For the design-sensitive technique, we created four observations from $k$  in a new data set. In the $i^{\text{th}}$ observation labeled $\text{C}=i$ in SAS, a class (categorical) variable added to the model statement, we created a dependent variable (D) equal to $y_{ik}$ in equation (2.1). We needed to add EVENT = “1” after D in the model statement because we were modeling when $\text{D}=1.$

SAS code for both estimation techniques are in the appendix. The NSDUH data set we used had 60 variance strata with two variance primary sampling units (PSUs) in each and analysis weights based on the probabilities of selection and unit response.

The parameter estimates from our pseudo-maximum-likelihood and design-sensitive SAS runs are displayed in Tables 2.1 and 2.2, respectively. In Table 2.1, Intercept  $=i$ is the pseudo-maximum-likelihood estimate of ${\alpha }_{ik}$ in equation (2.1). The sum of the Intercept and $\text{C}=i$  in Table 2.2 is the design-sensitive estimate for ${\alpha }_{ik}$ when $i=1,2,$ or $3,$ while the design-sensitive estimate for ${\alpha }_{4k}$ is the Intercept in Table 2.2 minus the sum: $\text{[C}=1]+[\text{C}=2]+[\text{C}=3].$ Finally (and more simply), meds in both tables estimates  $\beta \text{.}$

In all cases, estimates of the same parameter from the two tables are close. The percent increase in every level of satisfaction with treatment due to having taken drugs for depression (the estimate for $\beta )$ is roughly 45% (in our discussion of the results of the logistic regressions, we treat differences of the log odds as equal to percent differences in the odds, even though this is only approximately true). That near equality suggests that the parallel-lines assumption is not violated by our NSDUH data.

The parallel-lines assumption can be tested directly by adding a class variable M to the design-sensitive data set with

$$\begin{array}{l}\text{M} = 1\text{ when C} = 1\text{ and }meds=1,\hfill \\ \text{M} = 2\text{ when C} = 2\text{ and} meds=1,\hfill \\ \text{M} = 3\text{ when C} = 3\text{ and} meds=1,\text{ and}\hfill \\ \text{M} = 4\text{ otherwise}.\hfill \end{array}$$

When added to the model statement in SAS, the class variable M captures the differing impacts of taking medication for depression in the previous year on the levels of satisfaction with treatment. For example, the estimated percent increase in the odds of being extremely pleased by treatment due to having taken drugs for depression during the year is, according to Table 2.3, 0.3816 (from $meds)$ plus 0.0717 (from M = 1) or 45.33%. The other percent increases are lower, but none are significantly different from the others. We see that from the extremely low F value for M in Table 2.4. In addition, none of the $t$ -values for an M in Table 2.3 is significant even at the 0.5 level (10 times larger than the standard 0.05 level).

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