Testing the Parallel Lines Assumption

For an ordinal response, PROC SURVEYLOGISTIC performs a test of the parallel lines assumption. In the displayed output, this test is labeled Score Test for the Equal Slopes Assumption when the LINK= option is NORMIT or CLOGLOG. When LINK=LOGIT, the test is labeled as Score Test for the Proportional Odds Assumption in the output. This section describes the methods used to calculate the test.

For this test, the number of response levels, $D+1$, is assumed to be strictly greater than 2. Let Y be the response variable taking values $1, \ldots , D, D+1$. Suppose there are k explanatory variables. Consider the general cumulative model without making the parallel lines assumption:

\[  g(\mbox{Pr}(Y\leq d~ |~ \mb {x}))= (1,\mb {x})\btheta _ d, \quad 1 \leq d \leq D  \]

where $g(\cdot )$ is the link function, and ${\btheta }_ d=(\alpha _ d, \beta _{d1}, \ldots , \beta _{dk})^\prime $ is a vector of unknown parameters consisting of an intercept $\alpha _ d$ and k slope parameters $\beta _{k1}, \ldots , \beta _{kd}$. The parameter vector for this general cumulative model is

\[  \btheta =(\btheta ’_1,\ldots ,\btheta ’_ D)’  \]

Under the null hypothesis of parallelism $H_0\colon \beta _{1i}=\beta _{2i}=\cdots =\beta _{Di}, 1 \leq i \leq k $, there is a single common slope parameter for each of the s explanatory variables. Let $\beta _1,\dots ,\beta _ k$ be the common slope parameters. Let $\hat{\alpha }_1, \ldots , \hat{\alpha }_ D$ and $\hat{\beta }_1, \ldots , \hat{\beta }_ D$ be the MLEs of the intercept parameters and the common slope parameters. Then, under $H_0$, the MLE of $\btheta $ is

\[  \hat{\btheta }=(\hat{\btheta }’_1,\ldots ,\hat{\btheta }’_ D)’ \quad \mbox{with} \quad \hat{\btheta }_ d=(\hat{\alpha }_ d,\hat{\beta }_1,\ldots , \hat{\beta }_ k)’, \quad 1 \leq d \leq D  \]

and the chi-square score statistic ${\mb {g}}’(\hat{\btheta })\mb {I}^{-1}(\hat{\btheta }){\mb {g}}(\hat{\btheta })$ has an asymptotic chi-square distribution with $k(D-1)$ degrees of freedom. This tests the parallel lines assumption by testing the equality of separate slope parameters simultaneously for all explanatory variables.