The SURVEYLOGISTIC Procedure

Determining Observations for Likelihood Contributions

If you use the events/trials syntax, each observation is split into two observations. One has the response value 1 with a frequency equal to the value of the events variable. The other observation has the response value 2 and a frequency equal to the value of (trials – events). These two observations have the same explanatory variable values and the same WEIGHT values as the original observation.

For either the single-trial or the events/trials syntax, let j index all observations. In other words, for the single-trial syntax, j indexes the actual observations. And, for the events/trials syntax, j indexes the observations after splitting (as described previously). If your data set has 30 observations and you use the single-trial syntax, j has values from 1 to 30; if you use the events/trials syntax, j has values from 1 to 60.

Suppose the response variable in a cumulative response model can take on the ordered values $1, \ldots , k, k+1$ , where k is an integer $\geq 1$ . The likelihood for the jth observation with ordered response value $y_ j$ and explanatory variables vector ( row vectors) $\mb {x}_ j$ is given by

$\begin{eqnarray*} L_ j = & \left\{ \begin{array}{ll} F(\alpha _1+\mb {x}_ j\bbeta ) & y_ j=1 \\ F(\alpha _ i+\mb {x}_ j\bbeta )- F(\alpha _{i-1}+\mb {x}_ j\bbeta ) & 1<y_ j=i\leq k \\ 1-F(\alpha _ k+\mb {x}_ j\bbeta ) & y_ j=k+1 \end{array} \right. \end{eqnarray*}$

where $F(.)$ is the logistic, normal, or extreme-value distribution function; $\alpha _1,\ldots ,\alpha _ k$ are ordered intercept parameters; and $\bbeta$ is the slope parameter vector.

For the generalized logit model, letting the $k+1$ st level be the reference level, the intercepts $\alpha _1,\ldots ,\alpha _ k$ are unordered and the slope vector $\bbeta _ i$ varies with each logit. The likelihood for the jth observation with ordered response value $y_ j$ and explanatory variables vector $\mb {x}_ j$ (row vectors) is given by

$\begin{eqnarray*} L_ j & =& {\Pr }({Y}=y_ j|\mb {x}_ j) \\ & =& \left\{ \begin{array}{ll} \displaystyle \frac{ {e}^{\alpha _ i+\mb {x}_ j{\bbeta }_ i}}{1+\sum _{i=1}^{k} {e}^{\alpha _ i+\mb {x}_ j {\bbeta }_ i}} & 1\le y_ j=i\le k \\ \displaystyle \frac{1}{1+\sum _{i=1}^{k} {e}^{\alpha _ i+\mb {x}_ j {\bbeta }_ i}} & y_ j=k+1 \end{array} \right. \end{eqnarray*}$