1 Univariate Normal Distribution of Liability

One approach to the analysis of ordinal data is to assume that the ordered categories reflect imprecise measurement of an underlying normal distribution of liability. A second assumption is that the liability distribution has one or more threshold values that discriminate between the categories (see Figure 2.1). This model has been used widely in genetic applications (Falconer, 1960; Neale et al., 1986; Neale, 1988; Heath et al 1989a). As long as we consider one variable at a time, it is

Figure 2.1: Univariate normal distribution with thresholds distinguishing ordered response categories.

always possible to place the thresholds so that the proportion of the distribution lying between adjacent thresholds exactly matches the observed proportion of the sample that is found in each category. For example, suppose we had an item with four possible responses: `none', `a little', `quite a lot', and `a great deal'. In a sample of 200 subjects, 20 say `none', 80 say `a little', 98 say `quite a lot' and 2 say `a great deal'. If our assumed underlying normal distribution has mean 0 and variance 1, then placing thresholds at z-values of -1.282, 0.0 and 2.326 would partition the normal distribution as required. In mathematical terms, if there are $p$ categories, $p-1$ thresholds are needed to divide the distribution. The expected proportion lying in category $i$ is

$$\int_{t_{i-1}}^{t_{i}} \phi(x) \; dx$$

where $t_0=-\infty$, $t_p=\infty$, and $\phi(x)$ is the unit variance normal probability density function (pdf), given by

$$\phi(x) = \frac{e ^{-.5x^{2}}} {\sqrt{2\pi}}$$

This formulation is really a parametric model for the distribution of ordinal responses.