Fig. 3.

Illustration of an ordered probit model. The upper line represents a continuous latent attainment score. The expected latent attainment score for the ith participant is given by ${\mu }_i=X_i^T\beta$ and represented by the position of the black dot on the upper line. A latent attainment score $y_i^{*}$ is then sampled from a normal distribution (curved black line) with mean µ_i and standard deviation σ. The observed attainment score then depends on which of the thresholds C1, . . ., C_K–1 (gray dotted lines) $y_i^{*}$ falls between. Here, $y_i^{*}$ falls between the second and third thresholds, giving an observed attainment score of 3. Note that the threshold parameters will not necessarily be equally spaced.