Now, scalogram pattern data that fit a Guttman scale model are extremely rare. More often, item data follow a pattern that can be explained by the more lenient IRT model. Figure 1E shows the attenuation effect when the true model is a one-parameter IRT model with variance 1, where all items are endorsed by fewer than half the participants (i.e., all β parameters larger than the average latent score). Again we see that under the usual IRT model, the attenuation effect depends on the number of items and again we see that due to the skewness of the sum score distribution, the attenuation is not proportional to the true correlation. For example, with five items the simulated sum score correlation equals 0.591256 for true correlation 0.9 (66%), 0.316222 for true correlation 0.5 (63%), and 0.056567 for true correlation 0.1 (57%). When true correlations are again 0.9 and 0.5, the most likely model would be, when based on an analysis of the sum scores with 5 items, 5% dominance genetic variance, 61% additive genetic variance and 34% non-shared environmental variance.
