Bias-Variance Trade-Off in Continuous Test Norming

Abstract

In continuous test norming, the test score distribution is estimated as a continuous function of predictor(s). A flexible approach for norm estimation is the use of generalized additive models for location, scale, and shape. It is unknown how sensitive their estimates are to model flexibility and sample size. Generally, a flexible model that fits at the population level has smaller bias than its restricted nonfitting version, yet it has larger sampling variability. We investigated how model flexibility relates to bias, variance, and total variability in estimates of normalized z scores under empirically relevant conditions, involving the skew Student t and normal distributions as population distributions. We considered both transversal and longitudinal assumption violations. We found that models with too strict distributional assumptions yield biased estimates, whereas too flexible models yield increased variance. The skew Student t distribution, unlike the Box–Cox Power Exponential distribution, appeared problematic to estimate for normally distributed data. Recommendations for empirical norming practice are provided.