Reaction to “Sufficient statistics and insufficient explanations”: Use your information

In the reaction of van Breukelen (2019), the claim is made that the sum score is a sufficient statistic for the latent trait parameter (hij of person i at time point j) in a latent growth model analysis. The argumentation is given that with the sum score as the sufficient statistic, the estimated between-subject variance cannot be biased and the estimated within-subject variance is contaminated with a measurement error variance term. We agree that differences in item response patterns leading to the same sum score become irrelevant when the sum score is the sufficient statistic. However, we can show that the sum score is not the sufficient statistic for the latent trait parameter in the longitudinal IRT model (i.e. latent growth model with IRT measured longitudinal latent variables). First, we show that additional information in the data about hij is ignored, when considering the sum score as the sufficient statistic. Thus, the sum score is not sufficient. Second, we show that the estimated variance components (within-subject and between-subject) are contaminated with unexplained error variance, when using the sum score as the outcome variable instead of the item response data. This supports the conclusions of our paper. It is a common mistake to assume that the sum score is the sufficient statistic for the latent trait parameter, when the item responses are conditionally independently distributed given the latent trait (as in the Rasch model). This is only true when the data do not provide additional information about the latent trait. The longitudinal data consist of repeated measurements, where on each measurement occasion, a latent trait is measured and a latent growth model is assumed for the longitudinal latent traits. This latent growth model defines a distribution for the occasion-specific latent trait parameters. The data from the different measurement occasions for a subject are relevant for each occasion-specific latent trait measurement due to this distribution. Because data from the other measurement occasions provide information about each occasion-specific latent trait, the sum score is not the sufficient statistic. The influence of the additional information on the latent trait is easily illustrated by considering the posterior expected value for hij given the data. Consider quantitative item responses Zijk of persons i, measurement j and item k, let the Zijk be normally distributed, and we assume a linear trend for the latent trait parameter