Transforming the time scale in linear multivariate growth curve models

Latente Wachstumskurven-Modelle zeigen wiederholte Messungen von Ergebnis-Variablen als Funktionen von aufeinanderfolgenden Zeitpunkten und anderen Maßen. Einige Autoren bemerkten bereits, dass die Beziehung zwischen dem anfänglichen Status und der Wachstumsrate von der Zeitskala abhängt, die mit dem Modell verbunden ist. Verschiedene Zeitskalen führen zu verschiedenen Schätzungen dieser beiden Wachstumsparameter, wie auch ihre Varianzen und Ko-Varianzen. Im vorliegenden Beitrag betrachtet der Autor ein multivariates Wachstumskurven-Modell, in dem die Beziehung zwischen den Mustern des Wandels durch mehr als eine Ergebnis-Variable modelliert werden kann. Es wird gezeigt, dass die Abhängigkeit auch im multivariaten Fall in Erscheinung tritt. Es wird ein mathematischer Beweis erbracht, in dessen Rahmen eine Verbindung von anfänglichem Status und Wachstumsrate mit einer ausgewählten Zeitskala hergestellt wird. Das Wesen dieser Verbindung wird anhand von Modellen mit einer verschiedenen Zeitskala für dieselben empirischen Daten veranschaulicht. (ICIÜbers)'Latent growth curve models represent repeated measures of outcome variables as functions of consecutive time points and other measures. Already a few authors noticed that the relationship between the initial status and growth rate depends on the time scale involved in the model. Different time scales lead to different estimates of these two growth parameters, as well as their variances and covariances. In this article the author's consider the multivariate growth curve model, in which the relationship between patterns of change of more than one outcome variable can be modeled. The author's will show that the dependency also occurs in the multivariate case. Mathematical evidence will be presented in which the relationship will be established of initial status and growth rate with the selected time scale. The nature of the relationship will be illustrated an models with a different time scale for the same empirical data.' (author's abstract

On the Structure of Measurements in Facet Theory

facet theory, mapping sentence, multidimensional scaling, error, response function, principal components, homogeneous groups, inhomogeneous groups, dimensionality,

Systematic variations of 34 unbalanced 2 x 2 designs with 120 subjects.

<p>Systematic variations of 34 unbalanced 2 x 2 designs with 120 subjects.</p

The SS of example data, using (0, 1) coding for analysis.

<p>The SS of example data, using (0, 1) coding for analysis.</p

Number of unbalanced designs (percentages of all 34 designs) with rejection rates of H0 that are higher or lower when compared to the rejection rates of the balanced data (incongruous data).

<p>Number of unbalanced designs (percentages of all 34 designs) with rejection rates of H0 that are higher or lower when compared to the rejection rates of the balanced data (incongruous data).</p

The SS of Example data, using (-1, 1) coding for analysis.

<p>The SS of Example data, using (-1, 1) coding for analysis.</p

Venn diagram of the example data [13], using contrast coding (-1, 1).

<p>Venn diagram of the example data [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121412#pone.0121412.ref013" target="_blank">13</a>], using contrast coding (-1, 1).</p

Venn diagram of the example data [13], using treatment coding (0, 1).

<p>Venn diagram of the example data [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121412#pone.0121412.ref013" target="_blank">13</a>], using treatment coding (0, 1).</p

Mean rejection rates of H0 (congruous data analysis).

<p>Mean rejection rates of H0 (congruous data analysis).</p