11.7 Adding Explanatory Variables

Assessing the effects of an intervention, grouping variable or construct on change in another construct is a central goal in virtually all areas of the social, health and behavioral sciences.

• developmental researchers interested in how the occurrence of particular life events (e.g., job loss, marriage) affects changes in well-being or personality.
• educational psychology researchers investigate how teacher characteristics (e.g.,competence, motivation) predict changes in students’ achievement.

Most researchers rely on one of two basic strategies when analyzing the prospective effects of one construct on another construct in the context of two-wave data: (1) the ANCOVA model or (2) the difference score model.

For example, in this situation the autoregressive model becomes the traditional ANCOVA model

\[ y_{2i} = \beta_{0} + \beta_{1} x_{i} + \beta_{2} y_{1i} + \epsilon_{i} \] and we have the following difference score model

\[ \Delta_{i} = \beta_{0} + \beta_{1}x_{i} + \epsilon_{i} \] where \(x_{i}\) is either zero or one for the \(i^{th}\) individual in the sample. Although not immediately obvious, the difference between these two approaches is that \(\beta{2}\) is estimated in the ANCOVA model, and fixed to \(1\) in the difference score approach.

As pointed out by many different researchers over the year, it can be challenging to decide which of the two approaches is more appropriate. However, the decision is often critical as the two approaches can yield results that substantially differ concerning the magnitude, sign, and statistical significance of the estimated treatment effect.