# Formulating a hierarchical model in educational measurement

I'm having some trouble positing a model in levels for the question below.

Question: How does the percentage of students born in the United States at the school level moderate the effect of the use of applications or models while learning science (denoted as SCAPPLY) on PISA science scores (PISA - outcome variable)?

Level-1 Predictors: ESCS = Index of economic, social and cultural status (or socioeconomic status (SES)); SCAPPLY; SCINVEST = An index measuring the degree to which students report that they conduct investigations when learning science; SCINTACT = Student Interaction.

Level-2 Predictor: NATIVE = the percentage of students born in the US for each school in the sample

In the model, we allow the effect of ESCS to vary at level-2; all other slopes at level-1 remained fixed at level-2. Also, we allow the school-level intercepts to vary and the variability in the ESCS slopes to vary with the school-level intercepts.

So far I have the following.

----------------------------------------------------EDIT-------------------------------------------------------

Level 1: $$PISA_{ij} = \beta_{0j} + \beta_{1j}(ESCS_{ij}) + \beta_{2j}(SCAPPLY_{ij}) + \beta_{3j}(SCINTACT_{ij}) + \beta_{4j}(SCINVEST_{ij}) + e_{ij}$$

Level 2: \begin{align} \beta_{0j} &= \gamma_{00} + \mu_{0j}\\ \beta_{1j} &= \gamma_{10} + \mu_{1j}\\ \beta_{2j} &= \gamma_{20} + \gamma_{21}(NATIVE_j)\\ \beta_{3j} &= \gamma_{30}\\ \beta_{4j} &= \gamma_{40} \end{align}

Did I posit my models in levels correctly?

• One thing's for sure given the question, $ESCS$ is at level 1, not level 2 as it is now in your equations. – Patrick Coulombe May 18 '14 at 17:09
• @PatrickCoulombe Yes, that's what I thought. Thank you. Also, I revised the problem. Could you check my new models? – Dan May 18 '14 at 18:48