Note: For a fuller treatment, download our series of lectures Hierarchical Linear Models. The Empty Model As a first step, R&B begin with an empty model containing no covariates.
\[ \begin{equation}Y_{ij} = \beta_{0j} + e_{ij}\tag{1}\end{equation} \]
Each school’s intercept, \(\beta_{0j}\), is then set equal to a grand mean, \(\gamma_{00}\), and a random error \(u_{0j}\).
\[ \begin{equation}\beta_{0j} = \gamma_{00} + u_{0j}\tag{2}\end{equation} \]
Substituting (2) into (1) produces
\[ \begin{equation}Y_{ij} = \gamma_{00} + u_{0j} + e_{ij}\tag{3}\end{equation} \]

Note: For a fuller treatment, download our series of lectures Hierarchical Linear Models. Means-as-Outcomes Model After estimating the empty model, R&B develop a Means-as-Outcomes model in which a school-level variable, meanses, is added to the model for the intercept. This variable reflects the average student SES level in each school. Recall Equation (1):
\[ \begin{equation}Y_{ij} = \beta_{0j} + e_{ij}\tag{1}\end{equation} \]
The intercept can be modelled as a grand mean \(\gamma_{00}\), plus the effect of the average SES score \(\gamma_{01}\), plus a random error \(u_{0j}\).

Note: For a fuller treatment, download our series of lectures Hierarchical Linear Models. Random Coefficient Model Next, R&B present a model in which student-level SES is included instead of average SES, and they treat the slope of student SES as random. One complication is that R&B present results after group-mean centering student SES. Group-mean centering means that the average SES for each student’s school is subtracted from each student’s individual SES.

Conducting ANOVA in SPSS We want to study the effectiveness of different treatments on anxiety. We collect a sample of 75 subjects in the following categories:
No treatment (\(n_1\) = 27). Biofeedback (\(n_2\) = 24). Cognitive-behavioral Treatment (\(n_3\) = 24). The dependent variable is anxiety levels. In this analysis, we will include both planned contrasts and post hoc comparisons.
Steps to doing an ANOVA The steps to doing an ANOVA in SPSS are as follows:

One of the biggest benefits of using R is its flexibility in working with various types of data used by other statistical software. If you are collaborating with other researchers, they may be working with data produced by SAS (.sas7bdat), SPSS (.sav), or Stata (.dta). The haven package in R was developed specifically to import and export data in these formats. Similar to readr for rectangular text data, haven’s functions read files in these formats into a tibble object in R.

t-Tests in SPSS SPSS allows you to conduct one-sample, independent samples, and paired samples \(t\)-tests. This page demonstrates how to perform each using SPSS. The data used in this tutorial can be downloaded from here. The one-sample and independent samples examples will use the iq_long.sav data, and the paired samples example will use iq_wide.sav.
One Sample \(t\)-Test Say we have data from 200 subjects who have taken an IQ test. We know in the general population the mean IQ is 100.