An example of multilevel modeling :

Consider an educational study with data from students in many schools,predicting in each school the students’ grades y on a standardized test given their scores on a pre-test x and other information. Assume just one student-level predictor x (for example, a pre-test score) and one school-level predictor u (for example, average parents’ incomes).

We write the model in which the regressions have the same slope in each of the schools, and only the intercepts vary.

By varying intercept , I understood the deviation of jth school average score $(\bar y_j)$ from overall mean score $(\bar y)$ .

  • What is meant by same slope in this example , that is , what is the interpretation of same slope in this example ? Will the interpretation include group level variable ?

1 Answer 1


This looks like what my textbook mentions as Analysis of variance model.

Assume your observations belong to some groups (typically the levels of a categorical variable), but your response variable $y$ is also influenced by a continuous variable $x$. If your observations for the different groups are not equally spread over the values of $x$, how do you differentiate between the direct influence of the group itself and the indirect influence through different spreads over the domain of $x$ values.

From my textbook:

  • In general terms the model is

    observation = mean + group effect + covariance effect + error

  • More specifically, if $y$ij denotes the value of the response variable for the $j$th individual in the $i$th group and $x$ij the value of the covariate for this individual, then the model assumed is

    $y$ij$ = \mu + \alpha$i +$\beta(x$ij$ - \bar x$) + $\epsilon$ij

    where $\beta$ is the regression coefficient linking response variable and covariance; $\bar x$ is the grand mean of the covariate values.

  • The regression coefficient is assumed to be the same in each group.

  • The means of the response variable adjusted for the covariate are obtained simply as adjusted group mean, i.e.

    $= $group mean$ + \hat\beta($grand mean of covariate$ - $group mean of covariate)$

    where $\hat\beta$ is the estimate of the regression coefficient in the model.

An essential assumption in the model is, that your response depends approximately linearly on $x$, and this dependency is hardly different for the different groups.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.