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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 ?
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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.

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