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Say we're interested in how student exam grades are affected by the number of hours that those students study. To explore this relationship, we could run the following linear regression: $$ \text{exam.grades}_i = a + \beta_1 \times \text{hours.studied}_i + e_i $$

But if we sample pupils fron several different schools, we might expect pupils in the same school to more similar to each other than pupils from different schools. To deal with this dependency issue, the advice in many textbooks/on the web, is to run a mixed effects and enter school as a random effect. So the model would become: $$ \text{exam.grades}_i = a + \beta_1 \times \text{hours.studied}_i + \text{school}_j + e_i $$ But why does this solve the dependency problem that was present in the linear regression?

Please respond as if you're talking to a 12 year old

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  • $\begingroup$ Whether it "solves" the dependency problem is context specific. But you probably can see that now the extended model has a term which can, at least partially, account for an effect related to a particular school. $\endgroup$ – image_doctor Apr 3 '15 at 10:27
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Including random terms in the model is a way to induce some covariance structure bteween the grades. The random factor for the school induces a non zero covariance between different students from the same school, whereas it is $0$ when the school are different.

Let's write your model as $$ Y_{s,i} = \alpha + \text{hours}_{s,i} \beta + \text{school}_s + e_{s, i} $$ where $s$ indexes the school and $i$ indexes the students (in each school). The terms $\text{school}_s$ are independent random variables drawn in a $\mathcal N(0, \tau)$. The $e_{s, i}$ are independent random variables drawn in a $\mathcal N(0, \sigma^2)$.

This vector has expected value $$\left[ \alpha + \text{hours}_{s,i} \beta \right]_{s,i}$$ which is determined by the number of worked hours.

The covariance between $Y_{s,i}$ and $Y_{s',i'}$ is $0$ when $s \ne s'$, which means that the departure of the grades from the expected values are independent when the students are not in the same school.

The covariance between $Y_{s,i}$ and $Y_{s,i'}$ is $\tau$ when $i \ne i'$, and the variance of $Y_{s,i}$ is $\tau + \sigma^2$: grades of students from the same school will have correlated departures from their expected values.

Example and simulated data

Here is a short R simulation for fifty students from five schools (here I take $\sigma^2 = \tau = 1$); the names of the variable are self documenting:

set.seed(1)
school        <- rep(1:5, each=10)
school_effect <- rnorm(5)

school_effect_by_ind <- rep(school_effect, each=10)
individual_effect    <- rnorm(50)

We plot the departures from expected grade for each student, that is the terms $\text{school}_s + e_{s, i}$, together with (dotted line) the mean departure for each school:

plot(individual_effect + school_effect_by_ind, col=school, pch=19, 
     xlab="student", ylab="grades departure from expected value")
segments(seq(1,length=5,by=10), school_effect, seq(10,length=5,by=10), col=1:5, lty=3)

mixed model

Now let's comment on this plot. The level of each dotted line (corresponding to $\text{school}_s$) is drawn at random in a normal law. The student specific random terms are also drawn at random in a normal law, they correspond to the distance of the points from the dotted line. The resulting value is, for each student, the departure from $\alpha + \text{hours} \beta$, the grade determined by the time spent to work. As a result, pupils in the same school are more similar to each other than pupils from different schools, as you stated in your question.

The variance matrix for this example

In the above simulations we draw separately the school effects $\text{school}_s$ and the individual effects $e_{s,i}$, so the covariance considerations with which I began don’t appear clearly here. In fact, we would have obtained similar results by drawing a random normal vector of dimension 50 with block-diagonal covariance matrix $$\left[\begin{matrix} A & 0 & 0 & 0 & 0 \\ 0 & A & 0 & 0 & 0 \\ 0 & 0 & A & 0 & 0 \\ 0 & 0 & 0 & A & 0 \\ 0 & 0 & 0 & 0 & A \end{matrix}\right]$$ where each of the five $10\times 10$ blocks $A$ correspond to the covariance between the students of a same school: $$A = \left[\begin{matrix} 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 \end{matrix}\right].$$

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    $\begingroup$ Elvis: thats probably a great answer for people more versed in statistics than I. However I can extract little meaning from it. Could you edit your response in a way that a 12 year old might be able to understand? $\endgroup$ – luciano Apr 3 '15 at 10:57
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    $\begingroup$ A... 12 years old?! Wow! I will add some simulations, if this can help. $\endgroup$ – Elvis Apr 3 '15 at 11:49
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    $\begingroup$ Done. Hope this helps. If not, please be more specific about what you don’t get. Note that a 12 yo would not understand the question either... you can’t ask for an answer simpler than the question. $\endgroup$ – Elvis Apr 3 '15 at 12:16

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