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In a linear mixed model, you take the covariance between data into account by adding a random intercept per cluster.

For example, you measure the effect of a drug campaign over time on students, and add a random intercept per school and a random intercept per student within a school:

$Y_{ijk}=\beta_{0}+\beta_{1}*Time+b_k+b_{jk}+\epsilon_{ijk}$

With Y the outcome variable "how much does he knows with drugs. We assume that the random intercepts are normally distributed with mean and variance:

$b_k \sim N(0, \sigma_3^2)$, $b_{jk} \sim N(0, \sigma_2^2)$ and $\epsilon_{ijk} \sim N(0, \sigma_1^2)$

With these variances of random intercepts we can calculate the variance of the treatment effect on the different levels, for example on the school level.

But what if I want to model that the campaign has not the same average increase over time in every school? I add a random slope per school:

$Y_{ijk}=\beta_{0}+\beta_{1}*Time+b_{0k}+b_{1k}*Time+b_{jk}+\epsilon_{ijk}$

Is in this model the variation of $b_{0k}$ still meaningful? Is my intuition correct that the variation between different schools increases over time, as the random slopes drive the schools further apart?

And as a second question, without random slope we also know that the covariance between students within the same school is given by:

$Cov_{school}=\frac{sigma_3^2}{sigma_1^2+sigma_2^2+sigma_3^2}$

What is this covariance if a random intercept is included?

Thank you in advance!

UPDATE: included a plot of the data I am working with, level 1 are the 8 repeated measurements at each measurement occasion, level 2 are the difference measurement occasions, level 3 are the patients.

enter image description here

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    $\begingroup$ I'd say that $Cov_{school}$ a) is not the covariance between students within the same school, but one of the correlation coefficients between the random terms, b) it should be $\sigma^2_3/(\sigma^2_1+\sigma^2_2+\sigma^2_3)$. Am i wrong? $\endgroup$ – Sergio May 21 '14 at 16:32
  • $\begingroup$ @Sergio you are write about the squares in the formal, I will update it. The interpretation of is correlation is however correct I think, this is out of a similar example of our textbook: "The correlation among the scores for children from different classrooms within the same school is approximately...", which is similar to what I described. $\endgroup$ – Kasper May 22 '14 at 17:14
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    $\begingroup$ Where there is an interaction between a random slope and a predictor variable the correlation is more complex and there is not a costant correlation coefficient. I'm preparing my answer offline, because it's not brief. I hope I'll post it today or (more likely) tomorrow. $\endgroup$ – Sergio May 22 '14 at 17:46
  • $\begingroup$ Do you think this is an excuse to leave out the random slope and just work with the random intercept? $\endgroup$ – Kasper May 22 '14 at 17:58
  • $\begingroup$ One should work with a random slope or not after plots, pooled separate and mixed regressions, residual analysis, hypotesis testing etc. ;-) $\endgroup$ – Sergio May 22 '14 at 18:12
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Let me start with simpler models (detailed answers below), and let me use a slightly different notation (NB: random effecs at different levels assumed to be uncorrelated).

Model 1 (Two levels, random intercept, fixed slope). There are $N$ observations of $J$ students: $$\begin{align}y_{ij}&=\pi_{0j}+\beta_1 X_{ij}+\varepsilon_{ij},\quad&\varepsilon_{ij}\sim\mathcal{N}(0,\sigma^2_\varepsilon)\\ \pi_{0j}&=\beta_0+b_{0j},\quad& b_{0j}\sim\mathcal{N}(0,\sigma^2_0)\end{align}$$. If we substitute for $\pi_0$ and $\pi_1$ we get: $$y_{ij}=\beta_0+\beta_1X_{ij}+(b_{0j}+\varepsilon_{ij})$$ so: $$\begin{align}\text{Var}(y_{ij})&=\text{Var}(b_{0j}+\varepsilon_i)=\sigma^2_0+\sigma^2_\varepsilon \\ \text{Cov}(y_{ij},y_{i'j})&=\sigma^2_0 \quad& i\ne i' \\ \text{Cov}(y_{ij},y_{i'j'})&=0 \quad& i\ne i',\;j\ne j' \end{align}$$ The intraclass correlation coefficient: $$\rho=\frac{\sigma^2_0}{\sigma^2_0+\sigma^2_\varepsilon}$$ measures the proportion of the variance in the outcome that is between students. As to the overall covariance matrix, it is (Goldstein, Multilevel Statistical Models, pg. 20): $$ \begin{bmatrix} \sigma^2_0\mathbf{J}_{N_1}+\sigma^2_\varepsilon\mathbf{I}_{N_1} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ \dots & \dots & \sigma^2_0\mathbf{J}_{N_j}+\sigma^2_\varepsilon\mathbf{I}_{N_j} & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \sigma^2_0\mathbf{J}_{N_J}+\sigma^2_\varepsilon\mathbf{I}_{N_J} \end{bmatrix}$$ where $N_j$ is the number of observations for student $j$, $\mathbf{J}_{N_j}$ is the $(N_j\times N_j)$ matrix of ones and $\mathbf{I}_{N_j}$ is the $(N_j\times N_j)$ identity matrix. For example, if there are $J=2$ students and $N_1=3,N_1=2$ observations: $$\begin{bmatrix} \sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 & \sigma^2_0 & 0 & 0 \\ \sigma^2_0 & \sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 & 0 & 0 \\ \sigma^2_0 & \sigma^2_0 & \sigma^2_0+\sigma^2_\varepsilon & 0 & 0 \\ 0 & 0 & 0 & \sigma^2_0+\sigma^2_\varepsilon & \sigma^2_0 \\ 0 & 0 & 0 & \sigma^2_0 & \sigma^2_0+\sigma^2_\varepsilon \end{bmatrix}=(\sigma^2_0+\sigma^2_\varepsilon) \begin{bmatrix} 1 & \rho & \rho & 0 & 0 \\ \rho & 1 & \rho & 0 & 0 \\ \rho & \rho & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & \rho \\ 0 & 0 & 0 & \rho & 1 \end{bmatrix}$$

Model 2 (Three levels, random intercept, fixed slope). There are $N$ observations, $J$ students, and $K$ schools: $$\begin{align} y_{ijk}&=\pi_{0jk}+\beta_1 X_{ijk}+\varepsilon_{ijk},\quad&\varepsilon_{iij}\sim\mathcal{N}(0,\sigma^2_\varepsilon) \\ \pi_{0jk} &= \gamma_{0k}+b_{0jk},\quad& b_{0jk}\sim\mathcal{N}(0,\sigma^2_0) \\ \gamma_{0k} &= \beta_0 + b_{0k},\quad & b_{0k}\sim\mathcal{N}(0,\tau^2_0) \end{align}$$ The composite model is: $$y_{ijk}=\beta_0+\beta_1 X_{ijk}+ (b_{0k}+b_{0jk}+\varepsilon_{ijk})$$ and: $$\text{Var}(y_{ijk})=\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon$$ Instead of one intraclass correlation coefficient, several correlation coefficients may be defined:

  • $(\tau^2_0+\sigma^2_0)/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among students;
  • $\tau^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among schools;
  • $\sigma^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance among students within schools;
  • $\sigma^2_\varepsilon/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$: the proportion of variance within students.

The covariance matrix is complex. If there are $K=3$ schools, a covariance matrix looks like (Rodriguez): $$(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)\begin{bmatrix} 1 & \rho_1 & \rho_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \rho_1 & 1 & \rho_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \rho_2 & \rho_2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \rho_1 & \rho_2 & \rho_2 & 0 & 0 & 0\\ 0 & 0 & 0 & \rho_1 & 1 & \rho_2 & \rho_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \rho_2 & \rho_2 & 1 & \rho_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \rho_2 & \rho_2 & \rho_1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \rho_1 & \rho_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho_1 & 1 & \rho_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho_2 & \rho_2 & 1 \end{bmatrix}$$ where $\rho_1=(\tau^2_0+\sigma^2_0)/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$ and $\rho_2=\tau^2_0/(\tau^2_0+\sigma^2_0+\sigma^2_\varepsilon)$.

Model 3 (Two levels, random intercept and slope). We let $\pi_1$ vary across the students in Model 1: $$\begin{align}y_{ij}&=\pi_{0j}+\pi_1 X_{ij}+\varepsilon_{ij},\quad&\varepsilon_i\sim\mathcal{N}(0,\sigma^2_\varepsilon)\\ \pi_{0j}&=\beta_0+b_{0j},\quad& b_{0j}\sim\mathcal{N}(0,\sigma^2_0)\\ \pi_1 &= \beta_1+b_{1j},\quad& b_{1j}\sim\mathcal{N}(0,\sigma^2_1) \end{align}$$ but now we must represent the dispersion of the level-2 random effects as a covariance matrix: $$\text{Var}(\mathbf{b}_j)=\text{Var}\begin{bmatrix} b_{0j} \\ b_{1j} \end{bmatrix} =\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}$$ The combined model is: $$y_{ij}=\beta_0+\beta_1X_{ij}+(b_{0j}+b_{1j}X_{ij}+\varepsilon_{ij})$$ so (Goldstein, Multilevel Statistical Models, pg. 22): $$\begin{align}\text{Var}(y_{ij})&=\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}'+\sigma^2_\varepsilon \\ \text{Cov}(y_{ij},y_{i'j})&=\sigma^2_0+2\sigma_{01}(X_{ij}+X_{i'j})+\sigma^2_1 X_{ij} X_{i'j} \\ \text{Cov}(y_{ij},y_{i'j'})&=0 \end{align}$$

The overall covariance matrix is still block diagonal but the blocks are: $\left(\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}\text{Var}(\mathbf{b}_j)\begin{bmatrix} \mathbf{1} & \mathbf{X} \end{bmatrix}'\right)\mathbf{J}_{N_j}+\sigma^2_\varepsilon\mathbf{I}_{N_j}$.

We could define: $$\rho=\frac{\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2}{\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2+\sigma^2_\varepsilon}$$ but it is not a correlation coefficient like the previous ones. The correlation between two observations is given by: $$\frac{\sigma^2_0+2\sigma_{01}(X_{ij}+X_{i'j})+\sigma^2_1 X_{ij} X_{i'j}} {\sqrt{(\sigma^2_0+2\sigma_{01}X_{ij}+\sigma^2_1 X_{ij}^2+\sigma^2_\varepsilon)(\sigma^2_0+2\sigma_{01}X_{i'j}+\sigma^2_1 X_{i'j}^2+\sigma^2_\varepsilon)}}$$ Its value is a function of $X$, so there is not a unique correlation coefficient, but several coefficients that can be calculated for any value of the predictor variable(s) (Goldstein et al., Partitioning variation in multilevel models).

Model 4 (Three levels, random intercept and slope). We let $\pi_1$ vary across the students in Model 2: $$\begin{align} y_{ijk}&=\pi_{0jk}+\pi_{1jk} X_{ijk}+\varepsilon_{ijk},\quad&\varepsilon_{iij}\sim\mathcal{N}(0,\sigma^2_\varepsilon) \\ \pi_{0jk} &= \gamma_{0k}+b_{0jk},\quad& b_{0jk}\sim\mathcal{N}(0,\sigma^2_0) \\ \pi_{1jk} &= \gamma_{1k}+b_{1jk},\quad& b_{1jk}\sim\mathcal{N}(0,\sigma^2_1) \\ \gamma_{0k} &= \beta_0 + b_{0k},\quad & b_{0k}\sim\mathcal{N}(0,\tau^2_0) \\ \gamma_{1k} &= \beta_1 + b_{1k},\quad & b_{1k}\sim\mathcal{N}(0,\tau^2_1) \end{align}$$

The covariance matrices for level-2 and level-3 are:

$$\text{Var}(\mathbf{b}_j)=\begin{bmatrix} \sigma^2_0 & \sigma_{01} \\ \sigma_{01} & \sigma^2_1 \end{bmatrix}\qquad\qquad \text{Var}(\mathbf{b}_k)=\begin{bmatrix} \tau^2_0 & \tau_{01} \\ \tau_{01} & \tau^2_1 \end{bmatrix}$$

The composite model is:

$$y_{ijk}=\beta_0 + \beta_1 X_{ijk} + (b_{0jk}+b_{1jk} X_{ijk})+(b_{0k}+b_{1k} X_{ijk}) + \varepsilon_{ijk}$$

and the variance at level-1 is:

$$\text{Var}(y_{ijk})=\text{Var}(b_{0jk}+b_{1jk} X_{ijk})+\text{Var}(b_{0k}+b_{1k} X_{ijk})+\text{Var}(\varepsilon_{ijk})$$

where $\text{Var}(b_{0jk}+b_{1jk} X_{ijk})$ is the variance at level-2 and $\text{Var}(b_{0k}+b_{1k} X_{ijk})$ is the variance at level-3. They depend on the value of $X$ as in Model 3.

Now:

Multilevel models including random slopes: how to calculate variance

I hope it is somewhat clear now.

$Y_{ijk}=\beta_0+\beta_1*Time+b_k+b_{jk}+\epsilon_{ijk}$

I'write: $Y_{ijk}=\beta_0+\beta_1\text{Time}+b_{0k}+b_{0jk}$ (Model 2) to highlight that $b_{0k}$ and $b_{0jk}$ randomize $\beta_0$.

I add a random slope per school:
$Y_{ijk}=\beta_0+\beta_1* Time+b_{0k}+b_{1k}*Time+b_{jk}+\epsilon_{ijk}$
Is in this model the variation of $b_{0k}$ still meaningful?

I'd write: $Y_{ijk}=\beta_0+\beta_1\text{Time}+b_{0k}+b_{1k}\text{Time}+b_{0jk}+\epsilon_{ijk}$, i.e.: $$Y_{ijk}=(\beta_0+b_{0jk}+b_{0k})+(\beta_1+b_{1k})\text{Time}+\epsilon_{ijk}$$ where $(\beta_0+b_{0jk}+b_{0k})$ is the random intercept per student within a school and per school, and $(\beta_1+b_{1k})$ is the random slope per school (Model 4 with $b_{1jk}=E(b_{1jk})=0$, $\sigma^2_1=0$.)

As to $b_{0k}$, the random intercept per school, it is even more meaningful than $b_{1k}$! "Given that among multilevel modelers random slopes tend to be more popular than heteroscedasticity, unrecognized heteroscedasticity may show up in the form of a fitted model with a random slope [...] which then may disappear if the heteroscedasticity is modeled" (Snijders and Berkhof, Diagnostic Checks for Multilevel Models, §3.2.1).

Is my intuition correct that the variation between different schools increases over time, as the random slopes drive the schools further apart?

In all the above models I have assumed that all random effects are uncorrelated and that their variances are constant. Is these assumptions hold, then covariances increase quadratically with time (see conjectures' answer). However, if the $\tau$ variances and covariances decrease with time you could observe a somewhat different pattern.

BTW, your plot does show a different pattern and suggests a bit of heteroscedasticity more than a random slope.

the covariance between students within the same school is given by:
$\text{Cov}_{school}=\frac{\sigma^2_3}{\sigma^2_1+\sigma^2_2+\sigma^2_3}$ What is this covariance if a random intercept is included?

That is not a covariance, it is the $\rho_2$ correlation coefficient in Model 2. The covariance between two schools depends on Time.

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  • $\begingroup$ I feel like I am very close but I am overlooking something $\endgroup$ – Kasper May 29 '14 at 17:16
  • $\begingroup$ Excellent. Just a question, there is something I've seen in books, as well in your answer, which is very clear, that I do not understand. In your model 1, for example, how do you estimate $\sigma_0$? $\endgroup$ – chuse Apr 9 '18 at 15:10
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I believe your intuition is correct.

With the following model, ranging over schools $k$, individual within school $j$ and observation within individual within school $i$. (If I get your notation right).

$Y_{ijk}=\beta_0+\beta_1 t + b_{0k} + b_{1k}t + b_{jk} + ϵ_{ijk}$

With

$b_{1k} \sim N(0,\sigma_4^2)$, $b_{0k} \sim N(0,\sigma_3^2)$, $b_{jk} \sim N(0,\sigma_2^2)$, $\epsilon_{ijk} \sim N(0,\sigma_1^2)$

And assuming all random effects and error terms are pairwise independent.

Then for two two distinct individuals,$h$ and $j$, in the same school at the same time:

$\text{Cov}(Y_{ijk},Y_{ihk})=\text{Cov}(\beta_0+\beta_1 t + b_{0k} + b_{1k}t + b_{jk} + ϵ_{ijk}, \beta_0+\beta_1 t + b_{0k} + b_{1k}t + b_{hk} + ϵ_{ihk})$

$=\text{Var}(b_{0k}) + t^2 \text{Var}(b_{1k})$

So the covariance increases quadratically with time. The example in the text above looks like a correlation, but the question is about covariance. This also means the series are not stationary.

Also, to me the series in the plot look like you could argue they should be modelled with a single slope.

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  • $\begingroup$ Don't you forget the covariance between $b_{0k}$ and $b_{1k}$ in your last formula? $\endgroup$ – Kasper May 29 '14 at 14:49
  • $\begingroup$ We should do in general. I'd assumed they were independent, but in general they wouldn't be, and you'd be right. $\endgroup$ – conjectures May 29 '14 at 19:12

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