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Covariance matrix4/8/2023 ![]() ![]() So now we can see that for the single level model, two different pupils are unrelated, whether they go to the same school, or whether they go to different schools. So now if we look at the correlation structure, we need to divide the covariance by the total variance to get the correlation the total variance for a single level model is sigma squared_e, so all those diagonal terms are going to become 1 and the zeros will Now this covariance is the covariance between two different level one error terms, and we assumed that that was zero, so we just have zero.Īnd so all the off-diagonal terms in this matrix are zero. So now, for two different observations, again we subtract the predicted value from the response, and we end up again with just the error terms. And this is just the covariance of the level one error term with itself- you can see we've got e_i1 and e_i1, the same thing, and when we have the covariance of the same thing, that's just the variance, so we've got the variance of e_i1, and we've defined that to be sigma squared_e, so for the same observation, the covariance is sigma squared_e so down the diagonal of this matrix we have sigma squared_e So now, if we take the covariance of an observation with itself, so as we said we're subtracting from the response the predicted value (from the covariates), and we end up with the error terms, that's all that's left after we've taken the fixed part away. So how do we work out what these covariances actually are? Well, we need to go back to theĪssumptions of the single level model, and the one that chiefly concerns us here is this one, that the error terms for different observations are uncorrelated. Because we're interested in the dependency after we've controlled for the things we've put in the model. So notice that it's not the covariance between the responses for each pair of pupils: what we're actually taking is the covariance of the response after we've controlled for the covariates, so we're basically subtracting the predicted value from the regression line from the response. So let's look first of all at the covariance matrix for a single-level model, so here we have a model of exam results for pupils within schools the numbers along the top and down the left in red, green and blue are the numbers of the school, and just within those we have the numbers of the pupils, and then the entries of the matrix are the covariance between each pair of pupils. But a question of interest is how the multilevel model actually takes into account this dependency: how does it cope with the fact that our exam results are more similar for pupils in the same school, or our heights are more similar for children from the same family? Well, in order to understand how it does this, we can take a look at the structure of the model using the correlation matrix, and to do that, first of all we'll look at the covariance matrix. So for example if we have exam results, then exam results for pupils from the same school are likely to be more similar than exam results for pupils from different schools, or if we have heights, the heights of children in the same family are likely to be more similar than the heights of children in different families, and this is something that we saw in another audio recording, Measuring Dependency we also saw that we can measure the dependency using something called rho, or the variance partitioning coefficient. ![]() So we're using multilevel modelling because we have dependent data. To watch the presentation go to Covariance and correlation matrices - voice-over with slides and subtitles (If you experience problems accessing any videos, please email ).Covariance and correlation matrices A transcript of covariance and correlation matrices presentation, by Rebecca Pillinger ![]()
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