 Suomessa on mieltä eri mieltä, jota on tärkeää. Voi puolesta, jota on kovata mieltä eri mieltä, joka on matriakin koko. Toivon, että nämä matriakin ovat käyttäneet, se on käytännössä, mitä nämä matriakin ovat. Miltä eri mieltä eri matriakin koko on aika kovata, mutta jos en ole mitään yksitystä, jota on hienoa, on onnistuu. Eli esimerkiksi data-usio-manueella mieltä eri mieltä eri mieltä on kovata matriakin koko. The idea here is that we have a fixed part xb, xbeta, and then we have the random part consisting of z, u, and under error term, e. Let's take a look at this equation in more detail. So we have a few elements here. We have the y, this is the observed dependent variable values. It's a vector, so we have one dependent variable, there is one column, multiple values for each case. Then we have a matrix of the observed predictor variables and that has multiple columns and m multiple rows. Then we have the vector of the rigorous and coefficients. There's just one model and the multiple coefficients rows. Then we have the error term E and this Y equals X beta plus E is also just the linear regression model in matrix form. The multi-level model or mixed model adds this mixed effect or random effects here and we have Z which is a matrix that allocates the random effects to the cases and then we have the random effect vector U. Let's take a and these these are X and B are called design matrices because they specify how those effects that are estimated are allocated onto the cases. The reason why they're called design matrices is a bit historic column. It originates from applying regression analysis for analysis on data from experimental designs and these are beta and U are parameter vectors estimated by a computer and we don't actually estimate U directly but instead we'd estimate the variances of these random effects. Let's take a look at a numerical example to understand what goes into these matrices. So in our example we have nine observations in three groups with random intercepts so Y, X, beta, Z, U and E are the matrices and vectors and we have the dependent variable values here, Y's. We have the X's here and we also have one column that is for multiplying the intercept. So the intercept is multiplied by one before being added to the cases and then we have this random intercept part so intercepts how these U1, U2 and U3, three different values that are unobserved are allocated to the cases. We can see that Y1 is used on the first three cases, Y2 used for the fourth, fifth and sixth and the final three get Y3 and then we have the error terms here. If we start multiplying this out we can see that these are the fitted values of the regression analysis. How we get the fitted values is that we take the first cell of the design matrix, we multiply the first cell of the parameter vector, then we move on to the second cell in both and third cell and we just take a sum. So that gives us the fitted values. This has nine rows and this has one column so the resulting thing is a nine by one, nine rows, one column matrix. Then the same applies to U so we use U1 for the first three cases and so on and then the error terms are just added like they are. These models are not estimated by estimated random effect values. Instead we estimate what data calls g and this is a variance covariance matrix of the random effects and then we check what is the likelihood of obtaining the data or the residuals that we got from the model by comparing it against the variance covariance matrix. So in practice what we do is that we take the fitted values, we subtract the fitted values from the option values that gives us the residuals e and then we calculate the variance of these estimated random effects and error terms and this is the covariance matrix so they are the first three observations covariate because they share a random effect, their variances are the sums of the random effects plus the error terms, then the second three observations covariate because they share a random effect and the final three observations covariate because they too share a random effect and then we just find the parameter values that make these residuals e from this covariance matrix to be as likely as possible so that's a maximum likelihood estimation. In practice we don't take the full residual vector and compare it against the full error covariance matrix but we work one cluster at the time because these clusters are independent, they don't correlate the error terms we can just calculate the likelihood of the first three residuals from this three by three sub matrix then we calculate the log likelihood of the second cluster and the third cluster by looking at those sub matrices we take the sum of the log likelihoods and that's our full likelihood. The reason why we work with sub matrices instead of this full matrix is computational it's a lot simpler to work with smaller matrices a lot faster. So how about random slopes? Let's take a look at this example here so the random slope model means that the slope of xi x1 is the sum of the fixed part beta plus random effect so the slope of x1 varies between clusters and we can reorganize the equation a bit so we have the fixed part here and then we have the random part here and we just put those into the matrices the fixed part is the same xb there's no difference and now what we have here is that instead of simply adding each random effect by multiplying it by one we multiply this these u1i random effects with the values x11, x12, x13 and so on so instead of adding the random effects by multiplying them each by one we add them by multiplying them with data and then we work we get the covariance matrix similarly and the estimation proceeds similarly so this is the multi-level model in matrix format if you are interested in knowing how it works when we have generalized mixed effects models we simply add a link function here and that's it so there is not not much difference between estimating specifying normal glm in matrix form and specifying multiple glm in matrix form