The INITITER= option in the PROC GLIMMIX statement controls the number of iterations in this step. Further detail on the specification of \(\gamma _{k,j,i}\) can be found in Appendix (6.6.1). Under three different simulation settings, each with a different design structure, the algorithms are compared against one another, the lmer function from the R package lme4, and the baseline truth used to generate the simulations. Wiley, New York (2013), Dempster, A.P., Laird, N.M., Rubin, D.B. \end{aligned}$$, \({\hat{\eta }}^h=[{\hat{\sigma }}^2, \text {vech}({\hat{D}}_1)',\ldots ,\text {vech}({\hat{D}}_r)']'\), $$\begin{aligned}&\frac{d S^2({\hat{\eta }}^h)}{d {\hat{\sigma }}^2}=L(X'{\hat{V}}^{-1}X)^{-1}L',\\&\frac{d S^2({\hat{\eta }}^h)}{d \text {vech}({\hat{D}}_k)} = {\hat{\sigma }}^{2}{\mathcal {D}}_{q_k}'\bigg (\sum _{j=1}^{l_k}{\hat{B}}_{(k,j)}\otimes {\hat{B}}_{(k,j)}\bigg ), \end{aligned}$$, \({\hat{B}}_{(k,j)} = Z_{(k,j)}'{\hat{V}}^{-1}X(X'{\hat{V}}^{-1}X)^{-1}L'\), $$\begin{aligned} \text {MATH}_{i,j,k} = \beta _0 + \beta _1 \times \text {YEAR}_{i,j,k} + s_i + t_j + \epsilon _{i,j,k}, \end{aligned}$$, $$\begin{aligned} \begin{aligned}&s_i \sim N(0, \sigma ^2_s), \quad t_j \sim N(0, \sigma ^2_t), \quad \epsilon _{i,j,k} \sim N(0,\sigma ^2), \end{aligned} \end{aligned}$$, $$\begin{aligned} \text {ENG}_{k,j,i}= & {} \beta _0 + \beta _1 \times \text {AGE}_{i} + \beta _2 \times \text {SEX}_{i} + \ldots \beta _3\\&\times \text {AGE}_{i} \times \text {SEX}_{i} + \beta _4 \times \text {PSQI}_{i} + \gamma _{k,j,i} + \epsilon _{k,j,i}, \end{aligned}$$, \(D_k = \sigma ^{-2}_e(\sigma ^2_a{\mathbf {K}}^a_k + \sigma ^2_c{\mathbf {K}}^c_k)\), $$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial K} \bigg [ g'(A+\sum _t B_tKB_t')^{-1}g \bigg ]\\&\quad =-\sum _s B_s'\big (A'+\sum _t B_tKB_t'\big )^{-1}gg'\big (A'+\sum _t B_tKB_t'\big )^{-1}B_s. This result further highlights the importance of modelling all relevant variance terms. The results presented here exhibit strong agreement with those reported in West etal. Consistent estimation of the true parameter vector is shown to be important if a fast rate of convergence is to be achieved, but if this condition is met then the algorithm is very attractive. Stat. For this reason, simulation setting 3 is the most susceptible to numerical problems of the kind described by Pinheiro and Bates (1996). Analogous notation is used for the full and Cholesky representations. Counts are either 0 or a postive whole number, which means we need to use special . The prediction function of GLMs The GLM predict function has some peculiarities that should be noted. The variables 'Is_graduate', with label "Yes", and 'Credit_score', with label "Good", are the two most significant variables. \end{aligned}$$, $$\begin{aligned} \text {vec}(A)=K_{m,n}\text {vec}(A'). We obtain an expression for var\(({\hat{\eta }}^h)\) by noting that the asymptotic variance of \({\hat{\eta }}^h\) is given by \({\mathcal {I}}({\hat{\eta }}^h)^{-1}\) where \({\mathcal {I}}({\hat{\eta }}^h)\) is a sub-matrix of \({\mathcal {I}}({\hat{\theta }}^h)\), given by equations (8)(10). However, preliminary tests have indicated that the performance of such an approach, in terms of computation time, is significantly worse than the previously proposed algorithms. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. \(l(\theta ^f)=l(\theta ^h)=l(\theta ^c)\)). Fisher scoring is a hill-climbing algorithm for getting results - it maximizes the likelihood by getting successively closer and closer to the maximum by taking another step ( an iteration). In this post, I'll demonstrate how to estimate the coefficents of a Logistic Regression model using the Fisher Scoring algorithm. If small changes in \theta result in large changes in the likely values of x x, then the samples we observe tell us a lot about \theta . J. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. It won't do any good to add a fractional number to all data, as the result will depend on the number added (try 1.0, 0.5 and 0.1 to see this). \(N_k\) is defined as the unique matrix of dimension \((k^2 \times k^2)\) which implements symmetrization for any arbitrary square matrix A of dimension \((k \times k)\) in vectorized form, i.e. We use the LRT for negative binomial models. Why are taxiway and runway centerline lights off center? For example, if the family structure describes families containing one twin-pair and one half-sibling, we assume that the members of every family who exhibit such a structure are given in the same order: (twin, twin, half-sibling). \end{aligned} \end{aligned}$$, $$\begin{aligned} \frac{1}{2}N_{q_{k_1}}\sum _{j=1}^{l_{k_2}}\sum _{i=1}^{l_{k_1}}\bigg [(T_{(k_1,i)}T_{(k_2,j)}')\otimes (T_{(k_1,i)}T_{(k_2,j)}')\bigg ]. https://doi.org/10.2307/3314625, Pinheiro, J., Bates, D.: Mixed-effects models in S and S-PLUS. Learn how we and our ad partner Google, collect and use data. Assoc. \end{aligned}$$, $$\begin{aligned} \frac{\partial {\tilde{\tau }}}{\partial \tau } = \begin{bmatrix} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} &{} 1 \\ \end{bmatrix} = \mathbb {1}_{(1,r)} \otimes I_2. The degrees of freedom estimates, for each of the three simulation settings, are summarized in Table 3. Number of Fisher Scoring iterations: 5 Loglinear (Poisson) Regression Model of CHD on Smoking , Behavior Type and Blood Pressure > model2 <- glm ( chd ~ smoke + behavior + bp + offset( logpyrs ),. a Canine Atopic Dermatitis Extent and Severity Index, 4th iteration (CADESI4) score > 10], 10 and, the pruritus manifestations were of at least a moderate grade (i.e. \({\mathcal {D}}_k\) satisfies the following relation: \({\mathcal {L}}_k\) is the unique \(1-0\) elimination matrix of dimension \((k(k+1)/2 \times k^2)\), which maps the vectorization of any arbitrary lower triangular matrix A of dimension \((k \times k)\) to its half-vectorization, i.e. As the first term inside the brackets of (2) does not depend on \(D_k\), we need only consider the second and third term for differentiation. Within any individual family unit (e.g. We suggest this idea may form a potential basis for future investigation. The third line provides the information about the data, where the categorical variables have been converted to 'factor' type. Scheipl etal. The analysis of longitudinal, heterogeneous or unbalanced clustered data is of primary importance to a wide range of applications. For our purposes, "hit" refers to your favored outcome and "miss" refers to your unfavored outcome. This is one of the technique used in Random Forest algorithm for example and many other places in statistics. the number of columns in the random effects design matrix) is small, the results of these simulations demonstrate strong computational efficiency for the FS, FFS, SFS and FSFS methods. Please advice on the significance of this. The lines of code below fit the model and prints the result summary. Following this, an extension of this result is provided for models containing constrained covariance matrices of the form described in Sect. Dempster etal. Comput. Simple linear regression is the simplest form of regression which uses only one covariate for predicting the target variable. Wiley Series in Probability and Statistics. How can you prove that a certain file was downloaded from a certain website? Biometrics Bull. 76(374), 341353 (1981). However, in recent years, the focus of the LMM literature has moved towards the development of estimation and inference methods for more complex, multi-factored designs. : Statistical challenges in big data human neuroimaging. 2192 ## ## Number of Fisher Scoring iterations: 5 . It uses the inverse standard normal distribution as a linear combination of the predictors. In this section, we consider the following three representations for \(\theta \): where \(\Lambda _k\) represents the lower triangular Cholesky factor of \(D_k\), such that \(D_k=\Lambda _k\Lambda _k'\). The probit regression coefficients give the change in the z-score or probit index for a one unit change in the predictor. l o g ( X )= l o g ( n )+ 0 + iiXi. It can be seen from Table 4 that the Powell optimizer and Fisher Scoring method attained extremely similar optimized likelihood values, with Fisher Scoring converging notably faster. This method of evaluation provides notably faster computation time than the corresponding for loop evaluated over all values of i andj. All reported results were obtained using an Intel(R) Xeon(R) Gold 6126 2.60GHz processor with 16GB RAM. In all three settings, all parameter estimates and maximised likelihood criteria produced by the FS, FFS, SFS and FSFS methods were identical to those produced by lmer up to a machine error level of tolerance. Viewed 2k times 1 $\begingroup$ Lets say I run the following code: x <- seq(1, 1000) set 1388.3. l o g ( X )= l o g ( n )+ 0 + iiXi. To address this question, the covariance parameters \(\sigma ^2_a\), \(\sigma ^2_c\) and \(\sigma ^2_e\) must be estimated. Our response variable cannot contain negative values. t = t 1 ( 0) E [ ( 0)]. http://www.jstor.org/stable/2290128, Magnus, J.R., Neudecker, H.: The elimination matrix: some lemmas and applications. In traditional linear regression, the response variable consists of continuous data. There are two different algorithms: by linearization or by stochastic approximation. Econom. Since its conception in the seminal work of Laird and Ware (1982), the literature on linear mixed model (LMM) estimation and inference has evolved rapidly. Examples include compound symmetry, first-order auto-regression and a Toeplitz structure. I want to compare means of two groups of data but only with two What's the best way to measure growth rates in House sparrow chicks from day 2 to day 10? Sex: Whether the applicant is female (F) or male (M). While many of the examples presented in this paper were benchmarked against existing software, it is not the authors intention to suggest that the proposed methods are superior to existing software packages. For this reason, we only consider the simplified version of the Cholesky Fisher Scoring algorithm, analogous to the simplified approaches described in Sects. Through simulation and real data examples, we compare five variants of the Fisher Scoring algorithm with one another, as well as against a baseline established by the R package lme4, and find evidence of correctness and strong computational efficiency for four of the five proposed approaches. Also given are approximate T-statistics for each fixed effects parameter, alongside corresponding degrees of freedom estimates and p-values obtained via the methods outlined in Sect. 2.1.1. Fisher score is one of the most widely used supervised feature selection methods. An example of how this notation may be used in practice is given as follows. Analogous adjustments can be made for the algorithms presented in Sects. I included an offset term to account for varying "sample" effort. For numerical attributes, an excellent way to think about relationships is to calculate the correlation. : Maximum likelihood from incomplete data via the em algorithm. For the LMM, several different representations of the parameters of interest, \((\beta , \sigma ^2,D)\), can be used for numerical optimization and result in different Fisher Scoring iteration schemes. The output shows that the dataset has five categorical variables (labelled as 'chr') while the remaining four are numerical variables (labelled as 'int'). 2.1.12.1.5, and the baseline truth used for comparison was either the baseline truth used to generate the simulated data or the lmer computed estimates. You can also use predicted probabilities to help you understand the model. In this appendix, proof of the derivative result which was given in Sect. Typically, as the variance estimates obtained under ML estimation are biased downwards, ReML estimation is employed to obtain the estimate of \(\eta \) employed in the above expression. You also learned about building the correlation matrix for numerical variables and interpreting the output to identify statistically significant variables. : Essential formulae for restricted maximum likelihood and its derivatives associated with the linear mixed models. As a result, both computation time and memory consumption no longer scale with n. Another potential source of concern regarding computation speed arises from noting that the algorithms we have presented contain many summations of the following two forms: where matrices \(\{A_i\}\) and \(\{B_i\}\) are of dimension \((m_1 \times m_2)\), and matrices \(\{G_{i,j}\}\) and \(\{H_{i,j}\}\) are of dimension \((n_1 \times n_2)\). Intuitively, this is to be expected, as repeated entries in \(\theta ^f\) result in repeated rows in \({\mathcal {I}}(\theta ^f)\). The term "reweighted" refers to the fact that at each iterative step of the Fisher Scoring algorithm, we are using a new updated weight matrix. The Full Simplified Fisher Scoring algorithm (FSFS) combines the Full and Simplified approaches described in Sects. 4.1.2 are presented in Table 4. Cite this article. https://doi.org/10.1017/CBO9781139424400, Turnbull, B.J., Welsh, M.E., Heid, C.A., Davis, W., Ratnofsky, A.C.: The longitudinal evaluation of school change and performance (lescp) in title i schools. For the ith observation to belong to the jth level of the kth factor in the model may be interpreted as the ith subject belonging to the jth family exhibiting family structure of type k. The model employed for this example is given by: where both \(\gamma _{k,j,i}\) and \(\epsilon _{k,j,i}\) are mean-zero random variables. In this guide, we will be using the fictitious data of loan applicants containing 600 observations and nine variables, as described below: Marital_status: Whether the applicant is married ("Yes") or not ("No"). We denote the matrices formed from vertical concatenation of the \(\{A_i\}\) and \(\{B_i\}\) matrices as A and B, respectively, and G and H the matrices formed from block-wise concatenation of \(\{G_{i,j}\}\) and \(\{H_{i,j}\}\), respectively. 3.2. From the summary output above, we can see that SEX positively and significantly predicts a pupil's probability of repeating a grade, while PPED negatively and significantly so. Some software allows you to profile the likelihood to see a map of the surface in which you are trying to find the peak. The third Fisher Scoring algorithm proposed in this work relies on the half-representation of the parameters \((\beta ,\sigma ^2,D)\) and takes an approach, similar to that of coordinate ascent, which is commonly adopted in the single-factor setting (c.f. For all methods considered during simulation, the observed performance was unaffected by the choice of likelihood estimation criteria (i.e. Further detail on the constrained approach for the ACE model can be found in Appendix 6.6.2. 67(1), 148 (2015), Article Hi, I am trying to construct a multi-layer fibril structure from a single layer in PyMol by translating the layer along the fibril axis. \(\square \). We do not make use the imaging data in the HCP dataset but, instead, focus on the baseline variables for cognition and alertness. NeuroImage 123, 253268 (2015). Rather than estimate beta sizes, the logistic regression estimates the probability of getting one of your two outcomes (i.e., the probability of voting vs. not voting) given a predictor/independent variable (s). I see this as the effect of divergence in the iteratively reweighted least squares algorithm behind glm. 2.1.3 and uses the Cholesky parameterisation of \((\beta ,\sigma ^2,D)\), \(\theta ^c\). Comput. Many methods are available which provide approximate testing procedures for random effects covariance parameters (c.f. For some packages, this exceptional speed and efficiency arise from simplifying model assumptions, while for others, complex mathematical operations such as sparse matrix methodology and sweep operators are utilized to improve performance (Wolfinger etal. Summary An analysis is given of the computational properties of Fisher's method of scoring for maximizing likelihoods and solving estimating equations based on quasi-likelihoods. In this guide, the reader will learn how to fit and analyze statistical models on the quantitative (linear regression) and qualitative (logistic regression) target variables. The Pearson correlation coefficient, calculated using the cor function, is an indicator of the extent and strength of the linear relationship between the two variables. Springer Berlin Heidelberg, SAS Institute Inc, Cary, NC (2015), Satterthwaite, F.E. https://doi.org/10.1348/000711006X110562, Turkington, D.A. For any arbitrary integer k between 1 and r, the covariance of the total derivatives of \(l(\theta ^h)\) with respect to \(\beta \) and vech\((D_k)\) is given by: For any arbitrary integer k between 1 and r, the covariance of the total derivatives of \(l(\theta ^h)\) with respect to \(\sigma ^2\) and vech\((D_k)\) is given by: For any arbitrary integers \(k_1\) and \(k_2\) between 1 and r, the covariance of the total derivatives of \(l(\theta ^h)\) with respect to vech\((D_{k_1})\) and vech\((D_{k_2})\) is given by: We note that the results of Corollaries 4, 5 and 6 do not contain the matrix \(N_{q_k}\), which appears in the corresponding theorems (Theorems 2, 3 and 4). The Fisher Scoring algorithm update rule takes the following form: where \(\theta _s\) is the vector of parameter estimates given at iteration s, \(\alpha _s\) is a scalar step size, the score vector of \(\theta _s\), \(\frac{dl(\theta _s)}{d\theta }\), is the derivative of the log-likelihood with respect to \(\theta \) evaluated at \(\theta =\theta _s\), and \({\mathcal {I}}(\theta _{s})\) is the Fisher Information matrix of \(\theta _s\); A more general formulation of Fisher Scoring, which allows for low-rank Fisher Information matrices, is given by Rao and Mitra (1972): where superscript plus, \(^+\), is the MoorePenrose (or pseudo) inverse. The research question in this example concerns how well a students grade (i.e. Behav. (1994) also describe a Fisher Scoring algorithm. \end{aligned} \end{aligned}$$, $$\begin{aligned} \theta ^f_{s+1} = \theta ^f_{s} + \alpha _s F(\theta _{s}^f)^{-1}\frac{\partial l(\theta _s^f)}{\partial \theta }_, \end{aligned}$$, $$\begin{aligned} \begin{aligned}&F_{\text {vec}(D_{k_1}),\text {vec}(D_{k_2})}\\&\quad =\frac{1}{2}\sum _{j=1}^{l_{k_2}}\sum _{i=1}^{l_{k_1}}(Z'_{(k_1,i)}V^{-1}Z_{(k_2,j)}\otimes Z'_{(k_1,i)}V^{-1}Z_{(k_2,j)}) \end{aligned}, \end{aligned}$$, $$\begin{aligned} \beta _{s+1} = (X'V_s^{-1}X)^{-1}X'V_s^{-1}Y,\quad \sigma ^2_{s+1} = \frac{e_{s+1}'V^{-1}_{s}e_{s+1}}{n}.\nonumber \\ \end{aligned}$$, $$\begin{aligned}&\text {vech}(D_{k,s+1})\nonumber \\&\quad =\text {vech}(D_{k,s})+\alpha _s\big ({\mathcal {I}}^{h}_{\text {vech}(D_{k,s})}\big )^{-1}\frac{dl(\theta ^h_s)}{d\text {vech}(D_{k,s})}. For this reason, the relation \(T \sim t_{v}\) is not exact and is treated only as an approximating distribution (i.e. A probit regression is a version of the generalized linear model used to model dichotomous outcome variables. The Fisher Scoring update rule employed was of the form (3) with \(\theta \) substituted for \([\tau _a, \tau _c]'\). \end{aligned}$$, $$\begin{aligned} \frac{dl(\theta ^c)}{d\text {vech}(\Lambda _k)}=\frac{\partial \text {vech}(D_k)}{\partial \text {vech}(\Lambda _k)}\frac{\partial l(\theta ^c)}{\partial \text {vech}(D_k)}. the 'main effects' of period refer to the differences in habitat for the base level of period which seems to be period 2. In our case, the R-squared value of 0.68 means that 68 percent of the variation in the variable 'Income' is explained by the variable 'Investment'. Fisher Scoring Maximum Likelihood (ML) as Iterative Reweighted Least Squares (IRLS) Consider the generalized linear model (GLM) for N 1, 205 (1996). The larger \(D_k\) is in dimension, the greater the number of \(\Lambda _k\) that correspond to \(D_k\) there are. nursing schools in germany for international students. just enter his dataand run a model - it should not converge;both these data patterns do not admit to a solution; Put simply a frequency of zero in any single cell of the 2 by 2 table implies quasi-separation; two diagonally opposed zeros in the table means that the condition is complete - the slope is infinite - the maximum likelihood estimates simply do not exist. Let the probability of success equal \ (p= (1-x)p_0 + xp_1\), so that If \ (x=0\), then \ (p=0.4\) If \ (x=1\), then \ (p=0.6\) Copyright 2022 FAQS.TIPS. In this appendix, we provide detail on how the random effects vector, b, and covariance matrix, D, are defined for the ACE model. interim report to congress (1999), Van Essen, D., Smith, S., Barch, D., Behrens, T., Yacoub, E., Ugurbil, K.: The wu-minn human connectome project: an overview. 2.1.4, we believe this has not induced bias into the simulations as, as discussed in Sect. This term can be evaluated to \({\mathcal {L}}_{q_k}(\Lambda _k' \otimes I_{q_k})(I_{q_k^2}+K_{q_k}){\mathcal {D}}_{q_k}\), proof of which is provided in Appendix 6.4. However, the bias observed for the lmerTest estimates is notably more severe that of the direct-SW method, suggesting that the estimates produced by direct-SW have a higher accuracy than those produced by lmerTest. a 10-grade pruritus Visual Analog Scale [PVAS10] score >3.5) in the preceding 24 h. 11 Null deviance: 234.67 on 188 degrees of freedom Residual deviance: 234.67 on 188 degrees of freedom AIC: 236.67 Number of Fisher Scoring iterations: 4 Simulating Data for Count Models. We note that this approach may also be extended to models using the constrained covariance structures of Sect. Find centralized, trusted content and collaborate around the technologies you use most. The total derivative vector of the log-likelihood with respect to vech\((D_k)\) is given as: In this appendix, we provide derivations of the components of the Fisher Information matrix of \(\theta ^f\) which relate to \(D_k\), for some factor \(f_k\). Fisher's exact calculation gives a score of 4 but what does this number mean exactly?? 2.1 may be adapted to use an alternative likelihood criteria: the criteria employed by Restricted Maximum Likelihood (ReML) estimation. \end{aligned}$$, $$\begin{aligned} \frac{1}{4\sigma ^2}\text {cov}\bigg (u'u,\sum _{j=1}^{l_k}\bigg [(T_{(k,j)}u)\otimes (T_{(k,j)}u)\bigg ]\bigg ). 2013). This is due to another result of Magnus and Neudecker (1986), which states that \({\mathcal {D}}'_{k}N_k={\mathcal {D}}'_{k}\) for any integer k. This concludes the derivations of Fisher Information matrix expressions given in Sects. For large sparse LMMs that include a large number of random effects, without further adaptation of the methods presented in this work to employ sparse matrix methodology, it is expected that lmer will provide superior performance. apply to documents without the need to be rewritten? Stat. If you have rare events ( eg many people but few of them have died - lots of zeroes , not many ones) then you may wish to use other procedures to get less biased results; such as the Firth method which is apenalized likelihood approach to reducing small-sample bias in maximum likelihood estimation; see, https://www3.nd.edu/~rwilliam/stats3/RareEvents.pdf. 33(3), 333362 (2008). When this is the case you do not need to be concerned about it - accept what you have got. However, it is not immediately apparent to us whether the derivations we have provided may be used in conjunction with such methods to improve the testing procedures for random effects. \end{aligned}$$, $$\begin{aligned}&P=X'X,\quad Q=X'Y,\quad R=X'Z,\quad S=Y'Y,\quad T=Y'Z,\quad U=Z'Z. In some treatments (exactly 6) the values are "zero". I would be happy, if somebody could help! Using the Pr(>|z|) result above, we can conclude that the variable 'Credit_score' is an important predictor for 'diabetes', as the p-value is less than 0.05. For the SAT score model, the estimated parameters obtained using each of the methods detailed in Sects. Am. factor groupings are not nested inside one another). In this approach, for each factor, \(f_k\), the elements of vec\((D_k)\) are to be treated as distinct from one another with numerical optimization for \(D_k\) performed over the space of all \((q_k \times q_k)\) matrices. For me it seems as if habitat1 was the most popular, most frequently used habitat tpye?? Linear Regression is a type of regression models which assume the presence of linear relationship between the target and the predictor variables. Supplementary Material Section S13). To summarize, the ReML-based FS algorithm for the multi-factor LMM is almost identical to that given in Algorithm1. The accuracy and validity of the direct-SW degrees of freedom estimation method proposed in Sect. The random effects vector b was simulated according to a normal distribution with covariance D, where D was predefined, exhibited no particular constrained structure and contained a mixture of both zero and nonzero off-diagonal elements. - 150.95.25.133. We show how these expressions can be employed for LMM parameter estimation via the Fisher Scoring algorithm and can be further adapted for constrained optimization of the random effects covariance matrix. For each one unit increase in gpa, the z-score decreases by 0.478. There are a series of bumps each giving potentially different answers in terms of the estimates but with pretty similar likelihoods. Given \({\mathbf {K}}^a_k\) and \({\mathbf {K}}^c_k\), the covariance components \(\sigma ^2\) and \(\{D_k\}_{k \in \{1,\ldots ,r\}}\) are given as \(\sigma ^2=\sigma ^2_e\) and \(D_k = \sigma ^{-2}_e(\sigma ^2_a{\mathbf {K}}^a_k + \sigma ^2_c{\mathbf {K}}^c_k)\), respectively. Wiley, Probability and Statistics Series (1972), Raudenbush, S.W., Bryk, A.S.: Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd edn. Data Anal. We now note that, by the construction of the random effects design matrix, Z, and the block diagonal structure of D, it can be seen that: By substituting (33) into the second term of (2), and taking the partial derivative matrix with respect to \(D_k\) using Lemma 1, the below can be obtained: Similarly, by substituting (33) into the third term of (2), and taking the partial derivative matrix with respect to \(D_k\) using Lemma 2, the below can be obtained: By combining the previous two derivative expressions, (32) is obtained.\(\square \).
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