• Journal of the Korean Data and Information Science Society
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• v.25 no.2
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• pp.423-429
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• 2014

This paper discusses a method for getting Type I sums of squares by projections under a two-way fixed-effects model when variances of errors are not equal. The method of weighted least squares is used to estimate the parameters of the assumed model. The model is fitted to the data in a sequential manner by using the model comparison technique. The vector space generated by the model matrix can be composed of orthogonal vector subspaces spanned by submatrices consisting of column vectors related to the parameters. It is discussed how to get the Type I sums of squares by using the projections into the orthogonal vector subspaces.

• Journal of the Korean Data and Information Science Society
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• v.25 no.4
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• pp.799-805
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• 2014

This paper deals with a method for getting the Type III sums of squares on the basis of projections under the assumption of two-way fixed effects model. For unbalanced data in general total sum of squares is not equal to the sum of componentwise Type III sums of squares. There are some differencies between two quantities. The suggested method using projections can detect where the differences occur and how much they are different. The traditional ANOVA method could not explain clearly the differences. It also discusses how eigenvectors and eigenvalues of the projection matrices can be used to get the Type III sums of squares.

• Journal of the Korean Data and Information Science Society
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• v.26 no.1
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• pp.31-39
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• 2015

This paper deals with a method for estimating variance components on the basis of projections under the assumption of random effects model. It discusses how to use projections for getting sums of squares to estimate variance components. The use of projections makes the vector subspace generated by the model matrix to be decomposed into subspaces that are orthogonal each other. To partition the vector space by the model matrix stepwise procedure is used. It is shown that the suggested method is useful for obtaining Type I sum of squares requisite for the ANOVA method.

• The Korean Journal of Applied Statistics
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• v.24 no.2
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• pp.373-381
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• 2011

This paper discusses how to get the sums of squares due to treatment factors when Type I Analysis is used by projections for the analysis of data under the assumption of a two-way ANOVA model. The suggested method does not need to calculate the residual sums of squares for the calculation of sums of squares. There-fore, the calculation is easier and faster than classical ANOVA methods. It also discusses how eigenvectors and eigenvalues of the projection matrices can be used to get the calculation of sums of squares. An example is given to illustrate the calculation procedure by projections for unbalanced data.

• Journal of the Korean Data and Information Science Society
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• v.22 no.3
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• pp.381-387
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• 2011

This paper suggests a method for estimating components of variance in one-factor random model. Estimates of variance components are given by the method of moments. Sums of squares due to variance sources are obtained by projections. This paper also shows how to use eigenvalues for getting the coefficients of variance components in the expression of the expectations of the mean squares. The suggested method shows easier and faster than the method of Harley's synthesis.

• The Korean Journal of Applied Statistics
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• v.29 no.7
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• pp.1155-1163
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• 2016

This paper deals with an estimation procedure of variance components in a mixed effects model by projections. Projections are used to obtain sums of squares instead of using reductions in sums of squares due to fitting both the assumed model and sub-models in the fitting constants method. A projection matrix can be obtained for the residual model at each step by a stepwise procedure to test the hypotheses. A weighted least squares method is used for the estimation of fixed effects. Satterthwaite's approximation is done for the confidence intervals for variance components.

• The Korean Journal of Applied Statistics
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• v.28 no.6
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• pp.1093-1101
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• 2015

This paper discusses nested design models when nesting occurs in treatment structure and design structure. Some are fixed and others are random; subsequently, the fixed factors having a nested design structure are assumed to be nested in the random factors. The treatment structure can involve random and fixed effects as well as a design structure that can involve several sizes of experimental units. This shows how to use projections for sums of squares by fitting the model in a stepwise procedure. Expectations of sums of squares are obtained via synthesis. Variance components of the nested design model are estimated by the method of moments.

• Journal of the Korean Data and Information Science Society
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• v.23 no.6
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• pp.1155-1163
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• 2012

This paper suggests a method for getting sums of squares due to sources of variation under the assumption of two-way fixed effects model. The method used for calculating the quantities due to fixed-effects is based on the projections of an observation vector y on the column space generated by the model matrix X under the assumed model. The suggested method shows that the calculation of Type II sums of squares by projections is much easier than the classical Type II analysis.

• The Korean Journal of Applied Statistics
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• v.29 no.2
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• pp.291-299
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• 2016

This paper discusses how to establish estimable functions when there are fixed and random effects in design models. It proves that estimable functions of mixed models are not related to random effects. A fitting constants method is used to obtain sums of squares due to random effects and Hartley's synthesis is used to calculate coefficients of variance components. To test about the fixed effects the degrees of freedom associated with divisor are determined by means of the Satterthwaite approximation.

• Journal of the Korean Data and Information Science Society
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• v.25 no.3
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• pp.547-554
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• 2014

This paper discusses a method of estimating variance components for random effects model. Henderson's method I and III are discussed for the esimation of variance components. This paper shows how to use projections instead of using Henderson's methods for the calculation of sums of squares which are quadratic forms in the observations. It also discusses that eigenvalues can be used for getting the expectations of sums of squares in place of using the method of Hartley's synthesis. It shows the suggested method is much more effective than those methods.