This paper discusses the use of projections for the sums of squares in the analyses of variance for two-way nested design data. The model for this data is assumed to only have random effects. Two different sizes of experimental units are required for a given experimental situation, since nesting is assumed to occur both in the treatment structure and in the design structure. So, variance components are coming from the sources of random effects of treatment factors and error terms in different sizes of experimental units. The model for this type of experimental situation is a random effects model with more than one error terms and therefore estimation of variance components are concerned. A projection method is used for the calculation of sums of squares due to random components. Squared distances of projections instead of using the usual reductions in sums of squares that show how to use projections to estimate the variance components associated with the random components in the assumed model. Expectations of quadratic forms are obtained by the Hartley’s synthesis as a means of calculation.
Nesting can occur in certain experimental situations. When it occurs, it is possible to have nested effects in either the design structure or the treatment structure (or both). If it happens in the design structure of an experiment then there will be different sizes of experimental units where a smaller experimental unit is nested within a larger one. When nesting occurs in the treatment structure the levels of one factor occur with only one level of a second factor. Hence, the levels of the first factor are nested within the level of the second factor.
Models for different nested designs are discussed in the literature such as Milliken and Johnson (1984), Montgomery (1976) and Searle (1971). Some related topics about nested designs are shown in Khan
Henderson (1953) developed four methods of computing sums of squares depending on the types of models. For a random effects model Henderson’s Method 1 (or the analysis of variance method) can be used for the calculations of sums of squares. Variance components are estimated by equating sums of squares to their corresponding expected values. The calculations of the coefficients of the variance components can be obtained by Hartley’s (1967) method of synthesis.
Nesting can happen in either factors or design structures; therefore, this paper suggests a model and discuss a method using projections for analysing data in that situation because both types of nesting in nested designs are uncommon in the literature.
This paper discusses a method of using projections as a method of getting sums of squares for estimation of variance components in a two-way random model with nested effects in the design structure. The use of projections for calculations of sums of squares actually provides a different way of partioning the total sum of squares in the analysis of variance for unbalanced data. For finding appropriate projections it is necessary to decompose the vector space generated by the model matrix into the orthogonal vector subspaces at first. Establishing proper models then become important to understand the appropriate projections. The method is discussed for the estimation of variance components for nested design data.
Suppose that there are two factors A and B with the levels of B being nested within the levels of factor A in the treatment structure. Let
Let
where
The model has two sizes of experimental units and thus two error terms. The matrix form of the model can be expressed as
where
The parameters of the model are
To have R( ) terms in analyses of variance by Henderson’s Method 1 let
where
where
where
where
where
The covariance matrix of
Let
where tr(
where
Variance components can be estimated by equating the sums of squares to their expected values as follows.
Since the equations of (
where
From the normal equations, we get
Montgomery (1976)’s data from a two-way nested design is used as an example for the use of projections. Table 1 shows the coded purity data set.
For illustration, suppliers and batches are assumed to be random with batches nested within suppliers. The model for this two-stage nested design is a random model with just one error term and shown as
where
where
There are three variance components; therefore, sub-models should be derived from the projection of
The projection from the model (
where
where
Letting
The solutions for the system of linear equations in
This paper concerns the calculations of sums of squares due to random components in a two-stage nested design with which each classification has its own experimental unit. Consequently, a two-factor nested components-of-variance model with two types of random errors is proposed under the conditions.
Sums of squares in the analyses of variance are usually expressed in terms of quadratic forms and when each quadratic form is the squared distance of a projection. This idea came to the application of the projection to the calculation of sum of squares. Each projection is defined on the estimation space, spanned by the model matrix of the fitted model with the corresponding random vector. The discussion established appropriate model for the projection at each step.
This paper shows that reductions in sums of squares can be replaced by the squared distances of the projections and that the associated matrices with the quadratic forms are easily identified from the fitted models. These are quite different points of views from the traditional methods employed in ANOVA. However, the method can be used as a method for analyzing variance because it produces the same results as the analysis of variance method for the random effects model.
This research was supported by the Keimyung-Scholar Research Grant of Keimyung University in 2017.
Coded purity data from balanced two-stage nested data
Suppliers | 1 | 2 | 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | |
Batches | 1 | −2 | −2 | 1 | 1 | 0 | −1 | 0 | 2 | −2 | 1 | 3 |
−1 | −3 | 0 | 4 | −2 | 4 | 0 | 3 | 4 | 0 | −1 | 2 | |
0 | −4 | 1 | 0 | −3 | 2 | −2 | 2 | 0 | 2 | 2 | 1 | |
Batch totals | 0 | −9 | −1 | 5 | −4 | 6 | −3 | 5 | 6 | 0 | 2 | 6 |
Supplier totals | −5 | 4 | 14 |