Details


Missing Values

An observation with missing values for any of the variables used by PROC NESTED is omitted from the analysis. Blank values of CLASS character variables are treated as missing values.

Unbalanced Data

A completely nested design is defined to be unbalanced if the groups corresponding to the levels of some classification variable are not all of the same size . The NESTED procedure can compute unbiased estimates for the variance components in an unbalanced design, but because the sums of squares on which these estimates are based no longer have 2 distributions under a Gaussian model for the data, F tests for the significance of the variance components cannot be computed. PROC NESTED checks to see that the design is balanced. If it is not, a warning to that effect is placed on the log, and the columns corresponding to the F tests in the analysis of variance are left blank.

General Random Effects Model

A random effects model for data from a completely nested design with n factors has the general form

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where

is the value of the dependent variable observed at the r th replication with factor j at level i j , for j = 1 ,...,n .

¼

is the overall (fixed) mean of the sampled population.

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are mutually uncorrelated random effects with zero means and respective variances , , ..., .

Analysis of Covariation

When you specify more than one dependent variable, the NESTED procedure produces a descriptive analysis of the covariance between each pair of dependent variables in addition to a separate analysis of variance for each variable. The analysis of covariation is computed under the basic random effects model for each pair of dependent variables:

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where the notation is the same as that used in the preceding general random effects model.

There is an additional assumption that all the random effects in the two models are mutually uncorrelated except for corresponding effects, for which

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Error Terms in F Tests

Random effects ANOVAs are distinguished from fixed effects ANOVAs by which error mean squares are used as the denominator for F tests. Under a fixed effects model, there is only one true error term in the model, and the corresponding mean square is used as the denominator for all tests. This is how the usual analysis is computed in PROC ANOVA, for example. However, in a random effects model for a nested experiment, mean squares are compared sequentially. The correct denominator in the test for the first factor is the mean square due to the second factor; the correct denominator in the test for the second factor is the mean square due to the third factor; and so on. Only the mean square due to the last factor, the one at the bottom of the nesting order, should be compared to the error mean square.

Computational Method

The building blocks of the analysis are the sums of squares for the dependent variables for each classification variable within the factors that precede it in the model, corrected for the factors that follow it. For example, for a two-factor nested design, PROC NESTED computes the following sums of squares:

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where y ijr is the r th replication, n ij is the number of replications at level i of the first factor and level j of the second, and a dot as a subscript indicates summation over the corresponding index. If there is more than one dependent variable, PROC NESTED also computes the corresponding sums of crossproducts for each pair. The expected value of the sum of squares for a given classification factor is a linear combination of the variance components corresponding to this factor and to the factors that are nested within it. For each factor, the coefficients of this linear combination are computed. (The efficiency of PROC NESTED is partly due to the fact that these various sums can be accumulated with just one pass through the data, assuming that the data have been sorted by the classification variables.) Finally, estimates of the variance components are derived as the solution to the set of linear equations that arise from equating the mean squares to their expected values.

Displayed Output

PROC NESTED displays the following items for each dependent variable:

  • Coefficients of Expected Mean Squares, which are the coefficients of the n + 1 variance components making up the expected mean square. Denoting the element in the i th row and j th column of this matrix by C ij , the expected value of the mean square due to the i th classification factor is

    click to expand

    C ij is always zero for i > j , and if the design is balanced, C ij is equal to the common size of all classification groups of the j th factor for i j . Finally, the mean square for error is always an unbiased estimate of . In other words, C n +1 ,n +1 = 1.

For every dependent variable, PROC NESTED displays an analysis of variance table. Each table contains the following:

  • each Variance Source in the model (the different components of variance) and the total variance

  • degrees of freedom (DF) for the corresponding sum of squares

  • Sum of Squares for each classification factor. The sum of squares for a given classification factor is the sum of squares in the dependent variable within the factors that precede it in the model, corrected for the factors that follow it. (See the Computational Method section on page 2991.)

  • F Value for a factor, which is the ratio of its mean square to the appropriate error mean square. The next column, labeled PR > F, gives the significance levels that result from testing the hypothesis that each variance component equals zero.

  • the appropriate Error Term for an F test, which is the mean square due to the next classification factor in the nesting order. (See the Error Terms in F Tests section on page 2991.)

  • Mean Square due to a factor, which is the corresponding sum of squares divided by the degrees of freedom

  • estimates of the Variance Components. These are computed by equating the mean squares to their expected values and solving for the variance terms. (See the Computational Method section on page 2991.)

  • Percent of Total, the proportion of variance due to each source. For the i th factor, the value is

    click to expand
  • Mean, the overall average of the dependent variable. This gives an unbiased estimate of the mean of the population. Its variance is estimated by a certain linear combination of the estimated variance components, which is identical to the mean square due to the first factor in the model divided by the total number of observations when the design is balanced.

If there is more than one dependent variable, then the NESTED procedure displays an analysis of covariation table for each pair of dependent variables (unless the AOV option is specified in the PROC NESTED statement). See the Analysis of Covariation section on page 2990 for details. For each source of variation, this table includes the following:

  • Degrees of Freedom

  • Sum of Products

  • Mean Products

  • Covariance Component, the estimate of the covariance component

Items in the analysis of covariation table are computed analogously to their counterparts in the analysis of variance table. The analysis of covariation table also includes the following:

  • Variance Component Correlation for a given factor. This is an estimate of the correlation between corresponding effects due to this factor. This correlation is the ratio of the covariance component for this factor to the square root of the product of the variance components for the factor for the two different dependent variables. (See the Analysis of Covariation section on page 2990.)

  • Mean Square Correlation for a given classification factor. This is the ratio of the Mean Products for this factor to the square root of the product of the Mean Squares for the factor for the two different dependent variables.

ODS Table Names

PROC NESTED assigns a name to each table it creates. You can use these names to reference the table when using the Output Delivery System (ODS) to select tables and create output data sets. These names are listed in the following table. For more information on ODS, see Chapter 14, Using the Output Delivery System.

Table 49.1: ODS Tables Produced in PROC NESTED

ODS Table Name

Description

Statement

ANCOVA

Analysis of covariance

default with more than one dependent variable

ANOVA

Analysis of variance

default

EMSCoef

Coefficients of expected mean squares

default

Statistics

Overall statistics for fit

default




SAS.STAT 9.1 Users Guide (Vol. 4)
SAS.STAT 9.1 Users Guide (Vol. 4)
ISBN: N/A
EAN: N/A
Year: 2004
Pages: 91

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