Variance component

In most cases, the fixed effect parameters are the parameters of interest. However, adequate modeling of the variance-covariance structure is critical for assessment of the fixed effects and is useful in explaining the variability of the data. Indeed, sometimes the fixed effects are of little interest and the variance components are of primary importance. Covariance structures that are overparameterized may lead to poor estimation of the standard errors for estimates of the fixed effects (Altham, 1984). However, covariance matrices that are too restrictive may lead to invalid inferences about the fixed effects because the assumed covariance structure does not exist and is not valid. For this reason, methods need to be available for testing the significance of the variance components in a model. 

One method might be to use a Z-test, similar to the one developed for fixed effects. SAS, as well as many of other programs, present such estimates of the variance 

An alternative test would be to use the LRT comparing two models with the same mean structure, but with nested covariance structures since only newly added variance components are being added. Whatever the method used, testing for the significance of variance components is problematic since the null hypothesis that the estimate equals zero lies on the boundary of the parameter space for the alternative hypothesis. In other words, consider the hypothesis test H0 a2 0 versus. Ha j2 0. Notice that the alternative hypothesis is lower bounded by the null hypothesis. The LRT is not valid under these conditions because the test statistic is no longer distributed as a single chi-squared random variable, but becomes a mixture of chi-squared random variables (Stram and Lee, 1994). 

Reprinted with permission from Table C.l in Fitzmaurice, G.M., Laird, N.M., and Ware, J.H. Applied longitudinal analysis, Wiley, 2004. Copyright 2004. 

One method to adjust for boundary problems is to use simulation. First, replace the unknown parameters by their parameter estimates under the null model. Second, simulate data under the null model ignoring the variability in treating the unknown parameters as fixed. Third, the data is then fit to both the null and alternative model and the LRT statistic is calculated. This process is repeated many times and the empirical distribution of the test statistic under the null distribution is determined. This empirical distribution can then be compared to mixtures of chi-squared distributions to see which chi-squared distribution is appropriate. 

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Vmisc contains all other variance components, e.g., that stemming from the v-dependent Sy (heteroscedacity). (See Fig. 2.5.)... 

For each of the CS and the QC concentrations the overall mean and standard deviation are compared to the daily averages and SDs from this, variance components for the within-day and day-to-day effects are estimated by subtraction of variances. 

The terms and are called variance components. More compli-... 

Even If the cost of estimation error cannot be quantified as this model requires, effective allocation of resources may be possible when detailed knowledge of sources of variation Is available. In this case, a replication strategy can be based on variance component and cost Information. For example, consider the problem of deciding how many samples to collect and how many analyses to perform on each sample Let... 

The administrators or users of the study results must supply the objectives and required precision. Statisticians can develop the models for alternative sampling strategies. The estimates of variance components and costs can come from a number of places ... 

A slight modification of the pilot study is a double-sampling plan for estimation (9,10). Double-sampling plans were developed to provide estimates with a fixed precision using as few observations as possible. If the sources of variation and variance components are known prior to the study, then a fixed sample size plan is the best... 

In a double-sampling plan, a small initial sample (ni ) is selected. Variance components are estimated from the results of these samples and then these estimates are used to estimate the total sample required (n). An additional n-i samples are then collected to complete the study. 

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Table 5.7. Variance components in three-way ANOVA (abc model with repetitions)... 

Figure 4.13 Principal component analysis of the mean isotopic data for oceanic islands (courtesy of Vincent Salters). In the top left corner, the plane of the first two components (the Mantle Plane of Zindler et al, 1982) explains 93 percent of the variance. Component 1 is dominated by lead isotopes, component 2 by Sr and Nd isotopes. Other components are plotted for reference. In the top right corner, the Mantle Plane is viewed sideways along the direction of the second component, so the distance of each point to the plane can be easily seen. In the bottom left corner, it is viewed along the axis of the first component. The bottom right corner shows how little variance is left with components 3 and 4.

Figure 3. IQ variance component estimates for five age groups, derived from published IQ correlations using the Falconer heritability formula.

Ohnesorge, J., Saenger-van de Griend, C., and Waetzig, H. (2005). Quantification in capillary electrophoresis-mass spectrometry long- and short-term variance components and their compensation using internal standards. Electrophoresis 26, 2360—2375. 

Sampling error In surveys, investigators frequently take measurements (or samples) on the parameters of interest, from which inferences to the true but unknown population are inferred. The inability of the sample statistics to represent the true population statistics is called sample error. There are many reasons why the sample may be inaccurate, from the design of the experiment to the inability of the measuring device. In some cases, the sources of error may be separated (see Variance components). 

Variance components A statistical technique for factoring the total variance in a random parameter into its component parts. Typically, a model is defined that represents the experimenter s understanding of the variance components. This model is used to separate the variance components. The model is called a variance components model. 

From the mean square (MS) values and the formulas for the expected mean square (see Table 3.30) one can calculate the variance components. For the example of Figure 3.5 this gives ... 

To estimate the variance components for the random effects model, we also computed the group means regression. The sum of squared residuals from the LSDV estimator is 444,288. The sum of squares from the group means regression is 22382.1. The estimate of a,.2 is 444,288/93 = 4777.29. The estimate of a 2 is 22,382.1/2 - (1/20)4777.29 = 10,952.2. The model is then reestimated by FGLS using these estimates ... 

Substitution of variance components concept may be a hard sell degrees of freedom can be significantly reduced... 

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