VCA (version 1.3.4)

VCAinference: Inferential Statistics for VCA-Results.

Description

Function VCAinference constructs one- and two-sided confidence intervals, and performs Chi-Squared tests for total and error variance against claimed values for 'VCA' objects.

Usage

VCAinference(obj, alpha = 0.05, total.claim = NA, error.claim = NA,
  claim.type = "VC", VarVC = FALSE, excludeNeg = TRUE,
  constrainCI = TRUE, ci.method = "sas", quiet = FALSE)

Arguments

obj

(object) of class 'VCA' or, alternatively, a list of 'VCA' objects, where all other argument can be specified as vectors, where the i-th vector element applies to the i-th element of 'obj' (see examples)

alpha

(numeric) value specifying the significance level for \(100*(1-alpha)\)% confidence intervals.

total.claim

(numeric) value specifying the claim-value for the Chi-Squared test for the total variance (SD or CV, see claim.type).

error.claim

(numeric) value specifying the claim-value for the Chi-Squared test for the error variance (SD or CV, see claim.type).

claim.type

(character) one of "VC", "SD", "CV" specifying how claim-values have to be interpreted: "VC" (Default) = claim-value(s) specified in terms of variance(s), "SD" = claim-values specified in terms of standard deviations (SD), "CV" = claim-values specified in terms of coefficient(s) of variation (CV) and are specified as percentages. If set to "SD" or "CV", claim-values will be converted to variances before applying the Chi-Squared test (see examples).

VarVC

(logical) TRUE = if element "Matrices" exists (see anovaVCA), the covariance matrix of the estimated VCs will be computed (see vcovVC, which is used in CIs for intermediate VCs if 'method.ci="sas"'. Note, this might take very long for larger datasets, since there are many matrix operations involved. FALSE (Default) = computing covariance matrix of VCs is omitted, as well as CIs for intermediate VCs.

excludeNeg

(logical) TRUE = confidence intervals of negative variance estimates will not be reported. FALSE = confidence intervals for all VCs will be reported including those with negative VCs. See the details section for a thorough explanation.

constrainCI

(logical) TRUE = CI-limits for all variance components are constrained to be >= 0. FALSE = unconstrained CIs with potentially negative CI-limits will be reported. which will preserve the original width of CIs. See the details section for a thorough explanation.

ci.method

(character) string or abbreviation specifying which approach to use for computing confidence intervals of variance components (VC). "sas" (default) uses Chi-Squared based CIs for total and error and normal approximation for all other VCs (Wald-limits, option "NOBOUND" in SAS PROC MIXED); "satterthwaite" will approximate DFs for each VC using the Satterthwaite approach (see SattDF for models fitted by ANOVA) and all Cis are based on the Chi-Squared distribution. This approach is conservative but avoids negative values for the lower bounds.

quiet

(logical) TRUE = will suppress any warning, which will be issued otherwise

Value

(VCAinference) object, a list with elements:

ChiSqTest

(data.frame) with results of the Chi-Squared test

ConfInt

(list) with elements VC, SD, CV, all lists themselves containing (data.frame) objects OneSided and TwoSided

VCAobj

(VCA) object specified as input, if VarVC=TRUE, the 'aov.tab' element will have an extra column "Var(VC)" storing variances of VC-estimates"

Details

This function computes confidence intervals (CI) for variance components (VC), standard deviations (SD) and coefficients of variation (CV). VCs 'total' and 'error' can be tested against claimed values specifying parameters 'total.claim' and 'error.claim'. One can also specify claim-values in terms of SD or CV (see claim.type). Confidence intervals for VCs are constructed either following the same rules as in SAS 9.2 PROC MIXED with option 'method=type1' (ci.method="sas") or using Satterthwaite methodology throughout (ci.method="satterthwaite"). In the former approach for VC total and error, which are constrained to be \(>= 0\), CIs are based on the Chi-Squared distribution. Degrees of freedom (DF) for total variance are approximated using the Satterthwaite approximation (which is not available in either SAS procedure). For all other VCs, the CI is \([sigma^2-QNorm(alpha/2)*SE(sigma^2); sigma^2+QNorm(1-alpha/2)*SE(sigma^2)]\), where QNorm(x) indicates the x-quantile of the standard normal distribution. The second method approximates DFs for all VCs using the Satterthwaite approximation and CIs are based on the corresponding Chi-Squared distribution for all VCs (see examples). Note that in the computation of the covariance-matrix of the VCs, the estimated VCs will be used. If these are requested to be set to 0 (NegVC=FALSE in anovaVCA), the result might not be conformable with theory given in the first reference. The validity of this implementation was checked against SAS 9.2 PROC MIXED (method=type1), where VCs are not constrained to be >= 0. The sampling variances for VCs are obtained assuming normality throughout based on \(Var(\sigma^{2} = C^{-1} * Var(m_{SS} * (C^{-1})^{T}))\), where \(C^{-1}\) is the inverse of the coefficient matrix equating observed Sum of Squares (SS) to their expected values, and \((C^{-1})^{T}\) indicating the transpose of \(C^{-1}\) (see Searle et al. 1992, pg. 176).

An input VCA-object can be in one of three states:

State (1) corresponds to the situation, where all VC > 0. State (2) corresponds to the situation, where at least one VC < 0. State (3) corresponds to situations, where negative VC estimates occured but were set to 0, i.e. NegVC=FALSE - the Default.

State (2) occurs when parameter NegVC was set to TRUE in anovaVCA, state (3) represents the default-setting in function anovaVCA. If a VCA-object is in state (1), parameter excludeNeg has no effect (there are no negative VCs), only parameter constrainCI is evaluated. For VCA-objects in state(2), constrainCI has no effect, because constraining CIs for unconstrained VCs makes no sense. State (3) forces parameter constrainCI to be set to TRUE and one can only choose whether to exclude CIs of negative VC estimates or not. Whenever VCs have to be constrained, it is straight forward to apply constraining also to any CI. Note that situations outlined above only occur when parameter VarVC is set to TRUE, which causes estimation of the covariance-matrix of variance components. The default is only to compute and report CIs for total and error variance, which cannot become negative.

References

Searle, S.R, Casella, G., McCulloch, C.E. (1992), Variance Components., Wiley New York

Burdick, R., Graybill, F. (1992), Confidence Intervals on Variance Components. Marcel Dekker, Inc.

Satterthwaite, F.E. (1946), An Approximate Distribution of Estimates of Variance Components., Biometrics Bulletin 2, 110-114

See Also

print.VCAinference, getVCvar, anovaVCA

Examples

# NOT RUN {
# load data (CLSI EP05-A2 Within-Lab Precision Experiment) 
data(dataEP05A2_1)

# perform (V)variance (C)component (A)nalysis (also comute A-matrices)
res <- anovaVCA(y~day/run, dataEP05A2_1)

# get confidence intervals for total and error (VC, SD, CV)
VCAinference(res)

# additionally request CIs for all other VCs; default is to constrain 
# CI-limits to be >= 0
# first solve MME
res <- solveMME(res)
VCAinference(res, VarVC=TRUE)

# now using Satterthwaite methodology for CIs
VCAinference(res, VarVC=TRUE, ci.method="satt")

# request unconstrained CIs
VCAinference(res, VarVC=TRUE, constrainCI=FALSE)

# additionally request Chi-Squared Tests of total and error, default 
# is that claim values are specified as variances (claim.type="VC")
VCAinference(res, total.claim=4.5, error.claim=3.5)

# perform Chi-Squared Tests, where claim-values are given as SD, 
# compare p-values to former example
VCAinference(res, total.claim=sqrt(4.5), error.claim=sqrt(3.5), claim.type="SD")

# now using Satterthwaite methodology for CIs
VCAinference(res, total.claim=sqrt(4.5), error.claim=sqrt(3.5), 
				claim.type="SD", ci.method="satt")

# now add random error to example data forcing the ANOVA-estimate of the 
# day-variance to be negative
set.seed(121)
tmpData <- dataEP05A2_1
tmpData$y <- tmpData$y + rnorm(80,,3)
res2 <- anovaVCA(y~day/run, tmpData)

# call 'VCAinference' with default settings
VCAinference(res2)

# extract components of the returned 'VCAinference' object
inf <- VCAinference(res2, total.claim=12)
inf$ConfInt$VC$OneSided			# one-sided CIs for variance components
inf$ConfInt$VC$TwoSided			# two-sided CI for variance components
inf$ChiSqTest

# request CIs for all VCs, default is to exclude CIs of negative VCs (excludeNeg=TRUE) 
# solve MMEs first (or set MME=TRUE when calling anovaVCA)
res2 <- solveMME(res2)
VCAinference(res2, VarVC=TRUE)

# request CIs for all VCs, including those for negative VCs, note that all CI-limits 
# are constrained to be >= 0
VCAinference(res2, VarVC=TRUE, excludeNeg=FALSE)

# request unconstrained CIs for all VCs, including those for negative VCS
# one has to re-fit the model allowing the VCs to be negative
res3 <- anovaVCA(y~day/run, tmpData, NegVC=TRUE, MME=TRUE)
VCAinference(res3, VarVC=TRUE, excludeNeg=FALSE, constrainCI=FALSE)
 
### use the numerical example from the CLSI EP05-A2 guideline (p.25)
data(Glucose)
res.ex <- anovaVCA(conc~day/run, Glucose)

### also perform Chi-Squared tests
### Note: in guideline claimed SD-values are used, here, claimed variances are used
VCAinference(res.ex, total.claim=3.4^2, error.claim=2.5^2)


# load another example dataset and extract the "sample_1" subset
data(VCAdata1)
sample1 <- VCAdata1[which(VCAdata1$sample==1),]

# generate an additional factor variable and random errors according to its levels
sample1$device <- gl(3,28,252)                                      
set.seed(505)
sample1$y <- sample1$y + rep(rep(rnorm(3,,.25), c(28,28,28)),3)     

# fit a crossed-nested design with main factors 'lot' and 'device' 
# and nested factors 'day' and 'run' nested below, also request A-matrices 
res1 <- anovaVCA(y~(lot+device)/day/run, sample1) 

# get confidence intervals, covariance-matrix of VCs, ..., 
# explicitly request the covariance-matrix of variance components
# solve MMEs first
res1 <- solveMME(res1)
inf1 <- VCAinference(res1, VarVC=TRUE, constrainCI=FALSE)
inf1

# print numerical values with more digits
print(inf1, digit=12)

# print only parts of the 'VCAinference' object (see \code{\link{print.VCAinference}})
print(inf1, digit=12, what=c("VCA", "VC"))

# extract complete covariance matrix of variance components 
# (main diagonal is part of standard output -> "Var(VC"))
VarCovVC <- vcovVC(inf1$VCAobj)
round(VarCovVC, 12)

# use by-processing and specific argument-values for each level of the by-variable
data(VCAdata1)
fit.all <- anovaVCA(y~(device+lot)/day/run, VCAdata1, by="sample", NegVC=TRUE)
inf.all <- VCAinference(fit.all, total.claim=c(.1,.75,.8,1,.5,.5,2.5,20,.1,1))
print.VCAinference(inf.all, what="VC")
# }