VCAinference | R Documentation |

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.

```
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
)
```

`obj` |
(object) of class 'VCA' or, alternatively, a list of 'VCA' objects, where all other arguments 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 |

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

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

`claim.type` |
(character) one of "VC", "SD", "CV" specifying how claim-values have to be interpreted: |

`VarVC` |
(logical) TRUE = the covariance matrix of the estimated VCs will be computed (see |

`excludeNeg` |
(logical) TRUE = confidence intervals of negative variance estimates will not be reported. |

`constrainCI` |
(logical) TRUE = CI-limits for all variance components are constrained to be >= 0. |

`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 |

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

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.

(VCAinference) object, a list with elements:

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

`ConfInt` |
(list) with elements |

`VCAobj` |
(VCA) object specified as input, if |

Original CIs will always be available independent of parameter-settings of `excludeNeg`

and
`constrainCI`

. Original CIs are stored in attribute "CIoriginal" of the returned 'VCAinference'-object, e.g.
'attr(obj$ConfInt$SD$OneSided, "CIoriginal")' or 'attr(obj$ConfInt$CV$TwoSided, "CIoriginal")'.

Andre Schuetzenmeister andre.schuetzenmeister@roche.com

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

`print.VCAinference`

, `anovaVCA`

```
## Not run:
# load data (CLSI EP05-A2 Within-Lab Precision Experiment)
data(dataEP05A2_1)
# perform (V)variance (C)component (A)nalysis (also compute 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$ChiSqTest
# request CIs for all VCs, default is to exclude CIs of negative VCs (excludeNeg=TRUE)
# solve MMEs first (or set MME=TRUE when calling anovaVCA)
res2 <- solveMME(res2)
VCAinference(res2, VarVC=TRUE)
# request CIs for all VCs, including those for negative VCs, note that all CI-limits
# are constrained to be >= 0
VCAinference(res2, VarVC=TRUE, excludeNeg=FALSE)
# request unconstrained CIs for all VCs, including those for negative VCS
# one has to re-fit the model allowing the VCs to be negative
res3 <- anovaVCA(y~day/run, tmpData, NegVC=TRUE, MME=TRUE)
VCAinference(res3, VarVC=TRUE, excludeNeg=FALSE, constrainCI=FALSE)
### use the numerical example from the CLSI EP05-A2 guideline (p.25)
data(Glucose,package="VCA")
res.ex <- anovaVCA(result~day/run, Glucose)
### also perform Chi-Squared tests
### Note: in guideline claimed SD-values are used, here, claimed variances are used
VCAinference(res.ex, total.claim=3.4^2, error.claim=2.5^2)
# load another example dataset and extract the "sample_1" subset
data(VCAdata1)
sample1 <- VCAdata1[which(VCAdata1$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")
## End(Not run)
```

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