Robust Hotelling test using the S-estimator

Description

Robust one-sample and two-sample Hotelling test using the S-estimator and the Fast and Robust Bootstrap.

Usage

## S3 method for class 'formula'
FRBhotellingS(formula, data=NULL, ...)

## Default S3 method:
FRBhotellingS(X, Y=NULL, mu0 = 0, R = 999, bdp = 0.5, conf = 0.95,
method = c("HeFung", "pool"), control=Scontrol(...),
na.action=na.omit, ...)

Arguments

Details

The classical Hotelling test for testing if the mean equals a certain center or if two means are equal is modified into a robust one through substitution of the empirical estimates by the S-estimates of location and scatter. The S-estimator uses Tukey's biweight function where the constant is chosen to obtain the desired breakdown point as specified by bdp. One-sample S-estimates are computed by a call to the implementation of the fast-S algorithm in the rrcov package of Todorov and Filzmoser (2009). For two-sample S-estimates an adaptation of the fast-S algorithm is used. The tuning parameters of the algorithm can be changed via control.

The fast and robust bootstrap is used to mimic the distribution of the test statistic under the null hypothesis. For instance, the 5% critical value for the test is given by the 95% quantile of the recalculated statistics.

Robust simultaneous confidence intervals for linear combinations of the mean (or difference in means) are developed similarly to the classical case (Johnson and Wichern, 1988, page 239). The value CI is a matrix with the confidence intervals for each element of the mean (or difference in means), with level conf. It consists of two rows, the first being the lower bound and the second the upper bound. Note that these intervals are rather conservative in the sense that the simultaneous confidence level holds for all linear combinations and here only p of these are considered (with p the dimension of the data).

For the two-sample Hotelling test we assume that the samples have an underlying distribution with the same covariance matrix. This covariance matrix can be estimated in two different ways using the pooled covariance matrix or the two-sample estimator of He and Fung (He and Fung 2000), and argument method defaults to the second option. For more details see Roelant et al. (2008).

In the two-sample version, the null hypothesis always states that the two means are equal. For the one-sample version, the default null hypothesis is that the mean equals zero, but the hypothesized value can be changed and specified through argument mu0.

Bootstrap samples are discarded if the fast and robust covariance estimate is not positive definite, such that the actual number of recalculations used can be lower than R. This number is returned as ROK.

Value

An object of class FRBhot which extends class htest and contains at least the following components:

Author(s)

Ella Roelant, Stefan Van Aelst and Gert Willems

References

• X. He and W.K. Fung (2000) High breakdown estimation for multiple populations with applications to discriminant analysis. Journal of Multivariate Analysis, 72, 151–162.

• R.A. Johnson, D.W. Wichern (1988) Applied Multivariate Statistical Analysis, 2nd Edition, Prentice-Hall.

• E. Roelant, S. Van Aelst and G. Willems, (2008) Fast Bootstrap for Robust Hotelling Tests, COMPSTAT 2008: Proceedings in Computational Statistics (P. Brito, Ed.) Heidelberg: Physika-Verlag, 709–719.

• M. Salibian-Barrera, S. Van Aelst and G. Willems (2008) Fast and robust bootstrap. Statistical Methods and Applications, 17, 41–71.

• V. Todorov and P. Filzmoser (2009) An Object-Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32, 1–47. URL http://www.jstatsoft.org/v32/i03/.

• S. Van Aelst and G. Willems (2013). Fast and Robust Bootstrap for Multivariate Inference: The R Package FRB. Journal of Statistical Software, 53(3), 1–32. URL: http://www.jstatsoft.org/v53/i03/.

See Also

plot.FRBhot,summary.FRBhot, FRBhotellingMM, Scontrol

Examples

## One sample robust Hotelling test
data(delivery)
delivery.x <- delivery[,1:2]
FRBhotellingS(delivery.x,R=199)

## One sample robust Hotelling test
data(ForgedBankNotes)
samplemean <- apply(ForgedBankNotes, 2, mean)
res = FRBhotellingS(ForgedBankNotes, mu0=samplemean,R=199)
res
# Note that the test rejects the hypothesis that the true mean equals the
# sample mean; this is due to outliers in the data (i.e. the robustly estimated
# mean apparently significantly differs from the non-robust sample mean.

# Graphical display of the results:
plot(res)
# It is clear from the (scaled) simultaneous confidence limits that the rejection
# of the hypothesis is due to the differences in variables Bottom and Diagonal







## Two sample robust Hotelling test
data(hemophilia)
grp <-as.factor(hemophilia[,3])
x <- hemophilia[which(grp==levels(grp)[1]),1:2]
y <- hemophilia[which(grp==levels(grp)[2]),1:2]

#using the pooled covariance matrix to estimate the common covariance matrix
## Not run: res = FRBhotellingS(x,y,method="pool")

#using the estimator of He and Fung to estimate the common covariance matrix
res = FRBhotellingS(x,y,method="HeFung",R=199)

# or using the formula interface
## Not run: res = FRBhotellingS(as.matrix(hemophilia[,-3])~hemophilia[,3],method="HeFung",R=99)

# From the confidence limits it can be seen that the significant difference
# is mainly caused by the AHFactivity variable. The graphical display helps too:
plot(res)
# the red line on the histogram indicates the test statistic value in the original
# sample (it is omitted if the statistic exceeds 100)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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Type 'demo()' for some demos, 'help()' for on-line help, or
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Type 'q()' to quit R.

> library(FRB)
Loading required package: corpcor
Loading required package: rrcov
Loading required package: robustbase
Scalable Robust Estimators with High Breakdown Point (version 1.3-11)

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/FRB/FRBhotellingS.Rd_%03d_medium.png", width=480, height=480)
> ### Name: FRBhotellingS
> ### Title: Robust Hotelling test using the S-estimator
> ### Aliases: FRBhotellingS FRBhotellingS.default FRBhotellingS.formula
> ### Keywords: htest multivariate robust
> 
> ### ** Examples
> 
> ## One sample robust Hotelling test
> data(delivery)
> delivery.x <- delivery[,1:2]
> FRBhotellingS(delivery.x,R=199)

	One sample Hotelling test based on multivariate S-estimates (breakdown
	point = 0.5)

data:  delivery.x
T^2_R = 84.006, p-value < 2.2e-16
alternative hypothesis: true mean vector is not equal to (0,0) 

sample estimates:
              [,1]    [,2]
S-loc vector 5.809 282.428

> 
> ## One sample robust Hotelling test
> data(ForgedBankNotes)
> samplemean <- apply(ForgedBankNotes, 2, mean)
> res = FRBhotellingS(ForgedBankNotes, mu0=samplemean,R=199)
> res

	One sample Hotelling test based on multivariate S-estimates (breakdown
	point = 0.5)

data:  ForgedBankNotes
T^2_R = 132.85, p-value < 2.2e-16
alternative hypothesis: true mean vector is not equal to (214.823,130.3,130.193,10.53,11.133,139.45) 

sample estimates:
                [,1]    [,2]    [,3]   [,4]   [,5]    [,6]
S-loc vector 214.781 130.267 130.179 10.871 11.103 139.626

> # Note that the test rejects the hypothesis that the true mean equals the
> # sample mean; this is due to outliers in the data (i.e. the robustly estimated
> # mean apparently significantly differs from the non-robust sample mean.
> 
> # Graphical display of the results:
> plot(res)
> # It is clear from the (scaled) simultaneous confidence limits that the rejection
> # of the hypothesis is due to the differences in variables Bottom and Diagonal
> 
> 
> ## Two sample robust Hotelling test
> data(hemophilia)
> grp <-as.factor(hemophilia[,3])
> x <- hemophilia[which(grp==levels(grp)[1]),1:2]
> y <- hemophilia[which(grp==levels(grp)[2]),1:2]
> 
> #using the pooled covariance matrix to estimate the common covariance matrix
> ## Not run: res = FRBhotellingS(x,y,method="pool")
> 
> #using the estimator of He and Fung to estimate the common covariance matrix
> res = FRBhotellingS(x,y,method="HeFung",R=199)
> 
> # or using the formula interface
> ## Not run: res = FRBhotellingS(as.matrix(hemophilia[,-3])~hemophilia[,3],method="HeFung",R=99)
> 
> # From the confidence limits it can be seen that the significant difference
> # is mainly caused by the AHFactivity variable. The graphical display helps too:
> plot(res)
> # the red line on the histogram indicates the test statistic value in the original
> # sample (it is omitted if the statistic exceeds 100)
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>