aovp: Fitting and testing ANOVA using permutation tests

Description

aovp is aov modified to use permutation tests instead of normal theory tests. Like aov, the ANOVA model is fitted by a call to lmp for each stratum. Timing differences between aovp and aov are negligible.

Usage

aovp(formula, data = NULL, perm="Exact", seqs=FALSE,  center=TRUE, projections = FALSE, qr = TRUE, contrasts = NULL, ...)

Arguments

formula

A formula specifying the model.

data

A data frame in which the variables specified in the formula will be found. If missing, the variables are searched for in the standard way.

perm

"Exact", "Prob", "SPR" will produce permutation probabilities. Anything else, such as "", will produce F-test probabilities.

seqs

If TRUE, will calculate sequential SS. If FALSE, unique SS.

center

If TRUE, numerical variables will be centered.

projections

Logical flag: should the projections be returned?

Logical flag: should the QR decomposition be returned?

contrasts

A list of contrasts to be used for some of the factors in the formula. These are not used for any Error term, and supplying contrasts for factors only in the Error term will give a warning.

...

Arguments to be passed to aovp, such as those listed below.

Value

The usual output from aov, with permutation p-values instead of normal theory p-values.

Additional Parameters

These are the same as for lmp.

Details

The model Y=Xb+Zg+e is assumed, where X is the incidence matrix for fixed effects, and Z is an incidence matrix for random effects, with columns representing the several error strata. The aovp() algorithm projects Y into strata such that each stratum has a single error term, such as a stratum defined by whole blocks. X is also projected so that the model in this stratum becomes P(Y)=P(X)bi+ei.

The vector bi is divided into sources with dfj degrees of freedom for the jth source, and summary(aovp()) will produce an ANOVA table for these sources in the ith strata. See Venables and Ripley for details.

Either permutation test p-values or the usual F-test p-values will be output. Polynomial model terms are collected into sources, so that Y~A+B+I(A^2) will contain two sources, one for A with 2 df, and one for B with 1 df. Sources for factors are treated as usual, and polynomial terms and factors may be mixed in one model. The function poly.formula may be used to create polynomial models, and the function multResp may be used to create multiresponse matrices for the lhs from variables defined in data.

The Exact method will permute the values exactly. The Prob and SPR methods will approximate the permutation distribution by randomly exchanging pairs of Y elements. The Exact method will be used by default when the number of observations is less than or equal to maxExact, otherwise Prob will be used.

Prob: Iterations terminate when the estimated standard error of the estimated proportion p is less than p*Ca. The iteration continues until all sources and coefficients meet this criterion or until maxIter is reached. See Anscome(1953) for the origin of the criterion.

SPR: This method uses sequential probability ratio tests to decide between the hypotheses p0 and p1 for a strength (alpha, beta) test. The test terminates upon the acceptance or rejection of p0 or if maxIter is reached. See Wald (1947). The power of the SPR is beta at p0 and increases to 1-beta at p1. Placing p0 and p1 close together makes the cut off sharp.

Exact: This method generates all permutations of Y. It will generally be found too time consuming for more than 10 or 11 observations, but note that aovp may be used to divide the data into small enough blocks for which exact permutation tests may be possible.

For Prob and SPR, one may set nCycle to unity to exchange all elements instead of just pairs at each iteration, but there seems to be no advantage to doing this unless the number of iterations is small -- say less than 100.

The SS will be calculated sequentially, just as lm does; or they may be calculated uniquely, which means that the SS for each source is calculated conditionally on all other sources. This is SAS type III, which is also what drop1() produces, except that drop1() will not drop main effects when interactions are present. The parameter seqs may be used to override the default unique calculation behavior.

References

Examples

Run this code


## A simple randomized block example. 
# There are 7 blocks and 6 treatments. A first
# analysis with blocks as a factor shows block to be significant and treatments not. 

data(Hald17.4)
summary(aovp(Y~T+block,Hald17.4))

# Using the block to define a separate error strata tells a different story.

summary(aovp(Y~T+Error(block),Hald17.4))

# There appears to be a linear trend in the blocks. This may be investigated by
# extracting a linear component. The factor L was created by copying the block
# factor and assigning it a linear contrast, like this
# contrasts(L,1)<-contr.poly(7). The analysis then becomes.

summary(aovp(Y~T+L+Error(block),Hald17.4))

# The L factor is not significant under permutation. It is significant when aov()
# is used and the test is the usual F-test.


## From Venables and Ripley (2000)
# This is a 2^3 factorial in the variables N,P,K. It is fractioned by using the 
# three way interaction, NPK, into two fractions of 4. Each of these fractions is 
# allocated to 3 blocks, making 6 blocks in all. An analysis with block as a 
# variable is the following. As may be seen, aovp() discards the confounded NPK interaction.

data(NPK)
summary(aovp(yield ~ block + N*P*K, NPK))

# Since the NPK interaction was confounded with blocks, the experimenter no doubt judged 
# it of lesser interest. It may however be examined by including blocks as an additional
# error term as follows. The basic error level between blocks is of course larger than 
# that within blocks, so the NPK interaction would have to be substantially larger that
# it would have had to be were it tested within blocks.
summary(aovp(yield ~  N*P*K + Error(block), NPK))

# The SS calculated by aovp() are unique SS by default. That is,
# they are sums of squares for the difference of a model with and without the source. The
# resulting test is a test of the hypothesis that the source has no effect on the response.
# Sequential SS, which are those produced by aov() may be obtained by setting the
# parameter seqs=TRUE. simDesign is an unbalanced design created by the AlgDesign package.

data(simDesign)
summary(aovp(Y~.,simDesign))
summary(aovp(Y~.,simDesign,seqs=TRUE)) 

# Since there is only one stratum, these results are the same as would be obtained from
anova(lmp(Y~.,simDesign))

# ANOVA for numerical variables. First using contrasts, then numeric variables.
data(Federer276)
summary(aovp(Plants~Variety*Treatment+Error(Rep/Plot),Federer276))
data(Federer276Numeric)
summary(aovp(poly.formula(Plants~quad(Variety,Treatment)+Error(Rep/Plot)),Federer276Numeric))
# The coefficients and their p-values may be obtained by
summaryC(aovp(Plants~Variety*Treatment+Error(Rep/Plot),Federer276))

Run the code above in your browser using DataLab