lm modified to use permutation tests instead of normal
theory tests. Like
lm, it can be used to carry out regression,
single stratum analysis of variance and analysis of covariance . Timing
lm are negligible.
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The arguments are mostly the same as for
Additional parameters may be included. They are described in the
"Additional Parameters" section below. These additional parameters
are the same as for
a symbolic description of the model to be fit.
an optional data frame containing the variables
in the model. If not found in
"Exact", "Prob", "SPR" will produce permutation probabilities. Anyting else, such as "", will produce F-test probabilites.
If TRUE, will calculate sequential SS. If FALSE, unique SS.
If TRUE will center numerical variables
an optional vector specifying a subset of observations to be used in the fitting process.
an optional vector of weights to be used
in the fitting process. If specified, weighted least squares is used
a function which indicates what should happen
when the data contain
the method to be used; for fitting, currently only
an optional list. See the
this can be used to specify an a priori
known component to be included in the linear predictor
during fitting. An
additional arguments to be passed.
The usual regression model EY=Xb is assumed. The vector b is divided into sources
with dfi degrees of freedom for the ith source, and
anova(lmp()) will produce an
ANOVA table for these sources. 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
be used to create polynomial models, and the function
be used to create a multi-response matrix for the lhs from variables in
One may also use
summary(lm()) to obtain coefficient estimates and
estimates of the permutation test p-values. The
Exact method will permute
the values exactly. The
SPR methods will approximate
the permutation distribution by randomly exchanging pairs of Y elements. The
method will be used by default when the number of observations is less than
or equal to
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
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 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
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
produces, except that
drop1() will not drop main effects when interactions are present.
seqs may be used to override the default unique calculation behavior.
The usual output from
lm, with permutation p-values or F-test
p-values. The p-values for the coefficients are of necessity, two-sided.
For Prob and SPR: The number of iterations until the criterion is met.
For SPR: 1 the p0 hypothesis is accepted, and 0 for rejection or when no decision before
These are the same as for
If TRUE, settings such as sequential or unique will be printed. Default TRUE
If TRUE, SS/Resid SS will be used, otherwise SS. The default is TRUE
For Prob and SPR: The maximum number of iterations. Default 1000.
For Prob: Stop iterations when estimated standard error of the estimated p is less than Ca*p. Default 0.1
For SPR: Null hypothesis probability. Default 0.05
For SPR: Alternative hypothesis probability. Default 0.06
For SPR: Size of SPR test. Default 0.01
For SPR: Type II error for SPR test. Default 0.01
For Exact: maximum number of observations allowed. If data exceeds this, Prob is used. Default 10.
For Prob and SPR: Performs a complete random permutation, instead of pairwise exchanges, every nCycle cycles. Default 1000.
There is a vignette with more details and an example. To access it, type
The default contrasts are set internally to
(contr.sum, contr.poly), which means
that factor coefficients are either pairwise contrasts with the last level or polynomial contrasts.
Numerical variables should be centered in order to make them orthogonal to the constant when ANOVA is to be done.
This function will behave identically to
lm() if the following parameters are set:
perm="", seq=TRUE, center=FALSE. An exception for multiple responses is that an
ANOVA table for each response is output instead of a call to
Bob Wheeler [email protected]
Experimental Design, 2nd Ed.
John Wiley & Sons, New York.
Sequential analysis, Wiley, Sec. 5.3
"Product improvement by application of Taguchi methods." in American Supplier Institute News (special symposium ed.) Dearborn, MI. American Supplier Institute. 11-16.
Signal-to-noise ratios, performance criteria, and transformations. Technometics. 30-1. 1-17.
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# 3x3 factorial with ordered factors, each is average of 12. # This is a saturated design with no df for error. The results tend to support # Cochran and Cox who used a guessed residual SS for their analysis. The design # is balanced, so the sequential SS are the same as the unique SS. data(CC164) summary(lmp(y ~ N * P, data = CC164, perm="")) # F-value output as if lm() was used. summary(lmp(y ~ N * P, data = CC164,)) # Default, using "Exact" if possible. summary(lmp(y ~ N * P, data = CC164, perm="SPR")) anova(lmp(y ~ N * P, data = CC164)) # A two level factorial. The artificial data is N(0,1) with an effect of # 1.5 added to factor X4. When the number of iterations are small, as in # this case, using nCycle=1 is advantageous. X<-expand.grid(X1=1:2,X2=1:2,X3=1:2,X4=1:2) X$Y<-c(0.99,1.34,0.88,1.94,0.63,0.29,-0.78,-0.89,0.43,-0.03,0.50,1.66,1.65,1.78,1.31,1.51) summary(lmp(Y~(X1+X2+X3+X4)^2,X,"SP")) # The prob method is used because "SP" is not recognized. summary(lmp(Y~(X1+X2+X3+X4)^2,X,"SPR")) summary(lmp(Y~(X1+X2+X3+X4)^2,X,"SPR",nCycle=1)) #An additional parameter being passed. # A saturated design with 15 variables in 16 runs. The orginal analysis by Quinlan pooled the mean # squares from the 7 smallest effcts and found many variables to be significant. Box, reanalyzed # the data using half-normal plots and found only variables E and G to be important. The permutation # analysis agrees with this conclusion. data(Quinlan) summary(lmp(SN~.,Quinlan)) # A design containing both a polynomial variable and a factor data(simDesignPartNumeric) anova(lmp(poly.formula(Y~quad(A,B)+C),simDesignPartNumeric))
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