View source: R/summary.manylm.R
summary.manylm  R Documentation 
summary
method for class "manylm"  computes a table
summarising the statistical significance of different multivariate terms
in a linear model fitted to highdimensional data, such as multivariate
abundance data in ecology.
## S3 method for class 'manylm' summary(object, nBoot=999, resamp="residual", test="F", cor.type=object$cor.type, block=NULL, shrink.param=NULL, p.uni="none", studentize=TRUE, R2="h", show.cor = FALSE, show.est=FALSE, show.residuals=FALSE, symbolic.cor=FALSE, rep.seed=FALSE, tol=1.0e6, ... ) ## S3 method for class 'summary.manylm' print( x, digits = max(getOption("digits")  3, 3), signif.stars=getOption("show.signif.stars"), dig.tst=max(1, min(5, digits  1)), eps.Pvalue=.Machine$double.eps, ... )
object 
an object of class "manylm", usually, a result of a call to

nBoot 
the number of Bootstrap iterations, default is 
resamp 
the method of resampling used. Can be one of "case" (not yet available),"residual" (default), "perm.resid", "score" or "none". See Details. 
test 
the test to be used. Possible values are: "LR" = likelihood ratio statistic (default) and "F" = LawleyHotelling trace statistic. 
cor.type 
structure imposed on the estimated correlation matrix under the fitted model. Can be "I"(default), "shrink", or "R". See Details. 
block 
A factor specifying the sampling level to be resampled. Default is resampling rows. 
shrink.param 
shrinkage parameter to be used if 
p.uni 
whether to calculate univariate test statistics and their Pvalues, and if so, what type. 
studentize 
logical, whether studentized residuals or residuals should beused for simulation in the resampling steps. This option is not used in case resampling. 
R2 
the type of R^2 (correlation coefficient) that should be shown, can be one of: 
show.cor 
logical, if 
show.est 
logical. Whether to show the estimated model parameters. 
show.residuals 
logical. Whether to show residuals/a residual summary. 
symbolic.cor 
logical. If 
rep.seed 
logical. Whether to fix random seed in resampling data. Useful for simulation or diagnostic purposes. 
tol 
the tolerance used in estimations. 
x 
an object of class 
digits 
the number of significant digits to use when printing. 
signif.stars 
logical. If 
dig.tst 
the number of digits to round the estimates of the model parameters. 
eps.Pvalue 
a numerical tolerance for the formatting of p values. 
... 
for 
The summary.manylm
function returns a table summarising the statistical
significance of each multivariate term specified in the fitted manylm model.
For each model term, it returns a test statistic as determined by the argument
test
, and a Pvalue calculated by resampling rows of the data using a
method determined by the argument resamp
. The four possible resampling methods are residualpermutation (Anderson and Robinson (2001)), score resampling (Wu (1986)), case and residual resampling (Davison and Hinkley (1997, chapter 6)), and involve resampling under the alternative hypothesis. These methods ensure approximately valid inference even when the correlation between variables has been misspecified, and for case and score resampling, even when the equal variance assumption of linear models is invalid. By default, studentized residuals (r_i/sqrt(1h_ii)) are used in residual and score resampling, although raw residuals could be used via the argument studentize=FALSE
. If resamp="none"
, pvalues cannot be calculated, however the test statistics are returned.
If you have a specific hypothesis of primary interest that you want to test,
then you should use the anova.manylm
function, which can resample rows
of the data under the null hypothesis and so usually achieves a better
approximation to the true significance level.
To check model assumptions, use plot.manylm
.
The summary.manylm
function is designed specifically for highdimensional data (that, is when the number of variables p is not small compared to the number of observations N). In such instances a correlation matrix is computationally intensive to estimate and is numerically unstable, so by default the test statistic is calculated assuming independence of variables (cor.type="I"
). Note however that the resampling scheme used ensures that the Pvalues are approximately correct even when the independence assumption is not satisfied. However if it is computationally feasible for your dataset, it is recommended that you use cor.type="shrink"
to account for correlation between variables, or cor.type="R"
when p is small. The cor.type="R"
option uses the unstructured correlation matrix (only possible when N>p), such that the standard classical multivariate test statistics are obtained. Note however that such statistics are typically numerically unstable and have low power when p is not small compared to N. The cor.type="shrink"
option applies ridge regularisation (Warton 2008), shrinking the sample correlation matrix towards the identity, which improves its stability when p is not small compared to N. This provides a compromise between "R"
and "I"
, allowing us to account for correlation between variables, while using a numerically stable test statistic that has good properties. The shrinkage parameter by default is estimated by crossvalidation using the multivariate normal likelihood function, although it can be specified via shrink.param
as any value between 0 and 1 (0="I" and 1="R", values closer towards 0 indicate more shrinkage towards "I"). The validation groups are chosen by random assignment and so you may observe some slight variation in the estimated shrinkage parameter in repeat analyses. See ridgeParamEst
for more details.
Rather than stopping after testing for multivariate effects, it is often of interest to find out which response variables express significant effects. Univariate statistics are required to answer this question, and these are reported if requested. Setting p.uni="unadjusted"
returns resamplingbased univariate Pvalues for all effects as well as the multivariate Pvalues, whereas p.uni="adjusted"
returns adjusted Pvalues (that have been adjusted for multiple testing), calculated using a stepdown resampling algorithm as in Westfall & Young (1993, Algorithm 2.8). This method provides strong control of familywise error rates, and makes use of resampling (using the method controlled by resample
) to ensure inferences take into account correlation between variables.
A multivariate R^2 value is returned in output, but there are many ways to define a multivariate R^2. The type of R^2 used is controlled by the R2
argument. If cor.shrink="I"
then all variables are assumed independent, a special case in which Hooper's R^2 returns the average of all univariate R^2 values, whereas the vector R^2 returns their product.
print.summary.manylm
tries to be smart about formatting the coefficients, genVar
, etc. and additionally gives ‘significance stars’ if
signif.stars
is TRUE
.
summary.manylm returns an object of class "summary.manyglm", a list with components
call 
the component from 
terms 
the terms object used. 
show.residuals 
the supplied argument. 
show.est 
the supplied argument. 
p.uni 
the supplied argument. 
test 
the supplied argument. 
cor.type 
the supplied argument. 
resample 
the supplied argument. 
nBoot 
the supplied argument. 
rankX 
the rank of the design matrix 
residuals 
the model residuals 
genVar 
the estimated generalised variance 
est 
the estimated model coefficients 
shrink.param 
the shrinkage parameter. Either the value of the argument with the same name or if this was not supplied the estimated shrinkage parameter. 
aliased 
named logical vector showing if the original coefficients are aliased. 
df 
a 3vector of the rank of the model and the number of residual degrees of freedom, plus number of nonaliased coefficients. 
If the argument test
is not NULL
then the list also
included the components
coefficients 
a matrix containing the test statistics and the pvalues. 
n.iter.sing 
the number of iterations that were skipped due to singularity of the design matrix caused by case resampling. 
If furthermore the Design matrix is neither empty nor consists of the Intercept only, the following adddional components are included:
r.squared 
the calculated correlation coefficient. 
R2 
a character that describes which type of correlation coefficient was calculated. 
statistic 
a matrix containing the results of the overall test. 
cov.unscaled 
the unscaled ( 
If the argument show.cor
is TRUE
the following adddional
components are returned:
correlation 
the (p*q) by (p*q) correlation matrix, with p being the number of columns of the design matrix and q being the number of response variables. Note that this matrix can be very big. 
Yi Wang, Ulrike Naumann and David Warton <David.Warton@unsw.edu.au>.
Anderson, M.J. and J. Robinson (2001). Permutation tests for linear models. Australian and New Zealand Journal of Statistics 43, 7588.
Davison, A. C. and Hinkley, D. V. (1997) Bootstrap Methods and their Application. Cambridge University Press, Cambridge.
Warton D.I. (2008). Penalized normal likelihood and ridge regularization of correlation and covariance matrices. Journal of the American Statistical Association 103, 340349.
Warton D.I. and Hudson H.M. (2004). A MANOVA statistic is just as powerful as distancebased statistics, for multivariate abundances. Ecology 85(3), 858874.
Westfall, P. H. and Young, S. S. (1993) Resamplingbased multiple testing. John Wiley & Sons, New York.
Wu, C. F. J. (1986) Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis. The Annals of Statistics 14:4, 12611295.
manylm
, anova.manylm
, plot.manylm
data(spider) spiddat < log(spider$abund+1) ## Estimate the coefficients of a multivariate linear model: fit < manylm(spiddat~., data=spider$x) ## To summarise this multivariate fit, using score resampling to ## and F Test statistic to estimate significance: summary(fit, resamp="score", test="F") ## Instead using residual permutation with 2000 iteration, using the sum of F ## statistics to estimate multivariate significance, but also reporting ## univariate statistics with adjusted Pvalues: summary(fit, resamp="perm.resid", nBoot=2000, test="F", p.uni="adjusted") ## Obtain a summary of test statistics using residual resampling, accounting ## for correlation between variables but shrinking the correlation matrix to ## improve its stability and showing univariate pvalues: summary(fit, cor.type="shrink", p.uni="adjusted")
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