# nonpartest: Nonparametric Comparison of Multivariate Samples In npmv: Nonparametric Comparison of Multivariate Samples

## Description

Performs analysis of one-way multivariate data using nonparametric techniques developed since 2008. Allows for small samples and ordinal variables, or even mixture of the different variable types ordinal, quantitative, binary. Using F-approximations for ANOVA Type, Wilks' Lambda Type, Lawley Hotelling Type, and Bartlett Nanda Pillai Type test statics, as well as a permutation test for each, the package compares the multivariate distributions of the different samples. Also computes nonparametric relative effects.

## Usage

 ```1 2``` ```nonpartest(formula,data,permtest=TRUE,permreps=10000,plots=TRUE, tests=c(1,1,1,1),releffects=TRUE,...) ```

## Arguments

 `formula` an object of class "formula", with a single explanatory variable and multiple response variables (or one that can be coerced to that class). `data` an object of class "data.frame", containing the variables in the formula. `permtest` logical. If TRUE the p-values for the permutation test are returned `permreps` number of replications in permutation test `plots` logical. If TRUE box plots are produced for each response variable versus treatment `tests` vector of zeros and ones which specifies which test statistics are to be calculated. A 1 corresponds to the test statistics which are to be returned `releffects` logical. If TRUE the relative effects are returned `...` Graphical parameters to be passed to the boxplot function.

## Details

The nonparametric methods implemented in the code have been developed for complete data with no missing values. The code automatically produces a warning if there is missing data.

## Value

Returns a list of 2 data frames if relative effects are turned on, otherwise returns a single data frame. First data frame consist of p-values for test statistics and permutation test (if permutation test is turned on), second data frame consist of relative effects for each response variable.

## Warning

The nonparametric methods implemented in the code have been developed for complete data with no missing values. The code automatically produces a warning if there is missing data.

Under certain conditions, the matrices H and G are singular (See literature for explanation of H and G), for example when the number of response variables exceeds the sample size. When this happens, only the ANOVA type statistic can be computed. The code automatically produces a warning if H or G are singular.

## Note

We define (for simplicity, only the formula for the balanced case is given here, the unbalanced case is given in the literature): \[H=(1/(a-1))*sum_i=1^a n (Rbar_i .-Rbar_..)(Rbar_i.-Rbar_..)' \] \[G=(1/(N-1))*sum_i=1^a sum_j=1^n(R_ij-Rbar_i.)(R_ij-Rbar_i.)'\]

The ANOVA Type statistic is given by: \[T_A= (tr(H)/tr(G))\] The distribution of T_A is approximated by an F distribution with fhat_1 and fhat_2 where: \[fhat_1=(tr(G)^2/tr(G^2)) and fhat_2= (a^2)/((a-1)sum^a_i=1(1)/(n_i-1))* fhat_1

The Lawley Hotelling Type statistic is given by: \[U=tr[(a-1)H((N-a)G)^-1]\] Using the McKeon approximation the distribution of U is approximated by a "stretched" F distribution with degrees freedom K and D where: \[K=p(a-1) and D=4 + (K+2)/(B-1)\] and \[B = ((N-p-2)(N-a-1))/((N-a-p)(N-a-p-n))\]

The Bartlett Nanda Pillai Type statistic is given by: \[V= tr{(a-1)H[(a-1)H+(N-a)G]^-1}\] McKeon approximated the distribution of ((V/gamma)/nu_1)/((1-V/gamma)/nu_2) using an F distribution with degrees freedom nu_1 and nu_2 where: \[ gamma=min(a-1,p)\] \[ nu_1=(p(a-1))/(gamma(N-1))*[(gamma(N-a+gamma-p)(N-1))/((N-a)(N-p))-2]\] \[ nu_2=(N-a+gamma-p)/(N)*[(gamma(N-a+ gamma-p)(N-1))/((N-a)(N-p))-2]\]

The Wilks' Lambda Type Statistic is given by \[ lambda=det(((N-a)*G )/( (N-a)*G+(a-1)*H ) \] The F approximation statistic is given by \[F_lambda=[(1-lambda^1/t)/(lambda^1/t)](df_2/df_1)\] where \[df_1 = p(a - 1) and df_2 = r t - (p(a - 1) - 2)/2\] and \[r=(N-a)-(p-(a-1)+1)/2.\] If \[p(a-1)=2 then t=1, else t=sqrt (p^2(a-1)^2-4)/(p^2+(a-1)^2-5) \] Note that regarding the above formula, there is a typo in the article Liu, Bathke, Harrar (2011).

## Author(s)

Woodrow Burchett, Amanda Ellis, Arne Bathke

## References

Arne C. Bathke , Solomon W. Harrar, and Laurence V. Madden. "How to compare small multivariate samples using nonparametric tests," Computational Statistics and Data Analysis 52 (2008) 4951-4965

Brunner E, Domhof S, Langer F (2002), Nonparametric Analysis of Longitudinal Data in Factorial Experiments. Wiley, New York.

Chunxu Liu, Arne C. Bathke, Solomon W. Harrar. "A nonparametric version of Wilks' lambda-Asymptotic results and small sample approximations" Statistics and Probability Letters 81 (2011) 1502-1506

Horst, L.E., Locke, J., Krause, C.R., McMahaon, R.W., Madden, L.V., Hoitink, H.A.J., 2005. Suppression of Botrytis blight of Begonia by Trichoderma hamatum 382 in peat and compost-amended potting mixes. Plant Disease 89, 1195-1200.

## Examples

 ```1 2``` ```data(sberry) nonpartest(weight|bot|fungi|rating~treatment,sberry,permreps=1000) ```

### Example output

```\$results
Test Statistic    df1     df2
ANOVA type test p-value                                    2.984  6.836 27.3426
McKeon approx. for the Lawley Hotelling Test               5.769 12.000 12.0000
Muller approx. for the Bartlett-Nanda-Pillai Test          2.501 15.967 41.1641
Wilks Lambda                                               4.166 12.000 24.1033
P-value
ANOVA type test p-value                             0.019
McKeon approx. for the Lawley Hotelling Test        0.002
Muller approx. for the Bartlett-Nanda-Pillai Test   0.009
Wilks Lambda                                        0.001
Permutation Test p-value
ANOVA type test p-value                                              0.004
McKeon approx. for the Lawley Hotelling Test                         0.003
Muller approx. for the Bartlett-Nanda-Pillai Test                    0.003
Wilks Lambda                                                         0.001

\$releffects
weight     bot   fungi  rating
3 0.43750 0.59375 0.56250 0.53125
6 0.72656 0.15625 0.48438 0.30469
8 0.44531 0.37500 0.21875 0.53125
9 0.39062 0.87500 0.73438 0.63281
```

npmv documentation built on May 29, 2017, 10:35 p.m.