O título da minha tese é: “Revisão Taxonômica das Serpentes da Tribo Philodryadini Cope 1886 (Serpentes: Dipsadidae); Aplicação de Métodos Comparativos Usando Dados Morfológicos e Moleculares.” e sou orientado pelo Professor Dr. Hussam Zaher.

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Primeira versão das Propostas

Escalamento e eliminação do efeito do crescimento alométrico em variáveis morfométricas.

O emprego de variáveis morfométricas e matrizes de dados morfométricos multivariadas são de constante uso em análises de variação morfológica em diferentes áreas da biologia, como taxonomia, evolução, sistemática, e ecologia. No entanto, a natureza destas variáveis não é geralmente tomada em conta na hora de realizar análises que visem compreender a variação da forma e não a variação do tamanho.

A minha proposta consiste em criar uma função que escale variáveis morfométricas ¹⁾ ao aplicar uma técnica de normalização ²⁾ para escalar variáveis que exibem crescimento alométrico, produzindo um conjunto de dados pronto para análise, sem o efeito do tamanho (size-free). A função recebera uma matriz de dados morfométricos originais e um conjunto de argumentos e aplicara a transformação. No final, haverá dois conjuntos de dados, o original e o escalado. Como resultado, a função realizara um PCA para cada conjunto de dados e apresentará os gráficos de forma comparativa, o original do lado do escalado. Da mesma forma, e com a definição previa de um argumento que selecione alguma variável categórica que defina grupos na matriz morfométrica, uma tabela comparativa apresentará os resultados de um teste de MANOVA realizado a cada conjunto de dados (Original e Escalado).Lleonart et al. (2000)

size.free {unknown}							R Documentation

	Data transformation that removes the allometric effect of body size from
				morphometric data

Description:

Remove the effects of allometric growth on morphometric variables. This removal
is performed by applying the proposal of Lleonart et al (2000), using a modification
of the allometric growth equation.

Usage:

size.free(x,vpd=...,vf=...)

Arguments:

x	An R object of class “data.frame”. In the data set the rows must be
	the individuals and columns numeric variables, with at least one
	variable of class “factor”.

vpd	Column number (x[,i]) in the data set (data.frame) that indicates
	which variable will be used as the standard body size to calculate
	the constants of growth for each variable.

vf	Column number (x[,i]) in the data set (data.frame) that indicates
	the factor variable that will be used to performs an MANOVA on the
	original and scaled variables.

Details:

	The transformation is made by applying the equation 13 of Lleonart et al.
	(2000:page 88).
	Yi*=Yi(X0/Xi)^b
	Where Yi is the observed value for the individual i of the variable
	to be scaled Y, Xi is the observed value for the individual i of the
	standard body size variable X, X0 is the mean of the standard body
	size variable, and b is the growth constant.
	The growth constant is calculated using linear regression.

Value:


Graphics
	Returns a comparative plot (Scaled next to Original data) of principal
	component analysis (prcomp).

MANOVA
	Returns the summary of a MANOVA (manova) performed with the Original
	data followed by other MANOVA made with the Scaled data, both using
	the factor variable (vf) as factor.

File
	Saves a file in Comma-separated values format (.csv) with the scaled data
	in the working directory.


Warning:

	If there are any NAs or zeros in the input data set, an error message
	will indicate in which column are these.

Note:

	NAs and zero values are not allowed.


Author(s):

	Juan Camilo Arredondo	jcas36@gmail.com

References:

	Lleonart J., J. Salat, & G. T. Torres. 2000. Removing allometric effects
	of body size in morphological analysis. Journal of Theoretical Biology
	205:85–93.
	http: dx.doi.org/10.1006/jtbi.2000.2043

	Thorpe RS. 1976. Biometric analysis of geograph	ic variation and racial
	affinities. Biological Reviews 51: 407–452.
	http:dx.doi.org/10.1111/j.1469-185X.1976.tb01063.x


Examples:

## data.frame with NAs and zeros
(df<-data.frame(matrix(sample(c(0,NA,8:10),90,replace=TRUE),ncol=6)))

## Factor variable
(SP<-sample(paste("SP",1:3,sep=""),15,replace=TRUE))

## Including the factor variable into the data set
df$SP<-as.factor(SP)
head(df)

# Applying the function size.free
size.free(df,vpd=2,vf=7)

## Eliminating NAs
(jn<-df[sapply(df,is.numeric)])
(jn[is.na(jn)]<-sample((jn[!(is.na(jn))]),length(jn[is.na(jn)])))
df[,1:ncol(jn)]<-jn

# Applying the function size.free
size.free(df,vpd=2,vf=7)

## Eliminating zeros
(jn<-df[sapply(df,is.numeric)])
jn[jn==0]<-sample(jn[!jn==0],length(jn[df==0]))
df[,1:ncol(jn)]<-jn

# Apllying the function size.free
size.free(df,vpd=2,vf=7)

## vpd is used as argument to define the standard body size variable and vf is the factor variable
size.free<-function(x,vpd=...,vf=...)

## Extract the numeric variables and store it into a new data frame
{x1<-x[sapply(x,is.numeric)]

## Extracts the factor variable to a new object
Factor<-as.factor(x[,vf])

## Extracts the standard size variable to a new object
vpadrao<-x[,vpd]
  
#######  NAs and Zeros  #######

## Creates a logical object to determinated the presence of NAs at any position of the data frame and stores the column of each one
{Na<-sapply(x1,function(x)any(is.na(x)))
	
## Creates a logical object to determinated the presence of zeros at any position of the data frame and stores the column of each one
zero<-sapply(x1,function(x)any(x==0))

## Conditional statement that uses logical evaluation for determination of NAs
if(any(Na))

## Conditional statement that stops the process and shows an error message indicating treatment of NAs
{stop(paste("Please replace NA in column",paste(which(Na),collapse=", ")))}

## Conditional statement that uses logical evaluation for determination of zeros
else if(any(zero))

## Conditional statement that stops the process and shows an error message indicating treatment of zeros
{stop(paste("Please replace Zeros in column",paste(which(zero),collapse=", ")))}}

#######  Growth Constant  #######

## Copies the data into a new object
x2<-x

## Erase the standard body size variable from the data set
x2[,vpd]<-NULL

## Extract the numeric variables and store them into a new data frame, reducing to n-1 variables (data set without the standard size variable)
x3<-x2[sapply(x2,is.numeric)]

## Creates a matrix of NAs to later include the regression coefficients
{abs<-matrix(NA,ncol=2,nrow=ncol(x3))
	
## Creates a matrix of NAs to later include the growth constants
b<-rep(NA,ncol(x3))

## for-loop function that calculates the coefficients of each variable and stores into a new object
for (i in 1:ncol(x3))

## Calculates the regression between variables and includes its coefficients into the abs matrix
{abs[i,]<-coefficients(lm(log(vpadrao)~log(x3[,i])))

## Calculates the exponectial of each growth constant and include its value into the object b
b[i]<-abs[i,2]}}

#######  Transformation  #######

## Creates an object with the total number of elements in the data set to be scaled
{l<-nrow(x3)*ncol(x3)

## Transform the data set into a matrix
n.m<-as.matrix(x3)

## Creates a matrix with the values of the growth constants, repeated by columns.
n.b<-matrix(rep(b,rep(nrow(x3),ncol(x3))),ncol=ncol(x3))

## Creates a matrix with the values of the standard size variable, repeated by columns.
y<-matrix(rep(vpadrao,ncol(x3)),ncol=ncol(x3))

## for-loop function that calculates the scaled value of each value of the original matrix.
for(i in 1:l)

## Calculates the scaled value and include it in a new scaled matrix.
{n.m[i]<-n.m[i]*((mean(vpadrao)/y[i])^n.b[i])
	
## Tranform the scaled matrix into a data frame object
scaled<-as.data.frame(n.m)}

## Includes the standard size variable into the data set of scaled variables
scaled$VPD<-vpadrao

## Includes the factor into the data set of scaled variables
scaled$Factor<-Factor}

#######  PCA  #######

## Original Data ##

## Performs an analysis of principal components (using the covaration matrix) on the original data (only the numeric variables)
{pcaOrig<-prcomp(x1)

## Creates an object with the variances of each principal component (columns in pcaOrig$x)
vs<-apply(pcaOrig$x,2,var)

## Creates an object with the Proportion of variation of each principal component
povar<-vs/sum(vs)

## Saves the loadings of the principal components in a data frame
pcs<-as.data.frame(pcaOrig$x)

## Includes the Factor variable into the data set of loadings of the principal components of original data set
pcs$Factor<-Factor}

## Scaled Data ##

## Performs an analysis of principal components (using the covaration matrix) on the scaled data (only the numeric variables)
{pcaScal<-prcomp(scaled[sapply(scaled,is.numeric)])

## Creates an object with the variances of each principal component (columns in pcaScal$x)
vs2<-apply(pcaScal$x,2,var)

## Creates an object with the Proportion of variation of each principal component
povar2<-vs2/sum(vs2)

## Saves the loadings of the principal components in a data frame
pcs2<-as.data.frame(pcaScal$x)

## Includes the Factor variable into the data set of loadings of the principal components of scaled data set
pcs2$Factor<-Factor}

## Creates a graphic window to include two plots, arranged horizontally
par(mfrow=c(1,2))

## PCA Plots ##

## Plot the First Principal Component against the second, both from the original data
{plot(pcaOrig$x,main="PCA of Original Data",xlab=paste("First Principal Componet ",round((povar[1])*100,1),"%"),ylab=paste("Second Principal Componet ",round((povar[2])*100,1),"%"),bty="l",type="n")
	
## Plot the points as the their respective factor names, for the original variable
text(x=pcs[,1],y=pcs[,2],labels=pcs[,ncol(pcs)])

## Plot the First Principal Component against the second, both from the scaled data
plot(pcaScal$x,main="PCA of Scaled Data",xlab=paste("First Principal Componet ",round((povar2[1])*100,1),"%"),ylab=paste("Second Principal Componet ",round((povar2[2])*100,1),"%"),bty="l",type="n")

## Plot the points as the their respective factor names, for the scaled variable
text(x=pcs2[,1],y=pcs2[,2],labels=pcs2[,ncol(pcs2)])
}

#######  MANOVA  #######

## Performs a MANOVA with the original data
{manova.orig<-manova(as.matrix(x1)~Factor)
	
## Performs a MANOVA with the Scaled data
manova.scal<-manova(as.matrix(scaled[sapply(scaled,is.numeric)])~Factor)
 
## Prints a title previous to the summary of the Wilks MANOVA test for the orginal data
cat("Wilks MANOVA test for Original Data\n")

## Prints the summary table of the Wilks MANOVA test for the orginal data
print(summary(manova.orig,test="Wilks"))

## Prints a paragraph line
cat("\n")

## Prints a title previous to the summary of the Wilks MANOVA test for the Scaled data
cat("Wilks MANOVA test for Scaled Data\n")

## Prints the summary table of the Wilks MANOVA test for the Scaled data
print(summary(manova.scal,test="Wilks"))}

#######  File  #######

{

## Saves a file in the working directory with the scaled data
write.csv(scaled,"Data Scaled.csv",row.names=FALSE)

## Explanatory text for the function and the arragement of the columns in the scaled file
cat("\nExplanation\n\nThe File -Data Scaled.csv- was saved in your working\ndirectory, and contains your scaled variables as a\nproduct of the transformation of your original data,\nfollowing the proposal of Lleonart et al., (2000)*.\nThe file contains an equal number of variables that\nyour original data; however, the variables of the\nstandard size (vpd) and factor (vf) were named as\n-VPD- and -Factor-, respectively, and moved to the\nend of the data set.\n\n* Lleonart J., J. Salat, & G. T. Torres. 2000.\nRemoving allometric effects of body size in morphological analysis.\nJournal of Theoretical Biology 205:85–93.\nhttp://dx.doi.org/10.1006/jtbi.2000.2043")
}
}

size.free.R
example_size.free.R