Skip to contents

fn_ARSS {Morefi}

20 May 2024

fn_ARSS.Rmd

Coincident Curves Test

Description

The function fn_ARSS perform the Coincident Curves Test, to determine if there are significant differences between the fitted curves for each database. It is based on the Analysis of the Residual Sum of Squares (ARSS) (Chen et al. 1992).

\[F=\frac{\frac{RSS_{p}-RSS_{s}}{3\bullet \left( K-1 \right)}}{\frac{RSS_{s}}{N-3\bullet K}}\]

\(RSS_{p}\)= RSS of each regression fitted by pooled data, \(RSS_{s}\)= sum of the RSS of each regression fitted for each individual sample, N= total sample size, and K = number of samples in the comparison.

The residual sum of squares (RSS) and the degrees of freedom (DF) for each fitted regression are previously stored in the List_TCCT list. For each regression, the calculations are stored in a data frame T1, which is stored iteratively using a loop for in a list T.

Inside the function, the RSS and DF for the joined sample are calculated to perform the F test for two tails \(\alpha/2\). The decision criteria is performed: “*” if \(p-value\le \alpha\) or “NS” if the \(p-val\gt \alpha\).

The function requires defining:

  • List_TCCT: A list with fitted regression results.
  • i: An integer value indicating the ith regression analyzed.
  • alfa: A numerical value that defines the significance level. The default number is 0.05.

Seealso: FDist from the package stats (version 3.6.2)

fn_ARSS

The function is included in the Morefi package Morphological Relationships Fitted by Robust Regression.

The function is detailed below.

fn_ARSS <- function(List_TCCT, i,  alfa= 0.05){
  T  <- list()
  for (i in 1:length(List_TCCT)){
  
  T1 <- List_TCCT[[i]]
  lvar <- names(List_TCCT[i])

  T1[1,1] <- paste0(substr(lvar,1,2)," vs. ",
                    substr(lvar,3,nchar(lvar)))
  # RSSs sum of the RSS of each regression fitted
  T1[4,3] <- round(sum(as.numeric(T1[1,3]),
                       as.numeric(T1[2,3])),2)

  # DF Sum GL1 + GL2
  T1[4,4] <- sum(as.numeric(T1[1,4]),as.numeric(T1[2,4]))

  # F calculated (Fc)
  T1[1,5] <- round(abs((as.numeric(T1[3,3])-as.numeric(T1[4,3]))/
                         as.numeric(T1[4,3])),4)

  # F table (critical value)
  T1[1,6] <- round(qf(alfa, as.numeric(T1[3,4]),
                      as.numeric(T1[4,4]),
                      lower.tail= FALSE),4)

  # p-value for two tails
  P_value <- pf(as.numeric(T1[1,6]),
                as.numeric(T1[4,4]),
                as.numeric(T1[3,4]),
                lower.tail=FALSE)

  T1[1,7] <- if(P_value>1-alfa){
    "p>0.95"
  } else {
    format(P_value, format= "e", digits = 4)
  } # End if

  # Decision criteria
  T1[1,8] <-  if(P_value>alfa){
    "NS"
  } else {
    "*"
  } # End if
  T[[i]] <- as.data.frame(T1)
  } # End for
  return(T)
}  # End function

Examples

The length-weight was estimated for the Pacific Sierra Scomberomorus sierra by sex and the total sample (Unpublish data). The sex of 362 organisms was determined: 95 were females and 118 males. The adjusted models show the following data: Females SSR= 1557970.877 and DF= 93; Males SSR= 1188689.721 and DF= 116; and the total sample SSR= 2045176.876 and DF= 211. Values are stored in the table Table_CC, this is stored in a list, and the name of each item is built with the acronyms of the model variables (e.g. LTWT).

Table_CC <- data.frame(matrix(NA,nrow=4,ncol=8))
Table_CC[1,1] <- "Lt-WT" 
Table_CC[,2] <- c("Females","Males","Total","Joined")
colnames(Table_CC) <- c("Model","Category","RSS","DF","ARSS","F-table","p-value","Criteria")

Table_CC[1,3] <- 1557970.877
Table_CC[1,4] <-  93 
Table_CC[2,3] <- 1188689.721
Table_CC[2,4] <-  116 
Table_CC[3,3] <- 2045176.876
Table_CC[3,4] <-  211 


List_ARSS <- list(LTWT=Table_CC)

i <- 1

ARSS <- fn_ARSS(List_ARSS, i,  alfa= 0.05)
ARSS

[[1]] Model Category RSS DF ARSS F-table p-value Criteria 1 LT vs. WT Females 1557971 93 0.2554 1.2556 0.04988 * 2 Males 1188690 116 NA NA 3 Total 2045177 211 NA NA 4 Joined 2746661 209 NA NA

References

Chen Y., Jackson, D. A., & Harvey, H. H. 1992. A comparison of von Bertalanffy and polynomial functions in modelling fish growth data. Can. J. Fish. Aquat. Sci. 49(6): 1228–1235. https://doi.org/10.1139/f92-13