# title: "Replication script" # author: Robin Hankin # This R script aims at reproducing all figures, tables and other output # presented in the manuscript. It includes the R functions and usage # examples. We understand the likelihood functions to be defined only for # non-negative strengths. ################################################### ### code chunk number 1: setup ################################################### library("hyper2") ################################################### ### code chunk number 2: jss_hyper3.Rnw:220-224 ################################################### ## Illustration: creating a simple hyper3 object ## using named vectors. Here the likelihood function is a/(3a+2b+c), ## a+b+c = 1. LL <- hyper3() ## initialization: LL is an empty hyper3 object. LL[c(a = 1)] <- 1 ## First term: numerator of 'a' LL[c(a = 3, b = 2, c = 1)] <- -1 ## Second term: denominator (3a+2b+c), with power (-1) LL ## illustration of print method ################################################### ### code chunk number 3: jss_hyper3.Rnw:234-236 ################################################### ## Use case of log-likelihood as implemented by function loglik(). ## We will evaluate log-likelihood function LL at two points on the ## two-simplex: loglik(c(a = 0.01, b = 0.01, c = 0.98), LL) # L(0.01, 0.01, 0.98) loglik(c(a = 0.90, b = 0.05, c = 0.05), LL) # L(0.90, 0.05, 0.05) ################################################### ### code chunk number 4: jss_hyper3.Rnw:251-252 ################################################### ## Illustration of an order statistic, with preferred interpretation ## of a race between three clones of strength "a", two of strength ## "b", and a singleton of strength "c". (H <- ordervec2supp3(c("a", "c", "b", "a", "a", "b"))) ################################################### ### code chunk number 5: maxexamp ################################################### ## function maxp() is S3 generic, here returning the maximum ## likelihood estimate for competitors a, b, c (mH <- maxp(H)) ################################################### ### code chunk number 6: testequality ################################################### ## builtin function equalp.test() is one of a family of functions for ## testing a range of interesting nulls for compositional data equalp.test(H) ################################################### ### code chunk number 7: maxpaba ################################################### ## function maxp() used to illustrate a simple use-case of two twins ## and a singleton: maxp(ordervec2supp3(c("a", "b", "a"))) ################################################### ### code chunk number 8: jss_hyper3.Rnw:346-348 ################################################### ## likelihood function for one-simplex {a, b|a+b = 1}, in a form suitable ## for plotting a <- 1/2 # null hypothesis H0 (S_delta <- log(a * (1 - a)/(1 + a)) - log(3 - 2 * sqrt(2))) ################################################### ### code chunk number 9: jss_hyper3.Rnw:358-359 ################################################### ## calculate the p-value of H0 above, using the asymptotic ## distribution of the log-likelihood: pchisq(-2 * S_delta, df = 1, lower.tail = FALSE) ################################################### ### code chunk number 10: figaba ################################################### ## plot a support function for the observation a>b>a over the ## one-simplex {a, b|a+b = 1} [figure 1] a <- seq(from = 0, by = 0.005, to = 1) # specify horizontal axis S <- function(a){log(a * (1 - a) / ((1 + a) * (3 - 2 * sqrt(2))))} # likelihood function for a>b>a plot(a, S(a), type = "b", xlab = expression(p[a]), ylab = "support") # plot abline(h = c(0, -2)) # annotations [two units-of-support] abline(v = c(0.02438102, 0.9524271), col = "red") # annotations [credible interval] abline(v = sqrt(2) - 1) # annotations [evaluate] ################################################### ### code chunk number 11: figabbabb ################################################### ## plot harmonised likelihood functions for the three possible order ## statistics [viz a>a>b, a>b>a, b>a>a] f_aab <- function(a){a^2 / (1 + a)} # L(a>a>b) f_aba <- function(a){a * (1 - a) / (1 + a)^2} # L(a>b>a) f_baa <- function(a){(1 - a) / (1 + a)} # L(b>a>a) p <- function(f, ...){ # generic plot routine a <- seq(from = 0, by = 0.005, to = 1) points(a, f(a) / max(f(a)), ...) } plot(0:1, 0:1, xlab = expression(p[a]), ylab = "Likelihood", type = "n") # empty plot p(f_aab, type = "l", col = "black") # L(a>a>b) p(f_aba, type = "l", col = "red") # L(a>b>a) p(f_baa, type = "l", col = "blue") # L(b>a>a) text(0.8, 0.8, "AAB") # annotation text(0.8, 0.5, "ABA", col = "red") # annotation text(0.8, 0.15, "BAA", col = "blue") # annotation abline(h = exp(-2), lty = 2) # two units-of-support criterion ################################################### ### code chunk number 12: jss_hyper3.Rnw:453-454 ################################################### ## illustrate a more general observation, here a>b>>{a, b}; function ## ordervec2supp3() allows the user to specify competitors who did not ## finish. maxp(ordervec2supp3(c("a", "b"), nonfinishers = c("a", "b"))) ################################################### ### code chunk number 13: define_xy_wilcox ################################################### ## placenta dataset as used in base::wilcox.Rd, here used to ## illustrate Plackett-Luce approach to nonparametric tests: x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46) y <- c(1.15, 0.88, 0.90, 0.74, 1.21) ################################################### ### code chunk number 14: hyper3osdef ################################################### ## package idiom to test null of equal Plackett-Luce strength of x and ## y: names(x) <- rep("x", length(x)) names(y) <- rep("y", length(y)) (os <- names(sort(c(x, y)))) # here "os" means "order statistic" ################################################### ### code chunk number 15: hyper3xytest ################################################### ## Again use package idiom ordervec2supp3() but wit the the order ## statistic specified above for the placenta dataset: Hxy <- ordervec2supp3(os) # create the likelihood function Hxy... equalp.test(Hxy) # ... and test the null that p_x = p_y ################################################### ### code chunk number 16: plotwilcoxlike ################################################### ## show the likelihood function for the Plackett-Luce strength of "a" a <- seq(from = 0.02, to = 0.8, len = 40) # horizontal axis L <- sapply(a, function(p){loglik(p, Hxy)}) # vectorized idiom for loglikelihood function plot(a, L - max(L), type = "b", xlab = expression(p[a]), ylab = "support") # plot normalized loglikelihood abline(h = c(0, -2)) # two-units-of-support criterion abline(v = c(0.24)) # evaluate abline(v = c(0.5), lty = 2) # null ################################################### ### code chunk number 17: javelintable ################################################### ## Show explicitly the dataset used for the Plackett-Luce strength of ## the javelin competitors: javelin_table ################################################### ### code chunk number 18: converttosupp3 ################################################### ## Use more sophisticated bespoke package idiom, here ## attemptstable2supp3(), which returns a hyper3 likelihood function ## for the entire dataset, javelin_vector <- attemptstable2supp3( javelin_table, # primary dataset decreasing = TRUE, # decreasing = TRUE specifies that high # numerical values win [compare race times, # where low numerical values win] give.supp = FALSE) # return the order statistic, not its support function options(width = 60) # formatting for output javelin_vector # return order statistic for inspection ################################################### ### code chunk number 19: dothething2 ################################################### ## Now use ordervec2supp3() with the javelin order statistic ## calculated above to give a support function; discard no-throws javelin <- ordervec2supp3(v = names(javelin_vector)[!is.na(javelin_vector)]) ################################################### ### code chunk number 20: setdigits ################################################### ## formatting options(digits = 3) ################################################### ### code chunk number 21: testthejav ################################################### ## Now maximize the likelihood over the 7-simplex corresponding to the ## javelin throwers' Plackett-Luce strengths: (mj <- maxp(javelin)) # use optimization to find evaluate dotchart(mj, pch = 16, xlab = "Estimated Bradley-Terry strength") # visual plot of evaluate ################################################### ### code chunk number 22: havealook ################################################### ## generate a log-contrast plot for LC = log(p_Vadlejch / p_Vesely), ## using a bespoke function f(), which leverages output from ## specificp.test() when used to assess a null of Vesely having a ## particular strength. f <- function(s){ jj <- specificp.test(javelin, "Vesely", s, n = 2) p <- jj\$null_estimate return( # return a vector of two numeric values. The first is the # log-contrast LC [used as a horizontal axis in the next # chunk] and the second is the maximum support for the # particular value of p_Vesely c(log(p[6] / p[7]), jj\$null_support) ) } Ves <- seq(from = 0.0199, to = 0.33, len = 16) # specify Vesely's Plackett-Luce strength M <- sapply(Ves, f) # apply function f() defined above to return LC and its support M[2, ] <- M[2, ] - max(M[2, ]) # normalize rownames(M) <- c("logcontrast", "support") # cosmetic ################################################### ### code chunk number 23: plottheloglikcont ################################################### ## Plot the log-contrast dataset calculated in the previous chunk colnames(M) <- as.character(Ves) plot(t(M), type = "b") # plot the figure abline(h = c(0, -2)) # two-units-of support criterion abline(v = 0, lty = 2) # null abline(v = log(0.32062833 / 0.11402735)) # evaluate ################################################### ### code chunk number 24: showconstructortable ################################################### ## show the constructors' championship dataset used in the manuscript constructor_2021_table[, 1:9] ################################################### ### code chunk number 25: maxpconstructor2021 ################################################### ## Now Assess whether Mercedes have in fact decreased in strength ## between 2020 and 2021. Determine hyper3 likelihood functions # for the two years: const2020 <- ordertable2supp3(constructor_2020_table) # likelihood function for constructors 2020 const2021 <- ordertable2supp3(constructor_2021_table) # likelihood function for constructors 2021 options(digits = 4) # formatting maxp(const2020, n = 1) # show maximum likelihood estimate for 2020 maxp(const2021, n = 1) # show maximum likelihood estimate for 2021 ################################################### ### code chunk number 26: definecombinedlikelihoodfunction ################################################### ## Now use psubs() to distinguish 2020 results from 2021. Effectively ## define two teams, "Merc2020" for Mercedes 2020, and "Merc2021" for ## 2021. Note that the resulting likelihood function is very long and ## difficult to interpret, which is why it is not printed in the ## manuscript. H <- ( psubs(constructor_2020, "Merc", "Merc2020") ## psubs() substitutes "Merc" for "Merc2020" ## "+" is overloaded in the package. Here, it + ## corresponds to addition of (independent) log-likelihood ## functions. psubs(constructor_2021, "Merc", "Merc2021") # psubs() used again but for 2021 ) ################################################### ### code chunk number 27: usesamep ################################################### ## Test the null that Mercedes had the same strength in 2020 as 2021, viz ## H0:p_Merc2020 == p_Merc2021: options(digits = 4) samep.test(H, c("Merc2020", "Merc2021")) date()