Permutation test for group differences using 3D coordinate data

centroid_perm.R

# ===============================================================================
# Name               : centroid_perm
# Original author    : Steven Worthington ([email protected])
# Affiliation        : IQSS, Harvard University
# Date (mm/dd/yyyy)  : 06/14/2012 
# Version            : v0.8
# Aim                : exact permutation test for group differences
# ===============================================================================

# Goal:
# To demonstrate that the points (i.e., the trabecular orientation)
# of each species are occupying (or not) a different area on the sphere.

# values in both datasets are in radians (required for the Haversine formula)

# A = chimps, B = humans, C = fossils

# dataset #1
pts1 <- data.frame(
    Long = c(1.0727,1.5966,0.8448,1.1049,0.7834,0.8565,1.8754,1.4727,1.6216,
        1.5203,0.7419,0.3658,1.6343,1.3046,1.5075),
    Lat = c(1.3990,1.2087,1.4028,1.3570,1.3266,1.3806,1.4814,1.3943,1.4372,
        1.3573,1.4939,1.4475,1.4156,1.2779,1.4175),
    lvl = c( rep('A', 6), rep('B', 6), rep('C', 3) )
    )

# dataset #1 with humans and fossils lumped together
pts11 <- data.frame(
    Long = c(1.0727,1.5966,0.8448,1.1049,0.7834,0.8565,1.8754,1.4727,1.6216,
        1.5203,0.7419,0.3658,1.6343,1.3046,1.5075),
    Lat = c(1.3990,1.2087,1.4028,1.3570,1.3266,1.3806,1.4814,1.3943,1.4372,
        1.3573,1.4939,1.4475,1.4156,1.2779,1.4175),
    lvl = c( rep('A', 6), rep('B', 6), rep('B', 3) )
    )

# dataset #2
pts2 <- data.frame(
    Long = c(2.5952,1.3934,1.8102,1.7794,5.0217,1.4840,6.1798,1.1547,1.6422,
        5.1650,0.4375,0.5192,0.8254,6.027,1.9699),
    Lat = c(1.4742,1.2668,1.2611,1.4101,1.5240,1.3420,1.5194,1.4619,1.5554,
        1.5452,1.4370,1.4864,1.5296,1.4265,1.4940),
    lvl = c( rep('A', 6), rep('B', 6), rep('C', 3))
    )

# dataset #2 with humans and fossils lumped together
pts22 <- data.frame(
    Long = c(2.5952,1.3934,1.8102,1.7794,5.0217,1.4840,6.1798,1.1547,1.6422,
        5.1650,0.4375,0.5192,0.8254,6.027,1.9699),
    Lat = c(1.4742,1.2668,1.2611,1.4101,1.5240,1.3420,1.5194,1.4619,1.5554,
        1.5452,1.4370,1.4864,1.5296,1.4265,1.4940),
    lvl = c( rep('A', 6), rep('B', 6), rep('B', 3))
    )


# utility functions
# ---------------------------------------------------
# rad <- function(x) x * (pi / 180) # convert degrees into radians
# deg <- function(x) x * (180 / pi) # convert radians into degrees
# ---------------------------------------------------
long_antepodes <- function(points) {
    # convert longitudinal radians larger than pi to their antepodal values
    points$Long <- sapply(points$Long, function(x) { if (x > pi) { x - pi } else { x } })
    return(points)
}
# ---------------------------------------------------
LatLong2cartesian <- function(long, lat) {
    # convert lat/long to cartesian (x,y,z) coordinates
    r <- 1
    x <- r * cos(lat) * cos(long)
    y <- r * cos(lat) * sin(long)
    z <- r * sin(lat) 
    return(data.frame(x = x, y = y, z = z))
}
# ---------------------------------------------------
xyz_surface <- function(coords_ave) {
    # project centroids onto the surface of the sphere
    x_ave <- coords_ave[1]
    y_ave <- coords_ave[2]
    z_ave <- coords_ave[3]
    L <- sqrt( x_ave^2 + y_ave^2 + z_ave^2)
    x <- x_ave / L
    y <- y_ave / L
    z <- z_ave / L
    return(data.frame(x = x, y = y, z = z))
}
# ---------------------------------------------------
cartesian2LatLong <- function(coordsSum) {
    # convert cartesian (x,y,z) coordinates to lat/long
    X <- coordsSum[1]
    Y <- coordsSum[2]
    Z <- coordsSum[3]
    Long <- atan2(Y, X)
    Hyp <- sqrt(X^2 + Y^2)
    Lat <- atan2(Z, Hyp)
    # r <- sqrt(X^2 + Y^2 + Z^2)
    # Lat2 <- asin(Z / r)
    return(c(Long, Lat))
}
# ---------------------------------------------------
hf <- function(longs, lats, cent_long, cent_lat) {
    # function to calculate distance between 2 points on a sphere
    r <- 1 # sphere radius
    delta.long <- (longs - cent_long)
    delta.lat <- (lats - cent_lat)
    b <- sin(delta.lat/2)^2 + cos(lats) * cos(cent_lat) * sin(delta.long/2)^2
    p <- 2 * asin(pmin(1, sqrt(b)))
    dis <- r * p
    return(dis) 
}
# ---------------------------------------------------
upermn <- function(x) {
    # function to calculate number of unique permutations
    duplicates <- as.numeric(table(x))
    factorial(length(x)) / prod(factorial(duplicates))
}
# ---------------------------------------------------
uniqueperm <- function(group1, group2, group1n, group2n) {
    # function to calculate all unique permutations of pts$lvl
    totaln <- sum(group1n, group2n)
    x <- rep(group1, totaln) # vector of pts$lvl for group 1
    position <- combn(totaln, group2n) # all possible ways to place group2 lvls among group1 lvls
    perms <- as.data.frame( apply(position, 2, function(p) replace(x, p, group2)) ) # all perms
    return(perms) # returns a df of factors, with each column a unique permutation of lvls
}
# ---------------------------------------------------
dens <- function (nullDist, obs) {
    # function to plot kernal density
    d <- density(nullDist, na.rm = TRUE)
    r <- dist( range(nullDist) )
    plot(d, type = "n", main = "Null distribution and observed value",
         xlim = c( min(nullDist)-(r/10),
                   pmax( obs+(obs/10), max(nullDist)+(r/10) ) ) )
    polygon(d, col = "cornflowerblue", border = NA) 
    rug(nullDist, col = "darkred", lwd = 0.7)
}
# ---------------------------------------------------


# -----------------------------------------------------------------------
latlong_centroids <- function(LL, print = FALSE){
    # function to calculate centroids of lat/long coordinates
  
    # convert longitudinal radians larger than pi to their antepodal values
    LL <- long_antepodes(LL)
  
    # rows are x,y,z, and each column corresponds to a row in pts
    coords <- mapply(LatLong2cartesian, long = LL$Long, lat = LL$Lat)
    coords <- matrix(unlist(coords), nrow = nrow(coords), ncol = ncol(coords))
    if (print == TRUE) {
        cat("\n")
        cat("x, y, z Cartesian coordinates:", "\n")
        print(coords)
    }
    # calculate centroids as row means
    coordsCentroids <- data.frame(A = rowMeans(coords[, LL$lvl == "A"]),
                                  B = rowMeans(coords[, LL$lvl == "B"]), 
                                  C = rowMeans(coords[, LL$lvl == "C"]) )
    if (print == TRUE) {
        cat("\n")
        cat("x, y, z Cartesian centroids within sphere:", "\n")
        print(coordsCentroids)
    }

    surfaceCentroids <- sapply(coordsCentroids, xyz_surface) 
    surfaceCentroids <- as.data.frame( matrix(unlist(surfaceCentroids),
        nrow = nrow(surfaceCentroids), ncol = ncol(surfaceCentroids)) )
    if (print == TRUE) {
        cat("\n")
        cat("x, y, z Cartesian centroids on sphere surface:", "\n")
        print(surfaceCentroids)
    }

    centroids <- t( sapply(surfaceCentroids, cartesian2LatLong) )
    centroids <- data.frame(Long = centroids[, 1], Lat = centroids[, 2],
                            lvl = colnames(coordsCentroids))
    if (print == TRUE) {
        cat("\n")
        cat("Lat/Long centroids:", "\n")
        print(centroids)
    }
    return(centroids)
}
# -----------------------------------------------------------------------


# -----------------------------------------------------------------------
centroid_perm <- function(pt, comparison = c("AB", "AC", "BC")) {
    # function for performing permutation test on group centroid differences
  
    if (!is.data.frame(pt))
      stop(deparse(substitute(pt)), " is not a data frame.")

    if (any(pt$Long | pt$Lat) > pi*2)
      stop(deparse(substitute(pt)), " does not contain Lat/Long in radians.")

    comparison <- match.arg(comparison)
    
    if (comparison == "AB") {
      points <- pt[pt$lvl %in% c("A", "B"), ] # subset pt by groups of interest
        calc.diff <- function(c){
            centDistAB <- hf(c$Long[c$lvl == "A"], c$Lat[c$lvl == "A"],
                c$Long[c$lvl == "B"], c$Lat[c$lvl == "B"])
            return(centDistAB)
        }
    } 

    else if (comparison == "AC") {
      points <- pt[pt$lvl %in% c("A", "C"), ] # subset pt by groups of interest
        calc.diff <- function(c){
            centDistAC <- hf(c$Long[c$lvl == "A"], c$Lat[c$lvl == "A"],
                c$Long[c$lvl == "C"], c$Lat[c$lvl == "C"])
            return(centDistAC)
        }
    } 

    else if (comparison == "BC") {
      points <- pt[pt$lvl %in% c("B", "C"), ] # subset pt by groups of interest
        calc.diff <- function(c){
            centDistBC <- hf(c$Long[c$lvl == "B"], c$Lat[c$lvl == "B"],
                c$Long[c$lvl == "C"], c$Lat[c$lvl == "C"])
            return(centDistBC)
        }
    }

    # for observed (unpermuted) groups
    centroids <- latlong_centroids(points, print = TRUE) # calculate Lat/Long centroids
    t.obs <- calc.diff(centroids) # calculate distance between centroids

    # calculate all unique permutations for null distribution
    lvl_points <- droplevels(points) # drop redundent factor level
    perms <- uniqueperm(levels(lvl_points$lvl)[1], levels(lvl_points$lvl)[2], 
                        table(lvl_points$lvl)[1], table(lvl_points$lvl)[2])
           
    # permute group assignments accross coordinates
    perm.vals <- vector(mode = "numeric", length = ncol(perms))
    for (i in seq_along(perms)){
        points$lvl <- perms[, i]
        centroids <- latlong_centroids(points) # recalculate centroids for each group assignment
        perm.vals[i] <- calc.diff(centroids)
    }  
    perm.ecdf <- ecdf(perm.vals) # cumulative distribution function of null hypothesis
    p.val <- 1 - perm.ecdf(t.obs) # p-value
    
    # pdf plot
    dens(perm.vals, t.obs)
    arrows(t.obs, 4, t.obs, 0, col = "darkred", lwd = 1.5)
    text(t.obs, 6.5, labels = paste("Data set:", deparse(substitute(pt))), cex = 0.8)
    text(t.obs, 6, labels = paste("Comparison:", comparison), cex = 0.8)
    text(t.obs, 5.5, labels = paste("Permutations:", upermn(lvl_points$lvl)), cex = 0.8)
    text(t.obs, 5, labels = paste("P-value =", round(p.val, 4)), cex = 0.8)

    # the p-value and other info
    cat("\n")
    cat("Data set:", deparse(substitute(pt)), "\n")
    cat("Comparison:", comparison, "\n")
    cat("Permutations:", upermn(lvl_points$lvl), "\n")
    cat("P-value =", p.val, "\n")
    return(p.val)
}
# ------------------------------------------------------------------


# ------------------------------------------------------------------
# function calls:
pts1_ab <- centroid_perm(pts1, "AB")
pts1_ac <- centroid_perm(pts1, "AC")
pts1_bc <- centroid_perm(pts1, "BC")
#
pts2_ab <- centroid_perm(pts2, "AB")
pts2_ac <- centroid_perm(pts2, "AC")
pts2_bc <- centroid_perm(pts2, "BC")
#
pts11_ab <- centroid_perm(pts11, "AB") # humans and fossils merged
pts22_ab <- centroid_perm(pts22, "AB") # humans and fossils merged

p_values <- data.frame(comparison = c("AB", "AC", "BC", "new_AB"),
                       pts1 = c(pts1_ab, pts1_ac, pts1_bc, pts11_ab),
                       pts2 = c(pts2_ab, pts2_ac, pts2_bc, pts22_ab))
p_values # ~2 minutes

# adjust p-values for multiple comparisons
p_values$pts1_adj <- p.adjust(p = p_values$pts1, method = "holm") # sequential Bonferroni
p_values$pts2_adj <- p.adjust(p = p_values$pts2, method = "holm") # sequential Bonferroni
# ------------------------------------------------------------------






# ------------------------------------------------------------------
dens_bw<- function (nullDist, obs, p) {
    # function to plot kernal density in black and white
    d <- density(nullDist, na.rm = TRUE)
    r <- dist( range(nullDist) )
    plot(d, main = "Null distribution and observed value",
         xlim = c( min(nullDist)-(r/10),
                   pmax( obs+(obs/10), max(nullDist)+(r/10) ) ) ) 
    rug(nullDist, col = "gray30", lwd = 0.1)
    arrows(obs, 4, obs, 0, lwd = 1.2)
    text(obs, 5, labels = paste("P-value =", round(p, 4)), cex = 0.8)
}

# need to run the analysis outside of the centroid_perm() function
# to get values for perm.vals and t.obs
dens_bw(nullDist = perm.vals, obs = t.obs, p = 0.0043956)
# ------------------------------------------------------------------








# ===================================================================
# Plotting the points in 3D
# ===================================================================

# data for pts2
# ------------------------------------------

# Cartesian coordinates
coords <- structure(c(-0.0824040452325179, 0.05011425118923, 0.995338201395415, 
0.0528229766618636, 0.294638046695378, 0.954147868297211, -0.0722679466402008, 
0.296077263247749, 0.952426164107293, -0.033136207288388, 0.15653684561824, 
0.987116106509479, -0.01423971677629, 0.0445592704815626, 0.998905251703224, 
0.0196611658175792, 0.22595158795114, 0.973940100037498, -0.0510993918414077, 
0.00530182936886244, 0.99867949951863, 0.0439281880452844, 0.0994078687663381, 
0.994076651935047, -0.00109837695599064, 0.0153564877827651, 
0.999881478901895, -0.0111924388220814, 0.0230164764392326, 0.999672431912342, 
0.120833273964134, 0.0565173503067401, 0.991062616093158, 0.0731873209047366, 
0.0418265579026938, 0.996440743402637, 0.0279340451173512, 0.030263287830029, 
0.99915155133398, -0.139103121978486, 0.0364368128034391, 0.989607336335259, 
-0.0298131604990723, 0.0706913445215261, 0.997052611084689), .Dim = c(3L, 
15L))

# Cartesian centroids
coordsCentroids <- structure(list(A = c(-0.0215939622429923, 0.177979544197217, 
0.97697894867502), B = c(0.0290930958824458, 0.0402377617611054, 
0.996635570293951), C = c(-0.0469940791200691, 0.0457971483849981, 
0.995270499584642)), .Names = c("A", "B", "C"), row.names = c(NA, 
-3L), class = "data.frame")

# Cartesian centroids projected to sphere surface
surfaceCentroids <- structure(list(V1 = c(-0.0217397719917556, 0.179181322376338, 
0.983575841521747), V2 = c(0.0291551465469009, 0.0403235821173697, 
0.998761225796763), V3 = c(-0.0471151038906367, 0.0459150906764468, 
0.997833639157123)), .Names = c("V1", "V2", "V3"), row.names = c(NA, 
-3L), class = "data.frame")

# lat/long coordinates for dataset #2
pts2 <- data.frame(
    Long = c(2.5952,1.3934,1.8102,1.7794,5.0217,1.4840,6.1798,1.1547,1.6422,
        5.1650,0.4375,0.5192,0.8254,6.027,1.9699),
    Lat = c(1.4742,1.2668,1.2611,1.4101,1.5240,1.3420,1.5194,1.4619,1.5554,
        1.5452,1.4370,1.4864,1.5296,1.4265,1.4940),
    lvl = c( rep('A', 6), rep('B', 6), rep('C', 3))
    )

# lat/long centroids
LatLong_centroids <- structure(list(Long = c(1.69153452750487, 0.944780830082602, 
2.36909295471935), Lat = c(1.38930629806737, 1.52101620931788, 
1.50496102499399), lvl = structure(1:3, .Label = c("A", "B", 
"C"), class = "factor")), .Names = c("Long", "Lat", "lvl"), row.names = c("V1", 
"V2", "V3"), class = "data.frame")


# 3D plots
# ------------------------------------------
library(rgl)
# rgl.open()

# Cartesian coordinates
open3d()
plot3d(coords[1, ], coords[2, ], coords[3, ], col = as.numeric(pts2$lvl), size = 10)
text3d(coordsCentroids[1, ], coordsCentroids[2, ], coordsCentroids[3, ],
       texts = c("A", "B", "C"), col = c(1, 2, 3), adj = 0)
text3d(surfaceCentroids[1, ], surfaceCentroids[2, ], surfaceCentroids[3, ],
       texts = c("a", "b", "c"), col = c(1, 2, 3), adj = 0)

# Lat/Long plot on sphere
library(sm)
sm.sphere(lat = pts2$Lat * (180 / pi), long = pts2$Long * (180 / pi),
    theta = 30, phi = 120, pch = as.numeric(pts2$lvl), col = as.numeric(pts2$lvl))

sm.sphere(lat = LatLong_centroids$Lat * (180 / pi), long = LatLong_centroids$Long * (180 / pi),
    theta = 30, phi = 120, pch = c(1,2,3), col = c(1,2,3), add = TRUE)