R/triad_uman.R
In anespinosa/netmem: Social Network Measures using Matrices
Defines functions
triad_uman
Documented in
triad_uman
#' Triad census analysis assuming U|MAN
#'
#' Considering the triad census of Davis and Leinhardt (1972) for vector A, B, and C:
#'
#' 003 = A,B,C, empty triad
#'
#' 012 = A -> B, C, triad with a single directed edge
#'
#' 102 = A <-> B, C, triad with a reciprocated connection between two vertices
#'
#' 021D = A <-B-> C, triadic out-star
#'
#' 021U = A -> B <- C triadic in-star
#'
#' 021C = A-> B-> C, directed line
#'
#' 111D = A <-> B <-C
#'
#' 111U = A <-> B-> C
#'
#' 030T = A-> B <-C, A-> C
#'
#' 030C = A <-B <-C, A-> C
#'
#' 201 = A <-> B <-> C
#'
#' 120D = A <-B-> C, A <-> C
#'
#' 120U = A-> B <-C, A <->C
#'
#' 120C = A-> B-> C, A <-> C
#'
#' 210 = A-> B <-> C, A <-> C
#'
#' 300 = A <-> B <-> C, A <->C, complete triad.
#'
#' @param A A symmetric matrix object
#' @param ztest Return Z and p-value
#' @param covar Return the covarianc matrix for triadic analysis
#'
#' @return This function gives the counts of the triad census, the expected counts,
#' assuming that U|MAN distribution (Holland and Leinhardt, 1975, 1976) is operating,
#' and the standard deviations of these counts.
#'
#' @references
#'
#' Davis, J.A. and Leinhardt, S. (1972). The Structure of Positive Interpersonal Relations in Small Groups. In J. Berger (Ed.), Sociological Theories in Progress, Volume 2, 218-251. Boston: Houghton Mifflin.
#'
#' Holland, P. W. and Leinhardt, S. (1975). The statistical analysis of local structure in social networks. In D. R. Heise (Ed.), Sociological Methodology, 1976 (Jossey-Bass, pp. 1–45).
#'
#' Holland, P. W. and Leinhardt, S. (1976). Local Structure in Social Networks. Sociological Methodology, 7, 1–45. doi: \url{https://doi.org/10.2307/270703}
#'
#' Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.
#'
#' @author Alejandro Espinosa-Rada
#'
#' @importFrom stats pnorm
#' @importFrom Matrix Matrix diag t
#'
#' @examples
#'
#' data(krackhardt_friends)
#' triad_uman(krackhardt_friends)
#' \dontrun{
#' triad_uman(krackhardt_friends, ztest = TRUE, covar = TRUE)
#' }
#'
#' @export
triad_uman <- function(A, ztest = FALSE, covar = FALSE) {
# elements ----
A <- as.matrix(A)
if (any(abs(A > 1), na.rm = TRUE)) stop("The matrix should be binary")
if (any(is.na(A) == TRUE)) {
A <- ifelse(is.na(A), 0, A)
}
A <- Matrix::Matrix(A)
g <- dim(A)[1]
m <- (1 / 2) * sum(Matrix::diag(A %*% A)) # mutual
a <- sum(Matrix::diag(A %*% Matrix::t(A))) - sum(Matrix::diag(A %*% A)) # asymmetric
n <- ((g * (g - 1)) / 2) - (sum(Matrix::diag(A %*% Matrix::t(A))) - sum(Matrix::diag(A %*% A))) - ((1 / 2) * sum(Matrix::diag(A %*% A))) # null
g2 <- (1 / 2) * (g * (g - 1))
g3 <- (1 / 6) * (g * (g - 1) * (g - 2))
# d1 ----
d1_out <- d1(g2, n, m, a)
# d2 ----
d2_out <- d2(g2, n, m, a)
# d3 ----
d3_out <- d3(g2, n, m, a)
# expected value ----
EXP <- c(
# 003
"E003" <- g3 * d1_out$d1p1t003,
# 012
"E012" <- g3 * d1_out$d1p1t012,
# 102
"E102" <- g3 * d1_out$d1p1t102,
# 021D
"E021D" <- g3 * d1_out$d1p1t021D,
# 021U
"E021U" <- g3 * d1_out$d1p1t021U,
# 021C
"E021C" <- g3 * d1_out$d1p1t021C,
# 111D
"E111D" <- g3 * d1_out$d1p1t111D,
# 111U
"E111U" <- g3 * d1_out$d1p1t111U,
# 030T
"E030T" <- g3 * d1_out$d1p1t030T,
# 030C
"E030C" <- g3 * d1_out$d1p1t030C,
# 201
"E201" <- g3 * d1_out$d1p1t201,
# 120D
"E120D" <- g3 * d1_out$d1p1t120D,
# 120U
"E120U" <- g3 * d1_out$d1p1t120U,
# 120C
"E120C" <- g3 * d1_out$d1p1t120C,
# 210
"E210" <- g3 * d1_out$d1p1t210,
# 300
"E300" <- g3 * d1_out$d1p1t300
)
# variance ----
VAR <- c(
# 003
"Var003" <- (g3 * d1_out$d1p1t003 * (1 - d1_out$d1p1t003)) +
(3 * (g - 3) * g3) * (d3_out$man003_man003b - d2_out$man003_man003) +
g3 * (g3 - 1) * (d2_out$man003_man003 - (d1_out$d1p1t003^2)),
# 012
"Var012" <- (g3 * d1_out$d1p1t012 * (1 - d1_out$d1p1t012)) +
(3 * (g - 3) * g3) * (d3_out$man012_man012b - d2_out$man012_man012) +
g3 * (g3 - 1) * (d2_out$man012_man012 - (d1_out$d1p1t012^2)),
# 102
"Var102" <- (g3 * d1_out$d1p1t102 * (1 - d1_out$d1p1t102)) +
(3 * (g - 3) * g3) * (d3_out$man102_man102b - d2_out$man102_man102) +
g3 * (g3 - 1) * (d2_out$man102_man102 - (d1_out$d1p1t102^2)),
# 021D
"Var021D" <- (g3 * d1_out$d1p1t021D * (1 - d1_out$d1p1t021D)) +
(3 * (g - 3) * g3) * (d3_out$man021D_man021Db - d2_out$man021D_man021D) +
g3 * (g3 - 1) * (d2_out$man021D_man021D - (d1_out$d1p1t021D^2)),
# 021U
"Var021U" <- (g3 * d1_out$d1p1t021U * (1 - d1_out$d1p1t021U)) +
(3 * (g - 3) * g3) * (d3_out$man021U_man021Ub - d2_out$man021U_man021U) +
g3 * (g3 - 1) * (d2_out$man021U_man021U - (d1_out$d1p1t021U^2)),
# 021C
"Var021C" <- (g3 * d1_out$d1p1t021C * (1 - d1_out$d1p1t021C)) +
(3 * (g - 3) * g3) * (d3_out$man021C_man021Cb - d2_out$man021C_man021C) +
g3 * (g3 - 1) * (d2_out$man021C_man021C - (d1_out$d1p1t021C^2)),
# 111D
"Var021C" <- (g3 * d1_out$d1p1t111D * (1 - d1_out$d1p1t111D)) +
(3 * (g - 3) * g3) * (d3_out$man111D_man111Db - d2_out$man111D_man111D) +
g3 * (g3 - 1) * (d2_out$man111D_man111D - (d1_out$d1p1t111D^2)),
# 111U
"Var021C" <- (g3 * d1_out$d1p1t111U * (1 - d1_out$d1p1t111U)) +
(3 * (g - 3) * g3) * (d3_out$man111U_man111Ub - d2_out$man111U_man111U) +
g3 * (g3 - 1) * (d2_out$man111U_man111U - (d1_out$d1p1t111U^2)),
# 030T
"Var030T" <- (g3 * d1_out$d1p1t030T * (1 - d1_out$d1p1t030T)) +
(3 * (g - 3) * g3) * (d3_out$man030T_man030Tb - d2_out$man030T_man030T) +
g3 * (g3 - 1) * (d2_out$man030T_man030T - (d1_out$d1p1t030T^2)),
# 030C
"Var021C" <- (g3 * d1_out$d1p1t030C * (1 - d1_out$d1p1t030C)) +
(3 * (g - 3) * g3) * (d3_out$man030C_man030Cb - d2_out$man030C_man030C) +
g3 * (g3 - 1) * (d2_out$man030C_man030C - (d1_out$d1p1t030C^2)),
# 201
"Var021C" <- (g3 * d1_out$d1p1t201 * (1 - d1_out$d1p1t201)) +
(3 * (g - 3) * g3) * (d3_out$man201_man201b - d2_out$man201_man201) +
g3 * (g3 - 1) * (d2_out$man201_man201 - (d1_out$d1p1t201^2)),
# 120D
"Var120D" <- (g3 * d1_out$d1p1t120D * (1 - d1_out$d1p1t120D)) +
(3 * (g - 3) * g3) * (d3_out$man120D_man120Db - d2_out$man120D_man120D) +
g3 * (g3 - 1) * (d2_out$man120D_man120D - (d1_out$d1p1t120D^2)),
# 120U
"Var120U" <- (g3 * d1_out$d1p1t120U * (1 - d1_out$d1p1t120U)) +
(3 * (g - 3) * g3) * (d3_out$man120U_man120Ub - d2_out$man120U_man120U) +
g3 * (g3 - 1) * (d2_out$man120U_man120U - (d1_out$d1p1t120U^2)),
# 120C
"Var021C" <- (g3 * d1_out$d1p1t120C * (1 - d1_out$d1p1t120C)) +
(3 * (g - 3) * g3) * (d3_out$man120C_man120Cb - d2_out$man120C_man120C) +
g3 * (g3 - 1) * (d2_out$man120C_man120C - (d1_out$d1p1t120C^2)),
# 210
"Var021C" <- (g3 * d1_out$d1p1t210 * (1 - d1_out$d1p1t210)) +
(3 * (g - 3) * g3) * (d3_out$man210_man210b - d2_out$man210_man210) +
g3 * (g3 - 1) * (d2_out$man210_man210 - (d1_out$d1p1t210^2)),
# 300
"Var300" <- (g3 * d1_out$d1p1t300 * (1 - d1_out$d1p1t300)) +
(3 * (g - 3) * g3) * (d3_out$man300_man300b - d2_out$man300_man300) +
g3 * (g3 - 1) * (d2_out$man300_man300 - (d1_out$d1p1t300^2))
)
# covariance ----
# tips para los reemplazos
# ctrl + f + opcion regex en rstudio
# man[0-9] -> busca cualquier cosa de la forma man1, man2, etc...
# man[0-9][0-9] -> busca cualquier cosa de la forma man11, man12, etc...
# man[0-9][0-9][0-9]_man[0-9][0-9][0-9]b -> reemplaza todos los de la forma
# man123_man123b, etc
COVAR <- c(
# ----------- #
## 1
# 012 - 003
COVAR012_003 <- (-g3 * d1_out$d1p1t012 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man012_man003b - d2_out$man012_man003) +
g3 * (g3 - 1) * (d2_out$man012_man003 - (d1_out$d1p1t012 * d1_out$d1p1t003)),
# ----------- #
## 2
# 102 - 003
COVAR102_003 <- (-g3 * d1_out$d1p1t102 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man102_man003b - d2_out$man102_man003) +
g3 * (g3 - 1) * (d2_out$man102_man003 - (d1_out$d1p1t102 * d1_out$d1p1t003)),
# 102 - 012
COVAR102_012 <- (-g3 * d1_out$d1p1t102 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man102_man012b - d2_out$man102_man012) +
g3 * (g3 - 1) * (d2_out$man102_man012 - (d1_out$d1p1t102 * d1_out$d1p1t012)),
# ----------- #
## 3
# 021D - 003
COVAR021D_003 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021D_man003b - d2_out$man021D_man003) +
g3 * (g3 - 1) * (d2_out$man021D_man003 - (d1_out$d1p1t021D * d1_out$d1p1t003)),
# 021D - 012
COVAR021D_012 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021D_man012b - d2_out$man021D_man012) +
g3 * (g3 - 1) * (d2_out$man021D_man012 - (d1_out$d1p1t021D * d1_out$d1p1t012)),
# 021D - 102
COVAR021D_102 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021D_man102b - d2_out$man021D_man102) +
g3 * (g3 - 1) * (d2_out$man021D_man102 - (d1_out$d1p1t021D * d1_out$d1p1t102)),
# ----------- #
## 4
# 021U - 003
COVAR021U_003 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021U_man003b - d2_out$man021U_man003) +
g3 * (g3 - 1) * (d2_out$man021U_man003 - (d1_out$d1p1t021U * d1_out$d1p1t003)),
# 021U - 012
COVAR021U_012 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021U_man012b - d2_out$man021U_man012) +
g3 * (g3 - 1) * (d2_out$man021U_man012 - (d1_out$d1p1t021U * d1_out$d1p1t012)),
# 021U - 102
COVAR021U_102 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021U_man102b - d2_out$man021U_man102) +
g3 * (g3 - 1) * (d2_out$man021U_man102 - (d1_out$d1p1t021U * d1_out$d1p1t102)),
# 021U - 021D
COVAR021U_021D <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man021U_man021Db - d2_out$man021U_man021D) +
g3 * (g3 - 1) * (d2_out$man021U_man021D - (d1_out$d1p1t021U * d1_out$d1p1t021D)),
# ----------- #
## 5
# 021C - 003
COVAR021C_003 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021C_man003b - d2_out$man021C_man003) +
g3 * (g3 - 1) * (d2_out$man021C_man003 - (d1_out$d1p1t021C * d1_out$d1p1t003)),
# 021C - 012
COVAR021C_012 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021C_man012b - d2_out$man021C_man012) +
g3 * (g3 - 1) * (d2_out$man021C_man012 - (d1_out$d1p1t021C * d1_out$d1p1t012)),
# 021C - 102
COVAR021C_102 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021C_man102b - d2_out$man021C_man102) +
g3 * (g3 - 1) * (d2_out$man021C_man102 - (d1_out$d1p1t021C * d1_out$d1p1t102)),
# 021C - 021D
COVAR021C_021D <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man021C_man021Db - d2_out$man021C_man021D) +
g3 * (g3 - 1) * (d2_out$man021C_man021D - (d1_out$d1p1t021C * d1_out$d1p1t021D)),
# 021C - 021U
COVAR021C_021U <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man021C_man021Ub - d2_out$man021C_man021U) +
g3 * (g3 - 1) * (d2_out$man021C_man021U - (d1_out$d1p1t021C * d1_out$d1p1t021U)),
# ----------- #
## 6
# 111D - 003
COVAR111D_003 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man111D_man003b - d2_out$man111D_man003) +
g3 * (g3 - 1) * (d2_out$man111D_man003 - (d1_out$d1p1t111D * d1_out$d1p1t003)),
# 111D - 012
COVAR111D_012 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man111D_man012b - d2_out$man111D_man012) +
g3 * (g3 - 1) * (d2_out$man111D_man012 - (d1_out$d1p1t111D * d1_out$d1p1t012)),
# 111D - 102
COVAR111D_102 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man111D_man102b - d2_out$man111D_man102) +
g3 * (g3 - 1) * (d2_out$man111D_man102 - (d1_out$d1p1t111D * d1_out$d1p1t102)),
# 111D - 021D
COVAR111D_021D <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Db - d2_out$man111D_man021D) +
g3 * (g3 - 1) * (d2_out$man111D_man021D - (d1_out$d1p1t111D * d1_out$d1p1t021D)),
# 111D - 021U
COVAR111D_021U <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Ub - d2_out$man111D_man021U) +
g3 * (g3 - 1) * (d2_out$man111D_man021U - (d1_out$d1p1t111D * d1_out$d1p1t021U)),
# 111D - 021C
COVAR111D_021C <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Cb - d2_out$man111D_man021C) +
g3 * (g3 - 1) * (d2_out$man111D_man021C - (d1_out$d1p1t111D * d1_out$d1p1t021C)),
# ----------- #
## 7
# 111U - 003
COVAR111U_003 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man111U_man003b - d2_out$man111U_man003) +
g3 * (g3 - 1) * (d2_out$man111U_man003 - (d1_out$d1p1t111U * d1_out$d1p1t003)),
# 111U - 012
COVAR111U_012 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man111U_man012b - d2_out$man111U_man012) +
g3 * (g3 - 1) * (d2_out$man111U_man012 - (d1_out$d1p1t111U * d1_out$d1p1t012)),
# 111U - 102
COVAR111U_102 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man111U_man102b - d2_out$man111U_man102) +
g3 * (g3 - 1) * (d2_out$man111U_man102 - (d1_out$d1p1t111U * d1_out$d1p1t102)),
# 111U - 021D
COVAR111U_021D <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Db - d2_out$man111U_man021D) +
g3 * (g3 - 1) * (d2_out$man111U_man021D - (d1_out$d1p1t111U * d1_out$d1p1t021D)),
# 111U - 021U
COVAR111U_021U <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Ub - d2_out$man111U_man021U) +
g3 * (g3 - 1) * (d2_out$man111U_man021U - (d1_out$d1p1t111U * d1_out$d1p1t021U)),
# 111U - 021C
COVAR111U_021C <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Cb - d2_out$man111U_man021C) +
g3 * (g3 - 1) * (d2_out$man111U_man021C - (d1_out$d1p1t111U * d1_out$d1p1t021C)),
# 111U - 111D
COVAR111U_111D <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man111U_man111Db - d2_out$man111U_man111D) +
g3 * (g3 - 1) * (d2_out$man111U_man111D - (d1_out$d1p1t111U * d1_out$d1p1t111D)),
# ----------- #
## 8
# 030T - 003
COVAR030T_003 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man030T_man003b - d2_out$man030T_man003) +
g3 * (g3 - 1) * (d2_out$man030T_man003 - (d1_out$d1p1t030T * d1_out$d1p1t003)),
# 030T - 012
COVAR030T_012 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man030T_man012b - d2_out$man030T_man012) +
g3 * (g3 - 1) * (d2_out$man030T_man012 - (d1_out$d1p1t030T * d1_out$d1p1t012)),
# 030T - 102
COVAR030T_102 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man030T_man102b - d2_out$man030T_man102) +
g3 * (g3 - 1) * (d2_out$man030T_man102 - (d1_out$d1p1t030T * d1_out$d1p1t102)),
# 030T - 021D
COVAR030T_021D <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Db - d2_out$man030T_man021D) +
g3 * (g3 - 1) * (d2_out$man030T_man021D - (d1_out$d1p1t030T * d1_out$d1p1t021D)),
# 030T - 021U
COVAR030T_021U <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Ub - d2_out$man030T_man021U) +
g3 * (g3 - 1) * (d2_out$man030T_man021U - (d1_out$d1p1t030T * d1_out$d1p1t021U)),
# 030T - 021C
COVAR030T_021C <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Cb - d2_out$man030T_man021C) +
g3 * (g3 - 1) * (d2_out$man030T_man021C - (d1_out$d1p1t030T * d1_out$d1p1t021C)),
# 030T - 111D
COVAR030T_111D <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man030T_man111Db - d2_out$man030T_man111D) +
g3 * (g3 - 1) * (d2_out$man030T_man111D - (d1_out$d1p1t030T * d1_out$d1p1t111D)),
# 030T - 111U
COVAR030T_111U <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man030T_man111Ub - d2_out$man030T_man111U) +
g3 * (g3 - 1) * (d2_out$man030T_man111U - (d1_out$d1p1t030T * d1_out$d1p1t111U)),
# ----------- #
## 9
# 030C - 003
COVAR030C_003 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man030C_man003b - d2_out$man030C_man003) +
g3 * (g3 - 1) * (d2_out$man030C_man003 - (d1_out$d1p1t030C * d1_out$d1p1t003)),
# 030C - 012
COVAR030C_012 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man030C_man012b - d2_out$man030C_man012) +
g3 * (g3 - 1) * (d2_out$man030C_man012 - (d1_out$d1p1t030C * d1_out$d1p1t012)),
# 030C - 102
COVAR030C_102 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man030C_man102b - d2_out$man030C_man102) +
g3 * (g3 - 1) * (d2_out$man030C_man102 - (d1_out$d1p1t030C * d1_out$d1p1t102)),
# 030C - 021D
COVAR030C_021D <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Db - d2_out$man030C_man021D) +
g3 * (g3 - 1) * (d2_out$man030C_man021D - (d1_out$d1p1t030C * d1_out$d1p1t021D)),
# 030C - 021U
COVAR030C_021U <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Ub - d2_out$man030C_man021U) +
g3 * (g3 - 1) * (d2_out$man030C_man021U - (d1_out$d1p1t030C * d1_out$d1p1t021U)),
# 030C - 021C
COVAR030C_021C <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Cb - d2_out$man030C_man021C) +
g3 * (g3 - 1) * (d2_out$man030C_man021C - (d1_out$d1p1t030C * d1_out$d1p1t021C)),
# 030C - 111D
COVAR030C_111D <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man030C_man111Db - d2_out$man030C_man111D) +
g3 * (g3 - 1) * (d2_out$man030C_man111D - (d1_out$d1p1t030C * d1_out$d1p1t111D)),
# 030C - 111U
COVAR030C_111U <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man030C_man111Ub - d2_out$man030C_man111U) +
g3 * (g3 - 1) * (d2_out$man030C_man111U - (d1_out$d1p1t030C * d1_out$d1p1t111U)),
# 030C - 030T
COVAR030C_030T <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man030C_man030Tb - d2_out$man030C_man030T) +
g3 * (g3 - 1) * (d2_out$man030C_man030T - (d1_out$d1p1t030C * d1_out$d1p1t030T)),
# ----------- #
## 10
# 201 - 003
COVAR201_003 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man201_man003b - d2_out$man201_man003) +
g3 * (g3 - 1) * (d2_out$man201_man003 - (d1_out$d1p1t201 * d1_out$d1p1t003)),
# 201 - 012
COVAR201_012 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man201_man012b - d2_out$man201_man012) +
g3 * (g3 - 1) * (d2_out$man201_man012 - (d1_out$d1p1t201 * d1_out$d1p1t012)),
# COVARIANCE DIFFER WITH WASSERMAN AND FAUST
# 1994 (pp. 582-583)
# (i.e. TRIADS computer program of Walker and Wasserman, 1987)
# 201-102: we use the equation of Leinhardt and Holland (1975)
# P0(u,v): 9m^(3)n^(3); P1(u,v): 2m^(2)m^(2)(m+4-n)
# results: W-F: -18.4; here: 7.06
# 300-030T: original: -0.01; here:0.011
# 201 - 102
COVAR201_102 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man201_man102b - d2_out$man201_man102) +
g3 * (g3 - 1) * (d2_out$man201_man102 - (d1_out$d1p1t201 * d1_out$d1p1t102)),
# 201 - 021D
COVAR201_021D <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man201_man021Db - d2_out$man201_man021D) +
g3 * (g3 - 1) * (d2_out$man201_man021D - (d1_out$d1p1t201 * d1_out$d1p1t021D)),
# 201 - 021U
COVAR201_021U <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man201_man021Ub - d2_out$man201_man021U) +
g3 * (g3 - 1) * (d2_out$man201_man021U - (d1_out$d1p1t201 * d1_out$d1p1t021U)),
# 201 - 021C
COVAR201_021C <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man201_man021Cb - d2_out$man201_man021C) +
g3 * (g3 - 1) * (d2_out$man201_man021C - (d1_out$d1p1t201 * d1_out$d1p1t021C)),
# 201 - 111D
COVAR201_111D <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man201_man111Db - d2_out$man201_man111D) +
g3 * (g3 - 1) * (d2_out$man201_man111D - (d1_out$d1p1t201 * d1_out$d1p1t111D)),
# 201 - 111U
COVAR201_111U <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man201_man111Ub - d2_out$man201_man111U) +
g3 * (g3 - 1) * (d2_out$man201_man111U - (d1_out$d1p1t201 * d1_out$d1p1t111U)),
# 201 - 030T
COVAR201_030T <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man201_man030Tb - d2_out$man201_man030T) +
g3 * (g3 - 1) * (d2_out$man201_man030T - (d1_out$d1p1t201 * d1_out$d1p1t030T)),
# 201 - 030C
COVAR201_030C <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man201_man030Cb - d2_out$man201_man030C) +
g3 * (g3 - 1) * (d2_out$man201_man030C - (d1_out$d1p1t201 * d1_out$d1p1t030C)),
# ----------- #
## 11
# 120D - 003
COVAR120D_003 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120D_man003b - d2_out$man120D_man003) +
g3 * (g3 - 1) * (d2_out$man120D_man003 - (d1_out$d1p1t120D * d1_out$d1p1t003)),
# 120D - 012
COVAR120D_012 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120D_man012b - d2_out$man120D_man012) +
g3 * (g3 - 1) * (d2_out$man120D_man012 - (d1_out$d1p1t120D * d1_out$d1p1t012)),
# 120D - 102
COVAR120D_102 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120D_man102b - d2_out$man120D_man102) +
g3 * (g3 - 1) * (d2_out$man120D_man102 - (d1_out$d1p1t120D * d1_out$d1p1t102)),
# 120D - 021D
COVAR120D_021D <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Db - d2_out$man120D_man021D) +
g3 * (g3 - 1) * (d2_out$man120D_man021D - (d1_out$d1p1t120D * d1_out$d1p1t021D)),
# 120D - 021U
COVAR120D_021U <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Ub - d2_out$man120D_man021U) +
g3 * (g3 - 1) * (d2_out$man120D_man021U - (d1_out$d1p1t120D * d1_out$d1p1t021U)),
# 120D - 021C
COVAR120D_021C <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Cb - d2_out$man120D_man021C) +
g3 * (g3 - 1) * (d2_out$man120D_man021C - (d1_out$d1p1t120D * d1_out$d1p1t021C)),
# 120D - 111D
COVAR120D_111D <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120D_man111Db - d2_out$man120D_man111D) +
g3 * (g3 - 1) * (d2_out$man120D_man111D - (d1_out$d1p1t120D * d1_out$d1p1t111D)),
# 120D - 111U
COVAR120D_111U <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120D_man111Ub - d2_out$man120D_man111U) +
g3 * (g3 - 1) * (d2_out$man120D_man111U - (d1_out$d1p1t120D * d1_out$d1p1t111U)),
# 120D - 030T
COVAR120D_030T <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120D_man030Tb - d2_out$man120D_man030T) +
g3 * (g3 - 1) * (d2_out$man120D_man030T - (d1_out$d1p1t120D * d1_out$d1p1t030T)),
# 120D - 030C
COVAR120D_030C <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120D_man030Cb - d2_out$man120D_man030C) +
g3 * (g3 - 1) * (d2_out$man120D_man030C - (d1_out$d1p1t120D * d1_out$d1p1t030C)),
# 120D - 201
COVAR120D_201 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120D_man201b - d2_out$man120D_man201) +
g3 * (g3 - 1) * (d2_out$man120D_man201 - (d1_out$d1p1t120D * d1_out$d1p1t201)),
# ----------- #
## 12
# 120U - 003
COVAR120U_003 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120U_man003b - d2_out$man120U_man003) +
g3 * (g3 - 1) * (d2_out$man120U_man003 - (d1_out$d1p1t120U * d1_out$d1p1t003)),
# 120U - 012
COVAR120U_012 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120U_man012b - d2_out$man120U_man012) +
g3 * (g3 - 1) * (d2_out$man120U_man012 - (d1_out$d1p1t120U * d1_out$d1p1t012)),
# 120U - 102
COVAR120U_102 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120U_man102b - d2_out$man120U_man102) +
g3 * (g3 - 1) * (d2_out$man120U_man102 - (d1_out$d1p1t120U * d1_out$d1p1t102)),
# 120U - 021D
COVAR120U_021D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Db - d2_out$man120U_man021D) +
g3 * (g3 - 1) * (d2_out$man120U_man021D - (d1_out$d1p1t120U * d1_out$d1p1t021D)),
# 120U - 021U
COVAR120U_021U <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Ub - d2_out$man120U_man021U) +
g3 * (g3 - 1) * (d2_out$man120U_man021U - (d1_out$d1p1t120U * d1_out$d1p1t021U)),
# 120U - 021C
COVAR120U_021C <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Cb - d2_out$man120U_man021C) +
g3 * (g3 - 1) * (d2_out$man120U_man021C - (d1_out$d1p1t120U * d1_out$d1p1t021C)),
# 120U - 111D
COVAR120U_111D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120U_man111Db - d2_out$man120U_man111D) +
g3 * (g3 - 1) * (d2_out$man120U_man111D - (d1_out$d1p1t120U * d1_out$d1p1t111D)),
# 120U - 111U
COVAR120U_111U <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120U_man111Ub - d2_out$man120U_man111U) +
g3 * (g3 - 1) * (d2_out$man120U_man111U - (d1_out$d1p1t120U * d1_out$d1p1t111U)),
# 120U - 030T
COVAR120U_030T <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120U_man030Tb - d2_out$man120U_man030T) +
g3 * (g3 - 1) * (d2_out$man120U_man030T - (d1_out$d1p1t120U * d1_out$d1p1t030T)),
# 120U - 030C
COVAR120U_030C <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120U_man030Cb - d2_out$man120U_man030C) +
g3 * (g3 - 1) * (d2_out$man120U_man030C - (d1_out$d1p1t120U * d1_out$d1p1t030C)),
# 120U - 201
COVAR120U_201 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120U_man201b - d2_out$man120U_man201) +
g3 * (g3 - 1) * (d2_out$man120U_man201 - (d1_out$d1p1t120U * d1_out$d1p1t201)),
# 120U - 120D
COVAR120U_120D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man120U_man120Db - d2_out$man120U_man120D) +
g3 * (g3 - 1) * (d2_out$man120U_man120D - (d1_out$d1p1t120U * d1_out$d1p1t120D)),
# ----------- #
## 13
# 120C - 003
COVAR120C_003 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120C_man003b - d2_out$man120C_man003) +
g3 * (g3 - 1) * (d2_out$man120C_man003 - (d1_out$d1p1t120C * d1_out$d1p1t003)),
# 120C - 012
COVAR120C_012 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120C_man012b - d2_out$man120C_man012) +
g3 * (g3 - 1) * (d2_out$man120C_man012 - (d1_out$d1p1t120C * d1_out$d1p1t012)),
# 120C - 102
COVAR120C_102 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120C_man102b - d2_out$man120C_man102) +
g3 * (g3 - 1) * (d2_out$man120C_man102 - (d1_out$d1p1t120C * d1_out$d1p1t102)),
# 120C - 021D
COVAR120C_021D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Db - d2_out$man120C_man021D) +
g3 * (g3 - 1) * (d2_out$man120C_man021D - (d1_out$d1p1t120C * d1_out$d1p1t021D)),
# 120C - 021U
COVAR120C_021U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Ub - d2_out$man120C_man021U) +
g3 * (g3 - 1) * (d2_out$man120C_man021U - (d1_out$d1p1t120C * d1_out$d1p1t021U)),
# 120C - 021C
COVAR120C_021C <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Cb - d2_out$man120C_man021C) +
g3 * (g3 - 1) * (d2_out$man120C_man021C - (d1_out$d1p1t120C * d1_out$d1p1t021C)),
# 120C - 111D
COVAR120C_111D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120C_man111Db - d2_out$man120C_man111D) +
g3 * (g3 - 1) * (d2_out$man120C_man111D - (d1_out$d1p1t120C * d1_out$d1p1t111D)),
# 120C - 111U
COVAR120C_111U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120C_man111Ub - d2_out$man120C_man111U) +
g3 * (g3 - 1) * (d2_out$man120C_man111U - (d1_out$d1p1t120C * d1_out$d1p1t111U)),
# 120C - 030T
COVAR120C_030T <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120C_man030Tb - d2_out$man120C_man030T) +
g3 * (g3 - 1) * (d2_out$man120C_man030T - (d1_out$d1p1t120C * d1_out$d1p1t030T)),
# 120C - 030C
COVAR120C_030C <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120C_man030Cb - d2_out$man120C_man030C) +
g3 * (g3 - 1) * (d2_out$man120C_man030C - (d1_out$d1p1t120C * d1_out$d1p1t030C)),
# 120C - 201
COVAR120C_201 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120C_man201b - d2_out$man120C_man201) +
g3 * (g3 - 1) * (d2_out$man120C_man201 - (d1_out$d1p1t120C * d1_out$d1p1t201)),
# 120C - 120D
COVAR120C_120D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man120C_man120Db - d2_out$man120C_man120D) +
g3 * (g3 - 1) * (d2_out$man120C_man120D - (d1_out$d1p1t120C * d1_out$d1p1t120D)),
# 120C - 120U
COVAR120C_120U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man120C_man120Ub - d2_out$man120C_man120U) +
g3 * (g3 - 1) * (d2_out$man120C_man120U - (d1_out$d1p1t120C * d1_out$d1p1t120U)),
# ----------- #
## 14
# 210 - 003
COVAR210_003 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man210_man003b - d2_out$man210_man003) +
g3 * (g3 - 1) * (d2_out$man210_man003 - (d1_out$d1p1t210 * d1_out$d1p1t003)),
# 210 - 012
COVAR210_012 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man210_man012b - d2_out$man210_man012) +
g3 * (g3 - 1) * (d2_out$man210_man012 - (d1_out$d1p1t210 * d1_out$d1p1t012)),
# 210 - 102
COVAR210_102 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man210_man102b - d2_out$man210_man102) +
g3 * (g3 - 1) * (d2_out$man210_man102 - (d1_out$d1p1t210 * d1_out$d1p1t102)),
# 210 - 021D
COVAR210_021D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man210_man021Db - d2_out$man210_man021D) +
g3 * (g3 - 1) * (d2_out$man210_man021D - (d1_out$d1p1t210 * d1_out$d1p1t021D)),
# 210 - 021U
COVAR210_021U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man210_man021Ub - d2_out$man210_man021U) +
g3 * (g3 - 1) * (d2_out$man210_man021U - (d1_out$d1p1t210 * d1_out$d1p1t021U)),
# 210 - 021C
COVAR210_021C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man210_man021Cb - d2_out$man210_man021C) +
g3 * (g3 - 1) * (d2_out$man210_man021C - (d1_out$d1p1t210 * d1_out$d1p1t021C)),
# 210 - 111D
COVAR210_111D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man210_man111Db - d2_out$man210_man111D) +
g3 * (g3 - 1) * (d2_out$man210_man111D - (d1_out$d1p1t210 * d1_out$d1p1t111D)),
# 210 - 111U
COVAR210_111U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man210_man111Ub - d2_out$man210_man111U) +
g3 * (g3 - 1) * (d2_out$man210_man111U - (d1_out$d1p1t210 * d1_out$d1p1t111U)),
# 210 - 030T
COVAR210_030T <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man210_man030Tb - d2_out$man210_man030T) +
g3 * (g3 - 1) * (d2_out$man210_man030T - (d1_out$d1p1t210 * d1_out$d1p1t030T)),
# 210 - 030C
COVAR210_030C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man210_man030Cb - d2_out$man210_man030C) +
g3 * (g3 - 1) * (d2_out$man210_man030C - (d1_out$d1p1t210 * d1_out$d1p1t030C)),
# 210 - 201
COVAR210_201 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man210_man201b - d2_out$man210_man201) +
g3 * (g3 - 1) * (d2_out$man210_man201 - (d1_out$d1p1t210 * d1_out$d1p1t201)),
# 210 - 120D
COVAR210_120D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man210_man120Db - d2_out$man210_man120D) +
g3 * (g3 - 1) * (d2_out$man210_man120D - (d1_out$d1p1t210 * d1_out$d1p1t120D)),
# 210 - 120U
COVAR210_120U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man210_man120Ub - d2_out$man210_man120U) +
g3 * (g3 - 1) * (d2_out$man210_man120U - (d1_out$d1p1t210 * d1_out$d1p1t120U)),
# 210 - 120C
COVAR210_120C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120C) +
3 * (g - 3) * g3 * (d3_out$man210_man120Cb - d2_out$man210_man120C) +
g3 * (g3 - 1) * (d2_out$man210_man120C - (d1_out$d1p1t210 * d1_out$d1p1t120C)),
# ----------- #
## 15
# 300 - 003
COVAR300_003 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man300_man003b - d2_out$man300_man003) +
g3 * (g3 - 1) * (d2_out$man300_man003 - (d1_out$d1p1t300 * d1_out$d1p1t003)),
# 300 - 012
COVAR300_012 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man300_man012b - d2_out$man300_man012) +
g3 * (g3 - 1) * (d2_out$man300_man012 - (d1_out$d1p1t300 * d1_out$d1p1t012)),
# 300 - 102
COVAR300_102 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man300_man102b - d2_out$man300_man102) +
g3 * (g3 - 1) * (d2_out$man300_man102 - (d1_out$d1p1t300 * d1_out$d1p1t102)),
# 300 - 021D
COVAR300_021D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man300_man021Db - d2_out$man300_man021D) +
g3 * (g3 - 1) * (d2_out$man300_man021D - (d1_out$d1p1t300 * d1_out$d1p1t021D)),
# 300 - 021U
COVAR300_021U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man300_man021Ub - d2_out$man300_man021U) +
g3 * (g3 - 1) * (d2_out$man300_man021U - (d1_out$d1p1t300 * d1_out$d1p1t021U)),
# 300 - 021C
COVAR300_021C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man300_man021Cb - d2_out$man300_man021C) +
g3 * (g3 - 1) * (d2_out$man300_man021C - (d1_out$d1p1t300 * d1_out$d1p1t021C)),
# 300 - 111D
COVAR300_111D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man300_man111Db - d2_out$man300_man111D) +
g3 * (g3 - 1) * (d2_out$man300_man111D - (d1_out$d1p1t300 * d1_out$d1p1t111D)),
# 300 - 111U
COVAR300_111U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man300_man111Ub - d2_out$man300_man111U) +
g3 * (g3 - 1) * (d2_out$man300_man111U - (d1_out$d1p1t300 * d1_out$d1p1t111U)),
# COVARIANCE DIFFER WITH WASSERMAN AND FAUST
# 1994 (pp. 582-583)
# (i.e. TRIADS computer program of Walker and Wasserman, 1987)
# 201-102: we use the equation of Leinhardt and Holland (1975)
# P0(u,v): 9m^(3)n^(3); P1(u,v): 2m^(2)m^(2)(m+4-n)
# results: W-F: -18.4; here: 7.06
# 300-030T: original: -0.01; here:0.011
# 300 - 030T
COVAR300_030T <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man300_man030Tb - d2_out$man300_man030T) +
g3 * (g3 - 1) * (d2_out$man300_man030T - (d1_out$d1p1t300 * d1_out$d1p1t030T)),
# 300 - 030C
COVAR300_030C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man300_man030Cb - d2_out$man300_man030C) +
g3 * (g3 - 1) * (d2_out$man300_man030C - (d1_out$d1p1t300 * d1_out$d1p1t030C)),
# 300 - 201
COVAR300_201 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man300_man201b - d2_out$man300_man201) +
g3 * (g3 - 1) * (d2_out$man300_man201 - (d1_out$d1p1t300 * d1_out$d1p1t201)),
# 300 - 120D
COVAR300_120D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man300_man120Db - d2_out$man300_man120D) +
g3 * (g3 - 1) * (d2_out$man300_man120D - (d1_out$d1p1t300 * d1_out$d1p1t120D)),
# 300 - 120U
COVAR300_120U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man300_man120Ub - d2_out$man300_man120U) +
g3 * (g3 - 1) * (d2_out$man300_man120U - (d1_out$d1p1t300 * d1_out$d1p1t120U)),
# 300 - 120C
COVAR300_120C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120C) +
3 * (g - 3) * g3 * (d3_out$man300_man120Cb - d2_out$man300_man120C) +
g3 * (g3 - 1) * (d2_out$man300_man120C - (d1_out$d1p1t300 * d1_out$d1p1t120C)),
# 300 - 210
COVAR300_210 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t210) +
3 * (g - 3) * g3 * (d3_out$man300_man210b - d2_out$man300_man210) +
g3 * (g3 - 1) * (d2_out$man300_man210 - (d1_out$d1p1t300 * d1_out$d1p1t210))
)
# triads census ----
# E <- ifelse((A + Matrix::t(A)) > 0, 1, 0) # arcs to edges
E <- A + Matrix::t(A)
E[E > 0] <- 1
# Eb <- ifelse(E == 0, 1, 0)
Eb <- E
Eb[Eb == 0] <- -999
Eb[Eb != -999] <- 0
Eb[Eb == -999] <- 1
Matrix::diag(Eb) <- 0
# M <- ifelse((A + t(A)) > 1, 1, 0) # mutual edges only
M <- A + Matrix::t(A)
M[M == 1] <- -999
M[M > 1] <- 1
M[M < 1] <- 0
C <- A - M # asymmetric arcs only
t201 <- sum(M %*% M * Eb) # null dyad in a 201 triad
t021D <- sum(t(C) %*% C * Eb) # null dyad in an 021D triad
t021U <- sum(C %*% t(C) * Eb) # null dyad in an 021U triad
triad <- c(
"003" = sum(diag(Eb %*% Eb %*% Eb)) / 6,
"012" = sum((Eb %*% Eb) * (C + t(C))) / 2,
"102" = sum((Eb %*% Eb * M)) / 2,
"021D" = sum(t(C) %*% C * Eb) / 2,
"021U" = sum(C %*% t(C) * Eb) / 2,
"021C" = sum(C %*% C * Eb),
"111D" = (sum(A %*% t(A) * Eb) - t201 - t021U) / 2,
"111U" = (sum(t(A) %*% A * Eb) - t201 - t021D) / 2,
"030T" = sum((C %*% C) * C),
"030C" = sum(diag(C %*% C %*% C)) / 3,
"201" = sum(M %*% M * Eb) / 2,
"120D" = sum(t(C) %*% C * M) / 2,
"120U" = sum(C %*% t(C) * M) / 2,
"120C" = sum(C %*% C * M),
"210" = sum(M %*% M * (C + t(C))) / 2,
"300" = sum(diag(M %*% M %*% M)) / 6
)
label <- c(
"003", "012", "102", "021D", "021U",
"021C", "111D", "111U", "030T", "030C",
"201", "120D", "120U", "120C", "210", "300"
)
results <- as.data.frame(cbind(
label = as.character(label),
OBS = as.numeric(triad),
EXP = as.numeric(EXP),
VAR = as.numeric(VAR)
))
results$OBS <- round(as.numeric(as.character(results$OBS)), 0)
results$EXP <- round(as.numeric(as.character(results$EXP)), 3)
results$VAR <- round(as.numeric(as.character(results$VAR)), 3)
results$STD <- sqrt(results$VAR)
results$STD <- round(as.numeric(as.character(results$STD)), 3)
mempty <- matrix(0, nrow = 16, ncol = 16)
mindex <- matrix(1:256, nrow = 16, ncol = 16, byrow = T)
mempty[mindex[upper.tri(mindex)]] <- COVAR
diag(mempty) <- VAR # VARIANCE IN THE DIAGONAL
rownames(mempty) <- label
colnames(mempty) <- label
mempty <- round(mempty, 3)
z <- (sum(results$OBS) - sum(results$EXP)) / (sqrt(sum(results$VAR) + (2 * (sum(COVAR)))))
p <- 2 * pnorm(-abs(z))
res <- c(z = z, p = p)
res <- round(res, 3)
if (ztest & covar) {
results$Z <- (results$OBS - results$EXP) / results$STD
results$Z <- as.numeric(as.character(results$Z))
results$Z <- round(results$Z, 3)
results$P <- 2 * pnorm(-abs(results$Z))
results$P <- as.numeric(as.character(results$P))
results$P <- round(results$P, 3)
newlist <- list(results = results, z_test = res, covariance = mempty)
return(newlist)
}
if (covar) {
# FIXME: EXPERIMENTAL version. Use with caution... the covar 201-102 and 300-030T are under review
newlist <- list(results = results, covariance = mempty)
return(newlist)
}
if (ztest) {
results$Z <- (results$OBS - results$EXP) / results$STD
results$Z <- as.numeric(as.character(results$Z))
results$Z <- round(results$Z, 3)
results$P <- 2 * pnorm(-abs(results$Z))
results$P <- as.numeric(as.character(results$P))
results$P <- round(results$P, 3)
newlist <- list(results = results, z_test = res)
return(newlist)
} else {
return(results)
}
}
anespinosa/netmem documentation built on April 5, 2025, 5:02 p.m.
R Package Documentation
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Defines functions triad_uman
Documented in triad_uman
#' Triad census analysis assuming U|MAN
#'
#' Considering the triad census of Davis and Leinhardt (1972) for vector A, B, and C:
#'
#' 003 = A,B,C, empty triad
#'
#' 012 = A -> B, C, triad with a single directed edge
#'
#' 102 = A <-> B, C, triad with a reciprocated connection between two vertices
#'
#' 021D = A <-B-> C, triadic out-star
#'
#' 021U = A -> B <- C triadic in-star
#'
#' 021C = A-> B-> C, directed line
#'
#' 111D = A <-> B <-C
#'
#' 111U = A <-> B-> C
#'
#' 030T = A-> B <-C, A-> C
#'
#' 030C = A <-B <-C, A-> C
#'
#' 201 = A <-> B <-> C
#'
#' 120D = A <-B-> C, A <-> C
#'
#' 120U = A-> B <-C, A <->C
#'
#' 120C = A-> B-> C, A <-> C
#'
#' 210 = A-> B <-> C, A <-> C
#'
#' 300 = A <-> B <-> C, A <->C, complete triad.
#'
#' @param A A symmetric matrix object
#' @param ztest Return Z and p-value
#' @param covar Return the covarianc matrix for triadic analysis
#'
#' @return This function gives the counts of the triad census, the expected counts,
#' assuming that U|MAN distribution (Holland and Leinhardt, 1975, 1976) is operating,
#' and the standard deviations of these counts.
#'
#' @references
#'
#' Davis, J.A. and Leinhardt, S. (1972). The Structure of Positive Interpersonal Relations in Small Groups. In J. Berger (Ed.), Sociological Theories in Progress, Volume 2, 218-251. Boston: Houghton Mifflin.
#'
#' Holland, P. W. and Leinhardt, S. (1975). The statistical analysis of local structure in social networks. In D. R. Heise (Ed.), Sociological Methodology, 1976 (Jossey-Bass, pp. 1–45).
#'
#' Holland, P. W. and Leinhardt, S. (1976). Local Structure in Social Networks. Sociological Methodology, 7, 1–45. doi: \url{https://doi.org/10.2307/270703}
#'
#' Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.
#'
#' @author Alejandro Espinosa-Rada
#'
#' @importFrom stats pnorm
#' @importFrom Matrix Matrix diag t
#'
#' @examples
#'
#' data(krackhardt_friends)
#' triad_uman(krackhardt_friends)
#' \dontrun{
#' triad_uman(krackhardt_friends, ztest = TRUE, covar = TRUE)
#' }
#'
#' @export
triad_uman <- function(A, ztest = FALSE, covar = FALSE) {
# elements ----
A <- as.matrix(A)
if (any(abs(A > 1), na.rm = TRUE)) stop("The matrix should be binary")
if (any(is.na(A) == TRUE)) {
A <- ifelse(is.na(A), 0, A)
}
A <- Matrix::Matrix(A)
g <- dim(A)[1]
m <- (1 / 2) * sum(Matrix::diag(A %*% A)) # mutual
a <- sum(Matrix::diag(A %*% Matrix::t(A))) - sum(Matrix::diag(A %*% A)) # asymmetric
n <- ((g * (g - 1)) / 2) - (sum(Matrix::diag(A %*% Matrix::t(A))) - sum(Matrix::diag(A %*% A))) - ((1 / 2) * sum(Matrix::diag(A %*% A))) # null
g2 <- (1 / 2) * (g * (g - 1))
g3 <- (1 / 6) * (g * (g - 1) * (g - 2))
# d1 ----
d1_out <- d1(g2, n, m, a)
# d2 ----
d2_out <- d2(g2, n, m, a)
# d3 ----
d3_out <- d3(g2, n, m, a)
# expected value ----
EXP <- c(
# 003
"E003" <- g3 * d1_out$d1p1t003,
# 012
"E012" <- g3 * d1_out$d1p1t012,
# 102
"E102" <- g3 * d1_out$d1p1t102,
# 021D
"E021D" <- g3 * d1_out$d1p1t021D,
# 021U
"E021U" <- g3 * d1_out$d1p1t021U,
# 021C
"E021C" <- g3 * d1_out$d1p1t021C,
# 111D
"E111D" <- g3 * d1_out$d1p1t111D,
# 111U
"E111U" <- g3 * d1_out$d1p1t111U,
# 030T
"E030T" <- g3 * d1_out$d1p1t030T,
# 030C
"E030C" <- g3 * d1_out$d1p1t030C,
# 201
"E201" <- g3 * d1_out$d1p1t201,
# 120D
"E120D" <- g3 * d1_out$d1p1t120D,
# 120U
"E120U" <- g3 * d1_out$d1p1t120U,
# 120C
"E120C" <- g3 * d1_out$d1p1t120C,
# 210
"E210" <- g3 * d1_out$d1p1t210,
# 300
"E300" <- g3 * d1_out$d1p1t300
)
# variance ----
VAR <- c(
# 003
"Var003" <- (g3 * d1_out$d1p1t003 * (1 - d1_out$d1p1t003)) +
(3 * (g - 3) * g3) * (d3_out$man003_man003b - d2_out$man003_man003) +
g3 * (g3 - 1) * (d2_out$man003_man003 - (d1_out$d1p1t003^2)),
# 012
"Var012" <- (g3 * d1_out$d1p1t012 * (1 - d1_out$d1p1t012)) +
(3 * (g - 3) * g3) * (d3_out$man012_man012b - d2_out$man012_man012) +
g3 * (g3 - 1) * (d2_out$man012_man012 - (d1_out$d1p1t012^2)),
# 102
"Var102" <- (g3 * d1_out$d1p1t102 * (1 - d1_out$d1p1t102)) +
(3 * (g - 3) * g3) * (d3_out$man102_man102b - d2_out$man102_man102) +
g3 * (g3 - 1) * (d2_out$man102_man102 - (d1_out$d1p1t102^2)),
# 021D
"Var021D" <- (g3 * d1_out$d1p1t021D * (1 - d1_out$d1p1t021D)) +
(3 * (g - 3) * g3) * (d3_out$man021D_man021Db - d2_out$man021D_man021D) +
g3 * (g3 - 1) * (d2_out$man021D_man021D - (d1_out$d1p1t021D^2)),
# 021U
"Var021U" <- (g3 * d1_out$d1p1t021U * (1 - d1_out$d1p1t021U)) +
(3 * (g - 3) * g3) * (d3_out$man021U_man021Ub - d2_out$man021U_man021U) +
g3 * (g3 - 1) * (d2_out$man021U_man021U - (d1_out$d1p1t021U^2)),
# 021C
"Var021C" <- (g3 * d1_out$d1p1t021C * (1 - d1_out$d1p1t021C)) +
(3 * (g - 3) * g3) * (d3_out$man021C_man021Cb - d2_out$man021C_man021C) +
g3 * (g3 - 1) * (d2_out$man021C_man021C - (d1_out$d1p1t021C^2)),
# 111D
"Var021C" <- (g3 * d1_out$d1p1t111D * (1 - d1_out$d1p1t111D)) +
(3 * (g - 3) * g3) * (d3_out$man111D_man111Db - d2_out$man111D_man111D) +
g3 * (g3 - 1) * (d2_out$man111D_man111D - (d1_out$d1p1t111D^2)),
# 111U
"Var021C" <- (g3 * d1_out$d1p1t111U * (1 - d1_out$d1p1t111U)) +
(3 * (g - 3) * g3) * (d3_out$man111U_man111Ub - d2_out$man111U_man111U) +
g3 * (g3 - 1) * (d2_out$man111U_man111U - (d1_out$d1p1t111U^2)),
# 030T
"Var030T" <- (g3 * d1_out$d1p1t030T * (1 - d1_out$d1p1t030T)) +
(3 * (g - 3) * g3) * (d3_out$man030T_man030Tb - d2_out$man030T_man030T) +
g3 * (g3 - 1) * (d2_out$man030T_man030T - (d1_out$d1p1t030T^2)),
# 030C
"Var021C" <- (g3 * d1_out$d1p1t030C * (1 - d1_out$d1p1t030C)) +
(3 * (g - 3) * g3) * (d3_out$man030C_man030Cb - d2_out$man030C_man030C) +
g3 * (g3 - 1) * (d2_out$man030C_man030C - (d1_out$d1p1t030C^2)),
# 201
"Var021C" <- (g3 * d1_out$d1p1t201 * (1 - d1_out$d1p1t201)) +
(3 * (g - 3) * g3) * (d3_out$man201_man201b - d2_out$man201_man201) +
g3 * (g3 - 1) * (d2_out$man201_man201 - (d1_out$d1p1t201^2)),
# 120D
"Var120D" <- (g3 * d1_out$d1p1t120D * (1 - d1_out$d1p1t120D)) +
(3 * (g - 3) * g3) * (d3_out$man120D_man120Db - d2_out$man120D_man120D) +
g3 * (g3 - 1) * (d2_out$man120D_man120D - (d1_out$d1p1t120D^2)),
# 120U
"Var120U" <- (g3 * d1_out$d1p1t120U * (1 - d1_out$d1p1t120U)) +
(3 * (g - 3) * g3) * (d3_out$man120U_man120Ub - d2_out$man120U_man120U) +
g3 * (g3 - 1) * (d2_out$man120U_man120U - (d1_out$d1p1t120U^2)),
# 120C
"Var021C" <- (g3 * d1_out$d1p1t120C * (1 - d1_out$d1p1t120C)) +
(3 * (g - 3) * g3) * (d3_out$man120C_man120Cb - d2_out$man120C_man120C) +
g3 * (g3 - 1) * (d2_out$man120C_man120C - (d1_out$d1p1t120C^2)),
# 210
"Var021C" <- (g3 * d1_out$d1p1t210 * (1 - d1_out$d1p1t210)) +
(3 * (g - 3) * g3) * (d3_out$man210_man210b - d2_out$man210_man210) +
g3 * (g3 - 1) * (d2_out$man210_man210 - (d1_out$d1p1t210^2)),
# 300
"Var300" <- (g3 * d1_out$d1p1t300 * (1 - d1_out$d1p1t300)) +
(3 * (g - 3) * g3) * (d3_out$man300_man300b - d2_out$man300_man300) +
g3 * (g3 - 1) * (d2_out$man300_man300 - (d1_out$d1p1t300^2))
)
# covariance ----
# tips para los reemplazos
# ctrl + f + opcion regex en rstudio
# man[0-9] -> busca cualquier cosa de la forma man1, man2, etc...
# man[0-9][0-9] -> busca cualquier cosa de la forma man11, man12, etc...
# man[0-9][0-9][0-9]_man[0-9][0-9][0-9]b -> reemplaza todos los de la forma
# man123_man123b, etc
COVAR <- c(
# ----------- #
## 1
# 012 - 003
COVAR012_003 <- (-g3 * d1_out$d1p1t012 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man012_man003b - d2_out$man012_man003) +
g3 * (g3 - 1) * (d2_out$man012_man003 - (d1_out$d1p1t012 * d1_out$d1p1t003)),
# ----------- #
## 2
# 102 - 003
COVAR102_003 <- (-g3 * d1_out$d1p1t102 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man102_man003b - d2_out$man102_man003) +
g3 * (g3 - 1) * (d2_out$man102_man003 - (d1_out$d1p1t102 * d1_out$d1p1t003)),
# 102 - 012
COVAR102_012 <- (-g3 * d1_out$d1p1t102 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man102_man012b - d2_out$man102_man012) +
g3 * (g3 - 1) * (d2_out$man102_man012 - (d1_out$d1p1t102 * d1_out$d1p1t012)),
# ----------- #
## 3
# 021D - 003
COVAR021D_003 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021D_man003b - d2_out$man021D_man003) +
g3 * (g3 - 1) * (d2_out$man021D_man003 - (d1_out$d1p1t021D * d1_out$d1p1t003)),
# 021D - 012
COVAR021D_012 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021D_man012b - d2_out$man021D_man012) +
g3 * (g3 - 1) * (d2_out$man021D_man012 - (d1_out$d1p1t021D * d1_out$d1p1t012)),
# 021D - 102
COVAR021D_102 <- (-g3 * d1_out$d1p1t021D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021D_man102b - d2_out$man021D_man102) +
g3 * (g3 - 1) * (d2_out$man021D_man102 - (d1_out$d1p1t021D * d1_out$d1p1t102)),
# ----------- #
## 4
# 021U - 003
COVAR021U_003 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021U_man003b - d2_out$man021U_man003) +
g3 * (g3 - 1) * (d2_out$man021U_man003 - (d1_out$d1p1t021U * d1_out$d1p1t003)),
# 021U - 012
COVAR021U_012 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021U_man012b - d2_out$man021U_man012) +
g3 * (g3 - 1) * (d2_out$man021U_man012 - (d1_out$d1p1t021U * d1_out$d1p1t012)),
# 021U - 102
COVAR021U_102 <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021U_man102b - d2_out$man021U_man102) +
g3 * (g3 - 1) * (d2_out$man021U_man102 - (d1_out$d1p1t021U * d1_out$d1p1t102)),
# 021U - 021D
COVAR021U_021D <- (-g3 * d1_out$d1p1t021U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man021U_man021Db - d2_out$man021U_man021D) +
g3 * (g3 - 1) * (d2_out$man021U_man021D - (d1_out$d1p1t021U * d1_out$d1p1t021D)),
# ----------- #
## 5
# 021C - 003
COVAR021C_003 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man021C_man003b - d2_out$man021C_man003) +
g3 * (g3 - 1) * (d2_out$man021C_man003 - (d1_out$d1p1t021C * d1_out$d1p1t003)),
# 021C - 012
COVAR021C_012 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man021C_man012b - d2_out$man021C_man012) +
g3 * (g3 - 1) * (d2_out$man021C_man012 - (d1_out$d1p1t021C * d1_out$d1p1t012)),
# 021C - 102
COVAR021C_102 <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man021C_man102b - d2_out$man021C_man102) +
g3 * (g3 - 1) * (d2_out$man021C_man102 - (d1_out$d1p1t021C * d1_out$d1p1t102)),
# 021C - 021D
COVAR021C_021D <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man021C_man021Db - d2_out$man021C_man021D) +
g3 * (g3 - 1) * (d2_out$man021C_man021D - (d1_out$d1p1t021C * d1_out$d1p1t021D)),
# 021C - 021U
COVAR021C_021U <- (-g3 * d1_out$d1p1t021C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man021C_man021Ub - d2_out$man021C_man021U) +
g3 * (g3 - 1) * (d2_out$man021C_man021U - (d1_out$d1p1t021C * d1_out$d1p1t021U)),
# ----------- #
## 6
# 111D - 003
COVAR111D_003 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man111D_man003b - d2_out$man111D_man003) +
g3 * (g3 - 1) * (d2_out$man111D_man003 - (d1_out$d1p1t111D * d1_out$d1p1t003)),
# 111D - 012
COVAR111D_012 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man111D_man012b - d2_out$man111D_man012) +
g3 * (g3 - 1) * (d2_out$man111D_man012 - (d1_out$d1p1t111D * d1_out$d1p1t012)),
# 111D - 102
COVAR111D_102 <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man111D_man102b - d2_out$man111D_man102) +
g3 * (g3 - 1) * (d2_out$man111D_man102 - (d1_out$d1p1t111D * d1_out$d1p1t102)),
# 111D - 021D
COVAR111D_021D <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Db - d2_out$man111D_man021D) +
g3 * (g3 - 1) * (d2_out$man111D_man021D - (d1_out$d1p1t111D * d1_out$d1p1t021D)),
# 111D - 021U
COVAR111D_021U <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Ub - d2_out$man111D_man021U) +
g3 * (g3 - 1) * (d2_out$man111D_man021U - (d1_out$d1p1t111D * d1_out$d1p1t021U)),
# 111D - 021C
COVAR111D_021C <- (-g3 * d1_out$d1p1t111D * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man111D_man021Cb - d2_out$man111D_man021C) +
g3 * (g3 - 1) * (d2_out$man111D_man021C - (d1_out$d1p1t111D * d1_out$d1p1t021C)),
# ----------- #
## 7
# 111U - 003
COVAR111U_003 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man111U_man003b - d2_out$man111U_man003) +
g3 * (g3 - 1) * (d2_out$man111U_man003 - (d1_out$d1p1t111U * d1_out$d1p1t003)),
# 111U - 012
COVAR111U_012 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man111U_man012b - d2_out$man111U_man012) +
g3 * (g3 - 1) * (d2_out$man111U_man012 - (d1_out$d1p1t111U * d1_out$d1p1t012)),
# 111U - 102
COVAR111U_102 <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man111U_man102b - d2_out$man111U_man102) +
g3 * (g3 - 1) * (d2_out$man111U_man102 - (d1_out$d1p1t111U * d1_out$d1p1t102)),
# 111U - 021D
COVAR111U_021D <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Db - d2_out$man111U_man021D) +
g3 * (g3 - 1) * (d2_out$man111U_man021D - (d1_out$d1p1t111U * d1_out$d1p1t021D)),
# 111U - 021U
COVAR111U_021U <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Ub - d2_out$man111U_man021U) +
g3 * (g3 - 1) * (d2_out$man111U_man021U - (d1_out$d1p1t111U * d1_out$d1p1t021U)),
# 111U - 021C
COVAR111U_021C <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man111U_man021Cb - d2_out$man111U_man021C) +
g3 * (g3 - 1) * (d2_out$man111U_man021C - (d1_out$d1p1t111U * d1_out$d1p1t021C)),
# 111U - 111D
COVAR111U_111D <- (-g3 * d1_out$d1p1t111U * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man111U_man111Db - d2_out$man111U_man111D) +
g3 * (g3 - 1) * (d2_out$man111U_man111D - (d1_out$d1p1t111U * d1_out$d1p1t111D)),
# ----------- #
## 8
# 030T - 003
COVAR030T_003 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man030T_man003b - d2_out$man030T_man003) +
g3 * (g3 - 1) * (d2_out$man030T_man003 - (d1_out$d1p1t030T * d1_out$d1p1t003)),
# 030T - 012
COVAR030T_012 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man030T_man012b - d2_out$man030T_man012) +
g3 * (g3 - 1) * (d2_out$man030T_man012 - (d1_out$d1p1t030T * d1_out$d1p1t012)),
# 030T - 102
COVAR030T_102 <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man030T_man102b - d2_out$man030T_man102) +
g3 * (g3 - 1) * (d2_out$man030T_man102 - (d1_out$d1p1t030T * d1_out$d1p1t102)),
# 030T - 021D
COVAR030T_021D <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Db - d2_out$man030T_man021D) +
g3 * (g3 - 1) * (d2_out$man030T_man021D - (d1_out$d1p1t030T * d1_out$d1p1t021D)),
# 030T - 021U
COVAR030T_021U <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Ub - d2_out$man030T_man021U) +
g3 * (g3 - 1) * (d2_out$man030T_man021U - (d1_out$d1p1t030T * d1_out$d1p1t021U)),
# 030T - 021C
COVAR030T_021C <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man030T_man021Cb - d2_out$man030T_man021C) +
g3 * (g3 - 1) * (d2_out$man030T_man021C - (d1_out$d1p1t030T * d1_out$d1p1t021C)),
# 030T - 111D
COVAR030T_111D <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man030T_man111Db - d2_out$man030T_man111D) +
g3 * (g3 - 1) * (d2_out$man030T_man111D - (d1_out$d1p1t030T * d1_out$d1p1t111D)),
# 030T - 111U
COVAR030T_111U <- (-g3 * d1_out$d1p1t030T * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man030T_man111Ub - d2_out$man030T_man111U) +
g3 * (g3 - 1) * (d2_out$man030T_man111U - (d1_out$d1p1t030T * d1_out$d1p1t111U)),
# ----------- #
## 9
# 030C - 003
COVAR030C_003 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man030C_man003b - d2_out$man030C_man003) +
g3 * (g3 - 1) * (d2_out$man030C_man003 - (d1_out$d1p1t030C * d1_out$d1p1t003)),
# 030C - 012
COVAR030C_012 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man030C_man012b - d2_out$man030C_man012) +
g3 * (g3 - 1) * (d2_out$man030C_man012 - (d1_out$d1p1t030C * d1_out$d1p1t012)),
# 030C - 102
COVAR030C_102 <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man030C_man102b - d2_out$man030C_man102) +
g3 * (g3 - 1) * (d2_out$man030C_man102 - (d1_out$d1p1t030C * d1_out$d1p1t102)),
# 030C - 021D
COVAR030C_021D <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Db - d2_out$man030C_man021D) +
g3 * (g3 - 1) * (d2_out$man030C_man021D - (d1_out$d1p1t030C * d1_out$d1p1t021D)),
# 030C - 021U
COVAR030C_021U <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Ub - d2_out$man030C_man021U) +
g3 * (g3 - 1) * (d2_out$man030C_man021U - (d1_out$d1p1t030C * d1_out$d1p1t021U)),
# 030C - 021C
COVAR030C_021C <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man030C_man021Cb - d2_out$man030C_man021C) +
g3 * (g3 - 1) * (d2_out$man030C_man021C - (d1_out$d1p1t030C * d1_out$d1p1t021C)),
# 030C - 111D
COVAR030C_111D <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man030C_man111Db - d2_out$man030C_man111D) +
g3 * (g3 - 1) * (d2_out$man030C_man111D - (d1_out$d1p1t030C * d1_out$d1p1t111D)),
# 030C - 111U
COVAR030C_111U <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man030C_man111Ub - d2_out$man030C_man111U) +
g3 * (g3 - 1) * (d2_out$man030C_man111U - (d1_out$d1p1t030C * d1_out$d1p1t111U)),
# 030C - 030T
COVAR030C_030T <- (-g3 * d1_out$d1p1t030C * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man030C_man030Tb - d2_out$man030C_man030T) +
g3 * (g3 - 1) * (d2_out$man030C_man030T - (d1_out$d1p1t030C * d1_out$d1p1t030T)),
# ----------- #
## 10
# 201 - 003
COVAR201_003 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man201_man003b - d2_out$man201_man003) +
g3 * (g3 - 1) * (d2_out$man201_man003 - (d1_out$d1p1t201 * d1_out$d1p1t003)),
# 201 - 012
COVAR201_012 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man201_man012b - d2_out$man201_man012) +
g3 * (g3 - 1) * (d2_out$man201_man012 - (d1_out$d1p1t201 * d1_out$d1p1t012)),
# COVARIANCE DIFFER WITH WASSERMAN AND FAUST
# 1994 (pp. 582-583)
# (i.e. TRIADS computer program of Walker and Wasserman, 1987)
# 201-102: we use the equation of Leinhardt and Holland (1975)
# P0(u,v): 9m^(3)n^(3); P1(u,v): 2m^(2)m^(2)(m+4-n)
# results: W-F: -18.4; here: 7.06
# 300-030T: original: -0.01; here:0.011
# 201 - 102
COVAR201_102 <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man201_man102b - d2_out$man201_man102) +
g3 * (g3 - 1) * (d2_out$man201_man102 - (d1_out$d1p1t201 * d1_out$d1p1t102)),
# 201 - 021D
COVAR201_021D <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man201_man021Db - d2_out$man201_man021D) +
g3 * (g3 - 1) * (d2_out$man201_man021D - (d1_out$d1p1t201 * d1_out$d1p1t021D)),
# 201 - 021U
COVAR201_021U <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man201_man021Ub - d2_out$man201_man021U) +
g3 * (g3 - 1) * (d2_out$man201_man021U - (d1_out$d1p1t201 * d1_out$d1p1t021U)),
# 201 - 021C
COVAR201_021C <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man201_man021Cb - d2_out$man201_man021C) +
g3 * (g3 - 1) * (d2_out$man201_man021C - (d1_out$d1p1t201 * d1_out$d1p1t021C)),
# 201 - 111D
COVAR201_111D <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man201_man111Db - d2_out$man201_man111D) +
g3 * (g3 - 1) * (d2_out$man201_man111D - (d1_out$d1p1t201 * d1_out$d1p1t111D)),
# 201 - 111U
COVAR201_111U <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man201_man111Ub - d2_out$man201_man111U) +
g3 * (g3 - 1) * (d2_out$man201_man111U - (d1_out$d1p1t201 * d1_out$d1p1t111U)),
# 201 - 030T
COVAR201_030T <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man201_man030Tb - d2_out$man201_man030T) +
g3 * (g3 - 1) * (d2_out$man201_man030T - (d1_out$d1p1t201 * d1_out$d1p1t030T)),
# 201 - 030C
COVAR201_030C <- (-g3 * d1_out$d1p1t201 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man201_man030Cb - d2_out$man201_man030C) +
g3 * (g3 - 1) * (d2_out$man201_man030C - (d1_out$d1p1t201 * d1_out$d1p1t030C)),
# ----------- #
## 11
# 120D - 003
COVAR120D_003 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120D_man003b - d2_out$man120D_man003) +
g3 * (g3 - 1) * (d2_out$man120D_man003 - (d1_out$d1p1t120D * d1_out$d1p1t003)),
# 120D - 012
COVAR120D_012 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120D_man012b - d2_out$man120D_man012) +
g3 * (g3 - 1) * (d2_out$man120D_man012 - (d1_out$d1p1t120D * d1_out$d1p1t012)),
# 120D - 102
COVAR120D_102 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120D_man102b - d2_out$man120D_man102) +
g3 * (g3 - 1) * (d2_out$man120D_man102 - (d1_out$d1p1t120D * d1_out$d1p1t102)),
# 120D - 021D
COVAR120D_021D <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Db - d2_out$man120D_man021D) +
g3 * (g3 - 1) * (d2_out$man120D_man021D - (d1_out$d1p1t120D * d1_out$d1p1t021D)),
# 120D - 021U
COVAR120D_021U <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Ub - d2_out$man120D_man021U) +
g3 * (g3 - 1) * (d2_out$man120D_man021U - (d1_out$d1p1t120D * d1_out$d1p1t021U)),
# 120D - 021C
COVAR120D_021C <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120D_man021Cb - d2_out$man120D_man021C) +
g3 * (g3 - 1) * (d2_out$man120D_man021C - (d1_out$d1p1t120D * d1_out$d1p1t021C)),
# 120D - 111D
COVAR120D_111D <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120D_man111Db - d2_out$man120D_man111D) +
g3 * (g3 - 1) * (d2_out$man120D_man111D - (d1_out$d1p1t120D * d1_out$d1p1t111D)),
# 120D - 111U
COVAR120D_111U <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120D_man111Ub - d2_out$man120D_man111U) +
g3 * (g3 - 1) * (d2_out$man120D_man111U - (d1_out$d1p1t120D * d1_out$d1p1t111U)),
# 120D - 030T
COVAR120D_030T <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120D_man030Tb - d2_out$man120D_man030T) +
g3 * (g3 - 1) * (d2_out$man120D_man030T - (d1_out$d1p1t120D * d1_out$d1p1t030T)),
# 120D - 030C
COVAR120D_030C <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120D_man030Cb - d2_out$man120D_man030C) +
g3 * (g3 - 1) * (d2_out$man120D_man030C - (d1_out$d1p1t120D * d1_out$d1p1t030C)),
# 120D - 201
COVAR120D_201 <- (-g3 * d1_out$d1p1t120D * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120D_man201b - d2_out$man120D_man201) +
g3 * (g3 - 1) * (d2_out$man120D_man201 - (d1_out$d1p1t120D * d1_out$d1p1t201)),
# ----------- #
## 12
# 120U - 003
COVAR120U_003 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120U_man003b - d2_out$man120U_man003) +
g3 * (g3 - 1) * (d2_out$man120U_man003 - (d1_out$d1p1t120U * d1_out$d1p1t003)),
# 120U - 012
COVAR120U_012 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120U_man012b - d2_out$man120U_man012) +
g3 * (g3 - 1) * (d2_out$man120U_man012 - (d1_out$d1p1t120U * d1_out$d1p1t012)),
# 120U - 102
COVAR120U_102 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120U_man102b - d2_out$man120U_man102) +
g3 * (g3 - 1) * (d2_out$man120U_man102 - (d1_out$d1p1t120U * d1_out$d1p1t102)),
# 120U - 021D
COVAR120U_021D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Db - d2_out$man120U_man021D) +
g3 * (g3 - 1) * (d2_out$man120U_man021D - (d1_out$d1p1t120U * d1_out$d1p1t021D)),
# 120U - 021U
COVAR120U_021U <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Ub - d2_out$man120U_man021U) +
g3 * (g3 - 1) * (d2_out$man120U_man021U - (d1_out$d1p1t120U * d1_out$d1p1t021U)),
# 120U - 021C
COVAR120U_021C <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120U_man021Cb - d2_out$man120U_man021C) +
g3 * (g3 - 1) * (d2_out$man120U_man021C - (d1_out$d1p1t120U * d1_out$d1p1t021C)),
# 120U - 111D
COVAR120U_111D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120U_man111Db - d2_out$man120U_man111D) +
g3 * (g3 - 1) * (d2_out$man120U_man111D - (d1_out$d1p1t120U * d1_out$d1p1t111D)),
# 120U - 111U
COVAR120U_111U <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120U_man111Ub - d2_out$man120U_man111U) +
g3 * (g3 - 1) * (d2_out$man120U_man111U - (d1_out$d1p1t120U * d1_out$d1p1t111U)),
# 120U - 030T
COVAR120U_030T <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120U_man030Tb - d2_out$man120U_man030T) +
g3 * (g3 - 1) * (d2_out$man120U_man030T - (d1_out$d1p1t120U * d1_out$d1p1t030T)),
# 120U - 030C
COVAR120U_030C <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120U_man030Cb - d2_out$man120U_man030C) +
g3 * (g3 - 1) * (d2_out$man120U_man030C - (d1_out$d1p1t120U * d1_out$d1p1t030C)),
# 120U - 201
COVAR120U_201 <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120U_man201b - d2_out$man120U_man201) +
g3 * (g3 - 1) * (d2_out$man120U_man201 - (d1_out$d1p1t120U * d1_out$d1p1t201)),
# 120U - 120D
COVAR120U_120D <- (-g3 * d1_out$d1p1t120U * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man120U_man120Db - d2_out$man120U_man120D) +
g3 * (g3 - 1) * (d2_out$man120U_man120D - (d1_out$d1p1t120U * d1_out$d1p1t120D)),
# ----------- #
## 13
# 120C - 003
COVAR120C_003 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man120C_man003b - d2_out$man120C_man003) +
g3 * (g3 - 1) * (d2_out$man120C_man003 - (d1_out$d1p1t120C * d1_out$d1p1t003)),
# 120C - 012
COVAR120C_012 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man120C_man012b - d2_out$man120C_man012) +
g3 * (g3 - 1) * (d2_out$man120C_man012 - (d1_out$d1p1t120C * d1_out$d1p1t012)),
# 120C - 102
COVAR120C_102 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man120C_man102b - d2_out$man120C_man102) +
g3 * (g3 - 1) * (d2_out$man120C_man102 - (d1_out$d1p1t120C * d1_out$d1p1t102)),
# 120C - 021D
COVAR120C_021D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Db - d2_out$man120C_man021D) +
g3 * (g3 - 1) * (d2_out$man120C_man021D - (d1_out$d1p1t120C * d1_out$d1p1t021D)),
# 120C - 021U
COVAR120C_021U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Ub - d2_out$man120C_man021U) +
g3 * (g3 - 1) * (d2_out$man120C_man021U - (d1_out$d1p1t120C * d1_out$d1p1t021U)),
# 120C - 021C
COVAR120C_021C <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man120C_man021Cb - d2_out$man120C_man021C) +
g3 * (g3 - 1) * (d2_out$man120C_man021C - (d1_out$d1p1t120C * d1_out$d1p1t021C)),
# 120C - 111D
COVAR120C_111D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man120C_man111Db - d2_out$man120C_man111D) +
g3 * (g3 - 1) * (d2_out$man120C_man111D - (d1_out$d1p1t120C * d1_out$d1p1t111D)),
# 120C - 111U
COVAR120C_111U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man120C_man111Ub - d2_out$man120C_man111U) +
g3 * (g3 - 1) * (d2_out$man120C_man111U - (d1_out$d1p1t120C * d1_out$d1p1t111U)),
# 120C - 030T
COVAR120C_030T <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man120C_man030Tb - d2_out$man120C_man030T) +
g3 * (g3 - 1) * (d2_out$man120C_man030T - (d1_out$d1p1t120C * d1_out$d1p1t030T)),
# 120C - 030C
COVAR120C_030C <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man120C_man030Cb - d2_out$man120C_man030C) +
g3 * (g3 - 1) * (d2_out$man120C_man030C - (d1_out$d1p1t120C * d1_out$d1p1t030C)),
# 120C - 201
COVAR120C_201 <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man120C_man201b - d2_out$man120C_man201) +
g3 * (g3 - 1) * (d2_out$man120C_man201 - (d1_out$d1p1t120C * d1_out$d1p1t201)),
# 120C - 120D
COVAR120C_120D <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man120C_man120Db - d2_out$man120C_man120D) +
g3 * (g3 - 1) * (d2_out$man120C_man120D - (d1_out$d1p1t120C * d1_out$d1p1t120D)),
# 120C - 120U
COVAR120C_120U <- (-g3 * d1_out$d1p1t120C * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man120C_man120Ub - d2_out$man120C_man120U) +
g3 * (g3 - 1) * (d2_out$man120C_man120U - (d1_out$d1p1t120C * d1_out$d1p1t120U)),
# ----------- #
## 14
# 210 - 003
COVAR210_003 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man210_man003b - d2_out$man210_man003) +
g3 * (g3 - 1) * (d2_out$man210_man003 - (d1_out$d1p1t210 * d1_out$d1p1t003)),
# 210 - 012
COVAR210_012 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man210_man012b - d2_out$man210_man012) +
g3 * (g3 - 1) * (d2_out$man210_man012 - (d1_out$d1p1t210 * d1_out$d1p1t012)),
# 210 - 102
COVAR210_102 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man210_man102b - d2_out$man210_man102) +
g3 * (g3 - 1) * (d2_out$man210_man102 - (d1_out$d1p1t210 * d1_out$d1p1t102)),
# 210 - 021D
COVAR210_021D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man210_man021Db - d2_out$man210_man021D) +
g3 * (g3 - 1) * (d2_out$man210_man021D - (d1_out$d1p1t210 * d1_out$d1p1t021D)),
# 210 - 021U
COVAR210_021U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man210_man021Ub - d2_out$man210_man021U) +
g3 * (g3 - 1) * (d2_out$man210_man021U - (d1_out$d1p1t210 * d1_out$d1p1t021U)),
# 210 - 021C
COVAR210_021C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man210_man021Cb - d2_out$man210_man021C) +
g3 * (g3 - 1) * (d2_out$man210_man021C - (d1_out$d1p1t210 * d1_out$d1p1t021C)),
# 210 - 111D
COVAR210_111D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man210_man111Db - d2_out$man210_man111D) +
g3 * (g3 - 1) * (d2_out$man210_man111D - (d1_out$d1p1t210 * d1_out$d1p1t111D)),
# 210 - 111U
COVAR210_111U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man210_man111Ub - d2_out$man210_man111U) +
g3 * (g3 - 1) * (d2_out$man210_man111U - (d1_out$d1p1t210 * d1_out$d1p1t111U)),
# 210 - 030T
COVAR210_030T <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man210_man030Tb - d2_out$man210_man030T) +
g3 * (g3 - 1) * (d2_out$man210_man030T - (d1_out$d1p1t210 * d1_out$d1p1t030T)),
# 210 - 030C
COVAR210_030C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man210_man030Cb - d2_out$man210_man030C) +
g3 * (g3 - 1) * (d2_out$man210_man030C - (d1_out$d1p1t210 * d1_out$d1p1t030C)),
# 210 - 201
COVAR210_201 <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man210_man201b - d2_out$man210_man201) +
g3 * (g3 - 1) * (d2_out$man210_man201 - (d1_out$d1p1t210 * d1_out$d1p1t201)),
# 210 - 120D
COVAR210_120D <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man210_man120Db - d2_out$man210_man120D) +
g3 * (g3 - 1) * (d2_out$man210_man120D - (d1_out$d1p1t210 * d1_out$d1p1t120D)),
# 210 - 120U
COVAR210_120U <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man210_man120Ub - d2_out$man210_man120U) +
g3 * (g3 - 1) * (d2_out$man210_man120U - (d1_out$d1p1t210 * d1_out$d1p1t120U)),
# 210 - 120C
COVAR210_120C <- (-g3 * d1_out$d1p1t210 * d1_out$d1p1t120C) +
3 * (g - 3) * g3 * (d3_out$man210_man120Cb - d2_out$man210_man120C) +
g3 * (g3 - 1) * (d2_out$man210_man120C - (d1_out$d1p1t210 * d1_out$d1p1t120C)),
# ----------- #
## 15
# 300 - 003
COVAR300_003 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t003) +
3 * (g - 3) * g3 * (d3_out$man300_man003b - d2_out$man300_man003) +
g3 * (g3 - 1) * (d2_out$man300_man003 - (d1_out$d1p1t300 * d1_out$d1p1t003)),
# 300 - 012
COVAR300_012 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t012) +
3 * (g - 3) * g3 * (d3_out$man300_man012b - d2_out$man300_man012) +
g3 * (g3 - 1) * (d2_out$man300_man012 - (d1_out$d1p1t300 * d1_out$d1p1t012)),
# 300 - 102
COVAR300_102 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t102) +
3 * (g - 3) * g3 * (d3_out$man300_man102b - d2_out$man300_man102) +
g3 * (g3 - 1) * (d2_out$man300_man102 - (d1_out$d1p1t300 * d1_out$d1p1t102)),
# 300 - 021D
COVAR300_021D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021D) +
3 * (g - 3) * g3 * (d3_out$man300_man021Db - d2_out$man300_man021D) +
g3 * (g3 - 1) * (d2_out$man300_man021D - (d1_out$d1p1t300 * d1_out$d1p1t021D)),
# 300 - 021U
COVAR300_021U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021U) +
3 * (g - 3) * g3 * (d3_out$man300_man021Ub - d2_out$man300_man021U) +
g3 * (g3 - 1) * (d2_out$man300_man021U - (d1_out$d1p1t300 * d1_out$d1p1t021U)),
# 300 - 021C
COVAR300_021C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t021C) +
3 * (g - 3) * g3 * (d3_out$man300_man021Cb - d2_out$man300_man021C) +
g3 * (g3 - 1) * (d2_out$man300_man021C - (d1_out$d1p1t300 * d1_out$d1p1t021C)),
# 300 - 111D
COVAR300_111D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t111D) +
3 * (g - 3) * g3 * (d3_out$man300_man111Db - d2_out$man300_man111D) +
g3 * (g3 - 1) * (d2_out$man300_man111D - (d1_out$d1p1t300 * d1_out$d1p1t111D)),
# 300 - 111U
COVAR300_111U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t111U) +
3 * (g - 3) * g3 * (d3_out$man300_man111Ub - d2_out$man300_man111U) +
g3 * (g3 - 1) * (d2_out$man300_man111U - (d1_out$d1p1t300 * d1_out$d1p1t111U)),
# COVARIANCE DIFFER WITH WASSERMAN AND FAUST
# 1994 (pp. 582-583)
# (i.e. TRIADS computer program of Walker and Wasserman, 1987)
# 201-102: we use the equation of Leinhardt and Holland (1975)
# P0(u,v): 9m^(3)n^(3); P1(u,v): 2m^(2)m^(2)(m+4-n)
# results: W-F: -18.4; here: 7.06
# 300-030T: original: -0.01; here:0.011
# 300 - 030T
COVAR300_030T <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t030T) +
3 * (g - 3) * g3 * (d3_out$man300_man030Tb - d2_out$man300_man030T) +
g3 * (g3 - 1) * (d2_out$man300_man030T - (d1_out$d1p1t300 * d1_out$d1p1t030T)),
# 300 - 030C
COVAR300_030C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t030C) +
3 * (g - 3) * g3 * (d3_out$man300_man030Cb - d2_out$man300_man030C) +
g3 * (g3 - 1) * (d2_out$man300_man030C - (d1_out$d1p1t300 * d1_out$d1p1t030C)),
# 300 - 201
COVAR300_201 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t201) +
3 * (g - 3) * g3 * (d3_out$man300_man201b - d2_out$man300_man201) +
g3 * (g3 - 1) * (d2_out$man300_man201 - (d1_out$d1p1t300 * d1_out$d1p1t201)),
# 300 - 120D
COVAR300_120D <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120D) +
3 * (g - 3) * g3 * (d3_out$man300_man120Db - d2_out$man300_man120D) +
g3 * (g3 - 1) * (d2_out$man300_man120D - (d1_out$d1p1t300 * d1_out$d1p1t120D)),
# 300 - 120U
COVAR300_120U <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120U) +
3 * (g - 3) * g3 * (d3_out$man300_man120Ub - d2_out$man300_man120U) +
g3 * (g3 - 1) * (d2_out$man300_man120U - (d1_out$d1p1t300 * d1_out$d1p1t120U)),
# 300 - 120C
COVAR300_120C <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t120C) +
3 * (g - 3) * g3 * (d3_out$man300_man120Cb - d2_out$man300_man120C) +
g3 * (g3 - 1) * (d2_out$man300_man120C - (d1_out$d1p1t300 * d1_out$d1p1t120C)),
# 300 - 210
COVAR300_210 <- (-g3 * d1_out$d1p1t300 * d1_out$d1p1t210) +
3 * (g - 3) * g3 * (d3_out$man300_man210b - d2_out$man300_man210) +
g3 * (g3 - 1) * (d2_out$man300_man210 - (d1_out$d1p1t300 * d1_out$d1p1t210))
)
# triads census ----
# E <- ifelse((A + Matrix::t(A)) > 0, 1, 0) # arcs to edges
E <- A + Matrix::t(A)
E[E > 0] <- 1
# Eb <- ifelse(E == 0, 1, 0)
Eb <- E
Eb[Eb == 0] <- -999
Eb[Eb != -999] <- 0
Eb[Eb == -999] <- 1
Matrix::diag(Eb) <- 0
# M <- ifelse((A + t(A)) > 1, 1, 0) # mutual edges only
M <- A + Matrix::t(A)
M[M == 1] <- -999
M[M > 1] <- 1
M[M < 1] <- 0
C <- A - M # asymmetric arcs only
t201 <- sum(M %*% M * Eb) # null dyad in a 201 triad
t021D <- sum(t(C) %*% C * Eb) # null dyad in an 021D triad
t021U <- sum(C %*% t(C) * Eb) # null dyad in an 021U triad
triad <- c(
"003" = sum(diag(Eb %*% Eb %*% Eb)) / 6,
"012" = sum((Eb %*% Eb) * (C + t(C))) / 2,
"102" = sum((Eb %*% Eb * M)) / 2,
"021D" = sum(t(C) %*% C * Eb) / 2,
"021U" = sum(C %*% t(C) * Eb) / 2,
"021C" = sum(C %*% C * Eb),
"111D" = (sum(A %*% t(A) * Eb) - t201 - t021U) / 2,
"111U" = (sum(t(A) %*% A * Eb) - t201 - t021D) / 2,
"030T" = sum((C %*% C) * C),
"030C" = sum(diag(C %*% C %*% C)) / 3,
"201" = sum(M %*% M * Eb) / 2,
"120D" = sum(t(C) %*% C * M) / 2,
"120U" = sum(C %*% t(C) * M) / 2,
"120C" = sum(C %*% C * M),
"210" = sum(M %*% M * (C + t(C))) / 2,
"300" = sum(diag(M %*% M %*% M)) / 6
)
label <- c(
"003", "012", "102", "021D", "021U",
"021C", "111D", "111U", "030T", "030C",
"201", "120D", "120U", "120C", "210", "300"
)
results <- as.data.frame(cbind(
label = as.character(label),
OBS = as.numeric(triad),
EXP = as.numeric(EXP),
VAR = as.numeric(VAR)
))
results$OBS <- round(as.numeric(as.character(results$OBS)), 0)
results$EXP <- round(as.numeric(as.character(results$EXP)), 3)
results$VAR <- round(as.numeric(as.character(results$VAR)), 3)
results$STD <- sqrt(results$VAR)
results$STD <- round(as.numeric(as.character(results$STD)), 3)
mempty <- matrix(0, nrow = 16, ncol = 16)
mindex <- matrix(1:256, nrow = 16, ncol = 16, byrow = T)
mempty[mindex[upper.tri(mindex)]] <- COVAR
diag(mempty) <- VAR # VARIANCE IN THE DIAGONAL
rownames(mempty) <- label
colnames(mempty) <- label
mempty <- round(mempty, 3)
z <- (sum(results$OBS) - sum(results$EXP)) / (sqrt(sum(results$VAR) + (2 * (sum(COVAR)))))
p <- 2 * pnorm(-abs(z))
res <- c(z = z, p = p)
res <- round(res, 3)
if (ztest & covar) {
results$Z <- (results$OBS - results$EXP) / results$STD
results$Z <- as.numeric(as.character(results$Z))
results$Z <- round(results$Z, 3)
results$P <- 2 * pnorm(-abs(results$Z))
results$P <- as.numeric(as.character(results$P))
results$P <- round(results$P, 3)
newlist <- list(results = results, z_test = res, covariance = mempty)
return(newlist)
}
if (covar) {
# FIXME: EXPERIMENTAL version. Use with caution... the covar 201-102 and 300-030T are under review
newlist <- list(results = results, covariance = mempty)
return(newlist)
}
if (ztest) {
results$Z <- (results$OBS - results$EXP) / results$STD
results$Z <- as.numeric(as.character(results$Z))
results$Z <- round(results$Z, 3)
results$P <- 2 * pnorm(-abs(results$Z))
results$P <- as.numeric(as.character(results$P))
results$P <- round(results$P, 3)
newlist <- list(results = results, z_test = res)
return(newlist)
} else {
return(results)
}
}
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