Johannes Titz 04/27/2023
Last updated: 2024-07-04
This analysis accompanies the paper: Titz, J. The Relationship Between the Phi Coefficient and the Unidimensionality Index H: Improving Psychological Scaling From the Ground Up
I use librarian to manage dependencies, so please install it if you do not have it yet (uncomment the first line).
# install.packages("librarian")
library(librarian)
shelf(partitions, pbapply, tidyverse, psych, johannes-titz/zysno, xtable,
Matrix, plot.matrix, simpleCache, mokken, dplyr)
setCacheDir("cache")
# use multiple cores
cores <- round(parallel::detectCores() * 0.8)
# create vectors from phi matrix
vec_from_tbl <- function(a, b, c, d) {
mtrx <- matrix(c(a, b, c, d), ncol = 2)
v1 <- rep(c(0, 1), colSums(mtrx))
v2 <- rep(c(0, 1), rowSums(mtrx))
cbind(v1, v2)
}
# create vectors from phi matrix
vectors_from_crosstable <- function(freq_ct, n_cat = sqrt(length(freq_ct))) {
m <- matrix(freq_ct, ncol = n_cat)
#graph <- Kmisc::melt_(m)
graph <- reshape2::melt(m)
v1 <- rep(graph[,1], graph[,3])
v2 <- rep(graph[,2], graph[,3])
m2 <- cbind(v1, v2)
m2
}
# for trichotomous/polytomous
stats <- function(freq_ct) {
if (sum(freq_ct != 0) == 1) return(cbind(H = NA, r = NA))
m <- vectors_from_crosstable(freq_ct)
if (any(var(m) == 0)) return(cbind(H = NA, r = NA))
# avoid calculating H
if (cor(m)[1,2] < 0) return(cbind(H = NA, r = NA))
H <- suppressWarnings(mokken::coefH(m, nice.output = F, results = F))
cbind(H = as.numeric(H$H), phi = cor(m)[1,2])
}n <- 200
# all combos, even the ones where phi is NA
grid <- as.matrix(t(compositions(n = n, m = 4)))
colnames(grid) <- paste0("Var", 1:4)
# rowwise does not work with the following code, I do not know why
# multiple cores seem to give no advantage, so is not used (argument cl)
# we cache this manually as it takes some time
grid <- xfun::cache_rds({
grid <- as_tibble(grid) %>%
rowwise() %>%
mutate(errors = Var3,
expected_errors = (Var1 + Var3) * (Var3 + Var4) / (Var1+ Var2+ Var3+ Var4),
h = 1 - errors / expected_errors)
phi <- pbapply(grid[, 1:4], 1, function(x) phi(x, digits = 4))
cbind(grid, phi)
}, file = "sim1", rerun = F)
grid <- as_tibble(grid)grid_unidim <- grid %>%
filter(Var3 == 0) # only unidimensional
grid_row <- grid %>%
filter(phi >= 0, Var2 >= Var3) # error cell is bp <- ggplot(grid_row, aes(phi, h)) +
#geom_point(alpha = 0.05) +
geom_hex(bins = 100, show.legend = FALSE) +
theme_classic() +
scale_x_continuous("Phi", breaks = seq(0, 1, 0.1)) +
scale_y_continuous("H", breaks = seq(0, 1, 0.1) ) +
coord_fixed()
p
pdf("plots/phiH.pdf", width = 5, height = 5)
p
dev.off()## png
## 2
Note that the code is not very efficient and can likely be improved considerably.
The following two chunks are highly resource-intensive, demanding substantial RAM and time to recreate or load due to the objects’ uncompressed size of approximately 20 GB. Consequently, they are not executed by default. To recreate or load these chunks, adjust the chunk settings as needed. Ensure you have sufficient RAM available (e.g., on a server) before running them. If you prefer not to recreate the data, you can proceed to the subsequent chunk, which loads the final relevant statistics from the cache (default).
n <- 30
# all combos, even the ones where phi is NA
simpleCache("grid_poly", as.data.frame(as.matrix(compositions(n = n, m = 9))))simpleCache("poly", pbmcapply::pbmclapply(grid_poly, stats, mc.cores = 2))
poly <- data.frame(matrix(unlist(poly2), ncol = 2, byrow = T))
names(poly) <- c("H", "phi")simpleCache("grid_row_tri", poly[poly$phi >= 0, ])p <- ggplot(grid_row_tri, aes(phi, H)) +
#geom_point(alpha = 0.05) +
geom_hex(bins = 100, show.legend = FALSE) +
theme_classic() +
scale_x_continuous("Phi", breaks = seq(0, 1, 0.1)) +
scale_y_continuous("H", breaks = seq(0, 1, 0.1) ) +
coord_fixed()
p
pdf("plots/phiHPoly.pdf", width = 5, height = 5)
p
dev.off()## png
## 2
It is not hard to create unidimensional model that results in a perfect bifactor solution with factor analysis (for Pearson Correlation):
m1 <- vec_from_tbl(10, 80, 0, 10)
m2 <- vec_from_tbl(11, 78, 0, 11)
m3 <- vec_from_tbl(15, 70, 0, 15)
d <- cbind(m1, m2, m3[, -2])
# check that it is really unidimensional
zysnotize(d)## $error_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 0 0 0
## [2,] 0 0 0 0 0
## [3,] 0 0 0 0 0
## [4,] 0 0 0 0 0
## [5,] 0 0 0 0 0
##
## $expected_error_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 81.00 81.00 88.1100 88.1100 114.7500
## [2,] 81.00 81.00 88.1100 88.1100 114.7500
## [3,] 88.11 88.11 95.8441 95.8441 124.8225
## [4,] 88.11 88.11 95.8441 95.8441 124.8225
## [5,] 114.75 114.75 124.8225 124.8225 162.5625
##
## $scalability_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 1 1 1 1
## [2,] 1 1 1 1 1
## [3,] 1 1 1 1 1
## [4,] 1 1 1 1 1
## [5,] 1 1 1 1 1
##
## $scalability
## [1] 1
##
## $sum_errors
## [1] 0
##
## $sum_expected_errors
## [1] 2016.858
ncol <- ncol(d)
colnames(d) <- paste("Item", seq(ncol(d)))
f1 <- factanal(d, 1)
f2 <- factanal(d, 2)
f1$loadings##
## Loadings:
## Factor1
## Item 1 0.950
## Item 2 0.119
## Item 3 0.997
## Item 4 0.125
## Item 5 0.839
##
## Factor1
## SS loadings 2.632
## Proportion Var 0.526
f2$loadings##
## Loadings:
## Factor1 Factor2
## Item 1 0.950
## Item 2 0.946
## Item 3 0.997
## Item 4 0.993
## Item 5 0.837
##
## Factor1 Factor2
## SS loadings 2.617 1.886
## Proportion Var 0.523 0.377
## Cumulative Var 0.523 0.900
print(xtable::xtable(cbind(f1$loadings, f2$loadings), label = "tab:fa1b",
caption = "Cross tables of five unidimensional items"),
file = "tables/fa1b.tex", booktabs = TRUE)
combs <- Map(function(x, y) table(d[, x], d[, y]),
rep(1:ncol, each = ncol), rep(1:ncol, ncol))
lower <- seq(1, 21, 5)
upper <- seq(5, 25, 5)
res <- Map(function(x, y) Reduce(cbind, combs[x:y]),
lower, upper)
res2 <- Reduce(rbind, res)
colnames(res2) <- paste0("$", rep(1:5, each = 2), "_", c(0, 1), "$")
rownames(res2) <- paste0("$", rep(1:5, each = 2), "_", c(0, 1), "$")
res2## $1_0$ $1_1$ $2_0$ $2_1$ $3_0$ $3_1$ $4_0$ $4_1$ $5_0$ $5_1$
## $1_0$ 90 0 10 80 89 1 11 79 85 5
## $1_1$ 0 10 0 10 0 10 0 10 0 10
## $2_0$ 10 0 10 0 10 0 10 0 10 0
## $2_1$ 80 10 0 90 79 11 1 89 75 15
## $3_0$ 89 0 10 79 89 0 11 78 85 4
## $3_1$ 1 10 0 11 0 11 0 11 0 11
## $4_0$ 11 0 10 1 11 0 11 0 11 0
## $4_1$ 79 10 0 89 78 11 0 89 74 15
## $5_0$ 85 0 10 75 85 0 11 74 85 0
## $5_1$ 5 10 0 15 4 11 0 15 0 15
print(xtable::xtable(res2, label = "tab:fa1",
caption = "Example of factor analysis for five unidimensional items"),
booktabs = TRUE,
file = "tables/fa1.tex",
sanitize.text.function = function(x) {x})Load data
dorig <- read.csv2("data.csv")
d <- dorig %>%
dplyr::select(id, item_nmbr, correctness) %>%
pivot_wider(names_from = item_nmbr, values_from = correctness)Find order, calculate score, show df
``` r
o <- order(colMeans(d[,-1]), decreasing = TRUE)
d <- d[, c(1, o+1)]
d$score <- rowSums(d[, -1])
df <- as.data.frame(d)
df[order(df$score), ]
## id 1 4 3 2 5 score
## 13 13 FALSE FALSE FALSE FALSE FALSE 0
## 22 22 FALSE FALSE FALSE FALSE FALSE 0
## 46 46 FALSE FALSE FALSE FALSE FALSE 0
## 51 51 FALSE FALSE FALSE FALSE FALSE 0
## 54 54 FALSE FALSE FALSE FALSE FALSE 0
## 57 57 FALSE FALSE FALSE FALSE FALSE 0
## 64 64 FALSE FALSE FALSE FALSE FALSE 0
## 66 66 FALSE FALSE FALSE FALSE FALSE 0
## 73 73 FALSE FALSE FALSE FALSE FALSE 0
## 80 80 FALSE FALSE FALSE FALSE FALSE 0
## 85 85 FALSE FALSE FALSE FALSE FALSE 0
## 94 94 FALSE FALSE FALSE FALSE FALSE 0
## 97 97 FALSE FALSE FALSE FALSE FALSE 0
## 99 99 FALSE FALSE FALSE FALSE FALSE 0
## 101 101 FALSE FALSE FALSE FALSE FALSE 0
## 104 104 FALSE FALSE FALSE FALSE FALSE 0
## 116 116 FALSE FALSE FALSE FALSE FALSE 0
## 121 121 FALSE FALSE FALSE FALSE FALSE 0
## 123 123 FALSE FALSE FALSE FALSE FALSE 0
## 127 127 FALSE FALSE FALSE FALSE FALSE 0
## 17 17 TRUE FALSE FALSE FALSE FALSE 1
## 40 40 TRUE FALSE FALSE FALSE FALSE 1
## 41 41 FALSE FALSE FALSE FALSE TRUE 1
## 70 70 TRUE FALSE FALSE FALSE FALSE 1
## 100 100 TRUE FALSE FALSE FALSE FALSE 1
## 108 108 FALSE FALSE TRUE FALSE FALSE 1
## 2 2 TRUE TRUE FALSE FALSE FALSE 2
## 5 5 TRUE TRUE FALSE FALSE FALSE 2
## 7 7 TRUE TRUE FALSE FALSE FALSE 2
## 8 8 FALSE FALSE TRUE TRUE FALSE 2
## 9 9 TRUE TRUE FALSE FALSE FALSE 2
## 15 15 TRUE TRUE FALSE FALSE FALSE 2
## 18 18 TRUE TRUE FALSE FALSE FALSE 2
## 26 26 TRUE TRUE FALSE FALSE FALSE 2
## 29 29 TRUE TRUE FALSE FALSE FALSE 2
## 34 34 TRUE TRUE FALSE FALSE FALSE 2
## 37 37 TRUE TRUE FALSE FALSE FALSE 2
## 45 45 TRUE TRUE FALSE FALSE FALSE 2
## 56 56 TRUE TRUE FALSE FALSE FALSE 2
## 61 61 TRUE TRUE FALSE FALSE FALSE 2
## 65 65 TRUE TRUE FALSE FALSE FALSE 2
## 74 74 TRUE TRUE FALSE FALSE FALSE 2
## 76 76 FALSE FALSE TRUE TRUE FALSE 2
## 81 81 TRUE TRUE FALSE FALSE FALSE 2
## 83 83 TRUE TRUE FALSE FALSE FALSE 2
## 86 86 TRUE TRUE FALSE FALSE FALSE 2
## 89 89 TRUE FALSE TRUE FALSE FALSE 2
## 93 93 TRUE TRUE FALSE FALSE FALSE 2
## 107 107 TRUE TRUE FALSE FALSE FALSE 2
## 109 109 TRUE TRUE FALSE FALSE FALSE 2
## 112 112 TRUE TRUE FALSE FALSE FALSE 2
## 113 113 TRUE TRUE FALSE FALSE FALSE 2
## 120 120 TRUE FALSE FALSE TRUE FALSE 2
## 124 124 TRUE TRUE FALSE FALSE FALSE 2
## 30 30 FALSE TRUE TRUE TRUE FALSE 3
## 36 36 TRUE TRUE FALSE TRUE FALSE 3
## 47 47 TRUE TRUE TRUE FALSE FALSE 3
## 117 117 TRUE TRUE FALSE FALSE TRUE 3
## 129 129 TRUE TRUE TRUE FALSE FALSE 3
## 1 1 TRUE TRUE TRUE TRUE FALSE 4
## 4 4 TRUE TRUE TRUE TRUE FALSE 4
## 10 10 TRUE TRUE TRUE TRUE FALSE 4
## 12 12 TRUE TRUE TRUE TRUE FALSE 4
## 14 14 TRUE TRUE TRUE TRUE FALSE 4
## 20 20 TRUE TRUE TRUE TRUE FALSE 4
## 23 23 TRUE TRUE TRUE TRUE FALSE 4
## 24 24 TRUE TRUE TRUE TRUE FALSE 4
## 25 25 TRUE TRUE TRUE TRUE FALSE 4
## 31 31 TRUE TRUE TRUE TRUE FALSE 4
## 32 32 TRUE TRUE TRUE TRUE FALSE 4
## 33 33 TRUE TRUE TRUE TRUE FALSE 4
## 35 35 TRUE TRUE TRUE TRUE FALSE 4
## 42 42 TRUE TRUE TRUE TRUE FALSE 4
## 43 43 TRUE TRUE TRUE TRUE FALSE 4
## 49 49 TRUE TRUE TRUE TRUE FALSE 4
## 53 53 TRUE TRUE TRUE TRUE FALSE 4
## 55 55 TRUE TRUE TRUE TRUE FALSE 4
## 58 58 TRUE TRUE TRUE TRUE FALSE 4
## 62 62 TRUE TRUE TRUE TRUE FALSE 4
## 63 63 TRUE TRUE TRUE TRUE FALSE 4
## 68 68 TRUE TRUE TRUE TRUE FALSE 4
## 79 79 TRUE TRUE TRUE TRUE FALSE 4
## 84 84 TRUE TRUE TRUE TRUE FALSE 4
## 87 87 TRUE TRUE TRUE TRUE FALSE 4
## 95 95 TRUE TRUE TRUE TRUE FALSE 4
## 98 98 TRUE TRUE TRUE TRUE FALSE 4
## 102 102 TRUE TRUE TRUE TRUE FALSE 4
## 103 103 TRUE TRUE TRUE TRUE FALSE 4
## 106 106 TRUE TRUE TRUE TRUE FALSE 4
## 110 110 TRUE TRUE TRUE TRUE FALSE 4
## 118 118 TRUE TRUE TRUE TRUE FALSE 4
## 125 125 TRUE TRUE TRUE TRUE FALSE 4
## 128 128 TRUE TRUE TRUE TRUE FALSE 4
## 131 131 TRUE TRUE TRUE TRUE FALSE 4
## 132 132 TRUE TRUE TRUE TRUE FALSE 4
## 3 3 TRUE TRUE TRUE TRUE TRUE 5
## 6 6 TRUE TRUE TRUE TRUE TRUE 5
## 11 11 TRUE TRUE TRUE TRUE TRUE 5
## 16 16 TRUE TRUE TRUE TRUE TRUE 5
## 19 19 TRUE TRUE TRUE TRUE TRUE 5
## 21 21 TRUE TRUE TRUE TRUE TRUE 5
## 27 27 TRUE TRUE TRUE TRUE TRUE 5
## 28 28 TRUE TRUE TRUE TRUE TRUE 5
## 38 38 TRUE TRUE TRUE TRUE TRUE 5
## 39 39 TRUE TRUE TRUE TRUE TRUE 5
## 44 44 TRUE TRUE TRUE TRUE TRUE 5
## 48 48 TRUE TRUE TRUE TRUE TRUE 5
## 50 50 TRUE TRUE TRUE TRUE TRUE 5
## 52 52 TRUE TRUE TRUE TRUE TRUE 5
## 59 59 TRUE TRUE TRUE TRUE TRUE 5
## 60 60 TRUE TRUE TRUE TRUE TRUE 5
## 67 67 TRUE TRUE TRUE TRUE TRUE 5
## 69 69 TRUE TRUE TRUE TRUE TRUE 5
## 71 71 TRUE TRUE TRUE TRUE TRUE 5
## 72 72 TRUE TRUE TRUE TRUE TRUE 5
## 75 75 TRUE TRUE TRUE TRUE TRUE 5
## 77 77 TRUE TRUE TRUE TRUE TRUE 5
## 78 78 TRUE TRUE TRUE TRUE TRUE 5
## 82 82 TRUE TRUE TRUE TRUE TRUE 5
## 88 88 TRUE TRUE TRUE TRUE TRUE 5
## 90 90 TRUE TRUE TRUE TRUE TRUE 5
## 91 91 TRUE TRUE TRUE TRUE TRUE 5
## 92 92 TRUE TRUE TRUE TRUE TRUE 5
## 96 96 TRUE TRUE TRUE TRUE TRUE 5
## 105 105 TRUE TRUE TRUE TRUE TRUE 5
## 111 111 TRUE TRUE TRUE TRUE TRUE 5
## 114 114 TRUE TRUE TRUE TRUE TRUE 5
## 115 115 TRUE TRUE TRUE TRUE TRUE 5
## 119 119 TRUE TRUE TRUE TRUE TRUE 5
## 122 122 TRUE TRUE TRUE TRUE TRUE 5
## 126 126 TRUE TRUE TRUE TRUE TRUE 5
## 130 130 TRUE TRUE TRUE TRUE TRUE 5
## 133 133 TRUE TRUE TRUE TRUE TRUE 5
is_monotonic <- function(x) {
all(diff(as.numeric(x)) <= 0)
}
df$monotonic <- apply(df[, 2:6], 1, is_monotonic)
dg <- df[df$monotonic == FALSE, ]
dh <- dg[order(dg$score), ]
dh## id 1 4 3 2 5 score monotonic
## 41 41 FALSE FALSE FALSE FALSE TRUE 1 FALSE
## 108 108 FALSE FALSE TRUE FALSE FALSE 1 FALSE
## 8 8 FALSE FALSE TRUE TRUE FALSE 2 FALSE
## 76 76 FALSE FALSE TRUE TRUE FALSE 2 FALSE
## 89 89 TRUE FALSE TRUE FALSE FALSE 2 FALSE
## 120 120 TRUE FALSE FALSE TRUE FALSE 2 FALSE
## 30 30 FALSE TRUE TRUE TRUE FALSE 3 FALSE
## 36 36 TRUE TRUE FALSE TRUE FALSE 3 FALSE
## 117 117 TRUE TRUE FALSE FALSE TRUE 3 FALSE
print(
xtable::xtable(dh[, 2:7], caption = "Participant answers that violate unidimensionality",
label = "tab:viol", digits = 0),
booktabs = TRUE,
file = "tables/viol.tex"
)lv <- loevenize(as.matrix(d[, 2:6]))
lv## $error_matrix
## col
## row 1 2 3 4 5
## 1 0 1 4 3 1
## 2 1 0 4 3 1
## 3 4 4 0 2 2
## 4 3 3 2 0 2
## 5 1 1 2 2 0
##
## $expected_error_matrix
## col
## row 1 2 3 4 5
## 1 0.000000 19.360902 15.22556 14.84962 7.518797
## 2 19.360902 0.000000 18.27068 17.81955 9.022556
## 3 15.225564 18.270677 0.00000 30.88722 15.639098
## 4 14.849624 17.819549 30.88722 0.00000 16.240602
## 5 7.518797 9.022556 15.63910 16.24060 0.000000
##
## $h_matrix
## col
## row 1 2 3 4 5
## 1 0.0000000 0.9483495 0.7372840 0.7979747 0.8670000
## 2 0.9483495 0.0000000 0.7810700 0.8316456 0.8891667
## 3 0.7372840 0.7810700 0.0000000 0.9352483 0.8721154
## 4 0.7979747 0.8316456 0.9352483 0.0000000 0.8768519
## 5 0.8670000 0.8891667 0.8721154 0.8768519 0.0000000
##
## $h
## [1] 0.8604662
##
## $sum_errors
## [1] 23
##
## $sum_expected_errors
## [1] 164.8346
1-c(23, 18, 12)/ lv$sum_expected_errors## [1] 0.8604662 0.8907996 0.9271997
double check with mokken package:
d_numeric <- apply(d, 2, as.numeric)
d_numeric <- d_numeric[, 2:6]
mokken::coefH(d_numeric)## $Hij
## 1 se 4 se 3 se 2 se 5 se
## 1 0.948 (0.050) 0.737 (0.116) 0.798 (0.106) 0.867 (0.128)
## 4 0.948 (0.050) 0.781 (0.098) 0.832 (0.089) 0.889 (0.107)
## 3 0.737 (0.116) 0.781 (0.098) 0.935 (0.044) 0.872 (0.086)
## 2 0.798 (0.106) 0.832 (0.089) 0.935 (0.044) 0.877 (0.083)
## 5 0.867 (0.128) 0.889 (0.107) 0.872 (0.086) 0.877 (0.083)
##
## $Hi
## Item H se
## 1 0.842 (0.068)
## 4 0.860 (0.050)
## 3 0.850 (0.050)
## 2 0.875 (0.047)
## 5 0.876 (0.088)
##
## $H
## Scale H se
## 0.860 (0.047)
Explain violations, recalculate H:
d_cp <- d
d_cp[89, 2+1] <- TRUE
d_cp[30, 1+1] <- TRUE
d_cp[36, 3+1] <- TRUE
loevenize(as.matrix(d_cp[,2:6]))$h## [1] 0.8880157
d_cp[41, 5+1] <- FALSE
d_cp[117, 5+1] <- FALSE
loevenize(as.matrix(d_cp[,2:6]))$h## [1] 0.9242237
d_num <- apply(d[, c(-1, -7)], 2, as.numeric)
fa1 <- factanal(d_num, 1)
fa2 <- factanal(d_num, 2)
tab <- cbind(unclass(loadings(fa1)), unclass(loadings(fa2)))
print(
xtable::xtable(tab, caption = "Factor analysis",
label = "tab:fa2", digits = 3),
booktabs = TRUE,
file = "tables/fa2.tex"
)
tab## Factor1 Factor1 Factor2
## 1 0.5043532 0.2243658 0.9147548
## 4 0.5830152 0.3340612 0.8422924
## 3 0.9396848 0.9025577 0.2639481
## 2 0.9607688 0.9221507 0.2805721
## 5 0.4941331 0.4560955 0.1890938
tab <- round(cor(d[, -c(1, 7)]), 2)
# max corr, when H for item 5 is 1
round(cor(d[-c(41, 117), -c(1, 7)]), 2)## 1 4 3 2 5
## 1 1.00 0.84 0.44 0.46 0.30
## 4 0.84 1.00 0.53 0.54 0.34
## 3 0.44 0.53 1.00 0.90 0.50
## 2 0.46 0.54 0.90 1.00 0.52
## 5 0.30 0.34 0.50 0.52 1.00
loevenize(as.matrix(d[-c(41, 117), -c(1, 7)]))## $error_matrix
## col
## row 1 2 3 4 5
## 1 0 1 4 3 0
## 2 1 0 4 3 0
## 3 4 4 0 2 0
## 4 3 3 2 0 0
## 5 0 0 0 0 0
##
## $expected_error_matrix
## col
## row 1 2 3 4 5
## 1 0.000000 18.687023 14.83969 14.47328 6.961832
## 2 18.687023 0.000000 17.93130 17.48855 8.412214
## 3 14.839695 17.931298 0.00000 30.15267 14.503817
## 4 14.473282 17.488550 30.15267 0.00000 15.083969
## 5 6.961832 8.412214 14.50382 15.08397 0.000000
##
## $h_matrix
## col
## row 1 2 3 4 5
## 1 0.0000000 0.9464869 0.7304527 0.7927215 1
## 2 0.9464869 0.0000000 0.7769264 0.8284592 1
## 3 0.7304527 0.7769264 0.0000000 0.9336709 1
## 4 0.7927215 0.8284592 0.9336709 0.0000000 1
## 5 1.0000000 1.0000000 1.0000000 1.0000000 0
##
## $h
## [1] 0.8927677
##
## $sum_errors
## [1] 17
##
## $sum_expected_errors
## [1] 158.5344
a <- tril(lv$h_matrix, -1) # strict lower triangular matrix (omit diagonals)
b <- triu(tab, 1) # strict upper triangular matrix
c <- a + b
diag(c) <- NA
pdf("plots/itempairs.pdf", width = 7, height = 7)
plot(as.matrix(c), fmt.cell='%.2f', main = "", key = NULL,
xlab = "Item", ylab = "Item", col = sapply(seq(1, 0, -0.1), gray, alpha = 0.5))
dev.off()## png
## 2
plot(as.matrix(c), fmt.cell='%.2f', main = "", key = NULL,
xlab = "Item", ylab = "Item", col = sapply(seq(1, 0, -0.1), gray, alpha = 0.5))