ci.table: Test-Inversion CIs for Estimands in Contingency Tables
In cta: Contingency Table Analysis Based on ML Fitting of MPH Models
Description
Usage
Arguments
Value
Author(s)
References
See Also
Examples
Description
Constructs test-inversion approximate confidence intervals (CIs) for
estimands in contingency tables subject to equality constraints.
Test statistics include Wald-type statistics, and difference and
nested versions of power-divergence statistics. This program can also compute
test-inversion approximate confidence intervals for estimands in
contingency tables without additionally imposed equality constraints,
by setting the constraint function h.fct = 0.
Usage
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ci.table(y, h.fct = 0, h.mean = FALSE, S.fct, S.mean = FALSE, S.P = FALSE,
S.space.H0 = NULL, method = "all", cc = 0.95, pdlambda = 2/3,
trans.g = NULL, trans.g.epsilon = 0, trans.g.inv = NULL,
strata = rep(1, length(y)), fixed.strata = "all", delta = 0.5,
max.iter = 50, tol = 1e-2, tol.psi = 1e-4, adj.epsilon = 0.03,
iter.robust.max = 30, iter.robust.eff = 10, check.homog.tol = 1e-9,
check.zero.order.homog.tol = 1e-9, max.mph.iter = 1000, step = 1,
change.step.after = 0.25 * max.mph.iter, y.eps = 0, iter.orig = 5,
norm.diff.conv = 1e-6, norm.score.conv = 1e-6,
max.score.diff.iter = 10, h0.fct.deriv = NULL,
S0.fct.deriv = NULL, trans.g.deriv = NULL, plot.CIs = TRUE)
Arguments
y
Observed table counts in the contingency table(s), in vector form.
h.fct
The imposed equality constraint(s). Note that sampling constraints are not included in h.fct, and the imposed equality constraints should be non-redundant.
If h.mean = FALSE (default), h(p) should be the input, where p is the vector of data model probabilities, or it can be called the vector of table probabilities; If h.mean = TRUE, h(m) should be the input, where m is the vector of expected table counts, i.e. m = E(Y). In the case of h(m) being the input, the function h(\cdot) should be Z homogeneous, where Z is the population matrix. For the definition of Z homogeneity and the population matrix, see Lang (2004). Note that if there is no imposed equality constraint, we should input h.fct = 0 (real number 0). Please do not specify h.fct as a zero function in this case. On the contrary, if there is (are) imposed equality constraint(s), please specify h.fct as an R function. Another important note is that if there are multiple imposed equality constraints, please use rbind(), not c(), to concatenate the imposed equality constraints into a column vector.
By default, h.fct = 0.
h.mean
Logical argument, TRUE or FALSE. If h.mean = FALSE (default), the input h.fct is treated as a function of p; If h.mean = TRUE, the input h.fct is treated as a function of m.
S.fct
Parameter or estimand of interest. It should be an R function, which returns a real number. i.e. S(\cdot) is a real-valued function. If S.mean = FALSE and S.P = FALSE (default), S(p) should be the input; If S.mean = TRUE, S(m) should be the input; If S.P = TRUE, S(P) should be the input, where P is the vector of joint probabilities, or it can be called the vector of pre-data probabilities. In the case of S(m) or S(P) being the input, the function S(\cdot) should be zero-order Z homogeneous, then S(P) is Z estimable with S(P) = S(m). In addition, when we are in the process of computing test-inversion confidence intervals other than Wald intervals, we have to fit several models and obtain constrained MLEs of expected table counts. These models have equality constraints h_{0}^{*}(m) = 0, where h_{0}^{*}(m) = (h_{0}'(m), S_{0}(m) - ψ, samp_{0}'(m))'. Here h_{0}(m) = 0 is (are) the imposed equality constraint(s), written in terms of m; S_{0}(m) - ψ = 0 means that the estimand of interest is equal to ψ; samp_{0}(m) = 0 is (are) the sampling constraint(s), written in terms of m. Restriction of S(m) [or S(P)] to zero-order Z homogeneity guarantees the Z homogeneity of h_{0}^{*}(m).
S.mean, S.P
Logical argument, TRUE or FALSE. If S.mean = FALSE and S.P = FALSE (default), the input S.fct is treated as a function of p; If S.mean = TRUE, the input S.fct is treated as a function of m; If S.P = TRUE, the input S.fct is treated as a function of P.
S.space.H0
Restricted estimand space of S(\cdot) under H_{0}, i.e. subject to the imposed equality constraints along with sampling constraints. If S.space.H0 is not specified or the input S.space.H0 = NULL, the restricted estimand space is treated as (-∞, ∞), i.e. the whole real number line. If S.space.H0 is specified, it can either be input as a vector of length of an even number, or be input in class Intervals_full {intervals}. As an example, if the restricted estimand space is (-∞, -1] \cup [1, ∞), then the input S.space.H0 could be c(-Inf, -1, 1, Inf), or
Intervals_full(matrix(c(-Inf, -1, 1, Inf), ncol = 2, byrow = TRUE), closed = matrix(c(FALSE, TRUE, TRUE, FALSE), ncol = 2, byrow = TRUE), type = "R"). It is strongly recommended that S.space.H0 be specified, as it will improve the accuracy and (possibly) speed in interval estimation. However, it is often difficult to have an idea of the restricted estimand space exactly. In this scenario, specification of one (or several) possibly larger interval(s) that cover(s) the exact restricted estimand space is also helpful.
method
The test statistic(s) in constructing the test-inversion approximate confidence interval(s). There are eight different test statistics, and the user is allowed to choose any number of the test statistics out of the eight. The eight test statistics are listed as follows: "Wald", "trans.Wald" (need specification of the transformation g), "diff.Xsq", "nested.Xsq", "diff.Gsq" (same as "PL" or "LR"), "nested.Gsq", "diff.PD", "nested.PD" (need specification of the power-divergence index parameter λ). If the input method = "all" (default), all test statistics will be employed to compute confidence intervals.
cc
Confidence coefficient, or the nominal level of the confidence interval.
pdlambda
The index parameter λ in the power-divergence statistic.
trans.g
The transformation g used in the transformed Wald confidence interval. First, we construct a confidence interval for g(S(\cdot)), then we back-transform, i.e. apply g^{-1} to the endpoints in order to obtain a confidence interval for S(\cdot). There are several built-in options for the transformation: "Fisher's z", "log", "-log" (same as "negative log"), and "[A, B]". "[A, B]" refers to the reparameterization trick as stated in the Discussion part of Lang (2008). The user is also allowed to input their own choice of trans.g. Ordinarily, the transformation g should be a bijection. Ideally, g should be smooth, strictly monotonically increasing, and "to parameterize away the boundary" (Lang, 2008).
trans.g.epsilon
The small ε adjustment included in the transformation g. For example, the "[A, B]" transformation g with the small ε is
g(x) = \log(x - A + ε) - \log(B + ε - x).
By default, trans.g.epsilon = 0.
trans.g.inv
g^{-1} function used in back-transformation step in construction of the transformed Wald confidence interval. If trans.g is any one of the built-in options, then trans.g.inv is automatically specified accordingly.
strata
Vector of the same length as y that gives the stratum membership identifier. By default, strata = rep(1, length(y)) refers to the single stratum (non-stratified) setting. As another example, strata = c(1,1,2,2) means that the first and second table cells belong to the first stratum, and the third and fourth table cells belong to the second stratum.
fixed.strata
The object that gives information on which stratum (strata) has (have) fixed sample sizes. It can equal one of the keywords, fixed.strata = "all" or fixed.strata = "none", or it can be a vector of stratum membership identifiers, e.g. fixed.strata = c(1,3) or fixed.strata = c("pop1", "pop5").
delta
The constant δ that is in expressions of the moving critical values within each sliding quadratic step. By default, delta = 0.5.
max.iter
One of the stopping criteria. It is the maximum number of iterations in the sliding quadratic root-finding algorithm for searching the two roots to the test-inversion equation.
tol
One of the stopping criteria. In solving for the roots to the test-inversion equation, if the test statistic for testing H_{0}(ψ): S_{0}(m) = ψ vs. not H_{0}(ψ) under the general hypothesis H_{0}: (h_{0}'(m), samp_{0}'(m))' = 0, for a certain ψ, is within tol of the critical value, then we stop the iterations, and this current ψ is treated as one root. Note that since we are constructing approximate (contrary to exact) confidence intervals based on the asymptotic distribution under the null hypothesis, tol need not be too small.
tol.psi
One of the stopping criteria. In solving for the roots to the test-inversion equation, if the two ψ's that are in nearby iterates in the corresponding tests H_{0}(ψ) vs. not H_{0}(ψ) under the general hypothesis H_{0}, are less than tol.psi apart in distance, then we stop the iterations, and the current ψ is treated as one root. Note that we should specify tol.psi to be sufficiently small (compared with the size of the restricted estimand space) so that the iterations are to be terminated mainly because of closeness of the test statistic to the critical value.
adj.epsilon, iter.robust.max, iter.robust.eff
The parameters used in the robustifying procedure. First, we attempt to construct confidence intervals based on the original data y, but an error might occur during this process. The reason for occurrence of the error might be the non-existence of the constrained MLE subject to H_{0}, or it might be because of the fact that the ψ in the hypothesis test H_{0}(ψ) vs. not H_{0}(ψ) is, on some scale, too far away from \widehat{ψ} which is the constrained MLE of the estimand subject to H_{0}, although this ψ is still within the restricted estimand space. If an error, or non-convergence issue occurs, then the program will go through the robustifying procedure, with the goal of reporting a confidence interval anyway, even in the most extreme configuration and/or with the most "extreme" data.
In the robustifying procedure, we adjust the original data y by adding 1 * adj.epsilon to each original table count, and compute the confidence interval based on the adjusted data y + 1 * adj.epsilon. Note, however, that even the adjusted data may lead to non-convergence issue sometimes. We also adjust the original data by adding 2 * adj.epsilon, …, iter.robust.max * adj.epsilon, and compute confidence intervals based on these adjusted data, respectively. For computing purposes, as soon as iter.robust.eff confidence intervals based on the adjusted data have been successfully computed, we will not proceed further into adjustment and interval estimation based on adjusted data. Now, by exploiting the property that
\lim_{\texttt{adj.epsilon} \rightarrow 0+} CI(y + \texttt{adj.epsilon}; H_{0}) = CI(y; H_{0}) ,
we extrapolate using a polynomial fit of degree at most three based on lower and upper endpoints of the confidence intervals on adjusted data. It is advised that
adj.epsilon should not exceed 0.1, but it should not be too small. By default, adj.epsilon = 0.03.
check.homog.tol
Round-off tolerance for Z homogeneity check. If the function h(\cdot) with respect to m is not Z homogeneous, the algorithm will stop immediately and report an error.
check.zero.order.homog.tol
Round-off tolerance for zero-order Z homogeneity check. If the function S(\cdot) with respect to m or P is not zero-order Z homogeneous, the algorithm will stop immediately and report an error.
max.mph.iter, step, change.step.after, y.eps, iter.orig, norm.diff.conv, norm.score.conv, max.score.diff.iter
The parameters used in mph.fit.
h0.fct.deriv
The R function object that computes analytic derivative of the transpose of the constraint function h_{0}(\cdot) with respect to m. In this algorithm, if the input function h.fct is a function of p, then the algorithm automatically rewrites it into another function of m: h(p) = h(Diag^{-1}(ZZ'm)m) = h_{0}(m). If the input function h.fct is a function of m, then we let h_{0}(m) = h(m). h0.fct.deriv, if it is specified, equals \partial h_{0}'(m) / \partial m. Note that if h_{0}(\cdot) maps from R^p to R^q, i.e. there are q constraints, then h0.fct.deriv returns a p-by-q matrix of partial derivatives. If h0.fct.deriv is not specified or h0.fct.deriv = NULL, numerical derivatives will be used.
S0.fct.deriv
The R function object that computes analytic derivative of the estimand function S_{0}(\cdot) with respect to m. In this algorithm, if the input function S.fct is a function of p, then the algorithm automatically rewrites it into another function of m: S(p) = S(Diag^{-1}(ZZ'm)m) = S_{0}(m). If the input function S.fct is a function of m, then we let S_{0}(m) = S(m). If the input function S.fct is a function of P, since S(\cdot) is required to be zero-order Z homogeneous, in which case S(P) = S(m), we let S_{0}(m) =
S(P). S0.fct.deriv, if it is specified, equals \partial S_{0}(m) / \partial m. It is a column vector, whose length is the same as the length of m. If S0.fct.deriv is not specified or S0.fct.deriv = NULL, numerical derivatives will be used.
trans.g.deriv
The derivative function of the transformation g, i.e. d g(w) / d w. If it is specified, it should be an R function, even if the derivative function is a constant function.
plot.CIs
Logical argument, TRUE or FALSE. If plot.CIs = TRUE (default), a visual display of the computed confidence interval(s) will be created. If plot.CIs = FALSE, no plots will be created.
Value
ci.table returns a list, which includes the following objects:
result.table
A table that displays lower and upper endpoints of the computed confidence interval(s). The length(s) of the confidence interval(s) is (are) reported in the last column.
CIs
An object of class Intervals_full {intervals} that includes all of the computed confidence interval(s).
Shat
The constrained MLE of S(\cdot) subject to H_{0}. If there is an error or non-convergence issue during the process of fitting the model subject to H_{0} by mph.fit, Shat is set to be NA; or if the constrained MLE does not exist, Shat is also set to be NA.
ase.Shat
The asymptotic standard error, i.e. ase, of the constrained MLE of S(\cdot) subject to H_{0}. If there is an error or non-convergence issue during the process of fitting the model subject to H_{0} by mph.fit, ase.Shat is set to be NA; or if the constrained MLE does not exist, ase.Shat is also set to be NA.
S.space.H0
Restricted estimand space of S(\cdot) under H_{0}. It might be different from the input S.space.H0. If the input S.space.H0 is the union of at least two disjoint intervals, then the output S.space.H0 displays the particular interval in which Shat, the constrained MLE of S(\cdot) subject to H_{0}, lies. If the input S.space.H0 is an interval, then the output S.space.H0 is the same as the input. If S.space.H0 is unspecified or S.space.H0 = NULL in the input, then the output S.space.H0 = NULL.
cc
Confidence coefficient, or the nominal level of the confidence interval. It is the same as the cc in the input.
method
The test statistic(s) that is (are) actually used to construct the test-inversion approximate confidence interval(s).
pdlambda
The index parameter λ in the power-divergence statistic. It is the same as the pdlambda in the input.
warnings.collection
Includes all of the warning messages that occur during construction of the confidence
interval(s). They might be on evoking of the robustifying procedure: "xxx.CI: Adjustment used. Not on original data.\n", or they might be on unsuccessful construction of the confidence interval(s): "xxx.CI: NA.\n"
Author(s)
Qiansheng Zhu
References
Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.
Lang, J. B. (2008) Score and profile likelihood confidence intervals for contingency table parameters, Statistics in Medicine, 27, 5975–5990.
Zhu, Q. (2020) "On improved confidence intervals for parameters of discrete distributions." PhD dissertation, University of Iowa.
See Also
Examples
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### Construct test-inversion CIs subject to equality constraints.
# I. Mice-Fungicide data: Innes et al. (1969) conducted an experiment
# to test the possible carcinogenic effect of a fungicide Avadex on
# four subgroups of mice. The data is reproduced as a 2-by-2-by-4
# three-way contingency table. Within each of the four 2-by-2 two-way
# sub-tables, there is one fixed stratum for the treated group, and
# there is also one fixed stratum for the control group. Overall,
# the data was collected under the product-multinomial sampling scheme.
# We assume that the relative risks that correspond to the four 2-by-2
# two-way sub-tables are the same, and we construct 95% test-inversion
# confidence intervals for this common relative risk.
#
# For a detailed description of the Mice-Fungicide data set, see
# Gart (1971):
# Gart, J. J. (1971) The comparison of proportions: a review of
# significance tests, confidence intervals and adjustments for
# stratification. Revue de l'Institut International de Statistique,
# 39(2), pp. 148-169.
obs.y <- c(4, 12, 5, 74, 2, 14, 3, 84, 4, 14, 10, 80, 1, 14, 3, 79)
h.fct <- function(p) {
RR_1 <- p[1] / p[3]
RR_2 <- p[5] / p[7]
RR_3 <- p[9] / p[11]
RR_4 <- p[13] / p[15]
rbind(RR_1 - RR_2, RR_1 - RR_3, RR_1 - RR_4)
}
S.fct <- function(p) {
p[1] / p[3]
}
mice_result <- ci.table(obs.y, h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = rep(seq(1, 8), each = 2))
# II. Suppose there is a 3-by-4-by-2 three-way contingency table which
# cross-classifies three variables: X, Y, and Z. We assign scores
# {1,2,3}, {1,2,3,4}, and {1,2} to the variables X, Y, and Z,
# respectively. At each level of Z, there is a 3-by-4 two-way sub-table
# for variables X and Y, and the 3-by-4 sub-table forms a fixed
# stratum. We assume that the Pearson's correlation coefficient between
# X and Y when Z = 1 is the same as that when Z = 2. The observed table
# counts are (1,2,3,4,5,6,7,8,9,10,11,12) for the 3-by-4 sub-table when
# Z = 1, and (13,14,15,16,17,18,19,20,21,22,23,24) for the 3-by-4 sub-
# table when Z = 2. We construct a 95% profile likelihood confidence
# interval for this common Pearson's correlation coefficient.
corr_freq_prob <- function(freq, score.X, score.Y) {
# Compute the Pearson's correlation coefficient based on the vector
# of table (frequency) counts or the vector of underlying table
# probabilities.
# Note that the input freq is a vector.
c <- length(score.X)
d <- length(score.Y)
freq <- matrix(freq, nrow = c, ncol = d, byrow = TRUE)
P <- freq / sum(freq)
P.row.sum <- apply(P, 1, sum)
P.column.sum <- apply(P, 2, sum)
EX <- crossprod(score.X, P.row.sum)
EY <- crossprod(score.Y, P.column.sum)
EXsq <- crossprod(score.X^2, P.row.sum)
EYsq <- crossprod(score.Y^2, P.column.sum)
sdX <- sqrt(EXsq - EX^2)
sdY <- sqrt(EYsq - EY^2)
EXY <- 0
for (i in seq(1, c)) {
for (j in seq(1, d)) {
EXY <- EXY + score.X[i] * score.Y[j] * P[i, j]
}
}
Cov.X.Y <- EXY - EX * EY
if (Cov.X.Y == 0) {
corr <- 0
}
else {
corr <- as.numeric(Cov.X.Y / (sdX * sdY))
}
corr
}
h.fct <- function(p) {
corr_1 <- corr_freq_prob(p[seq(1, 12)], c(1, 2, 3), c(1, 2, 3, 4))
corr_2 <- corr_freq_prob(p[seq(13, 24)], c(1, 2, 3), c(1, 2, 3, 4))
corr_1 - corr_2
}
S.fct <- function(p) {
corr_freq_prob(p[seq(1, 12)], c(1, 2, 3), c(1, 2, 3, 4))
}
corr_result <- ci.table(y = seq(1, 24), h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(-1, 1), method = "LR",
trans.g = "Fisher's z", strata = rep(c(1, 2), each = 12),
plot.CIs = FALSE)
# III. Crying Baby data: Gordon and Foss (1966) conducted an experiment to
# investigate the effect of rocking on the crying of full term babies.
# The data set can be reproduced as a 2-by-2-by-18 three-way contingency
# table. Within each of the eighteen 2-by-2 two-way sub-tables, there is
# one fixed stratum for the experimental group and one fixed stratum for
# the control group. Overall, the data was collected under the product-
# multinomial sampling scheme. We assume common odds ratios among the
# eighteen two-way sub-tables, and we construct 95% test-inversion
# confidence intervals for this common odds ratio.
#
# For a detailed description of the Crying Baby data set, see Cox (1966):
# Cox, D. R. (1966) A simple example of a comparison involving quantal
# data. Biometrika, 53(1-2), pp. 213-220.
obs.y <- c(0,1,5,3,0,1,4,2,0,1,4,1,1,0,5,1,0,1,1,4,0,1,5,4,0,1,3,5,0,1,
4,4,0,1,2,3,1,0,1,8,0,1,1,5,0,1,1,8,0,1,3,5,0,1,1,4,0,1,2,4,
0,1,1,7,1,0,2,4,0,1,3,5)
strata <- rep(seq(1, 36), each = 2)
h.fct <- function(p) {
OR_1 <- p[1] * p[4] / (p[2] * p[3])
OR_2 <- p[5] * p[8] / (p[6] * p[7])
OR_3 <- p[9] * p[12] / (p[10] * p[11])
OR_4 <- p[13] * p[16] / (p[14] * p[15])
OR_5 <- p[17] * p[20] / (p[18] * p[19])
OR_6 <- p[21] * p[24] / (p[22] * p[23])
OR_7 <- p[25] * p[28] / (p[26] * p[27])
OR_8 <- p[29] * p[32] / (p[30] * p[31])
OR_9 <- p[33] * p[36] / (p[34] * p[35])
OR_10 <- p[37] * p[40] / (p[38] * p[39])
OR_11 <- p[41] * p[44] / (p[42] * p[43])
OR_12 <- p[45] * p[48] / (p[46] * p[47])
OR_13 <- p[49] * p[52] / (p[50] * p[51])
OR_14 <- p[53] * p[56] / (p[54] * p[55])
OR_15 <- p[57] * p[60] / (p[58] * p[59])
OR_16 <- p[61] * p[64] / (p[62] * p[63])
OR_17 <- p[65] * p[68] / (p[66] * p[67])
OR_18 <- p[69] * p[72] / (p[70] * p[71])
rbind(OR_1 - OR_2, OR_1 - OR_3, OR_1 - OR_4, OR_1 - OR_5, OR_1 - OR_6,
OR_1 - OR_7, OR_1 - OR_8, OR_1 - OR_9, OR_1 - OR_10, OR_1 - OR_11,
OR_1 - OR_12, OR_1 - OR_13, OR_1 - OR_14, OR_1 - OR_15,
OR_1 - OR_16, OR_1 - OR_17, OR_1 - OR_18)
}
S.fct <- function(p) {
p[1] * p[4] / (p[2] * p[3])
}
crying_baby_result <- ci.table(obs.y, h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = strata, fixed.strata = "all",
y.eps = 0.4)
# IV. Homicide data: Radelet & Pierce (1985) examined cases of 1017 homicide
# defendants in Florida between 1973 and 1977. Both the police department
# and prosecutors classified these cases into three mutually exclusive
# categories: 1 = "No Felony", 2 = "Possible Felony", 3 = "Felony".
# Three variables: police classification (P), court (i.e. prosecutors')
# classification (C), and race of defendant/victim (R) are cross-
# classified in a 3-by-3-by-4 three-way contingency table. The data
# was collected based on independent Poisson sampling, and the strata
# correspond to levels of the race combination (R).
#
# For a detailed description of the Homicide data set, see Agresti (1984)
# and Radelet & Pierce (1985):
# Agresti, A. (1984). Analysis of Ordinal Categorical Data. John Wiley &
# Sons.
# Radelet, M. L., & Pierce, G. L. (1985). Race and prosecutorial
# discretion in homicide cases. Law & Society Review, 19(4), pp. 587-622.
#
# To measure agreement between police and court classifications, the four
# estimands of interest are Cohen's unweighted kappa coefficients at four
# levels of R, respectively. We construct 95% test-inversion confidence
# intervals for the estimands subject to two sets of equality constraints,
# respectively.
# (1) WkW and BkB have the same unweighted kappa, and BkW and WkB have
# the same unweighted kappa.
# (2) A "row effects" model for the conditional R-C association:
# log mu_{ijk} = lambda + lambda_{i}^{R} + lambda_{j}^{P} + lambda_{k}^{C} +
# lambda_{ij}^{RP} + lambda_{jk}^{PC} + tau_{i}^{RC}(w_{k} - bar{w}),
# where race effects {tau_{i}^{RC}} that sum to zero are introduced for an
# R-C association. The variable C is viewed as being ordinal with integer
# monotonic scores {w_{k}}={1,2,3}.
BkW_v <- c(7, 1, 3, 0, 2, 6, 5, 5, 109)
WkW_v <- c(236, 11, 26, 7, 2, 21, 25, 4, 101)
BkB_v <- c(328, 6, 13, 7, 2, 3, 21, 1, 36)
WkB_v <- c(14, 1, 0, 6, 1, 1, 1, 0, 5)
obs.y <- c(BkW_v, WkW_v, BkB_v, WkB_v)
Unweighted.Kappa.BkW <- function(p) {
mat.p <- matrix(p[seq(1,9)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.WkW <- function(p) {
mat.p <- matrix(p[seq(10,18)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.BkB <- function(p) {
mat.p <- matrix(p[seq(19,27)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.WkB <- function(p) {
mat.p <- matrix(p[seq(28,36)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
# Constraints (1)
library(vcd)
WkW.BkB_BkW.WkB_cons <- function(p) {
mat.BkW <- matrix(p[seq(1,9)], nrow = 3, byrow = TRUE)
mat.WkW <- matrix(p[seq(10,18)], nrow = 3, byrow = TRUE)
mat.BkB <- matrix(p[seq(19,27)], nrow = 3, byrow = TRUE)
mat.WkB <- matrix(p[seq(28,36)], nrow = 3, byrow = TRUE)
rbind(Kappa(mat.BkW)$Unweighted[1] - Kappa(mat.WkB)$Unweighted[1],
Kappa(mat.WkW)$Unweighted[1] - Kappa(mat.BkB)$Unweighted[1])
}
homicide_kappa_same_fit <- mph.fit(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
homicide_kappa_same_fit$Gsq
pchisq(homicide_kappa_same_fit$Gsq, 2, lower.tail = FALSE) # p-value
BkW_kappa_same <- ci.table(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
S.fct = Unweighted.Kappa.BkW, S.space.H0 = c(0,1),
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none", trans.g = "[A,B]")
WkW_kappa_same <- ci.table(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
S.fct = Unweighted.Kappa.WkW, S.space.H0 = c(0,1),
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none", trans.g = "[A,B]")
# Constraints (2)
X_cond_RC_v <- c(1,1,0,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,-1,0,0,
1,1,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,
1,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,
1,1,0,0,0,1,1,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,
1,1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,
1,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,
1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,
1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
1,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,-1,0,
1,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,
1,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,
1,0,1,0,0,1,1,0,0,0,0,1,0,0,0,0,1,0,0,-1,0,
1,0,1,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,
1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,
1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,
1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
1,0,0,1,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,-1,
1,0,0,1,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,
1,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,
1,0,0,1,0,1,1,0,0,0,0,0,0,1,0,0,1,0,0,0,-1,
1,0,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,
1,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,
1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,1,1,
1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,
1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,
1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,1,1,1,
1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,
1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,
1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1)
X_cond_RC_mat <- matrix(X_cond_RC_v, ncol = 21, byrow = TRUE)
cond_RC_HLP_fit <- mph.fit(obs.y, L.fct = "logm", L.mean = TRUE,
X = X_cond_RC_mat,
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
mph.summary(cond_RC_HLP_fit)
library(MASS)
X_cond_RC_U <- Null(X_cond_RC_mat)
cond_RC_MPH_fit <- mph.fit(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
mph.summary(cond_RC_MPH_fit)
BkW_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.BkW,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
WkW_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.WkW,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
BkB_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.BkB,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
WkB_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.WkB,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
### Construct test-inversion CIs, without additionally imposed constraints.
# V. Binomial success rate parameter p.
# Model: 0 = x <- X | p ~ Bin(n = 5, p).
# Goal: Compute approximate 90% CIs for the success probability p.
bin_p_result <- ci.table(c(0, 5), h.fct = 0, S.fct = function(p) {p[1]},
S.space.H0 = c(0, 1), cc = 0.9, y.eps = 0.1)
# Example 2.1 in Lang (2008).
# Model: y = (39, 1) <- Y ~ mult(40, p1, p2).
# Goal: Compute approximate 95% CIs for the success probability p1.
bin_p_eg21_result <- ci.table(c(39,1), h.fct = 0, S.fct = function(p) {p[1]},
S.space.H0 = c(0,1), trans.g = "[A,B]")
# VI. Conditional probability.
# Model: y = (0, 39, 18, 11) <- Y ~ mult(68, p1, p2, p3, p4)
# Goal: Compute approximate 95% CIs for the conditional probability
# p1 / (p1 + p2).
cond_prob_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1] / (p[1] + p[2])},
S.space.H0 = c(0, 1), y.eps = 0.1)
# Model: y = (0, 39 // 18, 11) <- Y ~ prod mult(39, p1, p2 // 29, p3, p4).
# That is,
# y <- Y ~ MP(gamma, p | strata = c(1, 1, 2, 2), fixed = "all"),
# where gamma = (39, 29)'.
# Goal: Compute approximate 95% CIs for p1.
cond_prob_SS_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1]}, S.space.H0 = c(0, 1),
strata = c(1, 1, 2, 2), y.eps = 0.1)
# VII. Difference between conditional probabilities.
# Model: y = (0, 39, 18, 11) <- Y ~ mult(68, p1, p2, p3, p4)
# Goal: Compute approximate 95% CIs for the difference between conditional
# probabilities, p1 / (p1 + p2) - p3 / (p3 + p4).
diff_cond_prob_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1]/(p[1]+p[2]) - p[3]/(p[3]+p[4])},
S.space.H0 = c(-1, 1), trans.g = "[A,B]")
# VIII. Gamma variant.
# Example 2.3 in Lang (2008).
# Model: y = (25, 25, 12 // 0, 1, 3)
# ~ prod mult(62, p11, p12, p13 // 4, p21, p22, p23).
# Goal: Compute approximate 95% CIs for the Gamma* parameter as
# described in Lang (2008).
Gamma_variant_23 <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
P.case.gt.control <- (p[2, 2] + p[2, 3]) * p[1, 1] + p[2, 3] * p[1, 2]
P.case.lt.control <- p[1, 2] * p[2, 1] + p[1, 3] * (p[2, 1] + p[2, 2])
P.case.neq.control <- P.case.gt.control + P.case.lt.control
P.case.gt.control / P.case.neq.control
}
Gamma_variant_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = Gamma_variant_23, S.space.H0 = c(0, 1),
trans.g = "[A,B]", strata = c(1, 1, 1, 2, 2, 2))
### Alternative code...
gammastar.fct <- function(p) {
nr <- nrow(p)
nc <- ncol(p)
probC <- 0
probD <- 0
for (i in 1:(nr-1)) {
for (j in 1:(nc-1)) {
Aij <- 0
for (h in (i+1):nr) {
for (k in (j+1):nc) {
Aij <- Aij + p[h, k]
}
}
probC <- probC + p[i, j] * Aij
}
}
for (i in 1:(nr-1)) {
for (j in 2:nc) {
Aij <- 0
for (h in (i+1):nr) {
for (k in 1:(j-1)) {
Aij <- Aij + p[h, k]
}
}
probD <- probD + p[i, j] * Aij
}
}
probC / (probC + probD)
}
Gamma_variant_23_a <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
gammastar.fct(p)
}
Gamma_variant_a_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = Gamma_variant_23_a,
S.space.H0 = c(0, 1), trans.g = "[A,B]",
strata = c(1, 1, 1, 2, 2, 2))
# IX. Global odds ratio.
# Model: y = (25, 25, 12 // 0, 1, 3)
# ~ prod mult(62, p11, p12, p13 // 4, p21, p22, p23).
# Goal: Compute approximate 95% CIs for the first global odds ratio.
global_odds_ratio_23_11 <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
p[1, 1] * (p[2, 2] + p[2, 3]) / (p[2, 1] * (p[1, 2] + p[1, 3]))
}
global_odds_ratio_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = global_odds_ratio_23_11,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = c(1, 1, 1, 2, 2, 2))
# X. Difference between product-multinomial probabilities.
# Example 2.2 in Lang (2008).
# Source (secondary): Agresti 2002:65
# Early study of the death penalty in Florida (Radelet)
# Victim Black...
# White Defendant 0/9 received Death Penalty
# Black Defendant 6/103 received Death Penalty
#
# Model: y = (0, 9 // 6, 97) <- Y ~ prod mult(9, p1, p2 // 103, p3, p4).
# Goal: Compute approximate 95% CIs for the difference between
# product-multinomial probabilities, p1 - p3.
diff_prod_mult_prob_result <- ci.table(c(0, 9, 6, 97), h.fct = 0,
S.fct = function(p) {p[1] - p[3]},
S.space.H0 = c(-1, 1),
trans.g = "Fisher's z",
strata = c(1, 1, 2, 2))
### Alternative (artificial) data that is even more sparse...
diff_prod_mult_prob_a_result <- ci.table(c(0, 9, 0, 97), h.fct = 0,
S.fct = function(p) {p[1] - p[3]},
S.space.H0 = c(-1, 1),
trans.g = "Fisher's z",
strata = c(1, 1, 2, 2), y.eps = 0.4)
# XI. Kappa coefficient.
# Example 2.4 in Lang (2008).
# Model: y = (4, 0, 0, 0, 1, 0, 0, 0, 15)
# <- Y ~ mult(20, p11, p12, ..., p33).
# Goal: Compute approximate 95% CIs for the unweighted kappa coefficient.
Kappa_coeff_33 <- function(p) {
p <- matrix(p, 3, 3, byrow = TRUE)
s1 <- p[1, 1] + p[2, 2] + p[3, 3]
prow <- apply(p, 1, sum)
pcol <- apply(p, 2, sum)
s2 <- prow[1] * pcol[1] + prow[2] * pcol[2] + prow[3] * pcol[3]
(s1 - s2) / (1 - s2)
}
kappa_coeff_result <- ci.table(c(4, 0, 0, 0, 1, 0, 0, 0, 15), h.fct = 0,
S.fct = Kappa_coeff_33, S.space.H0 = c(-1, 1))
cta documentation built on Aug. 24, 2021, 1:06 a.m.
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Description Usage Arguments Value Author(s) References See Also Examples
Description
Constructs test-inversion approximate confidence intervals (CIs) for
estimands in contingency tables subject to equality constraints.
Test statistics include Wald-type statistics, and difference and
nested versions of power-divergence statistics. This program can also compute
test-inversion approximate confidence intervals for estimands in
contingency tables without additionally imposed equality constraints,
by setting the constraint function h.fct = 0.
Usage
1 2 3 4 5 6 7 8 9 10 11 | ci.table(y, h.fct = 0, h.mean = FALSE, S.fct, S.mean = FALSE, S.P = FALSE,
S.space.H0 = NULL, method = "all", cc = 0.95, pdlambda = 2/3,
trans.g = NULL, trans.g.epsilon = 0, trans.g.inv = NULL,
strata = rep(1, length(y)), fixed.strata = "all", delta = 0.5,
max.iter = 50, tol = 1e-2, tol.psi = 1e-4, adj.epsilon = 0.03,
iter.robust.max = 30, iter.robust.eff = 10, check.homog.tol = 1e-9,
check.zero.order.homog.tol = 1e-9, max.mph.iter = 1000, step = 1,
change.step.after = 0.25 * max.mph.iter, y.eps = 0, iter.orig = 5,
norm.diff.conv = 1e-6, norm.score.conv = 1e-6,
max.score.diff.iter = 10, h0.fct.deriv = NULL,
S0.fct.deriv = NULL, trans.g.deriv = NULL, plot.CIs = TRUE)
|
Arguments
y |
Observed table counts in the contingency table(s), in vector form. |
h.fct |
The imposed equality constraint(s). Note that sampling constraints are not included in If By default, |
h.mean |
Logical argument, |
S.fct |
Parameter or estimand of interest. It should be an R function, which returns a real number. i.e. S(\cdot) is a real-valued function. If |
S.mean, S.P |
Logical argument, |
S.space.H0 |
Restricted estimand space of S(\cdot) under H_{0}, i.e. subject to the imposed equality constraints along with sampling constraints. If |
method |
The test statistic(s) in constructing the test-inversion approximate confidence interval(s). There are eight different test statistics, and the user is allowed to choose any number of the test statistics out of the eight. The eight test statistics are listed as follows: |
cc |
Confidence coefficient, or the nominal level of the confidence interval. |
pdlambda |
The index parameter λ in the power-divergence statistic. |
trans.g |
The transformation g used in the transformed Wald confidence interval. First, we construct a confidence interval for g(S(\cdot)), then we back-transform, i.e. apply g^{-1} to the endpoints in order to obtain a confidence interval for S(\cdot). There are several built-in options for the transformation: |
trans.g.epsilon |
The small ε adjustment included in the transformation g. For example, the g(x) = \log(x - A + ε) - \log(B + ε - x). By default, |
trans.g.inv |
g^{-1} function used in back-transformation step in construction of the transformed Wald confidence interval. If |
strata |
Vector of the same length as |
fixed.strata |
The object that gives information on which stratum (strata) has (have) fixed sample sizes. It can equal one of the keywords, |
delta |
The constant δ that is in expressions of the moving critical values within each sliding quadratic step. By default, |
max.iter |
One of the stopping criteria. It is the maximum number of iterations in the sliding quadratic root-finding algorithm for searching the two roots to the test-inversion equation. |
tol |
One of the stopping criteria. In solving for the roots to the test-inversion equation, if the test statistic for testing H_{0}(ψ): S_{0}(m) = ψ vs. not H_{0}(ψ) under the general hypothesis H_{0}: (h_{0}'(m), samp_{0}'(m))' = 0, for a certain ψ, is within |
tol.psi |
One of the stopping criteria. In solving for the roots to the test-inversion equation, if the two ψ's that are in nearby iterates in the corresponding tests H_{0}(ψ) vs. not H_{0}(ψ) under the general hypothesis H_{0}, are less than |
adj.epsilon, iter.robust.max, iter.robust.eff |
The parameters used in the robustifying procedure. First, we attempt to construct confidence intervals based on the original data In the robustifying procedure, we adjust the original data \lim_{\texttt{adj.epsilon} \rightarrow 0+} CI(y + \texttt{adj.epsilon}; H_{0}) = CI(y; H_{0}) , we extrapolate using a polynomial fit of degree at most three based on lower and upper endpoints of the confidence intervals on adjusted data. It is advised that
|
check.homog.tol |
Round-off tolerance for Z homogeneity check. If the function h(\cdot) with respect to m is not Z homogeneous, the algorithm will stop immediately and report an error. |
check.zero.order.homog.tol |
Round-off tolerance for zero-order Z homogeneity check. If the function S(\cdot) with respect to m or P is not zero-order Z homogeneous, the algorithm will stop immediately and report an error. |
max.mph.iter, step, change.step.after, y.eps, iter.orig, norm.diff.conv, norm.score.conv, max.score.diff.iter |
The parameters used in |
h0.fct.deriv |
The R function object that computes analytic derivative of the transpose of the constraint function h_{0}(\cdot) with respect to m. In this algorithm, if the input function |
S0.fct.deriv |
The R function object that computes analytic derivative of the estimand function S_{0}(\cdot) with respect to m. In this algorithm, if the input function |
trans.g.deriv |
The derivative function of the transformation g, i.e. d g(w) / d w. If it is specified, it should be an R function, even if the derivative function is a constant function. |
plot.CIs |
Logical argument, |
Value
ci.table returns a list, which includes the following objects:
result.table |
A table that displays lower and upper endpoints of the computed confidence interval(s). The length(s) of the confidence interval(s) is (are) reported in the last column. |
CIs |
An object of class |
Shat |
The constrained MLE of S(\cdot) subject to H_{0}. If there is an error or non-convergence issue during the process of fitting the model subject to H_{0} by |
ase.Shat |
The asymptotic standard error, i.e. ase, of the constrained MLE of S(\cdot) subject to H_{0}. If there is an error or non-convergence issue during the process of fitting the model subject to H_{0} by |
S.space.H0 |
Restricted estimand space of S(\cdot) under H_{0}. It might be different from the input |
cc |
Confidence coefficient, or the nominal level of the confidence interval. It is the same as the |
method |
The test statistic(s) that is (are) actually used to construct the test-inversion approximate confidence interval(s). |
pdlambda |
The index parameter λ in the power-divergence statistic. It is the same as the |
warnings.collection |
Includes all of the warning messages that occur during construction of the confidence
interval(s). They might be on evoking of the robustifying procedure: |
Author(s)
Qiansheng Zhu
References
Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.
Lang, J. B. (2008) Score and profile likelihood confidence intervals for contingency table parameters, Statistics in Medicine, 27, 5975–5990.
Zhu, Q. (2020) "On improved confidence intervals for parameters of discrete distributions." PhD dissertation, University of Iowa.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 | ### Construct test-inversion CIs subject to equality constraints.
# I. Mice-Fungicide data: Innes et al. (1969) conducted an experiment
# to test the possible carcinogenic effect of a fungicide Avadex on
# four subgroups of mice. The data is reproduced as a 2-by-2-by-4
# three-way contingency table. Within each of the four 2-by-2 two-way
# sub-tables, there is one fixed stratum for the treated group, and
# there is also one fixed stratum for the control group. Overall,
# the data was collected under the product-multinomial sampling scheme.
# We assume that the relative risks that correspond to the four 2-by-2
# two-way sub-tables are the same, and we construct 95% test-inversion
# confidence intervals for this common relative risk.
#
# For a detailed description of the Mice-Fungicide data set, see
# Gart (1971):
# Gart, J. J. (1971) The comparison of proportions: a review of
# significance tests, confidence intervals and adjustments for
# stratification. Revue de l'Institut International de Statistique,
# 39(2), pp. 148-169.
obs.y <- c(4, 12, 5, 74, 2, 14, 3, 84, 4, 14, 10, 80, 1, 14, 3, 79)
h.fct <- function(p) {
RR_1 <- p[1] / p[3]
RR_2 <- p[5] / p[7]
RR_3 <- p[9] / p[11]
RR_4 <- p[13] / p[15]
rbind(RR_1 - RR_2, RR_1 - RR_3, RR_1 - RR_4)
}
S.fct <- function(p) {
p[1] / p[3]
}
mice_result <- ci.table(obs.y, h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = rep(seq(1, 8), each = 2))
# II. Suppose there is a 3-by-4-by-2 three-way contingency table which
# cross-classifies three variables: X, Y, and Z. We assign scores
# {1,2,3}, {1,2,3,4}, and {1,2} to the variables X, Y, and Z,
# respectively. At each level of Z, there is a 3-by-4 two-way sub-table
# for variables X and Y, and the 3-by-4 sub-table forms a fixed
# stratum. We assume that the Pearson's correlation coefficient between
# X and Y when Z = 1 is the same as that when Z = 2. The observed table
# counts are (1,2,3,4,5,6,7,8,9,10,11,12) for the 3-by-4 sub-table when
# Z = 1, and (13,14,15,16,17,18,19,20,21,22,23,24) for the 3-by-4 sub-
# table when Z = 2. We construct a 95% profile likelihood confidence
# interval for this common Pearson's correlation coefficient.
corr_freq_prob <- function(freq, score.X, score.Y) {
# Compute the Pearson's correlation coefficient based on the vector
# of table (frequency) counts or the vector of underlying table
# probabilities.
# Note that the input freq is a vector.
c <- length(score.X)
d <- length(score.Y)
freq <- matrix(freq, nrow = c, ncol = d, byrow = TRUE)
P <- freq / sum(freq)
P.row.sum <- apply(P, 1, sum)
P.column.sum <- apply(P, 2, sum)
EX <- crossprod(score.X, P.row.sum)
EY <- crossprod(score.Y, P.column.sum)
EXsq <- crossprod(score.X^2, P.row.sum)
EYsq <- crossprod(score.Y^2, P.column.sum)
sdX <- sqrt(EXsq - EX^2)
sdY <- sqrt(EYsq - EY^2)
EXY <- 0
for (i in seq(1, c)) {
for (j in seq(1, d)) {
EXY <- EXY + score.X[i] * score.Y[j] * P[i, j]
}
}
Cov.X.Y <- EXY - EX * EY
if (Cov.X.Y == 0) {
corr <- 0
}
else {
corr <- as.numeric(Cov.X.Y / (sdX * sdY))
}
corr
}
h.fct <- function(p) {
corr_1 <- corr_freq_prob(p[seq(1, 12)], c(1, 2, 3), c(1, 2, 3, 4))
corr_2 <- corr_freq_prob(p[seq(13, 24)], c(1, 2, 3), c(1, 2, 3, 4))
corr_1 - corr_2
}
S.fct <- function(p) {
corr_freq_prob(p[seq(1, 12)], c(1, 2, 3), c(1, 2, 3, 4))
}
corr_result <- ci.table(y = seq(1, 24), h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(-1, 1), method = "LR",
trans.g = "Fisher's z", strata = rep(c(1, 2), each = 12),
plot.CIs = FALSE)
# III. Crying Baby data: Gordon and Foss (1966) conducted an experiment to
# investigate the effect of rocking on the crying of full term babies.
# The data set can be reproduced as a 2-by-2-by-18 three-way contingency
# table. Within each of the eighteen 2-by-2 two-way sub-tables, there is
# one fixed stratum for the experimental group and one fixed stratum for
# the control group. Overall, the data was collected under the product-
# multinomial sampling scheme. We assume common odds ratios among the
# eighteen two-way sub-tables, and we construct 95% test-inversion
# confidence intervals for this common odds ratio.
#
# For a detailed description of the Crying Baby data set, see Cox (1966):
# Cox, D. R. (1966) A simple example of a comparison involving quantal
# data. Biometrika, 53(1-2), pp. 213-220.
obs.y <- c(0,1,5,3,0,1,4,2,0,1,4,1,1,0,5,1,0,1,1,4,0,1,5,4,0,1,3,5,0,1,
4,4,0,1,2,3,1,0,1,8,0,1,1,5,0,1,1,8,0,1,3,5,0,1,1,4,0,1,2,4,
0,1,1,7,1,0,2,4,0,1,3,5)
strata <- rep(seq(1, 36), each = 2)
h.fct <- function(p) {
OR_1 <- p[1] * p[4] / (p[2] * p[3])
OR_2 <- p[5] * p[8] / (p[6] * p[7])
OR_3 <- p[9] * p[12] / (p[10] * p[11])
OR_4 <- p[13] * p[16] / (p[14] * p[15])
OR_5 <- p[17] * p[20] / (p[18] * p[19])
OR_6 <- p[21] * p[24] / (p[22] * p[23])
OR_7 <- p[25] * p[28] / (p[26] * p[27])
OR_8 <- p[29] * p[32] / (p[30] * p[31])
OR_9 <- p[33] * p[36] / (p[34] * p[35])
OR_10 <- p[37] * p[40] / (p[38] * p[39])
OR_11 <- p[41] * p[44] / (p[42] * p[43])
OR_12 <- p[45] * p[48] / (p[46] * p[47])
OR_13 <- p[49] * p[52] / (p[50] * p[51])
OR_14 <- p[53] * p[56] / (p[54] * p[55])
OR_15 <- p[57] * p[60] / (p[58] * p[59])
OR_16 <- p[61] * p[64] / (p[62] * p[63])
OR_17 <- p[65] * p[68] / (p[66] * p[67])
OR_18 <- p[69] * p[72] / (p[70] * p[71])
rbind(OR_1 - OR_2, OR_1 - OR_3, OR_1 - OR_4, OR_1 - OR_5, OR_1 - OR_6,
OR_1 - OR_7, OR_1 - OR_8, OR_1 - OR_9, OR_1 - OR_10, OR_1 - OR_11,
OR_1 - OR_12, OR_1 - OR_13, OR_1 - OR_14, OR_1 - OR_15,
OR_1 - OR_16, OR_1 - OR_17, OR_1 - OR_18)
}
S.fct <- function(p) {
p[1] * p[4] / (p[2] * p[3])
}
crying_baby_result <- ci.table(obs.y, h.fct = h.fct, S.fct = S.fct,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = strata, fixed.strata = "all",
y.eps = 0.4)
# IV. Homicide data: Radelet & Pierce (1985) examined cases of 1017 homicide
# defendants in Florida between 1973 and 1977. Both the police department
# and prosecutors classified these cases into three mutually exclusive
# categories: 1 = "No Felony", 2 = "Possible Felony", 3 = "Felony".
# Three variables: police classification (P), court (i.e. prosecutors')
# classification (C), and race of defendant/victim (R) are cross-
# classified in a 3-by-3-by-4 three-way contingency table. The data
# was collected based on independent Poisson sampling, and the strata
# correspond to levels of the race combination (R).
#
# For a detailed description of the Homicide data set, see Agresti (1984)
# and Radelet & Pierce (1985):
# Agresti, A. (1984). Analysis of Ordinal Categorical Data. John Wiley &
# Sons.
# Radelet, M. L., & Pierce, G. L. (1985). Race and prosecutorial
# discretion in homicide cases. Law & Society Review, 19(4), pp. 587-622.
#
# To measure agreement between police and court classifications, the four
# estimands of interest are Cohen's unweighted kappa coefficients at four
# levels of R, respectively. We construct 95% test-inversion confidence
# intervals for the estimands subject to two sets of equality constraints,
# respectively.
# (1) WkW and BkB have the same unweighted kappa, and BkW and WkB have
# the same unweighted kappa.
# (2) A "row effects" model for the conditional R-C association:
# log mu_{ijk} = lambda + lambda_{i}^{R} + lambda_{j}^{P} + lambda_{k}^{C} +
# lambda_{ij}^{RP} + lambda_{jk}^{PC} + tau_{i}^{RC}(w_{k} - bar{w}),
# where race effects {tau_{i}^{RC}} that sum to zero are introduced for an
# R-C association. The variable C is viewed as being ordinal with integer
# monotonic scores {w_{k}}={1,2,3}.
BkW_v <- c(7, 1, 3, 0, 2, 6, 5, 5, 109)
WkW_v <- c(236, 11, 26, 7, 2, 21, 25, 4, 101)
BkB_v <- c(328, 6, 13, 7, 2, 3, 21, 1, 36)
WkB_v <- c(14, 1, 0, 6, 1, 1, 1, 0, 5)
obs.y <- c(BkW_v, WkW_v, BkB_v, WkB_v)
Unweighted.Kappa.BkW <- function(p) {
mat.p <- matrix(p[seq(1,9)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.WkW <- function(p) {
mat.p <- matrix(p[seq(10,18)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.BkB <- function(p) {
mat.p <- matrix(p[seq(19,27)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
Unweighted.Kappa.WkB <- function(p) {
mat.p <- matrix(p[seq(28,36)], nrow = 3, byrow = TRUE)
Kappa(mat.p)$Unweighted[1]
}
# Constraints (1)
library(vcd)
WkW.BkB_BkW.WkB_cons <- function(p) {
mat.BkW <- matrix(p[seq(1,9)], nrow = 3, byrow = TRUE)
mat.WkW <- matrix(p[seq(10,18)], nrow = 3, byrow = TRUE)
mat.BkB <- matrix(p[seq(19,27)], nrow = 3, byrow = TRUE)
mat.WkB <- matrix(p[seq(28,36)], nrow = 3, byrow = TRUE)
rbind(Kappa(mat.BkW)$Unweighted[1] - Kappa(mat.WkB)$Unweighted[1],
Kappa(mat.WkW)$Unweighted[1] - Kappa(mat.BkB)$Unweighted[1])
}
homicide_kappa_same_fit <- mph.fit(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
homicide_kappa_same_fit$Gsq
pchisq(homicide_kappa_same_fit$Gsq, 2, lower.tail = FALSE) # p-value
BkW_kappa_same <- ci.table(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
S.fct = Unweighted.Kappa.BkW, S.space.H0 = c(0,1),
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none", trans.g = "[A,B]")
WkW_kappa_same <- ci.table(obs.y, h.fct = WkW.BkB_BkW.WkB_cons,
S.fct = Unweighted.Kappa.WkW, S.space.H0 = c(0,1),
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none", trans.g = "[A,B]")
# Constraints (2)
X_cond_RC_v <- c(1,1,0,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,-1,0,0,
1,1,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,
1,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,
1,1,0,0,0,1,1,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,
1,1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,
1,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,
1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,
1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
1,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,-1,0,
1,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,
1,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,
1,0,1,0,0,1,1,0,0,0,0,1,0,0,0,0,1,0,0,-1,0,
1,0,1,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,
1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,
1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,
1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
1,0,0,1,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,-1,
1,0,0,1,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,
1,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,
1,0,0,1,0,1,1,0,0,0,0,0,0,1,0,0,1,0,0,0,-1,
1,0,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,
1,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,
1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,1,1,
1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,
1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,
1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,1,1,1,
1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,
1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,
1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1)
X_cond_RC_mat <- matrix(X_cond_RC_v, ncol = 21, byrow = TRUE)
cond_RC_HLP_fit <- mph.fit(obs.y, L.fct = "logm", L.mean = TRUE,
X = X_cond_RC_mat,
strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
mph.summary(cond_RC_HLP_fit)
library(MASS)
X_cond_RC_U <- Null(X_cond_RC_mat)
cond_RC_MPH_fit <- mph.fit(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, strata = rep(c(1,2,3,4), each = 9),
fixed.strata = "none")
mph.summary(cond_RC_MPH_fit)
BkW_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.BkW,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
WkW_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.WkW,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
BkB_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.BkB,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
WkB_cond_RC <- ci.table(obs.y, h.fct = function(m) {t(X_cond_RC_U) %*% log(m)},
h.mean = TRUE, S.fct = Unweighted.Kappa.WkB,
S.space.H0 = c(0,1), trans.g = "[A,B]",
strata = rep(c(1,2,3,4), each = 9), fixed.strata = "none")
### Construct test-inversion CIs, without additionally imposed constraints.
# V. Binomial success rate parameter p.
# Model: 0 = x <- X | p ~ Bin(n = 5, p).
# Goal: Compute approximate 90% CIs for the success probability p.
bin_p_result <- ci.table(c(0, 5), h.fct = 0, S.fct = function(p) {p[1]},
S.space.H0 = c(0, 1), cc = 0.9, y.eps = 0.1)
# Example 2.1 in Lang (2008).
# Model: y = (39, 1) <- Y ~ mult(40, p1, p2).
# Goal: Compute approximate 95% CIs for the success probability p1.
bin_p_eg21_result <- ci.table(c(39,1), h.fct = 0, S.fct = function(p) {p[1]},
S.space.H0 = c(0,1), trans.g = "[A,B]")
# VI. Conditional probability.
# Model: y = (0, 39, 18, 11) <- Y ~ mult(68, p1, p2, p3, p4)
# Goal: Compute approximate 95% CIs for the conditional probability
# p1 / (p1 + p2).
cond_prob_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1] / (p[1] + p[2])},
S.space.H0 = c(0, 1), y.eps = 0.1)
# Model: y = (0, 39 // 18, 11) <- Y ~ prod mult(39, p1, p2 // 29, p3, p4).
# That is,
# y <- Y ~ MP(gamma, p | strata = c(1, 1, 2, 2), fixed = "all"),
# where gamma = (39, 29)'.
# Goal: Compute approximate 95% CIs for p1.
cond_prob_SS_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1]}, S.space.H0 = c(0, 1),
strata = c(1, 1, 2, 2), y.eps = 0.1)
# VII. Difference between conditional probabilities.
# Model: y = (0, 39, 18, 11) <- Y ~ mult(68, p1, p2, p3, p4)
# Goal: Compute approximate 95% CIs for the difference between conditional
# probabilities, p1 / (p1 + p2) - p3 / (p3 + p4).
diff_cond_prob_result <- ci.table(c(0, 39, 18, 11), h.fct = 0,
S.fct = function(p) {p[1]/(p[1]+p[2]) - p[3]/(p[3]+p[4])},
S.space.H0 = c(-1, 1), trans.g = "[A,B]")
# VIII. Gamma variant.
# Example 2.3 in Lang (2008).
# Model: y = (25, 25, 12 // 0, 1, 3)
# ~ prod mult(62, p11, p12, p13 // 4, p21, p22, p23).
# Goal: Compute approximate 95% CIs for the Gamma* parameter as
# described in Lang (2008).
Gamma_variant_23 <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
P.case.gt.control <- (p[2, 2] + p[2, 3]) * p[1, 1] + p[2, 3] * p[1, 2]
P.case.lt.control <- p[1, 2] * p[2, 1] + p[1, 3] * (p[2, 1] + p[2, 2])
P.case.neq.control <- P.case.gt.control + P.case.lt.control
P.case.gt.control / P.case.neq.control
}
Gamma_variant_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = Gamma_variant_23, S.space.H0 = c(0, 1),
trans.g = "[A,B]", strata = c(1, 1, 1, 2, 2, 2))
### Alternative code...
gammastar.fct <- function(p) {
nr <- nrow(p)
nc <- ncol(p)
probC <- 0
probD <- 0
for (i in 1:(nr-1)) {
for (j in 1:(nc-1)) {
Aij <- 0
for (h in (i+1):nr) {
for (k in (j+1):nc) {
Aij <- Aij + p[h, k]
}
}
probC <- probC + p[i, j] * Aij
}
}
for (i in 1:(nr-1)) {
for (j in 2:nc) {
Aij <- 0
for (h in (i+1):nr) {
for (k in 1:(j-1)) {
Aij <- Aij + p[h, k]
}
}
probD <- probD + p[i, j] * Aij
}
}
probC / (probC + probD)
}
Gamma_variant_23_a <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
gammastar.fct(p)
}
Gamma_variant_a_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = Gamma_variant_23_a,
S.space.H0 = c(0, 1), trans.g = "[A,B]",
strata = c(1, 1, 1, 2, 2, 2))
# IX. Global odds ratio.
# Model: y = (25, 25, 12 // 0, 1, 3)
# ~ prod mult(62, p11, p12, p13 // 4, p21, p22, p23).
# Goal: Compute approximate 95% CIs for the first global odds ratio.
global_odds_ratio_23_11 <- function(p) {
p <- matrix(p, 2, 3, byrow = TRUE)
p[1, 1] * (p[2, 2] + p[2, 3]) / (p[2, 1] * (p[1, 2] + p[1, 3]))
}
global_odds_ratio_result <- ci.table(c(25, 25, 12, 0, 1, 3), h.fct = 0,
S.fct = global_odds_ratio_23_11,
S.space.H0 = c(0, Inf), trans.g = "log",
strata = c(1, 1, 1, 2, 2, 2))
# X. Difference between product-multinomial probabilities.
# Example 2.2 in Lang (2008).
# Source (secondary): Agresti 2002:65
# Early study of the death penalty in Florida (Radelet)
# Victim Black...
# White Defendant 0/9 received Death Penalty
# Black Defendant 6/103 received Death Penalty
#
# Model: y = (0, 9 // 6, 97) <- Y ~ prod mult(9, p1, p2 // 103, p3, p4).
# Goal: Compute approximate 95% CIs for the difference between
# product-multinomial probabilities, p1 - p3.
diff_prod_mult_prob_result <- ci.table(c(0, 9, 6, 97), h.fct = 0,
S.fct = function(p) {p[1] - p[3]},
S.space.H0 = c(-1, 1),
trans.g = "Fisher's z",
strata = c(1, 1, 2, 2))
### Alternative (artificial) data that is even more sparse...
diff_prod_mult_prob_a_result <- ci.table(c(0, 9, 0, 97), h.fct = 0,
S.fct = function(p) {p[1] - p[3]},
S.space.H0 = c(-1, 1),
trans.g = "Fisher's z",
strata = c(1, 1, 2, 2), y.eps = 0.4)
# XI. Kappa coefficient.
# Example 2.4 in Lang (2008).
# Model: y = (4, 0, 0, 0, 1, 0, 0, 0, 15)
# <- Y ~ mult(20, p11, p12, ..., p33).
# Goal: Compute approximate 95% CIs for the unweighted kappa coefficient.
Kappa_coeff_33 <- function(p) {
p <- matrix(p, 3, 3, byrow = TRUE)
s1 <- p[1, 1] + p[2, 2] + p[3, 3]
prow <- apply(p, 1, sum)
pcol <- apply(p, 2, sum)
s2 <- prow[1] * pcol[1] + prow[2] * pcol[2] + prow[3] * pcol[3]
(s1 - s2) / (1 - s2)
}
kappa_coeff_result <- ci.table(c(4, 0, 0, 0, 1, 0, 0, 0, 15), h.fct = 0,
S.fct = Kappa_coeff_33, S.space.H0 = c(-1, 1))
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