Package 'cta'

cta: Contingency Table Analysis Based on ML Fitting of MPH Models

Description

Contingency table analysis is performed based on maximum likelihood (ML) fitting of multinomial-Poisson homogeneous (MPH) models (Lang, 2004) and homogeneous linear predictor (HLP) models (Lang, 2005). Objects computed include model goodness-of-fit statistics; likelihood-based (cell- and link-specific) residuals; and cell probability and expected count estimates along with standard errors. This package can also compute test-inversion–e.g. Wald, profile likelihood, score, power-divergence–confidence intervals for contingency table estimands, when table probabilities are potentially subject to equality constraints. See Lang (2008) and Zhu (2020) for test-inversion intervals.

Details

Please call the following two R functions in this cta package.

mph.fit: Computes maximum likelihood estimates and fit statistics for MPH and HLP models for contingency tables.

ci.table: Constructs test-inversion approximate confidence intervals for estimands in contingency tables with or without equality constraints.

Author(s)

Joseph B. Lang, Qiansheng Zhu

References

Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.

Lang, J. B. (2005) Homogeneous linear predictor models for contingency tables, Journal of the American Statistical Association, 100, 121–134.

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.

---

Matrix Direct Sum

Description

Matrix direct sum function. Creates a block diagonal matrix.

Usage

block.fct(...)

Arguments

...	R matrices (matrix).

Value

block.fct returns a block diagonal matrix, where the direct sum is in the order of the input matrices (matrix).

Author(s)

Joseph B. Lang

Examples

A <- matrix(c(1, 2, 3, 4), nrow = 2, byrow = TRUE)
B <- matrix(c(5, 6, 7, 8, 9, 10), nrow = 2, byrow = TRUE)
C <- matrix(c(11, 12, 13, 14), nrow = 1, byrow = TRUE)
block.fct(A, B, C)

---

HLP Link Status Check

Description

Checks whether the link function $L(\cdot)$ is a candidate HLP link function.

Specifically, this program checks whether $L(\cdot)$ satisfies certain necessary conditions that follow from a sufficient condition for HLP link status.

If the necessary conditions are satisfied then there is corroborating evidence that $L(\cdot)$ has HLP link status. If the necessary conditions are not satisfied, then the sufficient condition for HLP link status is not satisfied, so $L(\cdot)$ may or may not have HLP link status.

Usage

check.HLP(L.fct, Z, tol = NULL)

Arguments

L.fct	An R function object, indicating the link $L(\cdot)$ for HLP link status check.
Z	Population (aka strata) matrix $Z$.
tol	The pre-set tolerance with which norm(diff) is to be compared with.

Details

The main idea:

The model $L(m) = X\beta$ is an HLP model if $L(\cdot)$ is a smooth link function that satisfies the HLP conditions with respect to $Z$ (i.e. strata $s$) and $X$. That is,

• (1) $L(\cdot)$ has HLP link status with respect to $Z$, and

• (2) The implied constraint function $h(m) = U'L(m)$ is $Z$ homogeneous. Here, $null(U') = span(X)$.

Here, (1) $L(\cdot)$ has HLP link status with respect to $Z$ if, for $m = Diag(Z\gamma)p$, equivalently, for $\gamma = Z'm$ and $p = Diag^{-1}(ZZ'm)m$,

• (1)(a) $L(m) = a(\gamma) + L(p)$, where $a(\gamma_{1}/\gamma_{2}) - a(1) = a(\gamma_{1}) - a(\gamma_{2})$, i.e. $a(\gamma)$ has the form $C \log\gamma + \texttt{constant}$; or

• (1)(b) $L(m) = G(\gamma) L(p)$, where $G(\gamma)$ is a diagonal matrix with diagonal elements that are powers of the $\gamma$ elements, i.e. $L(\cdot)$ is $Z$ homogeneous (see Lang (2004)); or

• (1)(c) The components of $L(\cdot)$ are a mixture of types (a) and (b): $L_{j}(m) = a_{j}(\gamma) + L_{j}(p)$ or $L_{j}(m) = G_{j}(\gamma) L_{j}(p)$, $j = 1, \ldots, l$.

N.B. Lang (2005) defined HLP models as those satisfying (1)(a) and (2). mph.fit uses a broader definition of HLP model. Specifically, models satisfying (1)(b) and (2) or (1)(c) and (2) are also considered HLP models.

Conditions (1)(b) and (2) can be checked using the check.homog function. Condition (1)(c) is not checked.

This function, check.HLP, is concerned with sufficient condition (1)(a) only. If $L(\cdot)$ satisfies (1)(a) then

• (i) $\texttt{diff1} = [L(Diag(Z\gamma_{1})p_{1}) - L(Diag(Z\gamma_{2})p_{1})] - [L(Diag(Z \gamma_{1}/\gamma_{2})p_{1}) - L(p_{1})] = 0$, and

• (ii) $\texttt{diff2} = [L(Diag(Z\gamma_{1})p_{1}) - L(Diag(Z\gamma_{1})p_{2})] - [L(p_{1}) - L(p_{2})] = 0$.

Here $p_{i} = Diag^{-1}(ZZ'm_{i})m_{i}$, where $m_{i} = Diag(Z\gamma_{i})p_{i}$, $i = 1, 2$.

This program randomly generates g1 ($\gamma_{1}$), g2 ($\gamma_{2}$), p1, p2, and computes norm(diff) = sqrt(norm(diff1)^2 + norm(diff2)^2). It returns a warning if norm(diff) is too far from $0$.

Value

check.HLP returns a character string chk. If chk = "", then there is corroborating evidence that $L(\cdot)$ has HLP link status. If chk = paste("L(m) may not be an HLP link [based on tol=",tol,"]!"), then the sufficient condition for HLP link status is not satisfied, so $L(\cdot)$ may or may not have HLP link status.

Author(s)

Joseph B. Lang

References

Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.

Lang, J. B. (2005) Homogeneous linear predictor models for contingency tables, Journal of the American Statistical Association, 100, 121–134.

See Also

mph.fit, check.homog, check.zero.order.homog

Examples

# 3-by-3-by-3 Table.
# For a description of the model, see Michael Haber's Example 2,
# p. 433, in  Biometrics (in Shorter Communications), Vol. 42,
# No. 2. (Jun., 1986), pp. 429-435.
A <- gl(3, 9, 27)
B <- gl(3, 3, 27)
C <- gl(3, 1, 27)
MAB <- kronecker(diag(9), matrix(1, 1, 3))
MAC <- kronecker(diag(3), kronecker(matrix(1, 1, 3), diag(3)))
MBC <- kronecker(matrix(1, 1, 3), diag(9))
M <- rbind(MAB, MAC, MBC)
Mr <- M[-c(3, 6, 7, 8, 9, 12, 15, 16, 17, 18, 21, 24,
           25, 26, 27), ]
C <- c(1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 1, -1, -1, 1, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1)
C <- matrix(C, 3, 12, byrow = TRUE)
L.fct <- function(m) {
  p <- m / sum(m)
  C %*% log(Mr %*% p)
}
Z <- matrix(rep(1, 27), ncol = 1)
check.HLP(L.fct, Z)

---

$Z$ Homogeneity Check

Description

Checks whether the constraint function $h(\cdot)$ satisfies a necessary condition for $Z$ homogeneity.

Usage

check.homog(h.fct, Z, tol = NULL)

Arguments

h.fct	An R function object, indicating the constraint function $h(\cdot)$ for $Z$ homogeneity check.
Z	Population (aka strata) matrix $Z$.
tol	The pre-set tolerance with which norm(diff) is to be compared with.

Details

The main idea:

$h(\cdot)$ is $Z$ homogeneous if $h(Diag(Z\gamma)x) = G(\gamma)h(x)$, where $G$ is a diagonal matrix with $\gamma$ elements raised to some power.

As a check, if $h(\cdot)$ is homogeneous then

$h(Diag(Z\gamma) x_{1}) / h(Diag(Z\gamma) x_{2}) = h(x_{1}) / h(x_{2});$

That is,

$\texttt{diff} = h(Diag(Z\gamma) x_{1}) h(x_{2}) - h(Diag(Z\gamma) x_{2}) h(x_{1}) = 0.$

Here, the division and multiplication are taken element-wise.

This program randomly generates gamma, x1, and x2, and computes norm(diff). It returns a warning if norm(diff) is too far from $0$.

Value

check.homog returns a character string chk that states whether $h(\cdot)$ is $Z$ homogeneous. If chk = "", it means that based on the necessary condition, we cannot state that $h(\cdot)$ is not $Z$ homogeneous.

Author(s)

Joseph B. Lang

References

Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.

See Also

check.zero.order.homog, mph.fit, check.HLP

Examples

# EXAMPLE 1
h.fct <- function(m) {m[1] - m[2]}
Z <- matrix(c(1, 1), nrow = 2)
check.homog(h.fct, Z)

# EXAMPLE 2
h.fct.2 <- function(m) {m[1]^2 - m[2]}
Z <- matrix(c(1, 1), nrow = 2)
check.homog(h.fct.2, Z)

---

Zero-Order $Z$ Homogeneity Check

Description

Checks whether the estimand function $S(\cdot)$ is zero-order $Z$ homogeneous.

Usage

check.zero.order.homog(S.fct, Z, tol = 1e-9)

Arguments

S.fct	An R function object, indicating the estimand function $S(\cdot)$ for zero-order $Z$ homogeneity check.
Z	Population (aka strata) matrix $Z$.
tol	The pre-set tolerance with which norm(diff.LRHS) is to be compared with.

Details

The main idea:

$S(\cdot)$ is zero-order $Z$ homogeneous if $S(Diag(Z\gamma) x) = S(x)$, for all $\gamma > 0$, and for all $x$ within its domain. This program randomly generates gam ($\gamma$) and x ($x$), and computes

$\texttt{diff.LRHS} = S(Diag(Z\gamma) x) - S(x).$

It returns a warning if norm(diff.LRHS) is too far from $0$.

Value

check.zero.order.homog returns a character string check.result that states whether $S(\cdot)$ is zero-order $Z$ homogeneous. If check.result = "", it means that we cannot state that $S(\cdot)$ is not zero-order $Z$ homogeneous based on the result of the check.

Author(s)

Qiansheng Zhu

References

Lang, J. B. (2004) Multinomial-Poisson homogeneous models for contingency tables, Annals of Statistics, 32, 340–383.

See Also

check.homog, check.HLP

Examples

# EXAMPLE 1
S.fct <- function(m) {(m[1] - m[2]) / (m[1] + m[2])}
Z <- matrix(c(1, 1, 1, 1), nrow = 4)
check.zero.order.homog(S.fct, Z)

# EXAMPLE 2
S.fct.2 <- function(m) {m[1] - m[2]}
Z <- matrix(c(1, 1, 1, 1), nrow = 4)
check.zero.order.homog(S.fct.2, Z)

---

Test-Inversion CIs for Estimands in Contingency Tables

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

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 $h_{0}^{*}(m) = (h_{0}'(m), S_{0}(m) - \psi, samp_{0}'(m))'$. Here $h_{0}(m) = 0$ is (are) the imposed equality constraint(s), written in terms of $m$; $S_{0}(m) - \psi = 0$ means that the estimand of interest is equal to $\psi$; $samp_{0}(m) = 0$ 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 $(-\infty, \infty)$, 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 $(-\infty, -1] \cup [1, \infty)$, 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 $\lambda$). 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 $\lambda$ 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 $\epsilon$ adjustment included in the transformation $g$. For example, the "[A, B]" transformation $g$ with the small $\epsilon$ is $g(x) = \log(x - A + \epsilon) - \log(B + \epsilon - x).$ 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 $\delta$ 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 $H_{0}(\psi): S_{0}(m) = \psi$ vs. not $H_{0}(\psi)$ under the general hypothesis $H_{0}: (h_{0}'(m), samp_{0}'(m))' = 0$, for a certain $\psi$, is within tol of the critical value, then we stop the iterations, and this current $\psi$ 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 $\psi$'s that are in nearby iterates in the corresponding tests $H_{0}(\psi)$ vs. not $H_{0}(\psi)$ under the general hypothesis $H_{0}$, are less than tol.psi apart in distance, then we stop the iterations, and the current $\psi$ 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 $\psi$ in the hypothesis test $H_{0}(\psi)$ vs. not $H_{0}(\psi)$ is, on some scale, too far away from $\widehat{\psi}$ which is the constrained MLE of the estimand subject to $H_{0}$, although this $\psi$ 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, $\ldots$, 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 $\lambda$ 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

mph.fit, mph.summary

Examples

### 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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