Computes the known correlation \rho between \widehat{\theta} and \widehat{\tau} from the p-vectors a and c and the design matrix X

Description

Computes the known correlation \rho between \widehat{\theta} and \widehat{\tau}. This correlation is computed from the p-vectors a and c and the n \times p design matrix X, with linearly independent columns, using the formula \rho=a^{\top}(X^{\top} X)^{-1} c /(v_{\theta} \, v_{\tau})^{1/2}, where v_{\theta} =a^{\top}(X^{\top} X)^{-1}a and v_{\tau} =c^{\top}(X^{\top} X)^{-1}c.

Usage

acX_to_rho(a, c, X)

Arguments

Value

The known correlation \rho between \widehat{\theta} and \widehat{\tau}.

Examples

a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
rho <- acX_to_rho(a, c, X)
print(rho)

The list that specifies the CIUUPI for the example

Description

In this example, the dataset described in Table 7.5 of Box et al. (1963) is used. The design matrix X is specified by the command X <- cbind(rep(1,4), c(-1, 1, -1, 1), c(-1, -1, 1, 1), c(1, -1, -1, 1)). A description of the parameter of interest is given in Discussion 5.8, p.3426 of Kabaila and Giri (2009). The parameter of interest is \theta = a^{\top} \beta, where the column vector a is specified by the command a <- c(0, 2, 0, -2). For this example, we have uncertain prior information that \tau = c^{\top} \beta = 0, where the column vector c is specified by the command c <- c(0, 0, 0, 1). The known correlation \rho between \widehat{\theta} and \widehat{\tau} is computed using the command rho <- acX_to_rho(a, c, X). The desired minimum coverage probability of the CIUUPI is 1 - \alpha, where \alpha = 0.05, which is specified by the command alpha <- 0.05. The CIUUPI is determined by \alpha and \rho and is found using the command bs.list.example <- bs_ciuupi(alpha, rho), which takes about 5 minutes to run.

Usage

bs.list.example

Format

An object of class list of length 8.

References

Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P. (1963) The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.

Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.

Computes the the functions b and s that specify the CIUUPI for all possible values of \sigma and the observed response vector

Description

Chooses the positive number d and the positive integer q, sets h=d/q, and then computes the (2q-1)-vector \big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h)\big) that determines, via cubic spline interpolation, the functions b and s which specify the confidence interval for \theta that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and the observed response vector. To an excellent approximation, this confidence interval has minimum coverage probability 1-\alpha.

Usage

bs_ciuupi(alpha, rho, natural = 1)

Arguments

Details

Suppose that

y = X \beta + \varepsilon

where y is a random n-vector of responses, X is a known n by p matrix with linearly independent columns, \beta is an unknown parameter p-vector and \varepsilon is the random error with components that are iid normally distributed with zero mean and known variance \sigma^2. The parameter of interest is \theta = a^{\top} \beta. Also let \tau = c^{\top}\beta -t, where a and c are specified linearly independent vectors and t is a specified number. The uncertain prior information is that \tau = 0.

Let rho denote the known correlation between the \widehat{\theta} and \widehat{\tau}. We can compute rho from given values of a, c and X using the function acX_to_rho.

The confidence interval for \theta, with minimum coverage probability 1-alpha, that utilizes the uncertain prior information that \tau = 0 belongs to a class of confidence intervals indexed by the functions b and s. The function b is an odd continuous function and the function s is an even continuous function. In addition, b(x)=0 and s(x) is equal to the 1-alpha/2 quantile of the standard normal distribution for all |x| \ge d, where d is a given positive number. Extensive numerical explorations have been used to find a formula (in terms of alpha and rho) for a 'goldilocks' value of d that is neither too large nor too small. Then let q=ceiling(d/0.75) and h=d/q. The values of the functions b and s in the interval [-d,d] are specified by the (2q-1)-vector

\big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h) \big).

The values of b(kh) and s(kh) for k=-q,...,q are deduced from this vector using the assumptions made about the functions b and s. The values of b(x) and s(x) for any x in the interval [-d, d] are then found using cube spline interpolation using the values of b(kh) and s(kh) for k=-q,...,q. For natural=1 (default) this is 'natural' cubic spline interpolation and for natural=0 this is 'clamped' cubic spline interpolation.

The vector \big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h)\big) is found by numerical nonlinear constrained optimization so that the confidence interval has minimum coverage probability 1-alpha and utilizes the uncertain prior information through its desirable expected length properties. This optimization is performed using the slsqp function in the nloptr package.

Value

A list with the following components.

alpha, rho, natural: the inputs

d: a 'goldilocks' value of d that is not too large and not too small

n.ints: number of equal-length consecutive intervals whose union is [0,d], this is the same as q

lambda.star: the computed value of \lambda^*

bsvec: the vector \big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h) \big) that determines the functions b and s that specify the CIUUPI for all possible values of \sigma and observed response vector

comp.time: the computation time in seconds

Examples

alpha <- 0.05
rho <- - 1 / sqrt(2)
bs.list <- bs_ciuupi(alpha, rho)

Evaluate the functions b and s at x

Description

Evaluate the functions b and s, as specified by bsvec, alpha, d, n.ints and natural, at x.

Usage

bsspline(x, bsvec, alpha, d, n.ints, natural)

Arguments

Value

A data frame containing x and the corresponding values of the functions b and s.

Examples

x <- seq(0, 8, by = 1)
alpha <- bs.list.example$alpha
natural <- bs.list.example$natural
d <- bs.list.example$d
n.ints <- bs.list.example$n.ints
bsvec <- bs.list.example$bsvec
bs <- bsspline(x, bsvec, alpha, d, n.ints, natural)

For given observed response vector y, compute the standard 1 - \alpha confidence interval

Description

If \sigma is provided then compute the standard 1 - \alpha confidence interval for \theta. If \sigma is not provided then, as long as n-p \ge 30, replace \sigma by its estimate to compute an approximate 1 - \alpha confidence interval for \theta.

Usage

ci_standard(a, X, y, alpha, sig = NULL)

Arguments

Details

Suppose that

y = X \beta + \varepsilon,

where y is a random n-vector of responses, X is a known n \times p matrix with linearly independent columns, \beta is an unknown parameter p-vector, and \varepsilon \sim N(0, \, \sigma^2 \, I), with \sigma^2 assumed known. Suppose that the parameter of interest is \theta = a^{\top} \beta. The R function ci_standard computes the standard 1 - \alpha confidence interval for \theta.

The example below is described in Discussion 5.8 on p.3426 of Kabaila and Giri (2009). This example is obtained by extracting a 2 \times 2 factorial data set from the 2^3 factorial data set described in Table 7.5 of Box et al. (1963).

Value

If \sigma is provided then a data frame of the lower and upper endpoints of the standard 1 - \alpha confidence interval for \theta. If \sigma is not provided then, as long as n-p \ge 30, a data frame of the lower and upper endpoints of an approximation to this confidence interval.

References

Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P. (1963) The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.

Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.

Examples

y <- c(87.2, 88.4, 86.7, 89.2)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
a <- c(0, 2, 0, -2)
ci_standard(a, X, y, 0.05, sig = 0.8)

For given observed response vector y, compute the confidence interval that utilizes the uncertain prior information (CIUUPI)

Description

If \sigma is provided then, for given observed response vector y, compute the confidence interval, with minimum coverage probability 1-\alpha, for the parameter \theta =a^{\top}\beta that utilizes the uncertain prior information that the parameter \tau=c^{\top} \beta-t (specified by the vector c and the number t) takes the value 0. If \sigma is not provided then, as long as n-p \ge 30, replace \sigma by its estimate to compute an approximation to the CIUUPI for \theta.

Usage

ciuupi_observed_value(a, c, X, alpha, bs.list, t, y, sig = NULL)

Arguments

Details

Suppose that

y = X \beta + \varepsilon

where y is a random n-vector of responses, X is a known n \times p matrix with linearly independent columns, \beta is an unknown parameter p-vector and \varepsilon has components that are iid normally distributed with zero mean and known variance. Suppose that \theta=a^{\top} \beta is the parameter of interest, where a is a specified vector. Let \tau=c^{\top} \beta -t, where c is a specified vector, t is a specified number and a and c are linearly independent vectors. Also suppose that we have uncertain prior information that \tau = 0. For given observed response vector y and a design matrix X, ciuupi_observed_value computes the confidence interval, with minimum coverage probability 1-alpha, for \theta that utilizes the uncertain prior information that \tau = 0.

The example below is described in Discussion 5.8 on p.3426 of Kabaila and Giri (2009). This example is obtained by extracting a 2 \times 2 factorial data set from the 2^3 factorial data set described in Table 7.5 of Box et al. (1963).

Value

If \sigma is provided then a data frame of the lower and upper endpoints of the confidence interval, with minimum coverage probability 1-\alpha, for the parameter \theta that utilizes the uncertain prior information that \tau = 0. If \sigma is not provided then, as long as n-p \ge 30, a data frame of the lower and upper endpoints of an approximation to this confidence interval.

References

Box, G.E.P., Connor, L.R., Cousins, W.R., Davies, O.L., Hinsworth, F.R., Sillitto, G.P. (1963) The Design and Analysis of Industrial Experiments, 2nd edition, reprinted. Oliver and Boyd, London.

Kabaila, P. and Giri, K. (2009) Confidence intervals in regression utilizing prior information. Journal of Statistical Planning and Inference, 139, 3419 - 3429.

Examples

a <- c(0, 2, 0, -2)
c <- c(0, 0, 0, 1)
x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)
alpha <- 0.05
t <- 0
y <- c(87.2, 88.4, 86.7, 89.2)
sig <- 0.8
ciuupi_observed_value(a, c, X, alpha, bs.list.example, t, y, sig=sig)

Compute the coverage probability of the CIUUPI

Description

Evaluate the coverage probability of the confidence interval that utilizes uncertain prior information (CIUUPI) at gam. The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector.

Usage

cpciuupi(gam, n.nodes, bs.list)

Arguments

Value

The value(s) of the coverage probability of the CIUUPI at gam.

Examples

gam <- seq(0, 10, by = 0.2)
n.nodes <- 10
cp <- cpciuupi(gam, n.nodes, bs.list.example)

Plot the graph of the odd function b used in the specification of the CIUUPI

Description

The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector. The R function plot_b plots the graph of the odd function b.

Usage

plot_b(bs.list)

Arguments

Value

A plot of the graph of the odd function b used in the specification of the CIUUPI.

Examples

plot_b(bs.list.example)

Plot the graph of the coverage probability of the CIUUPI

Description

The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector. The coverage probability of the CIUUPI is an even function of the unknown parameter \gamma = \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big). The R function plot_cp plots the graph of the coverage probability of the CIUUPI, as a function of |\gamma|. To provide a stringent assessment of this coverage probability, we use a fine equally-spaced grid seq(0, (d+4), by = 0.01) of values of \gamma and Gauss Legendre quadrature using 10 nodes in the relevant integrals. By contrast, for the computation of the CIUUPI, implemented in bs_ciuupi, we require that the coverage probability of this confidence interval is greater than or equal to 1-\alpha for the equally-spaced grid seq(0, (d+2), by = 0.05) of values of \gamma and we use Gauss Legendre quadrature with 5 nodes in the relevant integrals.

Usage

plot_cp(bs.list)

Arguments

Value

A plot of the graph of the coverage probability of the CIUUPI as a function of |\gamma|, where \gamma denotes the unknown parameter \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big).

Examples

plot_cp(bs.list.example)

Plot the graph of the even function s used in the specification of the CIUUPI

Description

The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector. The R function plot_s plots the graph of the odd function s.

Usage

plot_s(bs.list)

Arguments

Value

A plot of the graph of the even function s used in the specification of the CIUUPI.

Examples

plot_s(bs.list.example)

Plot the graph of the squared scaled expected length of the CIUUPI

Description

The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector. The scaled expected length of the CIUUPI is an even function of the unknown parameter \gamma = \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big). The R function plot_squared_sel plots the graph of the squared scaled expected length (i.e. squared SEL) of the CIUUPI, as a function of |\gamma|. To provide a stringent assessment of this squared SEL, we use a grid seq(0, (d+4), by = 0.01) of values of \gamma and Gauss Legendre quadrature with 10 nodes in the relevant integrals. By contrast, for the computation of the CIUUPI, implemented in bs_ciuupi, we use Gauss Legendre quadrature with 5 nodes in the relevant integrals.

Usage

plot_squared_sel(bs.list)

Arguments

Value

A plot of the graph of the squared scaled expected length (i.e. squared SEL) of the CIUUPI as a function of |\gamma|, where \gamma denotes the unknown parameter \tau \big/ \big(\sigma \, v_{\tau}^{1/2} \big).

Examples

plot_squared_sel(bs.list.example)

Compute the scaled expected length of the CIUUPI

Description

Evaluate the scaled expected length of the confidence interval that utilizes uncertain prior information (CIUUPI) at gam. This scaled expected length is defined to be the expected length of the CIUUPI divided by the expected length of the standard 1 - \alpha confidence interval. The input bs.list determines the functions b and s that specify the confidence interval that utilizes the uncertain prior information (CIUUPI), for all possible values of \sigma and observed response vector.

Usage

selciuupi(gam, n.nodes, bs.list)

Arguments

Value

The value(s) of the scaled expected length at gam.

Examples

gam <- seq(0, 10, by = 0.2)
n.nodes <- 10
sel <- selciuupi(gam, n.nodes, bs.list.example)