Description of ciuupi

Description of the confidence interval that utilizes uncertain prior information (CIUUPI)

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 \(\theta = a^{\top} \beta\). Let \(\tau = c^{\top} \beta - t = 0\), where \(a\) and \(c\) are specified linearly independent nonzero \(p\)-vectors and \(t\) is a specified number. Suppose that we have uncertain prior information that \(\tau = 0\). Define the scaled expected length of a confidence interval to be \[\frac{\text{expected length of this confidence interval}} {\text{expected length of the standard } 1-\alpha \text{ confidence interval for }\theta}.\]

This package computes a confidence interval, with minimum coverage \(1 - \alpha\), for \(\theta\) that utilizes the uncertain prior information that \(\tau = 0\) in the following sense. This confidence interval has scaled expected length that (a) is substantially less 1 when \(\tau=0\) and (b) has maximum value that is not too large. In addition, this confidence interval approaches the standard \(1-\alpha\) confidence interval for \(\theta\) when the data and the prior information are strongly discordant.

Let \(\widehat{\beta}\) denote the least squares estimator of \(\beta\). Then \(\widehat{\theta} = a^{\top} \widehat{\beta}\) and \(\widehat{\tau} = c^{\top} \widehat{\beta} - t\) are the least squares estimators of \(\theta\) and \(\tau\), respectively. Also let \(v_{\theta} = \text{Var}(\widehat{\theta})/\sigma^2 = a^{\top} (X^{\top}X)^{-1}a\) and \(v_{\tau} = \text{Var}(\widehat{\tau})/\sigma^2 = c^{\top} (X^{\top}X)^{-1}c\). The known correlation between \(\widehat{\theta}\) and \(\widehat{\tau}\) is \(\rho = a^{\top} (X^{\top}X)^{-1}c / (v_{\theta} \, v_{\tau})^{1/2}\). Let \(\gamma = \tau / (\sigma \, v_{\tau}^{1/2})\) and \(\hat{\gamma} = \hat{\tau}/(\sigma \, v_{\tau}^{1/2})\). The confidence interval for \(\theta\) that utilizes uncertain prior information that \(\tau=0\) has the form \[ \text{CI}(b,s) =\left[ \hat{\theta} - v_{\theta}^{1/2} \, \sigma \, b(\hat{\gamma}) - v_{\theta}^{1/2} \, \sigma \, s(\hat{\gamma}), \, \hat{\theta} - v_{\theta}^{1/2} \, \sigma \, b(\hat{\gamma}) + v_{\theta}^{1/2} \, \sigma \, s(\hat{\gamma}) \right], \] where \(b: \mathbb{R} \rightarrow \mathbb{R}\) is an odd continuous function and \(s: \mathbb{R} \rightarrow [0, \infty)\) is an even continuous function. In addition, \(b(x)=0\) and \(s(x)= z_{1-\alpha/2}\) for all \(|x| \ge d\), where \(z_{1-\alpha/2}\) denotes the \(1 - \alpha/2\) quantile of the standard normal distribution. For given \(d > 0\) and positive integer \(q\), let \(h = d / q\) and specify the functions \(b\) and \(s\) 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 \(k=-q,\dots,q\) 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 \(x \in [-d,d]\) are found via cubic spline interpolation using the values of \(b(kh)\) and \(s(kh)\) for \(k=-q,\dots,q\). This is through natural (default) or clamped cubic spline interpolation.

Version 1.0 of ciuupi was described by Mainzer and Kabaila (2019), who chose \(d = 6\) and \(q = 6\). However, it was subsequently found that this choice is no longer adequate when \(\alpha\) is very small and/or \(|\rho|\) is close to 1. 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. Version 1.2 uses this formula and sets \(q = \lceil d/0.75 \rceil\) and \(h = d/q\).

The confidence interval that utilizes the uncertain prior information (CIUUPI) is found by computing the value of the vector \(\big(b(h),...,b((q-1)h), s(0),s(h)...,s((q-1)h)\big)\), such that the confidence interval \(\text{CI}(b,s)\) has minimum coverage probability \(1 - \alpha\) and the desired scaled expected length properties. The method used is an obvious extension of that described by Mainzer and Kabaila (2019). However, version 1.2 employs a far more efficient algorithm for the computation of \(\lambda^*\), which forms part of the computation of the CIUUPI.

This confidence interval has the following three practical applications. Firstly, if \(\sigma^2\) has been accurately estimated from previous data then it may be treated as being effectively known. Secondly, for sufficiently large \(n - p\) (\(n-p \ge 30\), say), if we replace the assumed known value of \(\sigma^2\) by its usual estimator in the formula for the confidence interval then the resulting interval has, to a very good approximation, the same coverage probability and expected length properties as when \(\sigma^2\) is known. Thirdly, some more complicated models can be approximated by the linear regression model with \(\sigma^2\) known when certain unknown parameters are replaced by estimates (see e.g. Kabaila and Mainzer, 2019).

Functions in this package

We attach the package ciuupi using

library(ciuupi)

We illustrate the application of the functions in the package using the real-life example 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). The resulting design matrix \(X\) is specified by the command

x1 <- c(-1, 1, -1, 1)
x2 <- c(-1, -1, 1, 1)
X <- cbind(rep(1, 4), x1, x2, x1*x2)

A description of the parameter of interest and the uncertain prior information is given in Discussion 5.8 on 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)

The uncertain prior information is that the two-factor interaction term is zero i.e. \(\tau = 0\), where \(\tau = c^{\top} \beta\) and the column vector \(c\) is specified by the command

c <- c(0, 0, 0, 1)

# Compute rho
rho <- acX_to_rho(a, c, X)
print(rho)
#> [1] -0.7071068

plot_b(bs.list.example)

plot_s(bs.list.example)

plot_cp(bs.list.example)

plot_squared_sel(bs.list.example)

y <- c(87.2, 88.4, 86.7, 89.2)
sig <- 0.8

alpha <- 0.05
t <- 0
ciuupi_observed_value(a, c, X, alpha, bs.list.example, t, y, sig = sig)
#>             lower    upper
#> ciuupi -0.7696589 3.221789

alpha <- 0.05
ci_standard(a, X, y, alpha, sig)
#>              lower    upper
#> standard -1.017446 3.417446

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 Mainzer, R. (2019). An alternative to confidence intervals constructed after a Hausman pretest in panel data. doi:10.1111/anzs.12278

Mainzer, R. and Kabaila, P. (2019). ciuupi: An R package for computing confidence intervals that utilize uncertain prior information. doi:10.32614/RJ-2019-026