Description Usage Arguments Details Value References Examples
Compute the confidence interval that utilizes the uncertain prior information
1 |
alpha |
1 - |
a |
A vector used to specify the parameter of interest |
c |
A vector used to specify the parameter about which we have uncertain prior information |
x |
The n by p design matrix |
bsvec |
The vector (b(1),...,b(5),s(0),...,s(5)) that specifies the CIUUPI |
t |
A number used to specify the parameter about which we have uncertain prior information |
y |
The n-vector of observed responses |
natural |
Equal to 1 (default) if b and s functions are obtained by natural cubic spline interpolation or 0 if obtained by clamped cubic spline interpolation |
sig |
Standard deviation of the random error. If a value is not
specified then |
Suppose that
y = X β + ε
where y is a random n-vector of
responses, X is a known n by p matrix with linearly
independent columns, β is an unknown parameter p-vector and
ε with components that are iid normally distributed
with zero mean and known variance.
Then ciuupi
will compute a confidence interval for
θ=a
' β that utilizes the uncertain prior information that
c
' β - t
= 0, where a
and c
are specified
linearly independent vectors and t
is a specified number.
In the example below we use the data set described in Table 7.5 of Box et al. (1963). A description of the parameter of interest and the parameter about which we have uncertain prior information is given in Dicsussion 5.8, p.3426 of Kabaila and Giri (2009).
The confidence interval that utilizes uncertain prior information
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.
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 | # Specify alpha, a, c, x
alpha <- 0.05
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)
# Find the vector (b(1),b(2),...,b(5),s(0),s(1),...,s(5)) that specifies the
# CIUUPI: (this may take a few minutes to run)
bsvec <- bsciuupi(alpha, a = a, c = c, x = x)
# The result (to 7 decimal places) is
bsvec <- c(-0.03639701, -0.18051953, -0.25111411, -0.15830362, -0.04479113,
1.71997203, 1.79147968, 2.03881195, 2.19926399, 2.11845381,
2.00482563)
# Specify t and y
t <- 0
y <- c(87.2, 88.4, 86.7, 89.2)
# Find the CIUUPI
res <- ciuupi(alpha, a, c, x, bsvec, t, y, natural = 1, sig = 0.8)
res
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