Automatically solve an equation of `pt` for `ncp` - r

Question

I wonder if it is possible to efficiently change ncp in the below code such that x becomes .025 and .975 (within rounding error).
x <- pt(q = 5, df = 19, ncp = ?)
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Clarification
q = 5 and df = 19 (above) are just two hypothetical numbers, so q and df could be any other two numbers. What I expect is a function / routine, that takes q and df as input.

Answer 1

What is wrong with uniroot?
f <- function (ncp, alpha) pt(q = 5, df = 19, ncp = ncp) - alpha
par(mfrow = c(1,2))
curve(f(ncp, 0.025), from = 5, to = 10, xname = "ncp", main = "0.025")
abline(h = 0)
curve(f(ncp, 0.975), from = 0, to = 5, xname = "ncp", main = "0.975")
abline(h = 0)
So for 0.025 case, the root lies in (7, 8); for 0.975 case, the root lies in (2, 3).
uniroot(f, c(7, 8), alpha = 0.025)$root
#[1] 7.476482
uniroot(f, c(2, 3), alpha = 0.975)$root
#[1] 2.443316
---------
(After some discussion...)
OK, now I see your ultimate goal. You want to implement this equation solver as a function, with input q and df. So they are unknown, but fixed. They might come out of an experiment.
Ideally if there is an analytical solution, i.e., ncp can be written as a formula in terms of q, df and alpha, that would be so great. However, this is not possible for t-distribution.
Numerical solution is the way, but uniroot is not a great option for this purpose, as it relies on "plot - view - guess - specification". The answer by loki is also crude but with some improvement. It is a grid search, with fixed step size. Start from a value near 0, say 0.001, and increase this value and check for approximation error. We stop when this error fails to decrease.
This really initiates the idea of numerical optimization with Newton-method or quasi-Newton method. In 1D case, we can use function optimize. It does variable step size in searching, so it converges faster than a fixed step-size searching.
Let's define our function as:
ncp_solver <- function (alpha, q, df) {
## objective function: we minimize squared approximation error
obj_fun <- function (ncp, alpha = alpha, q = q, df = df) {
(pt(q = q, df = df, ncp = ncp) - alpha) ^ 2
}
## now we call `optimize`
oo <- optimize(obj_fun, interval = c(-37.62, 37.62), alpha = alpha, q = q, df = df)
## post processing
oo <- unlist(oo, use.names = FALSE) ## list to numerical vector
oo[2] <- sqrt(oo[2]) ## squared error to absolute error
## return
setNames(oo, c("ncp", "abs.error"))
}
Note, -37.62 / 37.62 is chosen as lower / upper bound for ncp, as it is the maximum supported by t-distribution in R (read ?dt).
For example, let's try this function. If you, as given in your question, has q = 5 and df = 19:
ncp_solver(alpha = 0.025, q = 5, df = 19)
# ncp abs.error
#7.476472e+00 1.251142e-07
The result is a named vector, with ncp and absolute approximation error.
Similarly we can do:
ncp_solver(alpha = 0.975, q = 5, df = 19)
# ncp abs.error
#2.443347e+00 7.221928e-07
----------
Follow up
Is it possible that in the function ncp_solver(), alpha takes a c(.025, .975) together?
Why not wrapping it up for a "vectorization":
sapply(c(0.025, 0.975), ncp_solver, q = 5, df = 19)
# [,1] [,2]
#ncp 7.476472e+00 2.443347e+00
#abs.error 1.251142e-07 7.221928e-07
How come 0.025 gives upper bound of confidence interval, while 0.975 gives lower bound of confidence interval? Should this relationship reversed?
No surprise. By default pt computes lower tail probability. If you want the "right" relationship, set lower.tail = FALSE in pt:
ncp_solver <- function (alpha, q, df) {
## objective function: we minimize squared approximation error
obj_fun <- function (ncp, alpha = alpha, q = q, df = df) {
(pt(q = q, df = df, ncp = ncp, lower.tail = FALSE) - alpha) ^ 2
}
## now we call `optimize`
oo <- optimize(obj_fun, interval = c(-37.62, 37.62), alpha = alpha, q = q, df = df)
## post processing
oo <- unlist(oo, use.names = FALSE) ## list to numerical vector
oo[2] <- sqrt(oo[2]) ## squared error to absolute error
## return
setNames(oo, c("ncp", "abs.error"))
}
Now you see:
ncp_solver(0.025, 5, 19)[[1]] ## use "[[" not "[" to drop name
#[1] 2.443316
ncp_solver(0.975, 5, 19)[[1]]
#[1] 7.476492
--------
Bug report and fix
I was reported that the above ncp_solver is unstable. For example:
ncp_solver(alpha = 0.025, q = 0, df = 98)
# ncp abs.error
#-8.880922 0.025000
But on the other hand, if we double check with uniroot here:
f <- function (ncp, alpha) pt(q = 0, df = 98, ncp = ncp, lower.tail = FALSE) - alpha
curve(f(ncp, 0.025), from = -3, to = 0, xname = "ncp"); abline(h = 0)
uniroot(f, c(-2, -1.5), 0.025)$root
#[1] -1.959961
So there is clearly something wrong with ncp_solver.
Well it turns out that we can not use too big bound, c(-37.62, 37.62). If we narrow it to c(-35, 35), it will be alright.
Also, to avoid tolerance problem, we can change objective function from squared error to absolute error:
ncp_solver <- function (alpha, q, df) {
## objective function: we minimize absolute approximation error
obj_fun <- function (ncp, alpha = alpha, q = q, df = df) {
abs(pt(q = q, df = df, ncp = ncp, lower.tail = FALSE) - alpha)
}
## now we call `optimize`
oo <- optimize(obj_fun, interval = c(-35, 35), alpha = alpha, q = q, df = df)
## post processing and return
oo <- unlist(oo, use.names = FALSE) ## list to numerical vector
setNames(oo, c("ncp", "abs.error"))
}
ncp_solver(alpha = 0.025, q = 0, df = 98)
# ncp abs.error
#-1.959980e+00 9.190327e-07
Damn, this is a pretty annoying bug. But relax now.
Report on getting warning messages from pt
I also receive some report on annoying warning messages from pt:
ncp_solver(0.025, -5, 19)
# ncp abs.error
#-7.476488e+00 5.760562e-07
#Warning message:
#In pt(q = q, df = df, ncp = ncp, lower.tail = FALSE) :
# full precision may not have been achieved in 'pnt{final}'
I am not too sure what is going on here, but meanwhile I did not observe misleading result. Therefore, I decide to suppress those warnings from pt, using suppressWarnings:
ncp_solver <- function (alpha, q, df) {
## objective function: we minimize absolute approximation error
obj_fun <- function (ncp, alpha = alpha, q = q, df = df) {
abs(suppressWarnings(pt(q = q, df = df, ncp = ncp, lower.tail = FALSE)) - alpha)
}
## now we call `optimize`
oo <- optimize(obj_fun, interval = c(-35, 35), alpha = alpha, q = q, df = df)
## post processing and return
oo <- unlist(oo, use.names = FALSE) ## list to numerical vector
setNames(oo, c("ncp", "abs.error"))
}
ncp_solver(0.025, -5, 19)
# ncp abs.error
#-7.476488e+00 5.760562e-07
OK, quiet now.

Answer 2

You could use two while loops like this:
i <- 0.001
lowerFound <- FALSE
while(!lowerFound){
x <- pt(q = 5, df = 19, ncp = i)
if (round(x, 3) == 0.025){
lowerFound <- TRUE
print(paste("Lower is", i))
lower <- i
} else {
i <- i + 0.0005
}
}
i <- 0.001
upperFound <- FALSE
while(!upperFound){
x <- pt(q = 5, df = 19, ncp = i)
if (round(x, 3) == 0.975){
upperFound <- TRUE
print(paste("Upper is ", i))
upper <- i
} else {
i <- i + 0.0005
}
}
c(Lower = lower, Upper = upper)
# Lower Upper
# 7.4655 2.4330
Of course, you can adapt the increment in i <- i + .... or change the check if (round(x,...) == ....) to fit this solution to your specific needs of accuracy.

Answer 3

I know this is an old question, but there is now a one-line solution to this problem using the conf.limits.nct() function in the MBESS package.
install.packages("MBESS")
library(MBESS)
result <- conf.limits.nct(t.value = 5, df = 19)
result
$Lower.Limit
[1] 2.443332
$Prob.Less.Lower
[1] 0.025
$Upper.Limit
[1] 7.476475
$Prob.Greater.Upper
[1] 0.025
$Lower.Limit is the result where pt = 0.975
$Upper.Limit is the result where pt = 0.025
pt(q=5,df=19,ncp=result$Lower.Limit)
[1] 0.975
> pt(q=5,df=19,ncp=result$Upper.Limit)
[1] 0.025