predIntNormTestPower(n, df = n - 1, n.mean = 1, k = 1, delta.over.sigma = 0,
pi.type = "upper", conf.level = 0.95)df=n-1.n.mean=1 (i.e., individual observations). Note that all
future averages must be based on the same sample size.conf.level. The
default value is k=1.pi.type="upper" (the default), and
pi.type="lower".conf.level=0.95.predIntNorm computes a standard prediction interval based on a
sample from a normal distribution. The function predIntNormTestPower
computes the probability that at least one out of $k$ future observations or
averages will not be contained in the prediction interval,
where the population mean for the future observations is allowed to differ from
the population mean for the observations used to construct the prediction interval.
The Form of a Prediction Interval
Let $\underline{x} = x_1, x_2, \ldots, x_n$ denote a vector of $n$
observations from a normal distribution with parameters
mean=$\mu$ and sd=$\sigma$. Also, let $m$ denote the
sample size associated with the $k$ future averages (i.e., n.mean=$m$).
When $m=1$, each average is really just a single observation, so in the rest of
this help file the term tolIntNorm).
Similarly, the form of a one-sided lower prediction interval is:
$$[\bar{x} - Ks, \infty] \;\;\;\;\;\; (4)$$
and the form of a one-sided upper prediction interval is:
$$[-\infty, \bar{x} + Ks] \;\;\;\;\;\; (5)$$
but $K$ differs for one-sided versus two-sided prediction intervals.
The derivation of the constant $K$ is explained in the help file for
predIntNormK.
Computing Power
The "power" of the prediction interval is defined as the probability that at
least one out of the $k$ future observations or averages
will not be contained in the prediction interval, where the population mean
for the future observations is allowed to differ from the population mean for the
observations used to construct the prediction interval. The probability $p$
that all $k$ future observations will be contained in a one-sided upper
prediction interval (pi.type="upper") is given in Equation (6) of the help
file for
predIntNormSimultaneousK, where $k=m$ and $r=1$:
$$p = \int_0^1 T(\sqrt{n}K; n-1, \sqrt{n}[\Phi^{-1}(v) + \frac{\Delta}{\sigma}]) [\frac{v^{k-1}}{B(k, 1)}] dv \;\;\;\;\;\; (6)$$
where $T(x; \nu, \delta)$ denotes the cdf of the
non-central Student's t-distribution with parameters
df=$\nu$ and ncp=$\delta$ evaluated at $x$;
$\Phi(x)$ denotes the cdf of the standard normal distribution
evaluated at $x$; and $B(\nu, \omega)$ denotes the value of the
beta function with parameters a=$\nu$ and
b=$\omega$.
The quantity $\Delta$ (upper case delta) denotes the difference between the
mean of the population that was sampled to construct the prediction interval, and
the mean of the population that will be sampled to produce the future observations.
The quantity $\sigma$ (sigma) denotes the population standard deviation of both
of these populations. Usually you assume $\Delta=0$ unless you are interested
in computing the power of the rule to detect a change in means between the
populations, as we are here.
If we are interested in using averages instead of single observations, with
$w \ge 1$ (i.e., n.mean$\ge 1$), the first
term in the integral in Equation (6) that involves the cdf of the
non-central Student's t-distribution becomes:
$$T(\sqrt{n}K; n-1, \frac{\sqrt{n}}{\sqrt{w}}[\Phi^{-1}(v) + \frac{\sqrt{w}\Delta}{\sigma}]) \;\;\;\;\;\; (7)$$
For a given confidence level $(1-\alpha)100%$, the power of the rule to detect
a change in means is simply given by:
$$Power = 1 - p \;\;\;\;\;\; (8)$$
where $p$ is defined in Equation (6) above using the value of $K$ that
corresponds to $\Delta/\sigma = 0$. Thus, when the argument
delta.over.sigma=0, the value of $p$ is $1-\alpha$ and the power is
simply $\alpha 100%$. As delta.over.sigma increases above 0, the power
increases.
When pi.type="lower", the same value of K is used as when
pi.type="upper", but Equation (4) is used to construct the prediction
interval. Thus, the power increases as delta.over.sigma decreases below 0.predIntNorm and
predIntNormSimultaneous.predIntNorm, predIntNormK,
plotPredIntNormTestPowerCurve, predIntNormSimultaneous,
predIntNormSimultaneousK,
predIntNormSimultaneousTestPower, Prediction Intervals,
Normal.# Show how the power increases as delta.over.sigma increases.
# Assume a 95% upper prediction interval.
predIntNormTestPower(n = 4, delta.over.sigma = 0:2)
#[1] 0.0500000 0.1743014 0.3990892
#----------
# Look at how the power increases with sample size for a one-sided upper
# prediction interval with k=3, delta.over.sigma=2, and a confidence level
# of 95%.
predIntNormTestPower(n = c(4, 8), k = 3, delta.over.sigma = 2)
#[1] 0.3578250 0.5752113
#----------
# Show how the power for an upper 95% prediction limit increases as the
# number of future observations k increases. Here, we'll use n=20 and
# delta.over.sigma=1.
predIntNormTestPower(n = 20, k = 1:3, delta.over.sigma = 1)
#[1] 0.2408527 0.2751074 0.2936486Run the code above in your browser using DataLab