predIntLnormAltTestPower {EnvStats} | R Documentation |

## Probability That at Least One Future Observation Falls Outside a Prediction Interval for a Lognormal Distribution

### Description

Compute the probability that at least one out of `k`

future observations
(or geometric means) falls outside a prediction interval for `k`

future
observations (or geometric means) for a normal distribution.

### Usage

```
predIntLnormAltTestPower(n, df = n - 1, n.geomean = 1, k = 1,
ratio.of.means = 1, cv = 1, pi.type = "upper", conf.level = 0.95)
```

### Arguments

`n` |
vector of positive integers greater than 2 indicating the sample size upon which the prediction interval is based. |

`df` |
vector of positive integers indicating the degrees of freedom associated with
the sample size. The default value is |

`n.geomean` |
positive integer specifying the sample size associated with the future
geometric means. The default value is |

`k` |
vector of positive integers specifying the number of future observations that the
prediction interval should contain with confidence level |

`ratio.of.means` |
numeric vector specifying the ratio of the mean of the population that will be
sampled to produce the future observations vs. the mean of the population that
was sampled to construct the prediction interval. See the DETAILS section below
for more information. The default value is |

`cv` |
numeric vector of positive values specifying the coefficient of variation for
both the population that was sampled to construct the prediction interval |

`pi.type` |
character string indicating what kind of prediction interval to compute.
The possible values are |

`conf.level` |
numeric vector of values between 0 and 1 indicating the confidence level of the
prediction interval. The default value is |

### Details

A prediction interval for some population is an interval on the real line
constructed so that it will contain `k`

future observations or averages
from that population with some specified probability `(1-\alpha)100\%`

,
where `0 < \alpha < 1`

and `k`

is some pre-specified positive integer.
The quantity `(1-\alpha)100\%`

is call the confidence coefficient or
confidence level associated with the prediction interval. The function
`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
a prediction interval based on the assumption of normally distributed observations,
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 function `predIntLnormAltTestPower`

assumes all observations are
from a lognormal distribution. The observations used to
construct the prediction interval are assumed to come from a lognormal distribution
with mean `\theta_2`

and coefficient of variation `\tau`

. The future
observations are assumed to come from a lognormal distribution with mean
`\theta_1`

and coefficient of variation `\tau`

; that is, the means are
allowed to differ between the two populations, but not the coefficient of variation.

The function `predIntLnormAltTestPower`

calls the function
`predIntNormTestPower`

, with the argument `delta.over.sigma`

given by:

`\frac{\delta}{\sigma} = \frac{log(R)}{\sqrt{log(\tau^2 + 1)}} \;\;\;\;\;\; (1)`

where `R`

is given by:

`R = \frac{\theta_1}{\theta_2} \;\;\;\;\;\; (2)`

and corresponds to the argument `ratio.of.means`

for the function
`predIntLnormAltTestPower`

, and `\tau`

corresponds to the argument
`cv`

.

### Value

vector of numbers between 0 and 1 equal to the probability that at least one of
`k`

future observations or geometric means will fall outside the prediction
interval.

### Note

See the help files for `predIntNormTestPower`

.

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### References

See the help files for `predIntNormTestPower`

and
`tTestLnormAltPower`

.

### See Also

`plotPredIntLnormAltTestPowerCurve`

,
`predIntLnormAlt`

,
`predIntNorm`

, `predIntNormK`

,

`plotPredIntNormTestPowerCurve`

,
`predIntLnormAltSimultaneous`

,

`predIntLnormAltSimultaneousTestPower`

, Prediction Intervals,
LognormalAlt.

### Examples

```
# Show how the power increases as ratio.of.means increases. Assume a
# 95% upper prediction interval.
predIntLnormAltTestPower(n = 4, ratio.of.means = 1:3)
#[1] 0.0500000 0.1459516 0.2367793
#----------
# Look at how the power increases with sample size for an upper one-sided
# prediction interval with k=3, ratio.of.means=4, and a confidence level of 95%.
predIntLnormAltTestPower(n = c(4, 8), k = 3, ratio.of.means = 4)
#[1] 0.2860952 0.4533567
#----------
# 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
# ratio.of.means=2.
predIntLnormAltTestPower(n = 20, k = 1:3, ratio.of.means = 2)
#[1] 0.1945886 0.2189538 0.2321562
```

*EnvStats*version 2.8.1 Index]