Ulf Olsson

Swedish University of Agricultural Sciences

Journal of Statistics Education Volume 13, Number 1 (2005), jse.amstat.org/v13n1/olsson.html

Copyright © 2005 by Ulf Olsson, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:**Generalized confidence interval.

Note that if *X* is log-normal, then the median of *Y* is equal to the log of
the median of *X*. In this paper we will assume that it is the arithmetic mean
of *X*, and not the median of *X*, that we want to make inference about.

It is a rather straight-forward task to use the log-transformed data *Y* to
calculate a confidence interval for the expected value (mean value) of
*Y*. We will discuss how this result can be used to calculate a confidence
interval for the expected value of *X*.

It holds (see e.g. Zhou and Gao, 1997) that

(1) |

This means that the mean value of *X* is not equal to the antilog of the mean
value of *Y*. An estimator of log() can be calculated
from sample data as

(2) |

An estimator of the variance of is given by

(3) |

see e.g. Zhou and Gao, (1997).

One sample of *n*=40 observations was generated, using SAS (1997) software,
from a log-normal distribution
with parameters = 5 and
= 1. The
population mean of *X* is = 244.69. The observations were transformed as
*Y=*log(*X*). The raw sample
data are given in Table 1. The sample data are summarized in
Table 2.

914.9 | 1568.3 | 50.5 | 94.1 | 199.5 | 23.8 | 70.5 | 213.1 |

44.1 | 331.7 | 139.3 | 115.6 | 38.4 | 357.1 | 725.9 | 253.2 |

905.6 | 155.4 | 138.1 | 95.2 | 75.2 | 275.0 | 401.1 | 653.8 |

390.8 | 483.5 | 62.6 | 128.5 | 81.5 | 218.5 | 308.2 | 41.2 |

60.3 | 506.9 | 221.8 | 112.5 | 93.7 | 199.3 | 210.6 | 39.2 |

Variable | Mean | Median | St. dev. |
---|---|---|---|

X | 274.963 | 177.350 | 310.343 |

Y = log(X) | 5.127 | 5.170 | 1.004 |

For our example data, the naïve approach would produce the point estimate
= e^{5.127}=168.51.
A standard 95% confidence interval for is calculated as
with limits
[4.806, 5.448]. This would give limits
for as e^{4.806} = 122.24 and e^{5.448} = 232.29.
Note that this confidence interval does not cover the population mean value, which is 244.69.
Of course, this can occur because of chance; after all, we have only studied
one single sample so far. However, it is noteworthy that the interval does not
even cover the *sample* mean, which is 275.0. This illustrates the fact
that the naïve method gives a biased estimator of .

Calculate a confidence interval for log() as

(4) |

where *z* is the appropriate percentage point of the standard Normal
distribution. The limits in this confidence interval are back-transformed to
give a confidence interval for . The method is valid for large
samples. A similar approach has been suggested by
Zhou, Gao, and Hui (1997) for the
two-sample case.

For the sample data, =5.127 and *s ^{2}*=1.010. The 95%
confidence interval for log(

For the sample data, = 5.127 and *s ^{2}* = 1.010. The 95%
confidence interval for log(

Calculate and s^{2} from the data.

For *i* = 1 to *m* (where *m* is large, for example *m*=10000)

GenerateZ~N(0, 1) and .For each

i, calculate .

(end *i* loop)

For a 95% confidence interval, the 2.5% and 97.5% percentiles for *T*_{2}
are calculated from the 10000 simulated values. These are the lower and upper
limits in a confidence interval for . This means
that a 95% confidence interval for the lognormal mean is obtained as
[exp(T_{2;0.025}), exp(T_{2;0.975})].

(5) |

In our example, the 95% confidence interval can be calculated as , which gives the limits as [178.84, 371.16].

CO level | Date |
---|---|

12.5 | 9/11/90 |

20 | 10/4/90 |

4 | 12/3/91 |

20 | 12/10/91 |

25 | 5/7/92 |

170 | 8/6/92 |

15 | 9/10/92 |

20 | 9/22/92 |

15 | 3/30/93 |

The 95% confidence intervals for the example data, using the different methods we have discussed, are given in Table 4. It may be noted that our modified Cox method gives a somewhat wider interval than the Cox method, as expected. The generalized confidence interval has an upper limit that is well above the others, for these data.

Method | Lower limit | Upper limit |
---|---|---|

Naïve approach | 9.15 | 40.95 |

Cox method | 14.15 | 68.49 |

Modified Cox method | 12.31 | 78.72 |

Large-sample approach | -6.11 | 73.11 |

Generalized confidence interval | 16.65 | 153.19 |

The confidence intervals included are:

- the naïve approach.
- the Cox approach (equation (4), using
*z*as multiplier. - the modified Cox method with
*t*instead of*z*as multiplier. - the generalized confidence intervals. The simulation of the sampling distribution was based on 10000 replications.
- the Large-sample approach, i.e.

Each interval was compared to the population mean value = 244.69, and the number of intervals below, covering, or above was calculated. The results that are summarized in Table 5 give the percentage of the samples that cover , and the percentage of the samples that produce intervals above or below .

Naïve approach | Cox method | Modified Cox method | ||||||||||

n | Below | Covering | Above | Below | Covering | Above | Below | Covering | Above | |||

5 | 13.5 | 86.2 | 0.3 | 10.6 | 87.2 | 2.2 | 5.9 | 93.5 | 0.6 | |||

10 | 31.3 | 68.5 | 0.0 | 8.2 | 91.1 | 0.7 | 5.9 | 93.9 | 0.2 | |||

20 | 54.8 | 45.2 | 0.0 | 4.8 | 94.2 | 1.0 | 3.6 | 95.7 | 0.7 | |||

30 | 75.9 | 24.1 | 0.0 | 6.5 | 92.6 | 0.9 | 5.4 | 93.9 | 0.7 | |||

50 | 94.3 | 5.7 | 0.3 | 4.0 | 95.4 | 0.6 | 3.9 | 95.5 | 0.6 | |||

100 | 99.9 | 0.1 | 0.0 | 3.3 | 95.5 | 1.2 | 3.2 | 95.7 | 1.1 | |||

200 | 100.0 | 0.0 | 0.0 | 2.6 | 95.2 | 2.2 | 2.6 | 95.2 | 2.2 | |||

500 | 100.0 | 0.0 | 0.0 | 3.0 | 95.1 | 1.9 | 3.0 | 95.1 | 1.9 | |||

1000 | 100.0 | 0.0 | 0.0 | 3.3 | 94.4 | 2.3 | 3.3 | 94.4 | 2.3 |

Large sample approach | Generalized C I | |||||||

n | Below | Covering | Above | Below | Covering | Above | ||

5 | 16.8 | 83.0 | 0.2 | 1.3 | 94.1 | 4.6 | ||

10 | 16.4 | 83.6 | 0.0 | 2.2 | 93.7 | 4.1 | ||

20 | 12.0 | 87.9 | 0.1 | 1.9 | 95.2 | 2.9 | ||

30 | 14.0 | 85.6 | 0.4 | 2.1 | 94.6 | 3.3 | ||

50 | 9.4 | 90.4 | 0.2 | 2.2 | 95.0 | 2.8 | ||

100 | 7.6 | 92.1 | 0.3 | 2.9 | 93.7 | 3.4 | ||

200 | 6.5 | 92.2 | 1.3 | 1.3 | 95.9 | 2.8 | ||

500 | 4.9 | 94.0 | 1.1 | 2.8 | 94.2 | 3.0 | ||

1000 | 4.8 | 93.8 | 1.4 | 2.3 | 95.8 | 1.9 |

The large-sample method, that is based on Central Limit Theorem arguments, gives a consistently lower coverage than 95%. Sample sizes of more than 200 seem to be needed to obtain a confidence level close to the nominal one. As expected, the intervals based on the naïve approach fail, since these intervals are intervals for some other parameter. The simulations were also run with standard deviations 0.5 and 2. All methods performed somewhat worse when the standard deviation increased but the relationships between methods remained unchanged.

It seems that the confidence intervals based on the modified Cox method work well for practical purposes. The calculations are simple and may be performed by hand, if desired. The generalized confidence interval approach also works well; a small disadvantage is that it requires a computer to simulate the sampling distribution.

Land, C. E. (1971), “Confidence intervals for linear functions of
the normal mean and variance,” *Annals of Mathematical Statistics*, 42, 1187-1205.

SAS Institute Inc. (1997), *SAS/STAT software: Changes
and enhancements through Release 6.12,* Cary, NC: SAS Institute Inc.

Weerahandi, S. (1993), “Generalized confidence intervals”.
*Journal of the American Statistical Association*, 88, 899-905.

Zhou, X-H., and Gao, S. (1997), “Confidence intervals for the
log-normal mean,” *Statistics in Medicine*, 16, 783-790.

Zhou, X-H., Gao, S., and Hui, S. L. (1997), “Methods for comparing
the means of two independent log-normal samples,” *Biometrics*,
53, 1129-1135.

Ulf Olsson

Department of Biometry and Engineering

Swedish University of Agricultural Sciences

Box 7032, S-75007

Uppsala

Sweden

*Ulf.Olsson@bt.slu.se*

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