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I have to compute the LCL95% and UCL95% using Land's "exact" method. I computed the LCL and UCL for this lognormal distribution using another technique and I can't find anything for Land's exact procedure.

My data set x = {0.043, 0.236, 0.057, 0.016}

Here is what I tried

$y =$ mean of $\ln{x}$

$s^2 =$ standard deviation of $\ln x$.

Confidence limits $ = \exp\left(y + s^2/2 \pm z\sqrt{s^2/n + s^4/2(n-1)}\right)$

and I got UCL: 2.98 and LCL: 0.139 but the answer using Land's exact is UCL95%:11.6 and LCL95%: 0.039

Here is what I have calculated already:

  • Mean: 0.088
  • Standard deviation: 0.1
  • Geometric mean: 0.0552
  • Geometric standard deviation: 3.04
  • Estimated arithmetic mean using MVUE: 0.085
  • 95th percentile: 0.343
  • Upper limit of tolerance: 16.8
  • mean of $\ln{x} = -2.898$
  • standard deviation of $\ln{x} = 1.112$

Can anyone please help me sketch out an algorithm for the formula when using Land's exact method?

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  • $\begingroup$ (i) when you say "I couldn't find anything" what did you try? (ii) When you say "I have to" ... from where does this compulsion arise? Is this for a class, say? $\endgroup$
    – Glen_b
    Commented Jul 23, 2014 at 0:00
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    $\begingroup$ If you're converting from an excel spreadsheet, you have a spreadsheet formula, do you not? The procedure is found in Land, C. E. (1988), Hypothesis tests and interval estimates, In Lognormal Distribution, (Crow and Shimizu eds), New York: Marcel Dekker, pp 87-112 ... but I am not sure you'll find it enlightening, since it takes about three pages to do it (p103-106) and it seems to assume you have tables that are in other documents. There's also R code for it here (see p 40) $\endgroup$
    – Glen_b
    Commented Jul 23, 2014 at 2:19
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    $\begingroup$ The table is on my Web site (as an Excel workbook) linkable through quantdec.com/envstats/software. $\endgroup$
    – whuber ♦
    Commented Jul 23, 2014 at 15:47
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    $\begingroup$ Averaging is not a good idea. Gilbert quotes Land in recommending that cubic interpolation in these tables should be adequate. A link on the web page I referenced ("Perform linear, quadratic, cubic, etc. interpolation (used to interpolate Land's H factor tables)") will send you to a spreadsheet showing how to do that. With a sample count of $4$ the results are heavily dependent on the assumption of lognormality, so you have much more serious issues to deal with in evaluating the sensitivity to that assumption. $\endgroup$
    – whuber ♦
    Commented Jul 23, 2014 at 18:00
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    $\begingroup$ Perhaps it is interpolating differently--but that's a pretty big difference. (The implied value of $H$ they are using is more than $1$ less than you are computing.) Because that workbook does not allow you to view the formulas or the macros, it cannot be considered authoritative or trustworthy. $\endgroup$
    – whuber ♦
    Commented Jul 23, 2014 at 20:27

2 Answers 2

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This report describes the Land Method on page 10.

page 10

The values for step 3 in Gilbert's Paper

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If you are looking for a statistical software implementation of Land's Exact method then in R this is provided by the EnvStats package available on CRAN.

The elnormAlt() function from this package can be used to calculate Land's confidence interval as well as other (Zou et al and Parkin et al and Cox) methods of confidence intervals for the lognormal distribution.

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  • $\begingroup$ Although implementation is often mixed with substantive content in questions, we are supposed to be a site for providing information about statistics, machine learning, etc., not code. It can be good to provide code as well, but please elaborate your substantive answer in text for people who don't read this language well enough to recognize & extract the answer from the code. $\endgroup$
    – gung - Reinstate Monica
    Commented Feb 28, 2019 at 14:25

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