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# Log normal confidence intervals

### Why do we need them

For certain applications, it is not possible to log transform the data, estimate the mean and confidence interval in the transformed scale, and then detransform to obtain the geometric mean with its asymmetric confidence interval. This is the case for example when density of an animal is being estimated using distance sampling.

In this situation all we have is an estimate of the density (D), together with an estimate of its variance (VAR). Note this is the variance of the statistic - hence the standard error is equal to the square root of the variance.

A normal approximation interval is therefore be given by:

95% CI (D)= D ± 1.96 × √VAR

But the distribution of D is positively skewed, so use of the normal approximation to obtain a confidence interval gives poor coverage.

### How to calculate them

Instead one can obtain a 100(1 − 2α)% log normal confidence interval as follows:

1. Estimate the variance of logeD:
 VAR (logeD) = loge [ 1 + VAR(D) ] D2
2. Estimate C as:

C   =    exp [zα . √VAR (logeD)]

where z is the value of the standardized normal deviate. If the estimate is based on only a small number of sightings, this is replaced by t. Full details are given in Buckland et al. (1993). 3. The lower and upper intervals are then given by

D/C and D×C 