Standard error

Deze pagina lezen in het Nederlands.

Author: dr. ir. N. van Geloven
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This page was translated from the Dutch-language original using AI and has not been fully reviewed. Where the two differ, the Dutch version takes precedence.

The standard error of an estimate indicates the precision of that estimate. The standard error (SE) is used to indicate how certain you are of an estimated value. Formally, the value represents the spread (standard deviation) of the estimate if there were multiple samples, with the estimate being made again for each sample.

How do I calculate the SE of a percentage increase?

I have a mean and SE in a control group (1) and an intervention group (2). As a summary measure, I calculate the percentage increase in the intervention group relative to the control group: (mean_2 - mean_1) / mean_1 * 100%. How can I calculate an SE for this percentage increase?

It is not customary to calculate an SE on this 'percentage' scale (relative change). The reason for this is that, for example, if you had a 30% decrease with an SE of 20%, converting this to a 95% confidence interval would give you 'odd' negative percentages (30% +/- 1.96*20%). In general, you use the SE on a scale where you can reasonably assume that a normal distribution applies (for the same reason, for example, in logistic regression the SE is reported for the regression coefficients and not for the odds ratio). You can calculate the SE on a transformed scale. In particular, you could use the natural logarithm of the fold increase (the ratio of the two means) here:

$ln(FI) = ln(\frac{mean_2}{mean_1}) = ln(mean_2) - ln(mean_1)$.

The corresponding SE on this scale is (you can calculate this using the delta method):

$SE_{ln(FI)} = \sqrt{\frac{SE_1^2}{mean_1^2} + \frac{SE_2^2}{mean_2^2}}$.

On this transformed scale, you could perform calculations (for example, pooling study results or calculating a 95% confidence interval for description).

From the numbers calculated on this scale, you can always transform back to a measure of relative change, using the following conversion:

$\frac{mean_2-mean_1}{mean_1} = e^{ln(\frac{mean_2}{mean_1})} - 1$

How do I calculate the standard error of a standard deviation?

The standard error of the estimate of a standard deviation can be calculated as:

$SE_s = \frac {s}{\sqrt{2*(n-1)}}$ ≈ $0.71 * SE_{mean}$,

where s is the sample standard deviation and n the number of observations. See also this tutorial and the paper by Harding et al. 2014.

References

• Douglas G. Altman, J Martin Bland, Statistical Notes, Standard deviations and standard errors, BMJ 2005;331:903
• Standard error versus standard deviation on TheStatsGeek.com A very clear English-language explanation of these two quantities.

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