Power calculations

Assessing the post-hoc power of a time series to detect a particular pattern of change has several challenges:

• lack of balance: the number of samples can vary between years
• the analytical variation can change over time and depends on the concentration in each sample (or at least it should)
• there are generally no closed-form solutions for the parameter estimates from a mixed model
• less-than measurements compound all the above!

These challenges can, in principle, be overcome by simulation, but would often require more information about sampling and analytic procedures than can be obtained directly from the data.

The compromise adopted here is to make some big simplifying assumptions:

• the same number of samples \(n\) are collected each year; \(n\) is taken to be the mean number of samples collected each year over the course of the time series (ignoring any years when there was no monitoring); for the purposes of the power calculations, it doesn’t matter if \(n\) is not a whole number
• the analytical standard deviation, \(\sigma_{\text{analytical}}\) is constant; this is taken to be the median analytical standard deviation in the time series
• all measurements are above the detection limit
• only linear trends in log concentration are considered

The problem then reduces to assessing the power of an F-test in a standard linear regression context.

Specifically, let \(y_i\) be the mean log concentration in year \(t_i, i = 1...N\). (Note that there is no requirement for the \(t_i\) to represent consecutive years, so gaps in the time series are accommodated.) Further, assume that log concentrations change linearly over time as

\[y_i = \mu + \beta t_i + \epsilon_i\]

where the random ‘errors’ \(\epsilon_i\) are assumed to be normally distributed with zero mean and constant variance

\[\psi^2 = \sigma^2_{\text{year}} + \frac{\sigma^2_{\text{sample}} + \sigma^2_{\text{analytical}}}{n}\]

and \(\sigma_{\text{year}}\) and \(\sigma_{\text{sample}}\) are the between-year and between-sample standard deviations respectively.

The null and alternative hypotheses \({\text H}_0: \beta = 0\) and \({\text H}_1: \beta \ne 0\) are compared by an F-test at the 100 \(\alpha\) % signficiance level. Let \({\text F}_{\text {crit}}\) be the critical value of the test that satisfies

\[\text{Prob} \left ( \text F(1, N-2, 0) > {\text F}_{\text {crit}} \right ) = \alpha \] where \(\text F(1, N-2, 0)\) is a (central) F-distribution on 1 and \(N-2\) degrees of freedom (and non-centrality parameter 0). The power of the test is then given by

\[\text{Prob} \left ( \text F(1, N-2, \delta) > {\text F}_{\text {crit}} \right )\] where

\[\delta = \frac{\beta^2 \sum_1^N (t_i-\bar t)^2 }{\psi^2} \] and

\[ \bar t = \frac 1 n \sum_1^N t_i\]

Remember that \(\beta\) is the change in log concentration each year. Some care is needed when back-transforming to the concentration scale. If \(\beta\) is small, say less than 0.05 in absolute value, then the percentage annual change in concentration is approximately \(100\beta\) %. More formally, a positive \(\beta\) is equivalent to an annual percentage increase in concentration of \(100\left( \exp(\beta) - 1 \right)\) % and a negative \(\beta\) is equivalent to an annual percentage decrease in concentration of \(- 100 \left( \exp(\beta) - 1 \right)\) %. For example, \(\beta = 0.05\) and \(\beta = - 0.05\) are equivalent to a percentage annual increase of 5.13 % and decrease of 4.88 % respectively. Whilst the powers to detect a change of \(\beta = 0.05\) and \(\beta = - 0.05\) are identical, the power to detact a percentage annual increase of 5 % is not the same as the power to detect a percentage annual decrease of 5 %.

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