*3.1. Example 1: Wyndham 1969 [14], Gold-Mining in South Africa*

Wyndham [14] presents fitted curves, but no data, of percentage performance (P) against natural wet bulb temperature (Tnw) for three wind speeds: 100, 400, and 800 ft/min (approximately 0.5, 2 and 4 ms−1). This temperature metric will typically (except in extremely humid conditions) take lower values than other scales such as Effective Temperature (ET), or WBGT. Wyndham's curves can be expressed as

$$\log\_{10} \mathbf{P} = 2 - \mathbf{b} \left( \mathbf{r}^{(\mathbf{T}\_{\text{new}} - 2\mathcal{T}\mathcal{I})} - 1 \right), \text{ Tnw} \ge 2\mathcal{T}\mathcal{I} \tag{4}$$

This ensures P = 100% when Tnw = 27.7 ◦C, which is the complement of how Wyndham defined the vertical axis of his graphs ("percentage falloff in productivity from the level at 27.7 ◦C wet bulb temperature"). We recovered Wyndham's fit by digitising values from the published figures and fitting this equation to them by least squares (on the P scale). We found that a common value of r = 1.880 provides an essentially perfect fit (data not shown), with b = 0.00460, 0.00245, and 0.00165, respectively, for wind speeds of 100, 400, and 800 ft/min. The exponential model chosen by Wyndham can be closely approximated by our preferred logistic model. Figure 1 shows logistic equations fitted by least squares to the same digitised values, with Wyndham's exponential curves for comparison.

**Figure 1.** The Wyndham's exponential model (solid lines) and our logistic curves (dashed lines).

The estimated logistic curves, shown as dashed lines in Figure 1, are


In terms of the model proposed here, these represent logistic curves for the mean μ of the beta distribution, with slope b that reduces with increasing wind speed. Wyndham gave no actual output data, so we have no direct information about the mean ψ and variance τ<sup>2</sup> of the maximum output.

However, we can infer something about θ. Wyndham adds 78% confidence intervals (±1.23 SE) to these graphs, stating that "where they do not overlap, there is a 95% chance that there is a real difference between the two curves", presumably on the basis that (1 − 0.78)<sup>2</sup> ≈ 0.05. Regardless of the statistical logic, when combined with the reported design points, and the sample size of 10 individuals tested in each environment, these intervals furnish information on the between-subject standard deviation, which simulations using nonlinear least squares modelling suggest was about 0.05 on the log10 scale or roughly ±12%. Constant variance of log-transformed data, as Wyndham assumed, implies that on the original productivity scale, the SD would be proportional to the mean; that is, the variance of Y falls at high temperatures in proportion to μ, and so has no contribution from the variation in P but only the original between-subject variation in Z. Under our model, this implies a degenerate beta distribution for P with zero variance, represented by an unbounded θ. Unfortunately, Wyndham did not present his original data, from which we might have tested this and estimated θ. Nevertheless, these data provide some support for the general applicability of our logistic model.

The estimated relative output (%) is shown in Table 1 for the range of temperatures in the Wyndham data, assuming a wind speed of 200 ft/min (approximately 1 m·s−1). As noted above, standard deviations for the between-worker variation around these means cannot be estimated from the data published.

