*3.2. Example 2: Nag 1992 [15], Light Manual Workers in India*

Reports of productivity in relation to heat stress do not always provide individual-level data, but may only give summary statistics such as the mean production in each environment and some measure of variation or uncertainty such as the standard error of each reported mean (SE) or the between-worker standard deviation (SD). The model parameters can be estimated from these summary statistics by maximising the sum of the two log-likelihood terms, one for the mean x ∼ N ψμ, SE2 and one for the standard deviation SD, where <sup>υ</sup>.SD<sup>2</sup> <sup>∼</sup> V.χ<sup>2</sup> <sup>ν</sup>, and V = ψ2σ<sup>2</sup> + μ2τ2. There are υ = n − 1 degrees of freedom for standard deviations estimated from groups of n workers. The likelihood contribution from each experimental environment is then

$$\log L = -\frac{1}{2} \left( \frac{\overline{\mathbf{x}} - \psi \mu}{\mathrm{SE}} \right)^2 - \frac{\upsilon}{2} \left( \ln V + \frac{\mathrm{SD}^2}{\mathrm{V}} \right) \tag{5}$$

This function is maximised over our model parameters T0, ψ, τ, b and θ, which jointly determine the mean and standard deviation of production in each environment.

Figure 2 of Nag and Nag (1992) [15] shows the means with error bars of actual production in the first, second, and third hour of 3-hour observation periods, at nine controlled heat stress levels from 26.0 ◦C to 35.8 ◦C effective temperature (ET), which is similar to WBGT. Each of the six workers was observed in each environment over nine days, experiencing one environment per day in a different order for each worker. Here, we ignore the first hour as representing a "run-in" period and average the output of the second and third hours. The data used here, digitised and summarised from the published graph, are shown in Table 2. The units are beedi/h.

**Table 2.** Production data digitised from Figure 2 of Nag and Nag (1992) [15]


**Figure 2.** The group data from Nag and Nag (1992) [15]: beedi/h (dots); Fitted marginal distribution of absolute production, with a 90% interval on the individual output T0 = 19.7 ◦C, mean ψ = 85.4, SDτ = 9.97, slope b = −0.285, scale θ = 13.6.

To estimate the between-worker variation, we can measure the width of the published error bars, averaged where distinguishable on the original graph, which is approximately 8 beedi/h. Assuming these are standard-error bars, the standard error of the means is about 4, which implies (since n = 6) a between-subject standard deviation in each hour of 4√<sup>6</sup> <sup>≈</sup> 9.8 beedi/h. The within-subject correlation between hours is unknown, but a correlation close to 1 seems likely in the workers' output in two consecutive hours, in which case the standard deviation of the individual output averaged across two hours remains 9.8 beedi/h.

We next use a logistic model for μ, and here we are able to estimate the optimal-conditions parameter T0 from the data using the maximum likelihood. Thus, we write logit(μ) = 2ln(θ) + b(T − T0), where θ, b and T0 are all parameters to be estimated. We also estimate the two parameters of the N(ψ, τ2) distribution of maximum potential output. The parameter estimates are T0 = 19.7 ◦C, ψ = 85.4, τ = 9.97, b = −0.285, and θ = 13.626. Thus, the optimal temperature is about 20 degrees Celsius, under which conditions workers produce about 85 ± 10 beedi/h. Production falls to about 65% when the temperature rises to 36 degrees, but relative production at that temperature varies between workers from about 50% to 80%. At T = 20 ◦C effective temperature, the relative production follows a Beta(178, 1.04) distribution, with a mean of 0.994 and a 90% probability interval (0.983, 0.9997), so that all workers are then*,* in effect*,* fully productive.
