*3.3. Example 3: Sahu 2013 [16], Rice Harvesters in India*

Sahu et al. (2013) [16] report the number of bundles of rice gathered over several days, by groups of 18 workers in the first and fifth hours of each day. Figure 3 shows the data: productivity strongly declines with temperature, and is lower for any given temperature towards the end of the working day, presumably due to tiredness and perhaps dehydration. By the end of a hot day, work proceeds at only about 60% of the rate at the start of a cool day.

**Figure 3.** The data from Sahu (2013) [16]: hourly rice production in bundles/h (dots); fitted model and approximate 90% confidence intervals (lines). (**a**) First hour: T0 = 0 ◦C, mean ψ = 100.0, SDτ = 1.04, slope b = −0.339, scale θ = 245; (**b**) Fifth hour: T0 = 0 ◦C, mean ψ = 90.2, SDτ = 0.545, slope b = −0.316, scale θ = 183.

Individual data are not available, so we model the output of a group as the dependent variable. There is no evidence of a threshold temperature even at the "coolest" conditions observed (which were nevertheless very hot), so we can reasonably set T0 to any sufficiently low value, well below the observed range. Figure 3a shows the fitted model for hour 1 and Figure 3b, for hour 5, using T0 = 0 ◦C on the WBGT scale.

These results agree well with those reported in Sahu et al. (2013) [16] as regards to the reduction in the mean output per degree increase in WBGT. Their linear model showed a mean reduction of 5.42 beed*i/h* per degree in the first hour, 5.14 in the fifth, where our model, illustrated in Figure 3, shows an average reduction of about 5.4 in the first hour, 5.3 in the fifth, but falling more steeply at higher temperatures. To these results, we can now add information about the between-worker variation in this reduction, finding that the variation itself strongly increases with the temperature, as shown in the last column of Table 1.

Estimating the parameters by maximum likelihood requires the marginal probability distribution of the product of the two random variables Z and P, which is not analytically tractable. The estimation may be done through approximate numerical quadrature, integrating over the distribution of Z using Simpson's rule. The calculations were done using the "solver" utility of Microsoft Excel. An approximate variance for the product is available as Var(Y) ≈ <sup>ψ</sup>2σ<sup>2</sup> <sup>+</sup> <sup>μ</sup>2τ2, and this can be used to generate probability intervals for individual output at a given temperature, as shown in Figure 3.
