Sample Size Requirements for Assessing Statistical Moments of Simulated Crop Yield Distributions
Abstract
:1. Introduction
2. Methods
- (1)
- 5000 samples of crop yields were drawn without replacement from the population for sample sizes i = 5,10,15,….,1500.
- (2)
- For each sample size i and realization j, mean (
), standard deviation (
) and skewness (
) were estimated based on the drawn sample.
- (3)
- Relative deviations of the individual estimates from their reference values were computed for all moments as:
is the relative difference of the mean yield
in the sample j = 1,2,3,…,5000 of size i = 5,10,15,…,1500 from the population’s mean yield μref.
- (4)
- Finally, upper 95%-quantiles for the distributions of
,
and
were computed for each sample size i. This measure can be used to determine the minimum sample size leading for 95% of all 5000 samples to a relative error smaller than a pre-defined level ∆ = 5%, 10%, 15% and 25%. The 95%-quantile has been chosen because it represents a robust measure and can directly be compared with the 95%-confidence interval usually applied in conjunction with Equations (1) and (2).
3. Results
Payerne | ||||
---|---|---|---|---|
Winterwheat Baseline | Winterwheat CC | Maize Baseline | Maize CC | |
Mean yield μref (t∙ha−1) | 8.393 | 7.109 | 11.221 | 9.318 |
Standard deviation σref (t∙ha−1) | 1.212 | 1.050 | 2.027 | 1.761 |
Coefficient of variation | 14.4% | 14.8% | 18.1% | 18.9% |
Skewness γref | −0.755 | −0.655 | −0.345 | 0.270 |
Uster | ||||
Winterwheat Baseline | Winterwheat CC | Maize Baseline | Maize CC | |
Mean yield μref (t∙ha−1) | 6.375 | 5.086 | 11.261 | 9.881 |
Standard deviation σref (t∙ha−1) | 0.515 | 0.469 | 1.066 | 1.386 |
Coefficient of variation | 8.1% | 9.2% | 9.5% | 14.0% |
Skewness γref | −1.220 | −0.050 | −2.892 | −1.890 |
Winterwheat Baseline | Winterwheat CC | Maize Baseline | Maize CC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Relative Error a,b | μ | σ | γ | μ | σ | γ | μ | σ | γ | μ | σ | γ |
<25% | 5 | 40 | 390 | 5 | 40 | 535 | 5 | 25 | 710 | 5 | 30 | 1020 |
<15% | 5 | 100 | 750 | 5 | 105 | 900 | 10 | 60 | 1060 | 10 | 75 | 1280 |
<10% | 10 | 205 | 1040 | 10 | 210 | 1155 | 15 | 120 | 1275 | 15 | 145 | 1400 |
<5% | 35 | 580 | 1355 | 35 | 585 | 1405 | 50 | 400 | 1440 | 55 | 450 | 1475 |
Winterwheat Baseline | Winterwheat CC | Maize Baseline | Maize CC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Relative Error a,b | μ | σ | γ | μ | σ | γ | μ | σ | γ | μ | σ | γ |
<25% | 5 | 135 | 1430 | 5 | 35 | 1485 | 5 | 165 | 315 | 5 | 90 | 190 |
<15% | 5 | 310 | 1430 | 5 | 85 | 1495 | 5 | 395 | 665 | 5 | 225 | 440 |
<10% | 5 | 525 | 1430 | 5 | 170 | 1500 | 5 | 670 | 965 | 10 | 425 | 715 |
<5% | 15 | 1040 | 1475 | 15 | 505 | 1500 | 15 | 1145 | 1330 | 30 | 915 | 1180 |
4. Discussion and Conclusions
- A sample size of 15 yield observations is sufficient to obtain estimates of mean yields with a relative error of less than 10% at 95% confidence.
- 200 realizations are in general sufficient to obtain estimates of the standard deviation of crop yields with a relative accuracy of better than 10%. The sample size should be increased to roughly 500 when it can be assumed that the crop yield distribution is strongly skewed (absolute skewness value > 1).
- At least 1000 realizations are needed in most cases to reliably characterize the skewness of the distribution. When a high degree of symmetry is suggested by the available information, much larger samples are needed. This implies that in the absence of prior information, risk analyses should always be based on very large sample sizes.
- In practice, simulating 1000 or more years of crop yields may not always be feasible. In these cases compromises between the computation time and the accuracy of the estimated statistical moments have to be made. For instance, the required sample size is reduced by a factor of about 5 with respect to the estimation of the standard deviation, if the allowable relative estimation error is increased from 10% to 25%. This can be meaningful in studies aiming e.g., at optimizing crop management (e.g., [11,16]) or in studies simulating crop yields in a spatially explicit manner (e.g., [15]).
Acknowledgments
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Lehmann, N.; Finger, R.; Klein, T.; Calanca, P. Sample Size Requirements for Assessing Statistical Moments of Simulated Crop Yield Distributions. Agriculture 2013, 3, 210-220. https://doi.org/10.3390/agriculture3020210
Lehmann N, Finger R, Klein T, Calanca P. Sample Size Requirements for Assessing Statistical Moments of Simulated Crop Yield Distributions. Agriculture. 2013; 3(2):210-220. https://doi.org/10.3390/agriculture3020210
Chicago/Turabian StyleLehmann, Niklaus, Robert Finger, Tommy Klein, and Pierluigi Calanca. 2013. "Sample Size Requirements for Assessing Statistical Moments of Simulated Crop Yield Distributions" Agriculture 3, no. 2: 210-220. https://doi.org/10.3390/agriculture3020210