# A.3.2 Results

Table A-3 summarizes the simulation results with 1000 Monte Carlo trials using 30 increments and 3, 5, and 7 replicates. The distribution is only mildly skewed (CV = 0.7) and the autocorrelation is high (Moran’s I z-score = 3.8). The following observations are noted:

- The spatial autocorrelation does not affect the coverage of either the simple random sampling or systematic grid sampling. With 30 increments and 3 replicates, the Chebyshev yields 96%–97% coverage, whereas the Student’s-
*t*yields 94% coverage. - As noted with the simulations using lognormal distributions, increasing the number of replicates results in a higher coverage for the Chebyshev UCL but generally no improvement in the Student’s-
*t*UCL. - The average RPD for the 95% UCL is lower by approximately a factor of 2 with systematic grid sampling. Introducing spatial autocorrelation tends to result in an improvement in the RPD metric. This is most likely because the autocorrelation affects the correlation between the sample mean and variance. For nonnormal distributions, simple random sampling yields a positive correlation between the sample mean and sample variance. When systematic grid sampling is applied to a scenario with high spatial autocorrelation, it is more likely that neighboring samples share similar values, thereby reducing the sample variance.
- Both sampling protocols yield relatively unbiased estimates in the mean; this is an expected result for simple random sampling but not necessarily for systematic grid sampling; however, even for a site with high spatial autocorrelation, the bias is negligible when the population has a very low CV.
- Splitting the DU into quadrants results in lower RPDs, mainly reflecting the increase in the total number of replicates.

Table A-3. Summary of simulation results for a site with high spatial autocorrelation (see map in Figure A-2)

**Decision Unit: Map with High Spatial Autocorrelation**

**Population Parameters**

mean: | 8,564 | median: | 6,476 |
---|---|---|---|

SD: | 6,087 | min: | 170 |

CV: | 0.71 | max: | 57,378 |

Sampling: | Simple Random Sampling (i.e., mimics no spatial autocorrelation) |
---|---|

Trials: | 1,000 |

Increments: | 30 |

**95UCL Coverage**

Student's t UCL | Chebyshev UCL | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 94.4% | NA | 97.0% | NA |

5 | 93.6% | NA | 99.3% | NA |

7 | 93.8% | NA | 99.4% | NA |

**Bias in Mean**

Grand Mean | Bias | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 8,561 | NA | -0.03% | NA |

5 | 8,563 | NA | -0.01% | NA |

7 | 8,564 | NA | 0.01% | NA |

**Average RPD between 95UCL and population mean**

Student's t UCL | Chebyshev UCL | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 19.1% | NA | 28.6% | NA |

5 | 11.6% | NA | 23.6% | NA |

7 | 9.1% | NA | 20.4% | NA |

**Decision Unit: Map with High Spatial Autocorrelation**

**Population Parameters**

mean: | 8,564 | median: | 6,476 |
---|---|---|---|

SD: | 6,087 | min: | 170 |

CV: | 0.71 | max: | 57,378 |

Sampling: | Systematic Grid and Random Start |
---|---|

Trials: | 1,000 |

Increments: | 30 |

**95UCL Coverage**

Student's t UCL | Chebyshev UCL | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 93.5% | 92.9% | 96.2% | 96.8% |

5 | 94.5% | 95.6% | 99.2% | 99.0% |

7 | 95.5% | 98.2% | 99.6% | 99.8% |

**Bias in Mean**

Grand Mean | Bias | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 8,602 | 8,612 | 0.4% | 0.6% |

5 | 8,604 | 8,615 | 0.5% | 0.6% |

7 | 8,605 | 8,616 | 0.5% | 0.6% |

**Average RPD between 95UCL and population mean**

Student's t UCL | Chebyshev UCL | |||
---|---|---|---|---|

Replicates | All Site ISM | Quad ISM | All Site ISM | Quad ISM |

3 | 10.1% | 4.9% | 14.8% | 7.1% |

5 | 6.3% | 3.2% | 12.4% | 6.0% |

7 | 5.1% | 2.7% | 10.8% | 5.2% |