                 #   Generating P-value grid from Mahalanobis Distance Grid: Calculating P-values for individual Mahalanobis Distance Grid cells: Generating P-value grid from Mahalanobis Distance Grid:

When the predictor variables used to generate the mean vector and covariance matrix are normally distributed, then Mahalanobis distances are distributed approximately according to a Chi-square distribution with n-1 degrees of freedom. In such cases it may be useful to convert the Mahalanobis distance grid into a grid of p-values. If the predictor variables are not normally distributed, it may still be useful to make this conversion because it rescales the unbounded Mahalanobis values such that they are all between 0 and 1. For more details on the uses of converting to p-values, see the discussion on Chi-Square values or refer to Clark et al. 1993.

The button provides an easy way to convert Mahalanobis grids into p-value grids. Click the button and you will be prompted to identify your Mahalanobis grid and the appropriate degrees of freedom: The degrees of freedom should be equal to n-1, where n = # predictor variables used to generate the original mean vector and covariance matrix. The Mahalanobis grid in this example was generated from 2 predictor variables (Slope and Elevation), so there would only be 1 degree of freedom. Click the OK button to generate the p-value grid: Due to a limitation in Spatial Analyst, this function does not generate exact p-values for each cell but rather classifies the grid into 26 p-value ranges. The p-value for each cell reflects the probability of seeing a Mahalanobis value as large or larger than the actual Mahalanobis value for that cell, assuming the vector of predictor values that produced that Mahalanobis value was sampled from a population with an ideal mean (i.e. equal to the vector of mean predictor variable values used to generate the original Mahalanobis grid). P-values close to 0 reflect high Mahalanobis distance values and are therefore very dissimilar to the ideal combination of predictor variables. P-values close to 1 reflect low Mahalanobis distances and are therefore very similar to the ideal combination of predictor variables.

Calculating P-values for individual Mahalanobis Distance Grid cells:

The tool allows you to calculate exact p-values for individual Mahalanobis surface grid cells, assuming a Chi-square distribution with n-1 degrees of freedom where n = # predictor variables used to generate the mean vector and covariance matrix. See the discussion of Chi-Square values for an explanation of this concept.

When you initially click the tool, you will be asked to identify the degrees of freedom to use in the calculations: After you enter a number, you can start clicking on the screen to calculate p-values. Your cursor should be represented with a symbol. As you click on different places on the grid, you will generate a running list of p-values:  Mahalanobis Intro  |  Mahalanobis Description  |  Generating Mahalanobis Grids  |  Mahalanobis Distances for Feature Themes  |  Mahalanobis Distances for Tables  |  Additional Mahalanobis Matrices  |  Mahalanobis References

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