What is a Z score What is a p-value

Most statistical tests begin by identifying a null hypothesis. The null hypothesis for pattern analysis tools essentially states that there is no spatial pattern among the features, or among the values associated with the features, in the study area -- said another way: the expected pattern is just one of the many possible versions of complete spatial randomness. The Z score is a test of statistical significance that helps you decide whether or not to reject the null hypothesis. The p-value is the probability that you have falsely rejected the null hypothesis.

Z scores are measures of standard deviation. For example, if a tool returns a Z score of +2.5 it is interpreted as "+2.5 standard deviations away from the mean". P-values are probabilities. Both statistics are associated with the standard normal distribution. This distribution relates standard deviations with probabilities and allows significance and confidence to be attached to Z scores and p-values.

Standard Normal Distribution

Very high or a very low (negative) Z scores, associated with very small p-values, are found in the tails of the normal distribution. When you perform a feature pattern analysis and it yields small p-values and either a very high or a very low (negative) Z score, this indicates it is very UNLIKELY that the observed pattern is some version of the theoretical spatial random pattern represented by your null hypothesis.

In order to reject the null hypothesis, you must make a subjective judgment regarding the degree of risk you are willing to accept for being wrong. This degree of risk is often given in terms of critical values and/or confidence levels.

To give an example: the critical Z score values when using a 95% confidence level are -1.96 and +1.96 standard deviations. The p-value associated with a 95% confidence level is 0.05. If your Z score is between -1.96 and +1.96, your p-value will be larger than 0.05, and you cannot reject your null hypothsis; the pattern exhibited is a pattern that could very likely be one version of a random pattern. If the Z score falls outside that range (for example -2.5 or +5.4), the pattern exhibited is probably too unusual to be just another version of random chance and the p-value will be small to reflect this. In this case, it is possible to reject the null hypothesis and proceed with figuring out what might be causing the statistically significant spatial pattern.

A key idea here is that the values in the middle of the normal distribution (Z scores like 0.19 or -1.2, for example), represent the expected outcome (the norm ...generally uninteresting). When the absolute value of the Z score is large (in the tails of the normal distribution) and the probabilities are small, you are seeing something unusual and generally very interesting. For the Hot Spot Analysis tool, for example, "unusual" means either a statistically significant hot spot or a statistically significant cold spot.

The Null Hypothesis

Many of the statistics in the spatial statistics toolbox are inferential spatial pattern analysis techniques (i.e., Global Moran's I, Local Moran's I, Gi*). Inferential statistics are grounded in probability theory. Probability is a measure of chance, and underlying all statistical tests (either directly or indirectly) are probability calculations that assess the role of chance on the outcome of your analysis. Typically, with traditional (non-spatial) statistics, you work with a random sample and try to determine the probability that your sample data is a good representation (is reflective) of the population at large. As an example, you might ask: "What are the chances that the results from my exit poll (showing candidate A will beat candidate B by a slim margin, perhaps) will reflect final election results?" But with many spatial statistics, including the spatial autocorrelation type statistics listed above, very often you are dealing with all available data for the study area (all crimes, all disease cases, attributes for every census block, and so on). When you compute a statistic (the mean, for example) for the entire population, you no longer have an estimate at all. You have a fact. Consequently, it makes no sense to talk about "likelihood" or "probabilities" any more. So what can you do in the case where you have all data values for a study area? You can only assess probabilities by postulating, via the null hypothesis, that your spatial data are, in fact, part of some larger population.

Where appropriate, the tools in the spatial statistics toolbox use the randomization null hypothesis as the basis for statistical significance testing. The randomization null hypothesis postulates that the observed spatial pattern of your data represents one of many (n!) possible spatial arrangements. If you could pick up your data values and throw them down onto the features in your study area, you would have one possible spatial arrangement. The randomization null hypothesis states that if you could do this exercise (pick them up, throw them down) infinite times, most of the time you would produce a pattern that would not be markedly different from the observed pattern (your real data). Once in a while you might accidentally throw all of the highest values into the same corner of your study area, but the probabilities of doing that are small. The randomization null hypothesis states that your data is one of many, many, many possible versions of complete spatial randomness. The data values are fixed; only their spatial arrangement could vary.

A common alternative null hypothesis, not implemented for the spatial statistics toolbox, is the normalization null hypothesis. The normalization null hypothesis postulates that the observed values are derived from an infinitely large, normally distributed population of values through some random sampling process. With a different sample you would get different values, but you would still expect those values to be representative of the larger distribution. The normalization null hypothesis states that the values represent one of many possible sample of values. If you could fit your observed data to a normal curve and then randomly select values to toss onto your study area, most of the time you would produce a pattern and distribution of values that would not be markedly different from the observed pattern/distribution (your real data). The normalization null hypothesis states that your data and their arrangement are one of many, many, many possible random samples. Neither the data values nor their spatial arrangment are fixed. The normalization null hypothesis is only appropriate when the data values are normally distributed.

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