What is a z-score? What is a p-value?
Most statistical tests begin by identifying a null hypothesis. The null hypothesis for the pattern analysis tools (Analyzing Patterns toolset and Mapping Clusters toolset) is Complete Spatial Randomness (CSR), either of the features themselves or of the values associated with those features. The z-scores and p-values returned by the pattern analysis tools tell you whether you can reject that null hypothesis or not. Often, you will run one of the pattern analysis tools, hoping that the z-score and p-value will indicate that you can reject the null hypothesis, because it would indicate that rather than a random pattern, your features (or the values associated with your features) exhibit statistically significant clustering or dispersion. Whenever you see spatial structure, like clustering, in the landscape (or in your spatial data), you are seeing evidence of some underlying spatial processes at work, and as a geographer or GIS analyst, this is often what you are most interested in.
The p-value is a probability. For the pattern analysis tools, it is the probability that the observed spatial pattern was created by some random process. When the p-value is very small, it means it is very unlikely (small probability) that the observed spatial pattern is the result of random processes, so you can reject the null hypothesis. You might ask: How small is small enough? Good question. See the table and discussion below.
Z-scores are simply standard deviations. If, for example, a tool returns a z-score of +2.5, you would say that the result is 2.5 standard deviations. Both z-scores and p-values are associated with the standard normal distribution as shown below.
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Very high or very low (negative) z-scores, associated with very small p-values, are found in the tails of the normal distribution. When you run a feature pattern analysis tool and it yields small p-values and either a very high or a very low z-score, this indicates it is unlikely that the observed spatial pattern reflects the theoretical random pattern represented by your null hypothesis (CSR).
To reject the null hypothesis, you must make a subjective judgment regarding the degree of risk you are willing to accept for being wrong (for falsely rejecting the null hypothesis). Consequently, before you run the spatial statistic, you select a confidence level. Typical confidence levels are 90, 95, or 99 percent. A confidence level of 99 percent would be the most conservative in this case, indicating that you are unwilling to reject the null hypothesis unless the probability that the pattern was created by random chance is really small (less than a 1 percent probability). The table below shows the critical p-values and z-scores for different confidence levels.
|
z-score (Standard Deviations) |
p-value (Probability) |
Confidence level |
|---|---|---|
|
< -1.65 or > +1.65 |
< 0.10 |
90% |
|
< -1.96 or > +1.96 |
< 0.05 |
95% |
|
< -2.58 or > +2.58 |
< 0.01 |
99% |
Consider an example. The critical z-score values when using a 95 percent confidence level are -1.96 and +1.96 standard deviations. The p-value associated with a 95 percent 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 hypothesis; the pattern exhibited could very likely be the result of random spatial processes. If the z-score falls outside that range (for example, -2.5 or +5.4 standard deviations), the observed spatial pattern is probably too unusual to be the result 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 structure in your data.
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. When the absolute value of the z-score is large and the probabilities are small (in the tails of the normal distribution), however, 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
Several statistics in the Spatial Statistics toolbox are inferential spatial pattern analysis techniques, for example, Spatial Autocorrelation (Global Moran's I), Cluster and Outlier Analysis (Anselin Local Moran's I), and Hot Spot Analysis (Getis-Ord 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 (nonspatial) 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) 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 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 how can the spatial pattern analysis tools, often applied to all data in the study area, legitimately report probabilities? The answer is that they can do this by postulating, via the null hypothesis, that the data is, in fact, part of some larger population. Consider this in more detail.
The Randomization Null Hypothesis: 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 of those values. (Note that picking up your data values and throwing them down arbitrarily is an example of a random spatial process). 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 the highest values into the same corner of your study area, but the probability of doing that is 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.
The Normalization Null Hypothesis: 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 samples of values. If you could fit your observed data to a normal curve and randomly select values from that distribution 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 arrangement are fixed. The normalization null hypothesis is only appropriate when the data values are normally distributed.
Additional Resources:
- Ebdon, David. Statistics in Geography. Blackwell, 1985.
- Mitchell, Andy. The ESRI Guide to GIS Analysis, Volume 2. ESRI Press, 2005.
- Goodchild, M.F., Spatial Autocorrelation. Catmog 47, Geo Books, 1986