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The test statistic for the χ2 test of independence involves comparing observed (sample data) and expected frequencies in each cell of the table. The expected frequencies are computed assuming that the null hypothesis is true. The null hypothesis states that the two variables (the grouping variable and the outcome) are independent. The definition of independence is as follows:

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Borer ET and Gruner DS (2009) Top‐down and bottom‐up regulation of communities. In: Levin SA, Carpenter SR, Godfray HCJ et al. (eds) Princeton Guide to Ecology, pp. 296–304. Princeton, NJ: Princeton University Press.


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Previously we used confidence intervals to estimate some unknown population parameter. For example, we constructed 1-proportion confidence intervals to estimate the true population proportion – this population proportion being the parameter of interest. We even went as far as comparing two intervals to see if they overlapped – if so we concluded that there was no difference between the population proportions for the two groups – or if the interval contained a specific parameter value.

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Turing's manifesto of 1948 distinguished two different approaches to AI, which may be termed "top down" and "bottom up". The work described so far in this article belongs to the top-down approach. In top-down AI, cognition is treated as a high-level phenomenon that is independent of the low-level details of the implementing mechanism--a brain in the case of a human being, and one or another design of electronic digital computer in the artificial case. Researchers in bottom-up AI, or , take an opposite approach and simulate networks of artificial neurons that are similar to the neurons in the human brain. They then investigate what aspects of cognition can be recreated in these artificial networks.

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The difference between the two approaches may be illustrated by considering the task of building a system to discriminate between W, say, and other letters. A bottom-up approach could involve presenting letters one by one to a neural network that is configured somewhat like a retina, and reinforcing neurons that happen to respond more vigorously to the presence of W than to the presence of aany other letter. A top-down approach could involve writing a computer program that checks inputs of letters against a description of W that is couched in terms of the angles and relative lengths of intersecting line segments. Simply put, the currency of the bottom-up approach is and of the top-down approach of relevant features of the task.