Pearson's chi-square test

Pearson's Chi-Square Test is a statistical test used to test the goodness of fit or to test if there is a difference between samples of data . A one-sample chi-square test is used to test the goodness of fit between observed frequencies and theoretical expected frequencies . An example of a one-sample in a chi-square is soil type on a farm. The farm is the single sample that is being tested. The soil types are the categories. The sample can have multiple categories but one sample is being tested. A two-or-more samples chi-square test is used to test differences between data samples . An example of a two-sample chi-square is testing the differences between soil types on two separate farms. The two farms are the two samples.

The Chi-Square statistical test can be used to asses geographic data. The one-sample test enables geographers to examine the differences between observed data and expected data. Two-or-more samples test enables geographers to examine the differences between samples.

Criteria for a Chi-Square Test

-In terms of the Scale of measurement, the data must be nominal.

-The data may also be "categorized" ordinal or interval data.

-The categories of data must be mutually exclusive.

-The "data must be in frequencies, i.e. the number of discrete objects occurring in different categories" . The data cannot be in percentages or proportions.

One-Sample Chi-Square Test

The chi-square statistic is a sum of differences between observed and expected outcome frequencies, each squared and divided by the expectation:

$\chi^2 = \sum_{i=1}^n {\frac{(O_i - E_i)}{E_i}^2}$

where:

$O_i$ = an observed frequency for the $i^{th}$ bin

$E_i$ = an expected (theoretical) frequency for the $i^{th}$ bin, asserted by the null hypothesis

The resulting value can be compared to the chi-square distribution to determine the goodness of fit.

In order to determine the degrees of Freedom of the Chi-Squared distribution, one takes the total number of observed frequencies and subtracts one. For example, if there are eight different frequencies, one would compare to a chi-squared with seven degrees of freedom.

Another way to describe the chi-squared statistic is with the differences weighted based on measurement error:

$\chi^2 = \sum {\frac{(O - E)^2}{\sigma^2}}$

where $\sigma^2$ is the variance of the observation . This definition is useful when one has estimates for the error on the measurements.

The reduced chi-squared statistic is simply the chi-squared divided by the number of degrees of freedom:

$\chi_{red}^2 = \frac{\chi^2}{\nu} = \frac{1}{\nu} \sum {\frac{(O - E)^2}{\sigma^2}}$

where $\nu$ is the number of degrees of freedom, usually given by $N-n-1$, where $N$ is the number of bins, and $n$ is the number of fit parameters. The advantage of the reduced chi-squared is that it already normalizes for the number of data points and model complexity. As a rule of thumb, a large $\chi_{red}^2$ indicates a poor model fit. However $\chi_{red}^2 < 1$ indicates that the model is 'over-fitting' the data (either the model is improperly fitting noise, or the error bars have been over-estimated). A $\chi_{red}^2 > 1$ indicates that the fit has not fully captured the data (or that the error bars have been under-estimated). In principle a $\chi_{red}^2 = 1$ is the best-fit for the given data and error bars.

Binomial Case

A binomial experiment is a sequence of independent trials in which the trials can result in one of two outcomes, success or failure. There are n trials each with probability of success, denoted by p. Provided that np_i ≫ 1 for every i (where i = 1, 2, ..., k), then

$\chi^2 = \sum_{i=1}^{k} {\frac{(N_i - np_i)^2}{np_i}} = \sum_{\mathrm{all\ cells}}^{} {\frac{(\mathrm{O} - \mathrm{E})^2}{\mathrm{E}}}.$

This has approximately a chi-squared distribution with k − 1 df. The fact that df = k − 1 is a consequence of the restriction $\sum N_i=n$. We know there are k observed cell counts, however, once any k − 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k − 1 freely determined cell counts, thus df = k − 1.

Further Reading

• Ebdon, David(1985). Oxford: Blackwell Publishing. ISBN 978-0-631-13688-0.

See Also

• Goodness of fit