Chi-Squared Test

The Chi-Squared test is used to determine if a sample comes from a population with a specific distribution. This test is applied to binned data, so the value of the test statistic depends on how the data is binned. Please note that this test is available for continuous sample data only.

Although there is no optimal choice for the number of bins (k), there are several formulas which can be used to calculate this number based on the sample size (N). For example, EasyFit employs the following empirical formula:

The data can be grouped into intervals of equal probability or equal width. The first approach is generally more acceptable since it handles peaked data much better (you can change the binning method in the Fitting Options dialog). Each bin should contain at least 5 or more data points, so certain adjacent bins sometimes need to be joined together for this condition to be satisfied.

Definition

The Chi-Squared statistic is defined as

,

where O_i is the observed frequency for bin i, and E_i is the expected frequency for bin i calculated by

,

where F is the CDF of the probability distribution being tested, and x₁, x₂ are the limits for bin i.

Hypothesis Testing

The null and the alternative hypotheses are:

• H₀: the data follow the specified distribution;
• H_A: the data do not follow the specified distribution.

The hypothesis regarding the distributional form is rejected at the chosen significance level () if the test statistic is greater than the critical value defined as

meaning the Chi-Squared inverse CDF with k-1 degrees of freedom and a significance level of . Though the number of degrees of freedom can be calculated as k-c-1 (where c is the number of estimated parameters), EasyFit calculates it as k-1 since this kind of test is least likely to reject the fit in error.

The fixed values of (0.01, 0.05 etc.) are generally used to evaluate the null hypothesis (H₀) at various significance levels. A value of 0.05 is typically used for most applications, however, in some critical industries, a lower value may be applied.

P-Value

The P-value, in contrast to fixed values, is calculated based on the test statistic, and denotes the threshold value of the significane level in the sense that the null hypothesis (H₀) will be accepted for all values of less than the P-value. For example, if P=0.025, the null hypothesis will be accepted at all significance levels less than P (i.e. 0.01 and 0.02), and rejected at higher levels, including 0.05 and 0.1.

The P-value can be useful, in particular, when the null hypothesis is rejected at all predefined significance levels, and you need to know at which level it could be accepted.

EasyFit displays the P-values based on the Chi-Squared test statistics (χ²) calculated for each fitted distribution.