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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.15.

Chi-Square Goodness-of-Fit Test
Purpose:
Test for distributional adequacy 		The chi-square test (Snedecor and Cochran, 1989) is used to test if a sample of data came from a population with a specific distribution.

An attractive feature of the chi-square goodness-of-fit test is that it can be applied to any univariate distribution for which you can calculate the cumulative distribution function. The chi-square goodness-of-fit test is applied to binned data (i.e., data put into classes). This is actually not a restriction since for non-binned data you can simply calculate a histogram or frequency table before generating the chi-square test. However, the value of the chi-square test statistic are dependent on how the data is binned. Another disadvantage of the chi-square test is that it requires a sufficient sample size in order for the chi-square approximation to be valid.

The chi-square test is an alternative to the Anderson-Darling and Kolmogorov-Smirnov goodness-of-fit tests. The chi-square goodness-of-fit test can be applied to discrete distributions such as the binomial and the Poisson. The Kolmogorov-Smirnov and Anderson-Darling tests are restricted to continuous distributions.

Additional discussion of the chi-square goodness-of-fit test is contained in the product and process comparisons chapter (chapter 7).

Definition The chi-square test is defined for the hypothesis:

H0: The data follow a specified distribution.
Ha: The data do not follow the specified distribution.
Test Statistic: For the chi-square goodness-of-fit computation, the data are divided into k bins and the test statistic is defined as
    CHI-SQUARE=SUM(O(i)-E(i))**2/E(i) where the summation is from 1 to k
where O(I) is the observed frequency for bin i and E(i) is the expected frequency for bin i. The expected frequency is calculated by
    E(i) = N*(F(Y(u) - F(Y(l))
where F is the cumulative Distribution function for the distribution being tested, Yu is the upper limit for class i, Yl is the lower limit for class i, and N is the sample size.

This test is sensitive to the choice of bins. There is no optimal choice for the bin width (since the optimal bin width depends on the distribution). Most reasonable choices should produce similar, but not identical, results. Dataplot uses 0.3*s, where s is the sample standard deviation, for the class width. The lower and upper bins are at the sample mean plus and minus 6.0*s, respectively. For the chi-square approximation to be valid, the expected frequency should be at least 5. This test is not valid for small samples, and if some of the counts are less than five, you may need to combine some bins in the tails.

Significance Level: alpha.
Critical Region: The test statistic follows, approximately, a chi-square distribution with (k - c) degrees of freedom where k is the number of non-empty cells and c = the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution + 1. For example, for a 3-parameter Weibull distribution, c = 4.

Therefore, the hypothesis that the data are from a population with the specified distribution is rejected if

    CHI-SQUARE > CHI-SQUARE(alpha,k-c)
where CHI-SQUARE(alpha,k-c) is the chi-square percent point function with k - c degrees of freedom and a significance level of alpha.

In the above formulas for the critical regions, the Handbook follows the convention that CHI-SQUARE(alpha) is the upper critical value from the chi-square distribution and CHI-SQUARE(1-alpha) is the lower critical value from the chi-square distribution. Note that this is the opposite of what is used in some texts and software programs. In particular, Dataplot uses the opposite convention.

Sample Output
 		Dataplot generated the following output for the chi-square test where 1,000 random numbers were generated for the normal, double exponential, t with 3 degrees of freedom, and lognormal distributions. In all cases, the chi-square test was applied to test for a normal distribution. The test statistics show the characteristics of the test; when the data are from a normal distribution, the test statistic is small and the hypothesis is accepted; when the data are from the double exponential, t, and lognormal distributions, the statistics are significant and the hypothesis of an underlying normal distribution is rejected at significance levels of 0.10, 0.05, and 0.01.

The normal random numbers were stored in the variable Y1, the double exponential random numbers were stored in the variable Y2, the t random numbers were stored in the variable Y3, and the lognormal random numbers were stored in the variable Y4. 

       *************************************************
       **  normal chi-square goodness of fit test y1  **
       *************************************************
  
  
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            NORMAL
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =     1000
    NUMBER OF NON-EMPTY CELLS   =       24
    NUMBER OF PARAMETERS USED   =        0
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    17.52155
    DEGREES OF FREEDOM          =       23
    CHI-SQUARED CDF VALUE       =    0.217101
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       32.00690               ACCEPT H0
             5%       35.17246               ACCEPT H0
             1%       41.63840               ACCEPT H0
  
       CELL NUMBER, BIN MIDPOINT, OBSERVED FREQUENCY,
       AND EXPECTED FREQUENCY
       WRITTEN TO FILE DPST1F.DAT
  
       *************************************************
       **  normal chi-square goodness of fit test y2  **
       *************************************************
  
  
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            NORMAL
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =     1000
    NUMBER OF NON-EMPTY CELLS   =       26
    NUMBER OF PARAMETERS USED   =        0
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    2030.784
    DEGREES OF FREEDOM          =       25
    CHI-SQUARED CDF VALUE       =    1.000000
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       34.38158               REJECT H0
             5%       37.65248               REJECT H0
             1%       44.31411               REJECT H0
  
       CELL NUMBER, BIN MIDPOINT, OBSERVED FREQUENCY,
       AND EXPECTED FREQUENCY
       WRITTEN TO FILE DPST1F.DAT
  
       *************************************************
       **  normal chi-square goodness of fit test y3  **
       *************************************************
  
  
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            NORMAL
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =     1000
    NUMBER OF NON-EMPTY CELLS   =       25
    NUMBER OF PARAMETERS USED   =        0
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    103165.4
    DEGREES OF FREEDOM          =       24
    CHI-SQUARED CDF VALUE       =    1.000000
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       33.19624               REJECT H0
             5%       36.41503               REJECT H0
             1%       42.97982               REJECT H0
  
       CELL NUMBER, BIN MIDPOINT, OBSERVED FREQUENCY,
       AND EXPECTED FREQUENCY
       WRITTEN TO FILE DPST1F.DAT
  
       *************************************************
       **  normal chi-square goodness of fit test y4  **
       *************************************************
  
  
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            NORMAL
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =     1000
    NUMBER OF NON-EMPTY CELLS   =       10
    NUMBER OF PARAMETERS USED   =        0
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    1162098.
    DEGREES OF FREEDOM          =        9
    CHI-SQUARED CDF VALUE       =    1.000000
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       14.68366               REJECT H0
             5%       16.91898               REJECT H0
             1%       21.66600               REJECT H0
  
       CELL NUMBER, BIN MIDPOINT, OBSERVED FREQUENCY,
       AND EXPECTED FREQUENCY
       WRITTEN TO FILE DPST1F.DAT
As we would hope, the chi-square test does not reject the normality hypothesis for the normal distribution data set and rejects it for the three non-normal cases.
Questions The chi-square test can be used to answer the following types of questions:
  • Are the data from a normal distribution?
  • Are the data from a log-normal distribution?
  • Are the data from a Weibull distribution?
  • Are the data from an exponential distribution?
  • Are the data from a logistic distribution?
  • Are the data from a binomial distribution?
Importance Many statistical tests and procedures are based on specific distributional assumptions. The assumption of normality is particularly common in classical statistical tests. Much reliability modeling is based on the assumption that the distribution of the data follows a Weibull distribution.

There are many non-parametric and robust techniques that are not based on strong distributional assumptions. By non-parametric, we mean a technique, such as the sign test, that is not based on a specific distributional assumption. By robust, we mean a statistical technique that performs well under a wide range of distributional assumptions. However, techniques based on specific distributional assumptions are in general more powerful than these non-parametric and robust techniques. By power, we mean the ability to detect a difference when that difference actually exists. Therefore, if the distributional assumption can be confirmed, the parametric techniques are generally preferred.

If you are using a technique that makes a normality (or some other type of distributional) assumption, it is important to confirm that this assumption is in fact justified. If it is, the more powerful parametric techniques can be used. If the distributional assumption is not justified, a non-parametric or robust technique may be required.

Related Techniques Anderson-Darling Goodness-of-Fit Test
Kolmogorov-Smirnov Test
Shapiro-Wilk Normality Test
Probability Plots
Probability Plot Correlation Coefficient Plot
Case Study Airplane glass failure times data.
Software Some general purpose statistical software programs, including Dataplot, provide a chi-square goodness-of-fit test for at least some of the common distributions.
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