One-Sample Chi-square (χ²) goodness-of-fit test

BRIEF DESCRIPTION

The Chi-square (χ²) goodness-of-fit test is a univariate measure for categorical scaled data, such as dichotomous, nominal, or ordinal data.  It tests whether the variable’s observed frequencies differ significantly from a set of expected frequencies. For example, is our observed sample’s age distribution of 20%, 40%, 40% significantly different from what we expect (e.g. the population age distribution) of 30%, 30%, 40%. Chi-square (χ²) is a non-parametric procedure.
 
SIMILAR STATISTICAL PROCEDURES:
  1. Binomial goodness-of-fit (for binary data)
  2. One-Sample Kolmogorov-Smirnov goodness-of-fit test (for continuous scaled data)
  3. Shapiro-Wilke (for smaller samples and continuous scaled data)
  4. Anderson-Darling goodness-of-fit test (a modified Kolmogorov-Smirnov test for continuous data that gives more weight to the tails of the distribution)
  5. Chi-square (χ²) test of independence (a bivariate test for differences between two categorical variables)
CHARACTERISTICS OF THE VARIABLE(S)
  • Dependent variable: categorical scales, such as dichotomous, nominal, or ordinal data.
  • Independent variable: not applicable
 
DATA ASSUMPTIONS
  1. The Chi-square (χ²) test makes no assumption about the data distribution as it is non-parametric and thus distribution free (Non-parametric tests do not require assumptions about the shape of the underlying distribution). 
  2. Size of expected frequency: When the number of cells is less than 10 and particularly with a small sample, it assumes that the expected value for each cell is five or higher (if not, select the Fishers Exact test). The observed frequencies can be any value, including zero.
  3. The expected frequencies for each category should be at least 1 and no more than 20% of the categories should have expected frequencies of less than.
  4. Random sampling from a defined population.
  5. Independence of observations (cells should be independent): Each observation is generated by a different subject and no subject is counted twice.
  6. Dichotomous, nominal, or ordinal data. Technically it can also be used with continuous data but then the preference will be for the Kolmogorov-Smirnov test.
 
WHERE TO FIND IN SPSS?
  1. Chi-square (χ²) : ANALYZSE / NON-PARA / CHI SQ
  2. Binomial goodness-of-fit test for binary data: ANALYZSE / NON-PARA / BINOMIAL
 
HOW TO REPORT THE FINDINGS?
If the Chi-square (χ²) goodness-of-fit test is significant (e.g. p<.05) then it indicates that our observed sample distribution is significantly different from what we expected (e.g. the population age distribution).
  • Reporting example: “A chi-square test of goodness-of-fit was performed to determine whether the three products were equally preferred. Preference for the three products was not equally distributed as expected, χ² (2, N = 200) = 6.55, p <.05″. NOTE: If p>.05 you don’t report the value (e.g. p>.05 or p=.129) but only indicate it as NS (non-significant).
  • Explanation of the above example: The χ² refers to our Chi-square (χ²) procedure, the 2 in brackets is the degrees of freedom (df), the N=200 is the sample size, the 6.55 is the actual Chi-square (χ²) test value, and the p<.05 indicates an actual p-value which is less (e.g. .023) than our chosen confidence level of .05.
 
FURTHER COMMENTS.
Reminder to first WEIGHT the cases by the FREQ counts data. Convert string variables to numeric variables by using the Automatic Recode function.
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