# Goodness of Fit

A goodness-of-fit test checks if a sample data fits a distribution from a certain po­­pulation (population with a Weibull distribution or one with a normal distribution). It tells us if the sample data represents the data you would expect to find in the actual population.

The most common goodness-of-fit tests used in statistics are:

• The Chi-square
• Anderson-Darling
• Kolmogorov-Smirnov
• Shapiro-Wilk

Chi-Square Goodness-of-Fit Test in SPSS

This is a single sample non-parametric test. The Chi-square goodness-of-fit test is also called the Pearson’s Chi-square goodness-of-fit test or the one-sample goodness-of-fit test. This test determines whether the distribution cases in a single categorical variable follows a hypothesized or known distribution. You can have either an equal or unequal proportion of cases expected in each group of the categorical variable. It is critical to know this when you are carrying out a Chi-square goodness-of-fit. This is both an important aspect of your research and determines how you will carry the test in SPSS statistical tool. From a practical perspective, it also determines how you interpret and write up your results.

Chi-square supports discrete distributions like the binomial distribution and the Poisson distribution. The Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests can only support continuous distributions.

Assumptions made in the Chi-square goodness-of-fit test

Before settling on the Chi-square goodness-of-fit test, you must first check and make sure that the data you want to analyze can be analyzed using this test. This is because it is only appropriate to use the Chi-square goodness of fit test if your data meets the four assumptions required. Only then will you get a valid result.

It is quite simple to check these assumptions when in practice. You are only required to use the SPSS Statistics package. Our online goodness-of-fit tutors have discussed the assumptions below:

1. You should have one categorical variable. This means that the variable can be
• Dichotomous variables – The examples of dichotomous variables are gender (female or male), treatment type (medication and no medication), education level (undergraduate and postgraduate. A dichotomous variable has only two groups.
• Ordinal variables – These include Likert scales. For example, a 7-point scale from “strongly agree” through to “strongly disagree”
• Nominal variables – examples of nominal variables include ethnicity ( Hispanic, Caucasian, and African American), profession ( Doctor, nurse, surgeon, accountant, and psychologist)
1. The observations should be independent – an independent observation means that there is no relationship between any of the cases.
2. The groups or levels of the categorical variable must be mutually exclusive. For example, suppose we have a variable physical activity with four groups, sedentary, low, moderate, and high. A participant in the study could only be one of the four groups. E.g. we can classify a participant as having high activity level but not high level and low level.
3. Each group of your categorical variable must have at least 5 expected frequencies. This assumption will be shown in your SPSS Statistics output when you run the test.

Disadvantages of the Chi-square goodness-of-fit test

• The Chi-square goodness-of-fit test is only good with data put in classes or groups. You will need to make a frequency table or a histogram before doing the test if your data is non-binned.
• The Chi-square test also requires a sufficient sample size for the approximation to be valid.

You should not confuse the Chi-square goodness-of-fit test with the Chi-square test for independence. The latter compares two sets of data to see if there is any relationship while the former fits one categorical variable to a distribution. However, these two tests both use the Chi-square statistic and distribution.

Running the Chi-square goodness-of-fit test

This test is usually run using software, like the SPSS Statistics package. The chi-square goodness-of-fit test’s null hypothesis is that the data comes from a specified distribution. On the other hand, the alternate hypothesis states that the data does not come from a specified distribution.

The analyst will need to choose an alpha level to interpret the test. The most common ones are 1%, 5%, and 10%. Also, the Chi-square goodness-of-fit test returns a p-value. You can reject the null hypothesis that says that the data comes from the specified distribution if the p-value is small (less than the level of significance).

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