Goodness Of Fit Test Example

Goodness of Fit Test: Examples and Applications

The goodness of fit test is a statistical method used to determine how well a sample data set matches a population. It assesses whether observed frequencies differ significantly from expected frequencies, allowing us to test hypotheses about the distribution of a variable. This article will delve into the goodness of fit test, providing numerous examples to illustrate its application across various scenarios. We will cover the underlying theory, step-by-step procedures, interpretation of results, and address frequently asked questions. Understanding this powerful tool is crucial for researchers and analysts in numerous fields, from biology and sociology to marketing and finance.

Understanding the Chi-Square Goodness of Fit Test

The most common type of goodness of fit test utilizes the chi-square (χ²) distribution. This test is particularly useful when dealing with categorical data. The null hypothesis (H₀) typically states that the observed data follows a specific distribution (e.g., uniform, normal, binomial). The alternative hypothesis (H₁) suggests that the observed data does not follow the specified distribution.

The test statistic, χ², is calculated as:

χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]

Where:

• Oᵢ represents the observed frequency for category i.
• Eᵢ represents the expected frequency for category i.

A larger χ² value indicates a greater discrepancy between observed and expected frequencies, suggesting a poorer fit. The degrees of freedom (df) are calculated as (k - p -1), where k is the number of categories and p is the number of parameters estimated from the sample data. For example, if we are testing against a uniform distribution with 5 categories, the df would be (5-1)=4. If we are testing against a normal distribution, we would estimate the mean and standard deviation from the sample, resulting in (k-3) degrees of freedom. We then compare the calculated χ² value to the critical χ² value from the chi-square distribution table at a chosen significance level (alpha, usually 0.05). If the calculated χ² exceeds the critical χ² value, we reject the null hypothesis.

Example 1: Testing for a Uniform Distribution of Dice Rolls

Let's say we roll a six-sided die 60 times. Under the null hypothesis of a fair die, we expect each face to appear 10 times (60 rolls / 6 faces = 10). However, our observed results are as follows:

Now, let's calculate the chi-square statistic:

χ² = [(8-10)²/10] + [(12-10)²/10] + [(9-10)²/10] + [(11-10)²/10] + [(10-10)²/10] + [(10-10)²/10] = 0.4 + 0.4 + 0.1 + 0.1 + 0 + 0 = 1.0

With 5 degrees of freedom (6 categories - 1), and a significance level of 0.05, the critical χ² value from the chi-square distribution table is approximately 11.07. Since our calculated χ² (1.0) is less than the critical χ² value (11.07), we fail to reject the null hypothesis. There is not enough evidence to suggest that the die is unfair.

Example 2: Testing for a Specific Proportional Distribution

A marketing company claims that their advertisement campaign results in customers choosing their product with specific proportions: 40% prefer Option A, 30% prefer Option B, and 30% prefer Option C. A survey of 200 customers reveals the following preferences:

Calculating the chi-square statistic:

χ² = [(70-80)²/80] + [(60-60)²/60] + [(70-60)²/60] = 1.25 + 0 + 1.67 = 2.92

With 2 degrees of freedom (3 categories - 1), and α = 0.05, the critical χ² value is approximately 5.99. Since our calculated χ² (2.92) is less than the critical χ² value (5.99), we fail to reject the null hypothesis. The observed data doesn't significantly contradict the company's claim.

Example 3: Goodness of Fit with a Normal Distribution

This example is more complex as it involves a continuous variable and requires estimating parameters from the sample data. Suppose we have the following sample data representing the heights of 100 students: (We'll simplify the calculation for demonstration purposes; a real-world application would utilize statistical software). Assume we calculate the sample mean (μ) as 170 cm and the sample standard deviation (σ) as 10 cm. We want to test if the height data follows a normal distribution. We need to divide the data into several intervals (bins) and determine the expected frequency for each interval based on the normal distribution's cumulative distribution function (CDF) using μ and σ. Suppose after binning our data, we observe the following:

The calculation of the chi-square statistic would then proceed as in the previous examples. The degrees of freedom would be reduced due to the estimation of μ and σ. This example illustrates that testing against a continuous distribution like the normal requires more steps and calculations, frequently necessitating the use of statistical software.

Step-by-Step Procedure for Performing a Chi-Square Goodness of Fit Test

1. State the Hypotheses: Define your null (H₀) and alternative (H₁) hypotheses clearly.
2. Determine the Expected Frequencies: Calculate the expected frequencies for each category based on the hypothesized distribution.
3. Calculate the Chi-Square Statistic: Use the formula χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ].
4. Determine the Degrees of Freedom: Calculate the degrees of freedom (df = k - p -1).
5. Find the Critical Chi-Square Value: Consult a chi-square distribution table using the calculated df and your chosen significance level (α).
6. Compare the Calculated and Critical Chi-Square Values: If the calculated χ² value is greater than the critical χ² value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
7. Interpret the Results: Draw conclusions based on your statistical findings in the context of the problem.

Limitations of the Chi-Square Goodness of Fit Test

• Sample Size: The chi-square test is most accurate with larger sample sizes. Expected frequencies in each category should ideally be at least 5. If this condition is violated, consider combining categories or using alternative tests.
• Independence: The observations should be independent of each other.
• Categorical Data: The test is primarily designed for categorical data. For continuous data, you'll need to categorize it into intervals first.
• Sensitivity to Small Differences: With large sample sizes, even minor deviations from the expected distribution can lead to a significant chi-square value.

Frequently Asked Questions (FAQ)

Q: What is the significance level (α)?

A: The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true (Type I error). It's typically set at 0.05 (5%), meaning there's a 5% chance of incorrectly rejecting a true null hypothesis.

Q: What if my expected frequencies are less than 5?

A: If expected frequencies are less than 5 in one or more categories, it can affect the accuracy of the chi-square test. Consider combining categories to meet the requirement, or use alternative tests like Fisher's exact test, particularly for small samples.

Q: Can I use the goodness of fit test for continuous data?

A: Yes, but you need to first categorize the continuous data into intervals. The choice of interval width affects the results.

Q: What software can I use to perform a chi-square goodness of fit test?

A: Many statistical software packages, including SPSS, R, SAS, and Python (with libraries like SciPy), can perform chi-square goodness of fit tests efficiently.

Conclusion

The goodness of fit test, particularly the chi-square test, is a valuable tool for assessing how well sample data conforms to a theoretical distribution. By following the steps outlined above and understanding its limitations, you can effectively apply this statistical method in various contexts. Remember that statistical significance does not necessarily equate to practical significance. Always interpret your results within the broader context of your research question and the limitations of the statistical test employed. Furthermore, it's crucial to visualize your data and consider alternative explanations before drawing strong conclusions solely based on the results of a statistical test. The examples provided offer practical insights into the application of the goodness-of-fit test in diverse scenarios. By carefully considering the assumptions and limitations of the test, researchers can make informed decisions about the distribution of their data and improve the validity of their findings.