Glossary

11.2 Goodness-of-Fit Test

2 min read•june 25, 2024

The helps determine if sample data follows a specific . It uses a chi-square statistic to compare observed frequencies with expected frequencies based on the hypothesized distribution.

Interpreting the test results involves comparing the calculated to a . If the statistic exceeds the , we reject the null hypothesis, suggesting the data doesn't fit the specified distribution.

Goodness-of-Fit Test

Goodness-of-fit test for distributions

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  • Determines if sample data comes from a population with a specific probability distribution (normal, binomial, Poisson)
  • Null hypothesis (H0H_0) states the data follows the specified distribution
  • () states the data does not follow the specified distribution
  • Steps to perform the test:
    1. State the hypotheses
    2. Calculate expected frequencies for each category based on the hypothesized distribution
    3. Calculate the using observed and expected frequencies
    4. Determine and critical value from the
    5. Compare test statistic to critical value and decide to reject or fail to reject H0H_0
    6. Calculate the to assess the strength of evidence against H0H_0

Test statistic calculation

  • Chi-square formula calculates the test statistic:
    • χ2=∑i=1k(Oi−Ei)2Ei\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}
      • represents the test statistic
      • represents for category ii
      • represents for category ii
      • kk represents the number of categories
  • Expected frequencies calculated by multiplying total sample size by probability of each category according to hypothesized distribution
  • Degrees of freedom for the test:
    • df=k−1−mdf = k - 1 - m
      • kk represents the number of categories
      • mm represents the number of parameters estimated from the data

Interpretation of chi-square results

  • is a
    • H0H_0 rejected if test statistic greater than critical value
  • Critical value determined using chi-square distribution table with calculated degrees of freedom and desired (typically α=0.05\alpha = 0.05)
  • If test statistic greater than critical value:
    • Reject H0H_0
    • Sufficient evidence suggests data does not follow specified distribution
  • If test statistic less than or equal to critical value:
    • Fail to reject H0H_0
    • Insufficient evidence to conclude data does not follow specified distribution

Additional Considerations

  • Goodness-of-fit test is a , meaning it does not assume a specific underlying distribution for the data
  • Used primarily for categorical data analysis
  • Can be extended to analyze contingency tables for independence or homogeneity tests

Key Terms to Review (27)

$\alpha$: $\alpha$ is a statistical significance level that represents the probability of making a Type I error, or rejecting the null hypothesis when it is actually true. It is a critical parameter used in hypothesis testing and is a fundamental concept in the context of goodness-of-fit tests and tests of independence.
$\chi^2$: $\chi^2$ (chi-squared) is a statistical test used to determine the goodness of fit between observed and expected frequencies in a dataset. It is a powerful tool for analyzing the relationship between categorical variables and assessing whether the differences between observed and expected values are statistically significant.
$E_i$: $E_i$ represents the expected frequency of a category in a goodness-of-fit test. It is calculated under the null hypothesis, which states that the observed distribution of data fits a specified theoretical distribution. The expected frequency is crucial because it provides a benchmark against which the observed frequencies can be compared to determine if there is a statistically significant difference between them.
$H_a$: $H_a$ is the alternative hypothesis in a statistical hypothesis test. It represents the statement that the researcher believes to be true, in contrast to the null hypothesis ($H_0$), which is the statement that the researcher is trying to disprove. The alternative hypothesis is the hypothesis that the researcher hopes to accept if the null hypothesis is rejected.
$O_i$: $O_i$ represents the observed frequency or count of observations in the $i$-th category or bin of a dataset. It is a crucial term in the context of the Goodness-of-Fit Test, which is used to determine whether a dataset follows a hypothesized probability distribution.
Alternative Hypothesis: The alternative hypothesis is a statement that suggests a potential outcome or relationship exists in a statistical test, opposing the null hypothesis. It indicates that there is a significant effect or difference that can be detected in the data, which researchers aim to support through evidence gathered during hypothesis testing.
Binomial probability distribution: A binomial probability distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by two parameters: the number of trials (n) and the probability of success (p).
Chi-Square Distribution: The chi-square distribution is a continuous probability distribution that arises when independent standard normal random variables are squared and summed. It is widely used in statistical hypothesis testing, particularly in evaluating the goodness-of-fit of observed data to a theoretical distribution and in testing the independence of two categorical variables.
Chi-Square Test: The chi-square test is a statistical hypothesis test that is used to determine if there is a significant difference between observed and expected frequencies in one or more categories. It is a versatile test that can be applied in various contexts, including contingency tables, goodness-of-fit, and tests for homogeneity.
Contingency table: A contingency table, also known as a cross-tabulation or crosstab, is a type of table in a matrix format that displays the frequency distribution of variables. It is commonly used to analyze the relationship between two categorical variables.
Contingency Table: A contingency table, also known as a cross-tabulation or a two-way table, is a type of table that displays the frequency distribution of two or more categorical variables. It is used to analyze the relationship between these variables and determine if they are independent or associated with each other.
Critical value: A critical value is a point on the scale of the standard normal distribution that is compared to a test statistic to determine whether to reject the null hypothesis. It separates the region where the null hypothesis is not rejected from the region where it is rejected.
Critical Value: The critical value is a threshold value in statistical analysis that is used to determine whether to reject or fail to reject a null hypothesis. It serves as a benchmark for evaluating the statistical significance of a test statistic and is a crucial concept across various statistical methods and hypothesis testing procedures.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. It plays a crucial role in determining the appropriate statistical tests and distributions used for hypothesis testing, estimation, and data analysis across various contexts.
Expected Frequency: The expected frequency is the anticipated or predicted frequency of an outcome in a statistical analysis, particularly in the context of contingency tables, goodness-of-fit tests, and tests of independence. It represents the expected number of observations in a particular cell or category under the null hypothesis.
Expected values: Expected values are the theoretical frequencies of outcomes in a distribution, calculated based on a specified model. They are used to determine how well observed data fits an expected distribution.
Goodness-of-fit test: A goodness-of-fit test is a statistical hypothesis test used to determine if a sample data matches a population with a specific distribution. It assesses how well the observed frequencies fit the expected frequencies under the null hypothesis.
Goodness-of-Fit Test: A goodness-of-fit test is a statistical method used to determine how well a sample of observed data matches a theoretical probability distribution. This test assesses whether the differences between observed and expected frequencies are significant enough to reject the hypothesis that the observed data follow a specified distribution. It plays a critical role in evaluating models based on probability distributions, such as discrete random variables and exponential distributions.
Nonparametric Test: A nonparametric test is a statistical hypothesis test that does not rely on the data following a specific probability distribution, such as the normal distribution. These tests are often used when the assumptions for parametric tests, like normality, are not met.
Observed Frequency: Observed frequency refers to the actual count or number of occurrences of a particular event or outcome in a dataset or sample. It represents the empirical or observed data, as opposed to the expected or theoretical frequency. This term is crucial in understanding and interpreting various statistical analyses, including contingency tables, goodness-of-fit tests, and tests of independence.
Observed values: Observed values are the actual data points collected from an experiment or survey. These values are used to compare against expected values in statistical tests.
P-value: The p-value is the probability of obtaining a test statistic at least as extreme as the one actually observed, assuming the null hypothesis is true. It is a crucial concept in hypothesis testing that helps determine the statistical significance of a result.
Probability Distribution: A probability distribution is a mathematical function that describes the likelihood or probability of different possible outcomes or events occurring in a given situation or experiment. It provides a comprehensive representation of the possible values a random variable can take on and their corresponding probabilities.
Right-Tailed Test: A right-tailed test is a statistical hypothesis test where the alternative hypothesis specifies that the parameter of interest is greater than the value stated in the null hypothesis. This type of test is used when the researcher is interested in determining if a particular characteristic or outcome exceeds a certain threshold or standard.
Significance Level: The significance level, denoted as α (alpha), is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. The significance level is a crucial concept in hypothesis testing and statistical inference, as it helps determine the strength of evidence required to draw conclusions about a population parameter or the relationship between variables.
Test statistic: A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis.
Test Statistic: A test statistic is a numerical value calculated from sample data that is used to determine whether to reject or fail to reject a null hypothesis in a hypothesis test. It serves as the basis for decision-making in statistical inference, providing a quantitative measure to evaluate the strength of evidence against the null hypothesis.
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