Inference for Categorical Data: Chi-Square · ⁨범주형 데이터에 대한 추론: 카이제곱⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.J: Identify questions suggested by variation between observed and expected counts in categorical data. [Skill 1.A]

• VAR-1.J.1 Variation between what we find and what we expect to find may be random or not.

Enduring Understanding (VAR-8): The chi-square distribution may be used to model variation.

Learning Objective VAR-8.A: Describe chi-square distributions. [Skill 3.C]

• VAR-8.A.1 Expected counts of categorical data are counts consistent with the null hypothesis. In general, an expected count is a sample size times a probability.

  The chi-square statistic measures the distance between observed and expected counts relative to expected counts.

  Chi-square distributions have positive values and are skewed right. Within a family of density curves, the skew becomes less pronounced with increasing degrees of freedom.

Learning Objective VAR-8.B: Identify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data. [Skill 1.F]

• VAR-8.B.1 For a chi-square goodness-of-fit test, the null hypothesis specifies null proportions for each category, and the alternative hypothesis is that at least one of these proportions is not as specified in the null hypothesis.

Learning Objective VAR-8.C: Identify an appropriate testing method for a distribution of proportions in a set of categorical data. [Skill 1.E]

• VAR-8.C.1 When considering a distribution of proportions for one categorical variable, the appropriate test is the chi-square test for goodness of fit.

Learning Objective VAR-8.D: Calculate expected counts for the chi-square test for goodness of fit. [Skill 3.A]

• VAR-8.D.1 Expected counts for a chi-square goodness-of-fit test are (sample size)(null proportion).

Learning Objective VAR-8.E: Verify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution. [Skill 4.C]

• VAR-8.E.1 In order to make statistical inferences for a chi-square test for goodness of fit we must check the following:
  • a. To check for independence:
    • i. Data should be collected using a random sample or randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$.
  • b. The chi-square test for goodness of fit becomes more accurate with more observations, so large counts should be used (shape).
    • i. A conservative check for large counts is that all expected counts should be greater than 5.

Enduring Understanding (VAR-8): The chi-square distribution may be used to model variation.

Learning Objective VAR-8.F: Calculate the appropriate statistic for the chi-square test for goodness of fit. [Skill 3.E]

• VAR-8.F.1 The test statistic for the chi-square test for goodness of fit is
  • Equation: $\chi^2 = \sum \dfrac{(Observed\ count - Expected\ count)^2}{Expected\ count}$, with $degrees\ of\ freedom = number\ of\ categories - 1$.
• VAR-8.F.2 The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be either a randomization distribution or, when a probability model is assumed to be true, a theoretical distribution (chi-square).

Learning Objective VAR-8.G: Determine the $p$-value for chi-square test for goodness of fit significance test. [Skill 3.E]

• VAR-8.G.1 The $p$-value for a chi-square test for goodness of fit for a number of degrees of freedom is found using the appropriate table or computer generated output.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.I: Interpret the $p$-value for the chi-square test for goodness of fit. [Skill 4.B]

• DAT-3.I.1 An interpretation of the $p$-value for the chi-square test for goodness of fit is the probability, given the null hypothesis and probability model are true, of obtaining a test statistic as, or more, extreme than the observed value.

Learning Objective DAT-3.J: Justify a claim about the population based on the results of a chi-square test for goodness of fit. [Skill 4.E]

• DAT-3.J.1 A decision to either reject or fail to reject the null hypothesis is based on comparison of the $p$-value to the significance level, $\alpha$.
• DAT-3.J.2 The results of a chi-square test for goodness of fit can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.

Enduring Understanding (VAR-8): The chi-square distribution may be used to model variation.

Learning Objective VAR-8.H: Calculate expected counts for two-way tables of categorical data. [Skill 3.A]

• VAR-8.H.1 The expected count in a particular cell of a two-way table of categorical data can be calculated using the formula:
  • Equation: $expected\ count = \dfrac{(row\ total)(column\ total)}{table\ total}$.

Enduring Understanding (VAR-8): The chi-square distribution may be used to model variation.

Learning Objective VAR-8.I: Identify the null and alternative hypotheses for a chi-square test for homogeneity or independence. [Skill 1.F]

• VAR-8.I.1 The appropriate hypotheses for a chi-square test for homogeneity are:

  $H_0$: There is no difference in distributions of a categorical variable across populations or treatments.

  $H_a$: There is a difference in distributions of a categorical variable across populations or treatments.

• VAR-8.I.2 The appropriate hypotheses for a chi-square test for independence are:

  $H_0$: There is no association between two categorical variables in a given population or the two categorical variables are independent.

  $H_a$: Two categorical variables in a population are associated or dependent.

Learning Objective VAR-8.J: Identify an appropriate testing method for comparing distributions in two-way tables of categorical data. [Skill 1.E]

• VAR-8.J.1 When comparing distributions to determine whether proportions in each category for categorical data collected from different populations are the same, the appropriate test is the chi-square test for homogeneity.
• VAR-8.J.2 To determine whether row and column variables in a two-way table of categorical data might be associated in the population from which the data were sampled, the appropriate test is the chi-square test for independence.

Learning Objective VAR-8.K: Verify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity. [Skill 4.C]

• VAR-8.K.1 In order to make statistical inferences for a chi-square test for two-way tables (homogeneity or independence), we must verify the following:
  • a. To check for independence:
    • i. For a test for independence: Data should be collected using a simple random sample.
    • ii. For a test for homogeneity: Data should be collected using a stratified random sample or randomized experiment.
    • iii. When sampling without replacement, check that $n \leq 10\%N$.
  • b. The chi-square tests for independence and homogeneity become more accurate with more observations, so large counts should be used (shape).
    • i. A conservative check for large counts is that all expected counts should be greater than 5.

This topic is intended to focus on the skill of selecting an appropriate inference procedure now that students have a range of options. Students should be given opportunities to practice when and how to apply all learning objectives relating to inference for categorical data.