Chi-Square Goodness of Fit Test Study Notes

Introduction to the Chi-Square Goodness of Fit Test

• Conceptual Overview: The Chi-Square Goodness of Fit test is presented in Chapter 15. It represents a significant departure from previous hypothesis tests covered in the curriculum.
• Progression of Statistical Tests: Statistics education often follows a logical, step-by-step progression:
  • One-sample $Z$-test to One-sample $t$-test (minimal changes).
  • One-sample tests to Two-sample tests (Independent or Repeated Measures).
  • Two-sample tests to Analysis of Variance (ANOVA), which handles three or more samples.
• The Shift to Correlation and Regression: A major break in the progression occurred with correlation and regression. In these tests, there is no independent variable defining groups; instead, there are two continuous variables (typically interval or ratio) that are measured rather than manipulated.
• The Chi-Square Departure: The Chi-Square test is another radical shift because it allows for the analysis of nominal and ordinal data, which have not been addressed previously in the course.

Scales of Measurement Refresher

• The Statistical Toolkit: Hypothesis tests are tools used for specific data situations. Knowing more tests broadens the ability to analyze different data structures.
• Nominal Data: Data that consists simply of names or labels with no inherent order. Examples include:
  • College Major: (Psychology, Sociology, Business, Nursing). There is no universally "right" order to list these.
  • Gender: (Male, Female, Non-binary).
  • Ethnicity, Religion, Nationality, Political Party Preference.
• Ordinal Data: Categorical data that has a logical rank or order. Any measure of "rank" is ordinal.
  • Example: Military Rank (General, Colonel, Private). While there is a clear order, the "distance" or interval between ranks is not quantifiable or consistent.
• Interval Data: Categories that have an order and consistent intervals (distances) between adjacent categories. Moving up one unit represents the same amount of change regardless of where you are on the scale.
• Ratio Data: Adds a "true zero point" to the interval scale. A value of zero represents a total absence of the variable being measured.
• Applicability: Chi-Square is primarily used for nominal data but can also be applied to ordinal data.

Proportions and Frequencies: Definitions and Distinctions

• Proportions (Strict Definition): A decimal number ranging from $0$ to $1$.
  • A proportion cannot be negative.
  • A proportion cannot exceed $1$.
  • It expresses "part of a whole" information where $0$ is none and $1$ is the whole.
• Proportions (Loose Definition): In practice, the term is often used interchangeably with fractions and percentages, as all three express part-of-a-whole information.
  • Fraction: $\frac{2}{5}$
  • Proportion (True): $0.4$
  • Percent: $40\%$
  • The percent symbol ($\%$) literally means "per hundred." The speaker offers a "fake history lesson" suggesting the symbol may be a corrupted way of writing the number $100$.
• Frequencies: A measure of how often something happens—a simple count.
  • Example: Taking a break five times during a two-hour lecture results in a frequency of $5$.

The Two Frequencies in Chi-Square

• Observed Frequency ($f_o$):
  • Obtained by counting actual occurrences in the real world.
  • Represents the actual data gathered during research.
  • Example: A geologist counting the number of earthquakes that actually occur in California over a century.
• Expected Frequency ($f_e$):
  • The number of times something is predicted or hypothesized to happen.
  • Obtained through calculation rather than direct observation.
  • Formula for Expected Frequency: $f_e = n \times \text{Expected Proportion}$
  • Example: If an instructor expects $60\%$ ($0.60$) of a $30$-student class to be female, the expected frequency ($f_e$) is $30 \times 0.60 = 18$.

Determining Expected Proportions ($H_0$)

• Origin of Expected Proportions: These proportions are placed in the Null Hypothesis ($H_0$) and come from one of two sources:
  1. Equal Distribution Assumption: Assuming all categories occur equally.
     • Formula: $\frac{1}{C}$, where $C$ is the number of categories.
     • Example: If testing if crimes occur equally across seven days of the week, the expected proportion for each day is $\frac{1}{7}$.
  2. Previous Research or Experience: Using data from past studies or historical records.
     • Example: A researcher investigating if the distribution of crimes has changed since $1980$ would use the specific percentages found in $1980$ (e.g., $21\%$ on Sunday, $12\%$ on Monday) as the expected proportions.

The General Purpose and Nature of the Chi-Square Test

• Purpose: To compare observed frequencies ($f_o$) to expected frequencies ($f_e$) to see if the difference is large enough to reject the null hypothesis.
  • If the fit is "good" (the observed frequencies are close to the hypothesized frequencies), $H_0$ is not rejected.
  • If the fit is "poor" (the observed frequencies differ significantly from hypothesized frequencies), $H_0$ is rejected.
• The Symbol: The Greek letter Chi squared ($\chi^2$). It is pronounced "chi" (like "sky" without the S), not "g-square" or "chi-square" with a soft 'ch'.
• Mathematical Properties:
  • $\chi^2$ can never be negative because the differences are squared.
  • $\chi^2$ would only be $0$ if the observed data perfectly matched the expected data, which is practically impossible in real research.
• Non-Parametric Nature: Chi-Square is a non-parametric test. Unlike the $\mu$ (mean) and $\sigma$ (standard deviation) used in parametric tests, non-parametric tests do not make assumptions about population parameters. This makes them more flexible and useful for categorical (nominal) variables.

The Four Steps of the Chi-Square Goodness of Fit Test

• Step 1: State the Hypotheses ($H_0$ and $H_1$):
  • $H_0$: Lists the expected proportions for every category.
  • If categories are assumed equal, you can use a shortcut: "All categories have a proportion of $\frac{1}{C}$." You must state the specific number.
  • If based on previous research, you must list every category and its specific proportion.
  • The textbook often uses the unhelpful definition $H_1$: "$H_0$ is false."
• Step 2: Calculate Expected Frequencies and Obtain Chi-Square:
  • First, calculate $f_e$ for every category using $f_e = n \times \text{Expected Proportion}$.
  • Then, calculate the Obtained Chi-Square: $\chi^2 = \sum \frac{(f_o - f_e)^2}{f_e}$.
  • The operation is performed for each category (e.g., each day of the week), then all resulting values are summed.
• Step 3: Determine Degrees of Freedom and Critical Value:
  • Degrees of Freedom ($df$) is based on categories, not sample size: $df = C - 1$.
  • Consult Table B.7: Use $df$ and the chosen alpha level ($\alpha = 0.05$ or $\alpha = 0.01$) to find the critical value.
• Step 4: Make a Decision and Report in APA Format:
  • Decision Rule: If $\chi^2_{\text{obtained}} > \chi^2_{\text{critical}}$, reject $H_0$.
  • Rejecting $H_0$ means the population proportions are not those stated in the null hypothesis.
  • APA Format: $\chi^2(df, n = \text{Total Sample Size}) = \text{Obtained Value}, p < \alpha$ (if rejected) or $p > \alpha$ (if failed to reject).