24 December, 2024

Chi-square !

 

Types of Chi-Square Tests

  1. Chi-Square Test of Independence:

    • Determines whether two categorical variables are independent of each other.
    • Example: Testing if gender and preference for a product are related.
  2. Chi-Square Goodness-of-Fit Test:

    • Compares the observed frequency distribution of a single categorical variable to an expected distribution.
    • Example: Checking if the roll of a die is fair (expected equal probability for all outcomes).

Formula

The Chi-square statistic (χ2\chi^2) is calculated as:

χ2=(OE)2E\chi^2 = \sum \frac{(O - E)^2}{E}

Where:

  • OO: Observed frequency
  • EE: Expected frequency

Steps for Conducting a Chi-Square Test

  1. State the hypotheses:

    • Null hypothesis (H0H_0): There is no association (independence) or the observed data fits the expected distribution.
    • Alternative hypothesis (HaH_a): There is an association (dependence) or the observed data does not fit the expected distribution.
  2. Calculate expected frequencies:

    • For independence: Use marginal totals.
    • For goodness-of-fit: Based on theoretical proportions.
  3. Compute the Chi-square statistic:

    • Apply the formula to calculate χ2\chi^2.
  4. Determine degrees of freedom (dfdf):

    • df=(r1)(c1)df = (r-1)(c-1) for independence tests, where rr and cc are the number of rows and columns in the contingency table.
    • df=number of categories1df = \text{number of categories} - 1 for goodness-of-fit tests.
  5. Compare χ2\chi^2 with the critical value:

    • Use a Chi-square distribution table or p-value with the chosen significance level (α\alpha).
  6. Make a decision:

    • If χ2\chi^2 is greater than the critical value or p-value < α\alpha, reject H0H_0.

Assumptions

  1. The data is categorical.
  2. Observations are independent.
  3. Expected frequencies are sufficiently large (typically E5E \geq 5).

Applications

  • Genetics: Testing Mendelian ratios.
  • Market Research: Analyzing consumer preferences.
  • Social Sciences: Assessing associations in survey data.
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