Lecture 7 - Hypothesis Testing for Nominal and Ordinal Variables (Chi Square)

Hypothesis Testing with Nominal and Ordinal Variables: Chi Square

• Hypothesis testing is an inferential statistical procedure used to determine if a relationship exists between variables or if there's a difference between groups.
• The logic involves comparing observed data from a sample to the expected data if there were no relationship between the variables.
• Hypothesis testing procedures are used when two categorical variables (nominal or ordinal) are analysed, assessing association between them in the context of bivariate tables and column percentages.
• A null hypothesis ("H₀") is a statement of "no difference" or "no relationship".
• A research hypothesis ("H₁") is the belief by the researcher that a difference exists between the groups or variables.
• The alpha level (α) represents the probability of rejecting the null hypothesis when it is actually true (a type I error).
• The five-step model is a systematic framework for conducting hypothesis tests in various contexts regardless of the unique characteristics of the test. The five steps are as follows:
  • Step 1: Make assumptions and satisfy all test requirements.
  • Step 2: Outline the null hypothesis.
  • Step 3: Select the sampling distribution and the critical region.
  • Step 4: Compute the test statistic (obtained) value.
  • Step 5: Make a decision, reject the null hypothesis or fail to reject it, and interpret results.
• Sample size affects the probability of rejecting the null hypothesis. Larger samples are more accurate.
• Statistical significance does not equal practical or theoretical importance.
• Chi-square tests measure the relationship between categorical variables in bivariate tables.
  • It's nonparametric and doesn't rely on assumptions about the shape of the population distribution.
  • It's applied to nominal or ordinal variables to find if there is a relationship.
  • Expected frequencies are calculated given that the null hypothesis is true
  • The observed frequencies are compared against expected frequencies
  • The difference between these frequencies yields the test statistic. The greater the difference, the more statistically significant the relationship is.

Computing Chi Square

• To compute chi-square, the test statistic (x²) is calculated from the sample data.
• The value of the obtained chi-square (x²) is compared with the critical value (x²) from the chi-square table for the specified alpha level and degrees of freedom.
• Degrees of freedom is calculated using the formula df=(r-1)(c-1), where r is the number of rows and c is the number of columns in the bivariate table.
• A computing table (e.g, Table 7.7) facilitates the multiple steps involved in calculating the chi-square test statistic.

Selecting an Alpha Level

• Researchers select an alpha level to define what constitutes an “unlikely” sample outcome, affecting the decision on the null hypothesis
• The alpha level (e.g., 0.05) refers to the probability.
• Lower alpha levels (e.g., 0.01) yield a smaller critical region and reduce type I errors (false positive).
• Higher alpha levels (e.g., 0.10) increase the likelihood of rejecting the null hypothesis even if there is no true relationship (increasing type II error- false negative).

The Five-Step Model for Hypothesis Testing: Application to Bivariate Tables

• Using the five-step model, ensure the sample data represents the population perfectly, and consider random sampling error as a potential factor.
• The Chi-Square Test for Independence is a nonparametric statistical test that determines if differences between two categorical variables in bivariate tables are significant
• This test does not establish the causality between the variables.