Chi-square test Knowledge-Based Study

Study notes

1. What Is the Chi-Square Test?

   In biological investigations, researchers frequently collect categorical count data, such as the numbers of offspring showing specific phenotypes or organisms choosing particular microhabitats. The chi-square ($\chi^2$) goodness-of-fit test is a statistical tool used to determine whether observed experimental outcomes differ significantly from expected theoretical predictions.

   Formulating Hypotheses

   Before applying the test, two mutually exclusive hypotheses are established:

   • Null Hypothesis ($H_0$): There is no significant difference between the observed and expected frequencies. Any observed deviation is due solely to random chance.
   • Alternative Hypothesis ($H_a$): There is a significant difference between the observed and expected frequencies. The observed deviation cannot be explained by chance alone, indicating an underlying biological factor or experimental effect.

   The Chi-Square Formula

   The test statistic is calculated by quantifying the deviation of observed counts from theoretical expectations across all outcome categories:

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

   Components of the Formula

   • $O$ (Observed value): The actual count recorded for each specific category during the experiment.
   • $E$ (Expected value): The theoretical count calculated based on the null hypothesis (e.g., Mendelian ratios multiplied by total sample size).
   • $\sum$ (Summation): The sum of the calculated values across all categories.

   Important Rule: Calculations must always use raw numerical counts (frequencies), never percentages or proportions.

   Degrees of Freedom

   To interpret the calculated $\chi^2$ value, we determine the degrees of freedom ($df$):

   $$df = n - 1$$

   where $n$ is the number of distinct phenotypic or outcome categories, not the total number of individuals sampled.

   Critical Value and Significance Level

   In biology, the standard significance threshold ($p$-value) is set at $\alpha = 0.05$. This means there is a 5% probability that the observed deviation occurred purely by chance.

   • If $\chi^2_{\text{calculated}} \le \chi^2_{\text{critical}}$, we fail to reject the null hypothesis. The observed data fit the expected model.
   • If $\chi^2_{\text{calculated}} > \chi^2_{\text{critical}}$, we reject the null hypothesis. A statistically significant difference exists.