Frequency Analysis 3-Contingency Tables and Chi-square Tests of Association

Study Example: Analyzing mortality in mice based on treatment with antiserum.

Sample Size: A total of 111 diseased mice were carefully selected for this study, ensuring a sufficient number of subjects for statistical analysis. These mice were divided into two groups based on their treatment:

• 57 mice received bacteria along with antiserum.

• 54 control mice were exposed to bacteria only without any treatment.

Observed Results:

Mortality Data: This section outlines the observed survival rates in both groups, which is critical for determining the effect of antiserum.

• With antiserum: 13 died, 44 alive, indicating a mortality rate of approximately 22.8%.

• Without antiserum: 25 died, 29 alive, resulting in a higher mortality rate of approximately 46.3%.

Contingency Table:

Statistical Analysis: In order to understand the relationship between antiserum treatment and mortality rates in mice, a statistical analysis was performed.

Test Choice: The Chi-square ($\chi^2$) test of association is widely used to evaluate if mortality rates significantly differ between the two treatment groups. This test is appropriate for categorical data and enables comparisons of observed versus expected frequencies.

Hypotheses:

• Null Hypothesis ($H_0$): The treatment group (G) and death outcome (D) are statistically independent, meaning that the antiserum treatment has no effect on mortality.

• Alternative Hypothesis ($H_1$): The treatment group (G) and death outcome (D) are not independent, indicating that antiserum treatment does influence mortality rates among the mice.

Expected Frequencies Calculation:

To interpret the Chi-square statistic, the expected frequencies for each group must be calculated based on the observed data.

• Formula: For a general r x c contingency table:
  $\chi^2k = \sum{i=1}^r \sum{j=1}^c \frac{(O{ij} - E{ij})^2}{E{ij}}$
  where:
  $O{ij}$ = observed counts $E{ij}$ = expected counts

• Degrees of Freedom: df = (r - 1) \times (c - 1) (In this case, where r=2 and c=2, df=1).

Calculation of Expected Frequencies:

An example calculated expected frequency table for this study is as follows:

• Expected frequency formulas used:
  $E_{ij} = \frac{row~total \times column~total}{total}$

Test Statistic Calculation:

To account for small sample sizes and ensure accuracy, the Chi-Square Statistic is computed using Yate’s Continuity Correction:

• $\chi^2{k} = \sum |O{ij} - E_{ij}|$ adjusted with -0.5 for each cell.

• After performing calculations, the yielded statistic was $\chi^2 = 5.77$.

Decision Making:

To determine statistical significance, the degrees of freedom and associated p-value were evaluated.

• Degrees of Freedom gives: (1 - 1) \times (1 - 1) = 1.

• P-Value Assessment: The p-value associated with $\chi^2 > 5.77$ lies between 0.02 and 0.01, indicating a significant result that suggests a rejection of the null hypothesis.

Conclusion:

Results Interpretation: Given the calculated p-value and the Chi-square statistic, we reject $H_0$. Our findings suggest strong evidence of a significant association between antiserum treatment and increased survival rates in mice, underscoring the potential impact of antiserum as a therapeutic measure.

Example of Color Patterns in Tiger Beetles

Data Collected regarding color patterns across seasonal changes indicates adaptations that may occur in response to environmental variables.
Results:

• Early spring: Bright red 29, Not bright red 11

• Late spring: Bright red 273, Not bright red 191

• Early summer: Bright red 8, Not bright red 31

• Late summer: Bright red 64, Not bright red 64

Statistical Test for Association:

Hypotheses Setup:

• Null Hypothesis ($H_0$): Color (C) and Season (S) are independent, indicating no relationship.

• Alternative Hypothesis ($H_1$): Color (C) and Season (S) are not independent, pointing to seasonal influences on coloration.

Expected Frequencies and Hypothesis Testing:

Similar methods apply for calculating expected frequencies based on observed counts. The analysis concludes with a test statistic that yields a p-value suggesting significance.
Example Degrees of Freedom: $df = (4 - 1)(2 - 1) = 3$

The final test results indicate significance, suggesting a potential relationship between color adaptations and seasonal changes, which could provide insights into the evolutionary strategies of tiger beetles.

Final Insights:

Statistical tests, particularly Chi-square tests, offer powerful tools to determine associations between two categorical variables, enhancing our understanding of biological phenomena. Such detailed analyses are essential in ecology and biology for interpreting data concerning treatment effects and adaptations in nature.