Biostatistics: Chi Square | Zoology Optional Notes for UPSC PDF Download

χ² = Σ [(O_i - E_i)² / E_i]

Where:

• χ² is the Chi-Square statistic.
• Σ represents the summation symbol, indicating that you sum the values for all cells in the contingency table.
• O_i stands for the observed frequency in each cell of the table.
• E_i represents the expected frequency in each cell of the table, which is usually calculated based on the null hypothesis of independence.

In this formula, you calculate the Chi-Square statistic by finding the squared differences between observed (O) and expected (E) frequencies in each cell, divide by the expected frequency, and sum these values across all cells. This statistic is then used to assess the degree of association or independence between the variables represented in the contingency table.

A Chi-Square Table: Unlocking Critical Values

A Chi-Square table is a valuable reference tool that provides critical values for the Chi-Square distribution. It aids in statistical analysis and hypothesis testing, allowing researchers to determine the significance of Chi-Square statistics.

1. Degree of Freedom (df):
   • To effectively use the Chi-Square table, you need to know two crucial values. The first is the degree of freedom (df), which is calculated as (r - 1) x (c - 1). Here, 'r' represents the number of rows, and 'c' represents the number of columns when data is organized in tabular form.
2. Alpha Level:
   • The second essential input is the alpha level, denoting the confidence level for the statistical test. Commonly used alpha levels include 0.01, 0.05, and 0.10, corresponding to 99%, 95%, and 90% confidence, respectively.

• The Chi-Square table displays critical values of the Chi-Square distribution, indicating the Chi-Square statistic required for a specific degree of freedom and alpha level. These values help researchers make informed decisions about the significance of their findings in Chi-Square tests.

• The table provides Chi-Square values with degrees of freedom listed along the left side and alpha levels displayed across the top. By locating the intersection of the degree of freedom and the chosen alpha level, you can identify the critical Chi-Square value. This value is then compared to the calculated Chi-Square statistic to determine statistical significance.

The Chi-Square table serves as an indispensable tool in the world of statistics, aiding in the interpretation of Chi-Square test results and guiding researchers in their decision-making processes.

Conditions for χ² Test:

1. Independence of Observations:

   • Each observation within the sample used for the Chi-Square test should be independent of each other. This ensures that the results are not influenced by previous observations.

2. Sufficient Sample Size:

   • The total number of observations used in the test must be sufficiently large, typically equal to or greater than 50, to ensure the validity of the Chi-Square test.

3. Dependence on Degree of Freedom:

   • The Chi-Square statistic (χ²) depends on the degree of freedom, which is calculated based on the number of rows and columns in the data table.

4. Random Sampling:

   • The sample collected for the Chi-Square test should be selected randomly to ensure that it is representative of the population or phenomenon under study.

1. Goodness of Fit (Pearson χ²):
   • Purpose: This test, also known as Pearson's Chi-Square, is used to assess whether the observed frequencies match the expected frequencies, indicating whether the data fits a specific theoretical distribution.
   • Formula: χ² = Σ [(O_i - E_i)² / E_i], where O is the observed frequency, and E is the expected frequency.
2. Contingency Chi-Square (Test of Independence of Attributes):
   • Purpose: This test evaluates the association between attributes when the sample data is presented as a contingency table with rows and columns. It helps determine if two or more categorical variables are independent or dependent.
   • Application: Contingency Chi-Square can be used to compare observations under different conditions to assess if they are dependent or independent.
   • Example: It can examine whether the height and weight of a person are associated.

3. Homogeneity Chi-Square:

   • Purpose: This test is utilized to assess the homogeneity or uniformity of two or more samples. It is used to determine if separate samples are sufficiently similar to be combined.

These types of Chi-Square tests serve different purposes in statistical analysis and hypothesis testing, helping researchers make inferences about the relationships between variables and the fit of data to theoretical distributions.

Applications of Chi-Square Test in Hypothesis Testing

The Chi-Square test is a powerful statistical tool used in various hypothesis testing scenarios. Its applications include:

• Purpose: The Chi-Square test of Goodness of Fit is employed to assess discrepancies between observed and expected frequencies, determining whether they align.
• Use: It measures the probabilities of association between two attributes, making it useful in assessing the fit of data to a specific theoretical distribution.

• Purpose: This test examines the relationship between attributes by categorizing them into a two-way table or contingency table.
• Application: It reveals whether there is an association or relationship between two or more variables in the contingency table, helping researchers understand the dependence or independence of these attributes.

• Purpose: The Chi-Square test of Homogeneity is used to assess the homogeneity of attributes with regard to specific characteristics.
• Applications:
  • It can determine the homogeneity of two samples taken from the same population, helping researchers understand the uniformity of these samples.
  • It can also be applied to test the homogeneity of population variances, which is essential in various statistical analyses.

In summary, the Chi-Square test is a versatile tool in hypothesis testing, with applications ranging from assessing data fit to a theoretical distribution to exploring the relationships between attributes and ensuring the homogeneity of samples and population variances.