F-test (one-way & two-way F-test) | Zoology Optional Notes for UPSC PDF Download

The F-Distribution

Statisticians have long recognized that when pairs of samples are drawn from a normal population, the ratio of their variances conforms to a consistent distribution, known as the F-distribution. Over the years, this distribution has proven its utility in various statistical tests, particularly in cases involving the ratio of variances.

F-Statistic: The F-statistic, which is at the core of the F-test, is a simple formula:

F = s₁² / s₂²

Here, s₁² represents the variance of sample 1, computed using the formula:

s² = ∑(x - x̄)² / (n - 1)

The shape of the F-distribution is noteworthy. If the variances s₁² and s₂² arise from samples of the same population, the majority of F-scores generated from numerous sample pairs will cluster around one. Given that variances are inherently positive, F-scores are also positive. However, the distribution is not symmetric; it has a long right tail and descends steeply toward zero on the left.

Applications of the F-Distribution: There are two primary applications of the F-distribution that we will explore:

1. A Simple Test: Assessing Equal Variances
   • The F-distribution's history and characteristics enable us to test whether two samples originate from populations with equal variances. By comparing the variances of two samples, you can hypothesize that they come from populations with the same variance. If the F-score is close to one, you accept this hypothesis. If it deviates significantly from one, it suggests differing population variances.
2. One-Way Analysis of Variance (ANOVA)
   • The F-distribution plays a pivotal role in one-way analysis of variance (ANOVA). This statistical technique determines if three or more samples stem from populations with the same mean. It allows for the exploration of variations between multiple sample groups.

Key Steps in Hypothesis Testing with the F-Test: For effective use of the F-test, two key steps are essential:

1. Formally Writing Hypotheses:
   • The null hypothesis (H₀) posits that there is no difference, while the alternative hypothesis (H₁ or Ha) suggests a difference. These hypotheses provide the foundation for the F-test.
2. Utilizing F-Distribution Tables:
   • To decide whether the F-score is close enough to one to accept the null hypothesis (fail to reject it), F-distribution tables come into play. Researchers must select a significance level (α), typically 0.05 (5%), before consulting the tables.

• In F-distribution tables, the degrees of freedom for each sample are crucial. These degrees of freedom correspond to the sample sizes minus one (n - 1). The table provides critical F-values for specific degrees of freedom and the chosen α level. By comparing the calculated F-score to the critical F-value, researchers can determine whether to accept or reject the null hypothesis.

In conclusion, the F-distribution, F-test, and one-way ANOVA are powerful tools in statistical analysis. They offer insights into variance ratios, equality of variances, and comparisons of means among multiple sample groups, allowing for robust hypothesis testing and data-driven decision-making.

F-Tables and Their Use

F-tables are typically presented as one-tail tables, focusing on the critical F-value that separates the right tail from the rest of the distribution. This format aligns with most statistical applications of the F-distribution, where the interest lies in assessing whether one variance is greater than another. To achieve this, researchers aim to determine if the F-score is greater than one.
When testing for equal variances, the goal is to assess if the F-score is close to one. Consequently, either a large F-score or a small F-score leads researchers to conclude that the variances are not equal. To adapt the test into a one-tail format, researchers often simply divide the larger sample variance by the smaller sample variance and use the printed tables to ascertain if the quotient is "larger than one."

Left-Tail Critical Value Calculation: For purists or specific cases, it is possible to compute the left-tail critical value. To find the left-tail critical value for x and y degrees of freedom (df), follow this straightforward process:

1. Identify the right-tail (table) critical value for y degrees of freedom and α - 0.05.
2. Locate the corresponding F-value for α - 0.05, with degrees of freedom reversed (x and y).
3. Calculate the inverse of this F-value, which gives the left-tail critical value.

Conducting the Test

To test whether two samples originate from populations with the same variance, the following steps are recommended:

1. Collect two samples and calculate the sample variance for each (s₁² and s₂²).
2. Formulate your null hypothesis (H₀) and alternative hypothesis (H₁) while selecting a significance level (α).
3. Compute the F-score from your samples, dividing the larger variance by the smaller, ensuring F > 1.
4. Refer to the F-distribution tables, specifically for α/2, and determine the critical (table) F-score based on the degrees of freedom (n-1 for both samples).
5. Compare the calculated F-score from your samples to the critical F-score:
   • If the samples' F is larger than the critical F, accept the alternative hypothesis (Ha) that the population variances are not equal.
   • If the samples' F is smaller than the critical F, accept the null hypothesis (Ho) that the population variances are equal.

Practical Example: Consider Lin Xiang, a banker managing a new branch in Winnipeg, Manitoba, and comparing it to her previous branch in Saskatoon, Saskatchewan. She wonders if the number of customers at her new branch is more variable than her previous one. To test this hypothesis, she collects data on the number of transactions in a half day from both branches:
Sol: Winnipeg branch: 156, 278, 134, 202, 236, 198, 187, 199, 143, 165, 223 Saskatoon branch: 345, 332, 309, 367, 388, 312, 355, 363, 381
Her hypotheses are as follows: H₀: σ²W = σ²S (Population variances are equal) Ha: σ²W ≠ σ²S (Population variances are not equal)
Lin chooses a significance level of α = 0.05. She computes the sample variances as s²W = 1828.56 and s²S = 795.19. Calculating the F-score:
F = s²W / s²S = 1828.56 / 795.19 = 2.30

She proceeds to find the critical F-value from the F-tables based on the degrees of freedom (10, 20) and significance level (α/2), and compares it to her calculated F-score to make a hypothesis testing decision.

The Significance of ANOVA and One-Way ANOVA: Analyzing Variance for Equal Means

The Role of ANOVA: One of the key applications of the F-distribution is in analyzing variance to determine if three or more samples are derived from populations with equal means. While this statistical test may not be used as frequently as some others, it holds significant importance due to its bridging role between univariate and multivariate statistics. The strategy employed in ANOVA is pivotal, as it forms the basis for many multivariate statistical tests and procedures commonly used in research.

One-Way ANOVA: Assessing Equality of Means in Multiple Samples: The concept of using variance analysis to investigate differences in means might initially seem counterintuitive. However, it represents a brilliant insight that has laid the groundwork for much of the multivariate statistical analysis employed by researchers today. ANOVA provides a framework for examining the relationships between two or more variables, which is often a primary focus of multivariate statistical analysis.

• Testing whether three or more samples originate from populations with the same mean can be thought of as a multivariate exercise. If these samples arise from distinct sources, such as different factories or varying treatments, the objective is to discern if the differences in outcomes are due to the factory or treatment applied—essentially exploring the relationship between the source and the results.
• Consider three distinct samples, each comprising data points denoted as x. Although these data points can be grouped into three sets (samples 1, 2, and 3), analyzing the total variance can provide valuable insights. By combining all the data points, you can calculate a grand mean and the total variance around this grand mean. Additionally, you can compute the mean and sample variance within each of the three groups. Finally, the sample means from the three groups can be assessed to find the variance between them. ANOVA's foundation lies in examining the origins of the total variance.
• Imagine selecting a single data point, x. The variance associated with this data point, which represents its deviation from the grand mean, can be deconstructed into two components: (1) the distance between the data point and the mean of its respective sample and (2) the distance between the mean of its sample and the grand mean. If the three samples indeed stem from populations with differing means, then, for most data points, the distance between the sample mean and the grand mean is likely to exceed the distance between the data point and its group mean. When these distances are aggregated and transformed into variances, it becomes evident that if the population means are distinct, the variance between the sample means is expected to surpass the variance within the individual samples.
• As you might have come to expect, statisticians have identified that if three or more samples are drawn from a normal population and the variance between the samples is divided by the variance within the samples, the resulting sampling distribution follows a known shape. In this case, the distribution follows an F-distribution with degrees of freedom (df) parameters of m-1 for the numerator and n-m for the denominator. Here, m represents the number of samples, and n denotes the total sample size. The variance between the samples can be calculated as follows:

Variance Between = Variance of the Grand Mean - (Sum of Within-Sample Variances)
By conducting ANOVA and observing the F-score, researchers can make informed decisions regarding the equality of means across multiple samples.

In summary, ANOVA serves as a fundamental tool for exploring the equality of means in multiple samples and plays a pivotal role in multivariate statistical analysis. This technique leverages the F-distribution to gain valuable insights into the relationships between variables and is a key component in statistical hypothesis testing.

Handling Double Sums in ANOVA

Dealing with double sums in the context of ANOVA requires a systematic approach. First, within the inner sum, calculate the mean of each sample and determine the sum of squares for the distances of each data point (x) from its respective sample mean. Then, in the outer sum, combine the results obtained from each of the individual samples.

Strategy for One-Way Analysis of Variance (ANOVA):
Conducting a one-way ANOVA involves a straightforward strategy:

1. Gather m samples, each representing a distinct group.
2. Calculate the variance between the samples and the variance within each sample.
3. Determine the ratio of the variance between the samples to the variance within the samples, which results in the F-score.
4. Evaluate the F-score to draw conclusions about the equality of means across the samples.

Interpreting the F-Score:

• If the F-score is less than one or only slightly greater than one, it suggests that the variance between the samples is not significantly different from the variance within the samples. In this scenario, it is likely that the samples originate from populations with the same mean.
• If the F-score is substantially greater than one, it indicates that the variance between the samples contributes significantly to the overall variance in the entire dataset. This suggests that the samples may come from populations with different means.

Details in Conducting a One-Way ANOVA

1. Writing Hypotheses:

   • The null hypothesis (Ho) posits that all population means are equal, while the alternative hypothesis (Ha) implies that not all means are equal. Often, only Ho is explicitly stated:

     H_o : m₁ = m₂… = m_m
2. Keeping Calculations Organized:
   • Organizing calculations for the variance within is crucial. Compute sample means and then find the squared distances for each data point within its respective sample. Sum the squared distances for each sample. Similarly, when calculating the F-score, remember that the goal is to assess if the variance between samples is large, which is achieved by dividing the variance between by the variance within.
3. Using F-Tables:
   • F-tables are tailored for one-tail tests, as ANOVA is a one-tail test. While the null hypothesis assumes equal means, the test is essentially checking if the variance between is less than or equal to the variance within. The degrees of freedom for ANOVA are m-1 for the numerator and n–m for the denominator, where m represents the number of samples and n is the total sample size.

Summary

The F-distribution is a crucial concept in statistics and hypothesis testing. It represents the sampling distribution of the ratio of variances between two samples drawn from a normal population. Its primary applications include testing for equality of variances and conducting analysis of variance (ANOVA) to examine whether three or more samples come from populations with the same mean.

1. Testing Equality of Variances:
   • The F-distribution is directly used to assess whether two samples originate from populations with equal variances. When the F-score is close to one, it suggests that the samples may have similar variances, supporting the hypothesis of equal variances.
2. Analysis of Variance (ANOVA):
   • ANOVA is a powerful statistical tool for testing whether multiple samples come from populations with the same mean. It does so by analyzing the sources of variance in the data.
   • ANOVA examines whether the variance in the dataset arises more from variations of individual observations around their respective sample means or from variations of the means of the samples around the grand mean. This helps determine if the samples have equal means or not.
3. Expanding into Multivariate Statistics:
   • In one-way ANOVA, each observation involves two variables: the primary variable (x) and the group to which the observation belongs. As you delve further into statistics, you will encounter multivariate analysis with observations containing multiple variables.
4. F-Test for Equality of Variances:
   • The F-test for equality of variances is often used in conjunction with the t-test for equality of means. This is because the t-test, as presented in this context, assumes that the samples come from populations with equal variances. Thus, the F-test is employed when researchers want to ensure equality of variances before proceeding with t-tests, especially in critical or rigorous research scenarios.

Overall, the F-distribution, F-test, and ANOVA play pivotal roles in hypothesis testing, statistical analysis, and decision-making, providing valuable insights into variance, means, and relationships between variables. These concepts form the foundation for more advanced statistical techniques and applications in the field of business and research.