Pooled Variance and Two-Sample Comparisons

Introduction

Key Concepts

Pooled Variance

Pooled variance is a method used to estimate the common variance of two or more populations when it is assumed that these populations share the same variance. This assumption is pivotal in conducting two-sample t-tests, particularly when comparing the means of two independent groups. By pooling the variances, we obtain a more accurate estimate, especially when dealing with small sample sizes.

Definition and Formula

The pooled variance ($S_p^2$) combines the variances of two independent samples to provide a single estimate of the population variance. It is calculated using the following formula:

$$ S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2} $$

Where:

• $n_1$, $n_2$ = sample sizes
• $S_1^2$, $S_2^2$ = sample variances

Assumptions for Pooled Variance

For the pooled variance to be a valid estimate, the following assumptions must hold:

• The two samples are independent.
• The populations from which the samples are drawn are normally distributed.
• The populations have equal variances (homogeneity of variance).

Two-Sample t-Test Using Pooled Variance

The two-sample t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. When the population variances are assumed to be equal, the pooled variance is used in the test statistic.

The test statistic is calculated as:

$$ t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$

Where:

• $\bar{X}_1$, $\bar{X}_2$ = sample means
• $S_p$ = pooled standard deviation
• $n_1$, $n_2$ = sample sizes

Degrees of Freedom

The degrees of freedom (df) for the two-sample t-test using pooled variance are calculated as:

$$ df = n_1 + n_2 - 2 $$

Example

Suppose we have two independent samples: Sample 1 with $n_1 = 10$, $S_1^2 = 4.5$ and Sample 2 with $n_2 = 12$, $S_2^2 = 5.2$. To calculate the pooled variance:

$$ S_p^2 = \frac{(10 - 1)(4.5) + (12 - 1)(5.2)}{10 + 12 - 2} = \frac{9 \times 4.5 + 11 \times 5.2}{20} = \frac{40.5 + 57.2}{20} = \frac{97.7}{20} = 4.885 $$

The pooled standard deviation ($S_p$) is:

$$ S_p = \sqrt{4.885} \approx 2.21 $$

Confidence Intervals for the Difference of Means

Using the pooled variance, we can construct confidence intervals for the difference between two population means. The confidence interval is given by:

$$ (\bar{X}_1 - \bar{X}_2) \pm t_{\alpha/2, df} \times S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} $$

Where:

• $t_{\alpha/2, df}$ = critical t-value
• Other symbols as previously defined.

Hypothesis Testing Steps

1. State the Hypotheses:
   • Null Hypothesis ($H_0$): $\mu_1 = \mu_2$
   • Alternative Hypothesis ($H_a$): $\mu_1 \neq \mu_2$ (two-tailed)
2. Set the Significance Level ($\alpha$): Commonly 0.05.
3. Calculate the Test Statistic: Using the two-sample t-test formula with pooled variance.
4. Determine the Critical Value: From the t-distribution table using $df$ and $\alpha$.
5. Make a Decision: Compare the test statistic with the critical value to accept or reject $H_0$.

Assumption Checking

Before applying the pooled variance method, it is crucial to verify that the assumptions hold. If the assumption of equal variances is violated, alternative methods such as Welch's t-test should be considered.

Advantages of Using Pooled Variance

• Provides a more accurate estimate of the population variance when assumptions are met.
• Increases the statistical power of the two-sample t-test.

Limitations of Pooled Variance

• Assumes equal variances, which may not hold in all scenarios.
• Sensitive to outliers, which can distort the pooled variance estimate.

Applications of Pooled Variance and Two-Sample Comparisons

• Comparing treatment effects in clinical trials.
• Assessing differences in educational outcomes between two groups.
• Evaluating product performance across different manufacturing batches.

Advanced Concepts

Mathematical Derivation of Pooled Variance

To derive the pooled variance, we start by considering two independent samples from populations with the same variance ($\sigma^2$). The unbiased estimator of $\sigma^2$ is the weighted average of the sample variances:

$$ S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2} $$

This formula ensures that the pooled variance is an unbiased estimator of the common population variance by appropriately weighting the sample variances based on their degrees of freedom.

Derivation of the Two-Sample t-Test Statistic

The two-sample t-test statistic using pooled variance is derived under the null hypothesis that the two population means are equal ($\mu_1 = \mu_2$). The test statistic is given by:

$$ t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$

This statistic follows a t-distribution with $df = n_1 + n_2 - 2$ degrees of freedom under the null hypothesis.

Proof of the t-Test Statistic's Distribution

Assuming normally distributed populations with equal variances, the difference between sample means ($\bar{X}_1 - \bar{X}_2$) is also normally distributed with mean zero and variance $\sigma^2\left(\frac{1}{n_1} + \frac{1}{n_2}\right)$. Dividing by the pooled standard deviation scales the difference to follow a t-distribution with $n_1 + n_2 - 2$ degrees of freedom:

$$ t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \sim t_{df} $$

Addressing Violation of Equal Variances

When the assumption of equal variances is violated, the pooled variance method becomes inappropriate. In such cases, Welch's t-test is preferred as it does not assume equal population variances and adjusts the degrees of freedom accordingly:

$$ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}} $$ $$ df = \frac{\left(\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}\right)^2}{\frac{\left(\frac{S_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{S_2^2}{n_2}\right)^2}{n_2 - 1}} $$

Power Analysis in Two-Sample Tests

Power analysis evaluates the probability of correctly rejecting the null hypothesis when it is false. In the context of two-sample t-tests, factors affecting power include sample size, effect size, significance level, and variance. Increasing the sample size or effect size, decreasing variance, or raising the significance level can enhance the test's power.

Confidence Interval Derivation

The confidence interval for the difference of means using pooled variance is derived from the distribution of the test statistic. Starting with the standard form:

$$ \bar{X}_1 - \bar{X}_2 \pm t_{\alpha/2, df} \times S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} $$

This interval estimates the range within which the true difference of population means lies with a specified level of confidence (e.g., 95%).

Interdisciplinary Connections

Pooled variance and two-sample comparisons extend beyond pure mathematics, finding applications in fields such as:

• Biology: Comparing the effectiveness of two different treatments.
• Economics: Analyzing wage differences between two sectors.
• Engineering: Assessing the reliability of two different manufacturing processes.

These applications demonstrate the versatility and practical importance of statistical inference in real-world scenarios.

Advanced Problem-Solving

Problem: A researcher conducts an experiment to compare the effectiveness of two fertilizers on plant growth. Fertilizer A is applied to 15 plants, resulting in a mean growth of 20 cm with a variance of 4 cm². Fertilizer B is applied to 18 plants, resulting in a mean growth of 22 cm with a variance of 5 cm². Conduct a two-sample t-test at the 5% significance level to determine if there is a significant difference in mean plant growth between the two fertilizers.

Solution:

1. State the Hypotheses:
   • Null Hypothesis ($H_0$): $\mu_A = \mu_B$
   • Alternative Hypothesis ($H_a$): $\mu_A \neq \mu_B$
2. Calculate the Pooled Variance ($S_p^2$): $$ S_p^2 = \frac{(15 - 1) \times 4 + (18 - 1) \times 5}{15 + 18 - 2} = \frac{14 \times 4 + 17 \times 5}{31} = \frac{56 + 85}{31} = \frac{141}{31} \approx 4.548 $$

   Thus, $S_p = \sqrt{4.548} \approx 2.133$

3. Calculate the Test Statistic: $$ t = \frac{20 - 22}{2.133 \times \sqrt{\frac{1}{15} + \frac{1}{18}}} = \frac{-2}{2.133 \times \sqrt{0.0667 + 0.0556}} = \frac{-2}{2.133 \times \sqrt{0.1223}} = \frac{-2}{2.133 \times 0.3497} \approx \frac{-2}{0.746} \approx -2.681 $$
4. Determine the Critical t-Value: For a two-tailed test with $df = 15 + 18 - 2 = 31$ at $\alpha = 0.05$, the critical t-values are approximately $\pm 2.040$.
5. Make a Decision: Since $-2.681 < -2.040$, we reject the null hypothesis.
6. Conclusion: There is significant evidence at the 5% level to conclude that there is a difference in mean plant growth between Fertilizer A and Fertilizer B.

Exploring Non-Parametric Alternatives

In cases where the assumptions for pooled variance and the two-sample t-test are not met, non-parametric tests such as the Mann-Whitney U test can be employed. These tests do not assume normality or equal variances and are based on the ranks of the data rather than their numerical values.

Bayesian Approaches to Two-Sample Comparisons

Beyond the frequentist framework, Bayesian statistics offers alternative methods for two-sample comparisons. In Bayesian t-tests, prior distributions are assigned to parameters, and posterior distributions are used to make inferences about the difference in means. This approach allows for incorporating prior knowledge and provides a probabilistic interpretation of the results.

Effect Size Measures

Effect size measures, such as Cohen's d, quantify the magnitude of the difference between two means, providing context beyond statistical significance. Cohen's d is calculated as:

$$ d = \frac{\bar{X}_1 - \bar{X}_2}{S_p} $$

This standardized measure facilitates comparisons across different studies and contexts.

Power and Sample Size Calculations

Determining the appropriate sample size to achieve a desired power level is crucial in study design. Power calculations consider the expected effect size, variability, significance level, and desired power to ensure that the study is adequately equipped to detect meaningful differences.

Multivariate Extensions

When dealing with more than two groups or multiple dependent variables, multivariate techniques such as ANOVA (Analysis of Variance) extend the principles of pooled variance and two-sample comparisons. These methods allow for the simultaneous assessment of multiple factors and their interactions.

Comparison Table

Aspect	Pooled Variance	Welch's t-Test
Assumptions	Equal population variances	Unequal population variances
Formula Complexity	Less complex	More complex due to variance adjustment
Degrees of Freedom	$n_1 + n_2 - 2$	Calculated using Welch–Satterthwaite equation
Power	Higher when equal variances assumption holds	More reliable when variances are unequal
Usage	When population variances are assumed equal	When population variances are assumed unequal

Summary and Key Takeaways