Understanding Bartlett's Test: Assessing Homogeneity of Variances in Combined Experiment Analysis

Summary:

This blog delves into the importance of Bartlett's test for validating homogeneity of error variances in pooled/combined experiments. It explains the test's significance, provides step-by-step calculations, and highlights its application in agricultural research. Practical examples and code snippets for various software are included for comprehensive understanding.

Estimated Reading Time: ~12 minutes.

 Introduction

In experimental research, especially in fields like agriculture, researchers often conduct experiments under varying conditions such as different times, locations, or environments. To draw more comprehensive and robust conclusions, combining or pooling the data from these experiments into a single analysis is a common practice.

Pooled analysis offers several benefits:

1. Increased Statistical Power: Pooling data increases the total sample size ($n$) and the degrees of freedom for error, thereby reducing the Mean Square Error (MSE). This leads to a smaller critical F-value in ANOVA, enhancing the ability to detect smaller treatment differences. For instance, pooling data from three completely randomized design (CRD) experiments, each with 10 replicates ($n = 30, df_{\text{error}} = 27$), results in a lower MSE compared to analyzing each experiment individually ($df_{\text{error}} = 9$ per experiment). This improvement allows for the detection of subtle treatment effects that might otherwise remain non-significant.

2. Interaction Analysis: Pooled analysis facilitates the identification of interactions between treatments and environments, locations, or years through the treatment-by-environment interaction term. This provides valuable insights into the consistency of treatment performance across varying conditions and broadens the applicability of the findings.

Despite these advantages, pooled analysis requires the error variances of the individual experiments to be homogeneous. This is a critical assumption to ensure the validity of the results and to avoid misleading conclusions.

This blog provides a detailed explanation of Bartlett's test, a statistical method used to assess the homogeneity of variances. It discusses the test's application in pooled experiments and guides researchers on how to perform it effectively.

The Importance of Homogeneous Error Variances in Pooled Analysis

For researchers conducting pooled analyses, ensuring homogeneity of error variances is paramount. Error variance refers to the portion of data variability that experimental factors cannot explain. In ANOVA-based pooled analyses, the assumption of homogeneous error variances across experiments underpins the validity of the F-statistic. When this assumption is violated, Mean Square Error (MSE) calculations may be distorted, undermining the reliability of results and increasing the likelihood of Type II errors—failing to detect genuine treatment effects. Addressing heterogeneous variances may require alternatives such as Welch’s ANOVA or variance-stabilizing transformations to ensure robust conclusions.

Illustrative Scenario:

Consider an experiment evaluating the effectiveness of foliar applications of fungicides to control Black Sigatoka disease in bananas under varying environmental conditions (e.g., different humidity levels). The study involves seven fungicides (Fungicides A, B, C, D, E, F, and G) applied to 21 banana plants (three replicates per treatment). Using a Completely Randomized Design (CRD), the treatments are randomly assigned to the plants. Below is the individual ANOVA for three distinct environments.

Bartlett's test checks if error variances across environments are homogeneous. In our pooled CRD experiment, it determines if the error variability in individual experiments is consistent. Homogeneity is crucial for pooling data; significant differences in variances mean the data cannot be pooled reliably. Let's proceed with Bartlett's test.

Hypotheses

The null hypothesis is that all the population variances (k populations being compared) are equal:
H₀: σ₁² = σ₂² = ... = σₖ²

The alternative hypothesis is that the population variances are not all equal, meaning at least one variance differs from the others. The test does not explicitly identify which one is different, only that at least one is different.

Formula for Bartlett's Test

The test statistic for Bartlett's test is calculated using the following formula:
χ² = [$N - K$ * ln(Sₚ²) - Σᵢ(nᵢ - 1) * ln(Sᵢ²)] / C

Where:

• N = Σᵢnᵢ: Total number of observations across all groups
• K: Number of groups (environments in our case)
• nᵢ: Number of observations in group i
• Sᵢ²: Sample variance of group i
• Sₚ² = [Σᵢ$(n_i-1)\text{<span style="color: black;">*</span> }{S}_i^2$ / (N - K): Pooled variance
• C = 1 + (1 / (3 * (K - 1))) * (Σᵢ(1 / (nᵢ - 1)) - 1 / (N - K)): Correction factor

Steps to Perform Bartlett's Test

We will calculate Bartlett's test step by step using our example:

1. Calculate the variances (Sᵢ²) for the residuals of each environment.
   The error variances for each environment can be obtained from the ANOVA table for each environment.

2. Compute the pooled variance:
   The pooled variance is calculated using the formula:
   Sₚ² = Σᵢ$(n_i-1)\text{<span style="color: black;">*</span>}{S}_i^2$ / (N - K)

   For our example:
   Sₚ² = [$(20\text{*}10.95)+(20\text{*}11.28)+(20\text{*}9.54)]$ / (63 - 3)
   Sₚ² = 635.37 / 60 = 10.589

3. Compute the Correction Factor (C):

   The correction factor $C$ is calculated using the formula:
   $C = 1 + \frac{1}{3(K-1)} \left( \sum_{i=1}^K \frac{1}{n_i - 1} - \frac{1}{N - K} \right)$
   

   For our example:
   $C = 1 + \frac{1}{3(2)} \left( \frac{1}{20} + \frac{1}{20} + \frac{1}{20} - \frac{1}{60} \right)$
   $C=1+\frac{1}{6}\cdot \frac{9}{60} \; =1+0.16\cdot \mathrm{0.15 }=1.024$

   Compute the Term $\sum_{i=1}^K (n_i - 1) \ln(S_i^2)$

   $\sum_{i=1}^K (n_i - 1) \ln(S_i^2) = 20 \cdot 2.39 + 20 \cdot 2.42 + 20 \cdot 2.55$
   $= 141.434$
   

   Plug in the Values to Calculate $\chi^2$:

   The test statistic ${\chi }^2$ is computed as:
   $\chi^2 = \frac{N - K \cdot \ln(S_P^2) - \sum_{i=1}^K (n_i - 1) \ln(S_i^2)}{C}$

   For our example:
   $\chi^2 = \frac{60 \cdot 2.359 - 141.434}{1.024}$
   ${\chi }^2=\frac{141.5917-141.434}{1.024}=\frac{0.157}{1.024}=0.157$

4. Compare the Computed $\chi^2$ Value

   To determine whether to reject the null hypothesis, compare the computed $\chi^2$ value with the critical value from the Chi-Square distribution table for $\text{df} = k - 1$ at the desired significance level ($\alpha$, usually 0.05).

   In our example, the calculated $\chi^2$ value (0.157) is less than the table $\chi^2$ value (5.99). In Excel, the critical $\chi^2$ value can be calculated using the formula:
   $\text{=CHISQ.INV.RT(Probability, Degrees\ of\ Freedom)}$
   

   At the 0.05 significance level, this result indicates insufficient evidence to reject the null hypothesis. While this does not confirm that the variances are equal, it suggests there is not enough data to conclude that at least one variance differs.

Codes with their package and respective software for performing bartletts test

Code	Package	Software
bartlett.test(values ~group)	stats	R
bartlett.test(values, grouping)	car	R
PROC GLM; CLASS group; MODEL value=group; TEST HOV;	-	SAS
Navigate to Analyze > Descriptive Statistics > Explore. Under "Plots," select "Test for Homogeneity of Variances (Bartlett's)."		SPSS
scipy.stats.bartlett(data1, data2)	scipy.stats	Python

Conclusion

Bartlett's test is a crucial step in validating the assumption of homogeneous error variances before pooling data in experimental analysis. In pooled experiments conducted across different environments or conditions, it ensures that variances are comparable, enabling the combined analysis to be both reliable and meaningful.

When Bartlett's test indicates homogeneous variances, pooling data enhances statistical power and provides a broader understanding of treatment effects and interactions. Conversely, if variances are significantly different, alternative approaches such as data transformations should be used to maintain the validity of conclusions. By carefully assessing variance homogeneity, researchers can confidently perform pooled analyses and draw robust inferences from their data.

The blog is written with great effort and due research by Jignesh Parmar

 

PhD Scholar, 

Department of Agricultural Statistics,

Anand Agricultural University