Zero Residual Sum of Square in ANOVA

 Summary:

This blog explains case when the residual sum of square is zero for two way ANOVA (RBD Design in terms of DoE). A sample data set is shared for detailed explanation. The goal is to share the observation. 

Reading time 5 mins

Introduction

In the field of Design of Experiments (DoE), the Randomized Block Design (RBD) is a commonly used layout where treatments are tested across blocks or replications. Typically, the ANOVA table in RBD presents some residual (error) variation that captures the unexplained variance. However, there are rare instances when the residual sum of squares (SS) turns out to be zero.

In this blog, we demonstrate one such case using a small dataset and provide insights into why this occurs, backed by observations.

Data set used in demonstration:

The data is of 6 treatments T1, T2, T3, T4, T5 and T6 with three replications R1, R2 and R3.

When we perform two way ANOVA the results is shown below:

As we can observe that difference between R1& R2 is constant and so is the case with R2, R3 and R1,R3. 

One more critical angle is order of ranking for values of treatment across replications. For every treatment the R3 values is highest followed by R1 and R2. Also the range of treatment data is constant 0.04 across all treatments.

Key Observation

The Error Sum of Squares (SS) is zero, which is unusual in practical data analysis. This means the variation in the data has been completely explained by the replication and treatment effects. In other words, the model fits the data perfectly, which is highly uncommon unless the data is structured with consistent internal patterns.

1. Difference between replications is constant for every treatment.

   • R1 - R2 = 0.02

   • R2 - R3 = -0.04

   • R1 - R3 = -0.02

2. The ranking of replications is consistent across all treatments:

   • R3 > R1 > R2

3. The range of treatment values across replications is always 0.04.

These regular patterns make each treatment a perfectly shifted version of each other across replications, allowing the ANOVA model to explain all variability — hence, zero residuals.

Takeaway

In statistical modeling, especially in designed experiments, a zero error term in the ANOVA table should prompt you to:

• Inspect the data for perfect patterns

• Understand the design structure

• Re-evaluate randomness and independence assumptions

This case is a great teaching example of how data structure influences ANOVA outputs, and why perfect symmetry in data may be more theoretical than practical.

Concluding Remark

While such datasets are rare in field experiments, they are crucial for educational understanding. This case clearly demonstrates how structured data patterns can eliminate the residual sum of squares in two-way ANOVA under RBD.