Three Factor ANOVA without Replication

Introduction

Just as for the Two Factor ANOVA there are both with replication and without replication versions of Three-Factor ANOVA. In ANOVA with More Than Two Factors and Three-Factor ANOVA Analysis we dealt with data with replications.

In Two Factor ANOVA without Replication, there is no interaction factor, while in the Three-Factor ANOVA without Replication there is no interaction of the three factors, but there are pairwise interactions. Thus, we have two possibilities: (1) pairwise interactions are separated and (2) pairwise interactions are included in the error.

Example

Example 1: Perform Three-Factor ANOVA on the data in Figure 1 using the Real Statistics Three-Factor ANOVA data analysis tool.

Three Factor ANOVA data

Figure 1 – Three-Factor ANOVA data

Note that there are no interactions among the three factors in the data in Figure 1. The Three-Factor ANOVA data analysis tool described in Three Factor ANOVA Analysis can be used with A3:D14 as the Input Range. The descriptive data output is shown in Figure 2 and the two forms of the ANOVA analysis are shown in Figure 3.

ANOVA3 descriptive stats 1

ANOVA3 descriptive stats 2

Figure 2 – Three-Factor ANOVA descriptive statistics

Note that in the first ANOVA output in Figure 2, SSW (in cell N6) is calculated using the formula =N7-N5-N4-N3 (and similarly for the dfW term). All the other cells are calculated in the usual way. In the second ANOVA output the A × B, B × C and A × C are calculated as in ANOVA with replication and only the A × B × C is omitted. Here SSW (cell N17) is calculated using the formula =N18-N16-N15-N14-N13-N12-N11.

ANOVA results

Figure 3 – Three-Factor ANOVA output

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Bao, J. (2021) Two-, three-, and four-way ANOVA 
https://sites.pitt.edu/~jub69/material/material/stat1301/Lec/SAS-ANOVA-2.pdf

Laerd (2024) Three-way ANOVA in SPSS statistics
https://statistics.laerd.com/spss-tutorials/three-way-anova-using-spss-statistics.php

Meier, L. (2023) Factorial treatment structure. ANOVA and mixed models: a short introduction using R.
https://people.math.ethz.ch/~meier/teaching/anova/index.html

8 thoughts on “Three Factor ANOVA without Replication”

  1. I have 7 years data of two companies on 10 different segments, Can you suggest which test is suitable for test the significance difference ?

    Reply
    • First things first. When you refer to a significant “difference”, you clearly mean a difference between A and B. What are A and B in this context? This is a non-trivial question.
      Charles

      Reply
  2. HI Mrs. Charles
    here I want to do three factorial research based on my title but, I am not clear with the treatment combination of the three factors. so, please try show me some examples..

    Best regards
    Safesa

    Reply
    • Hi Safesa,
      Can you explain further what you are not clear about? Do you want clarification about Three-Factor ANOVA without Replication or Three-Factor ANOVA with Replication?
      Charles

      Reply
  3. Hi!

    Thank you so much for your input.

    I have a question I was hoping to be answered in this page…

    How can I interpret the coefficients (with more than 2, lets say 3 levels as shown above) from a multiple linear regression model?

    Thank you,

    Reply
    • Hi Raj,
      I had never thought about this before, and so don’t know the answer to your question. Perhaps someone else from the community can answer your question.
      Charles

      Reply

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