Latin Squares with Replication

Introduction

We can augment the Latin Squares design to include replications. We deal with the case where the only difference between the replications is the randomization. For Example 1 of Latin Squares Design, this means that the same operators, machines and methods are modeled for each replication, except that the randomization may vary (i.e. the permutation of Latin letters may be different).

Example

Example 1: In Figure 1 we see the analysis for a 3 × 3 Latin Squares design with 3 replications. Note that the number of replications doesn’t need to be equal to the number of rows/columns/treatments.

Figure 1 – Latin Squares design with replication

The formulas used to compute the factor means (for the first replication, row, column and treatment) are shown in Figure 2.

Cell	Factor	Formula
J4	Replication	=AVERAGE(E4:G6)
K4	Row	=AVERAGE(IF(A4:A12=I4,E4:G12,””))
L4	Column	=AVERAGE(E4:E12)
M4	Treatment	=AVERAGE(IF($B$4:$D$12=CHAR(I4+64),$E$4:$G$12))

Figure 2 – Formulas for factor means

Note that the formulas used to calculate the row and treatment means are array formulas. The degrees of freedom for the treatment, rows and column factors are the same as in the no replications model. The degrees of freedom for the replications factor is one less than the number of replications; finally, the degrees of freedom for the error term is reduced by this amount.

The formulas used to calculate the sum of squares terms are shown in Figure 3.

Cell	Factor	Formula
P5	Treatment	=DEVSQ(M4:M6)*(Q5+1)*(Q8+1)
P6	Rows	=DEVSQ(K4:K6)*(Q6+1)*(Q8+1)
P7	Columns	=DEVSQ(L4:L6)*(Q7+1)*(Q8+1)
P8	Replications	=DEVSQ(J4:J6)*(Q5+1)^2
P9	Error	=P10-SUM(P5:P8)
P10	Total	=DEVSQ(E4:G12)

Figure 3 – Formulas for SS in Latin Squares with replication

Note that for this example, there is a significant difference between the treatments as well as between the rows.

References

Zhu, M. Y. (2005) Latin square and related design. Statistics 514: Design and Analysis of Experiments, Purdue University
http://www.stat.purdue.edu/~yuzhu/stat514s05/Lecnot/latinsquares05.pdf

Penn State (2025) The Latin Square design
https://online.stat.psu.edu/stat503/lesson/4/4.3

Montgomery, D. C. (2013) Design and analysis of experiments, 8th ed. Wiley
https://faculty.ksu.edu.sa/sites/default/files/douglas_c._montgomery-design_and_analysis_of_experiments-wiley_2012_edition_8.pdf