Latin Square Design Calculator

Latin Square ANOVA online with row, column and treatment SS, F tests, CV and post-hoc.

When to use it

Use to control TWO sources of variation simultaneously, typically row and column gradients in a field, soil fertility along a slope crossed with a moisture gradient, or operator by time-of-day effects in a lab. Each treatment appears exactly once per row and once per column.

When NOT to use it

Do not use when number of treatments differs from rows or columns (the design is square by definition). Avoid for very small (3 by 3) or very large squares: small squares have too few error df (only 2), large squares are impractical to lay out. For more than 8 treatments use a row-column lattice instead.

What you get

ANOVA table with sources Row, Column, Treatment, Error and the corresponding SS, df, MS, F, p. Treatment means with CD(5%), CV percent, residual diagnostics, and post-hoc letter display. Row and column F tests indicate whether blocking in either direction was worthwhile.

Worked example

Montgomery Design and Analysis of Experiments, Example 4.3, Rocket Propellant Problem (5 by 5 Latin square).

A, B, C, D, E
A:24, B:20, C:19, D:24, E:24
B:17, C:24, D:30, E:27, A:36
C:18, D:38, E:26, A:27, B:21
D:26, E:31, A:26, B:23, C:22
E:22, A:30, B:20, C:29, D:31

Expected output: F_treat = 7.73, p_treat approx 0.0025, SS_error = 128

Source: Montgomery, D. C. (2017). Design and Analysis of Experiments, 9th ed. Wiley, ch. 4, Example 4.3 (Rocket Propellant Problem).

How to interpret the output

Treatment F is the test of interest; rows and columns are nuisance directions. Error df equals (n-1)(n-2) for an n by n square, so a 4 by 4 has only 6 error df: low power unless treatment effects are large. If both row and column F are near 1, the Latin square gained nothing over CRD. Cross-checked against R 4.6.0 stats::aov on Montgomery (2017), Design and Analysis of Experiments, 9th ed., ch. 4, Example 4.3 (Rocket Propellant Problem).

Common pitfalls

Picking a Latin square when there is no real two-way gradient (waste of df).
Confusing Latin square with Greco-Latin (which adds a fourth orthogonal source).
Reading the row or column F as evidence about treatments.
Failing to randomise the assignment of treatments to letters (the standard square is not random).

Try it in StatVeda

The Latin Square Design engine runs entirely in the browser. No signup, no install, no data sent to a server. Paste your data, hit Run, copy the output.

Open Latin Square Design

References

Fisher, R. A. and Yates, F. (1938). Statistical Tables for Biological, Agricultural and Medical Research. Oliver and Boyd, Edinburgh.
Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd edition. John Wiley and Sons, New York. Chapter 4.
Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. John Wiley and Sons, Hoboken.