When to use CRD vs RBD vs Latin Square

Three single-factor designs cover most of the field experiments agronomists run. Completely Randomised Design (CRD) is for uniform material in a uniform environment. Randomised Block Design (RBD) blocks out one source of heterogeneity. Latin Square blocks out two. The difference is which sources of variation you can isolate from error, and that determines how small a real treatment difference you can detect.

The decision in one sentence

Use CRD when the experimental units are genuinely homogeneous, RBD when you have one obvious gradient (slope, fertility, age of plants), and Latin Square when there are two perpendicular gradients (slope plus sun, row plus column heterogeneity in a glasshouse).

Same data, three designs

Take a 4-treatment trial of nitrogen rates on grain yield, with 3 replications. Below is the data laid out in the format StatVeda accepts (long form), and the same data analysed three ways.

treatment,rep,row,col,yield_t_ha
N0,1,1,1,3.42
N0,2,2,2,3.61
N0,3,3,3,3.55
N50,1,1,2,4.71
N50,2,2,3,4.83
N50,3,3,1,4.69
N100,1,1,3,5.66
N100,2,2,1,5.72
N100,3,3,2,5.61
N150,1,2,1,5.91
N150,2,3,2,5.74
N150,3,1,3,5.85

1. CRD

Treat the 12 plots as fully exchangeable. Total variance gets split into Treatment and Error. df Treatment = 3, df Error = 8.

2. RBD

Treat the three reps as blocks (say each rep is one strip across a field gradient). Total variance gets split into Treatment, Block and Error. df Treatment = 3, df Block = 2, df Error = 6.

3. Latin Square

Treat row and column as two crossed blocking factors (say rows are a slope gradient and columns are a sun-shade gradient). Total variance gets split into Treatment, Row, Column and Error. df Treatment = 3, df Row = 3, df Column = 3, df Error = 6 (here a 4 by 4 arrangement, two plots per cell, hence df Error matches).

What changes in the output

On the same data, the F statistic for treatment, the residual MS, and therefore the Critical Difference at 5 percent move every time you change design. If the field actually had a row gradient and you ran a CRD, the residual MS absorbs that gradient and inflates. The treatment F drops, CD widens, and real differences look non-significant.

The pattern (illustrative)

Design          MS_error    F_treatment   CD(5%) t/ha
CRD             0.143        50.7         0.474
RBD             0.052       139.4         0.286
Latin Square    0.031       234.1         0.220

Each step (CRD to RBD, RBD to Latin Square) buys precision when there is a real gradient to absorb. If there is no gradient, the extra df spent on blocking is wasted and CD widens slightly. So the call is about whether the field actually carries the gradient you suspect, not about choosing the most complex design available.

How to pick before you plant

Walk the field. If you see one direction of variation (a slope, a boundary effect, a moisture line) use RBD with replications laid across that gradient. If you see two perpendicular sources, use Latin Square. If the area is small, flat and uniform, CRD is honest and spends fewer df on blocking.

Common mistakes

Using RBD when reps are placed at random across the field, with no gradient correspondence. Using Latin Square in a 3 by 3 arrangement (df Error becomes 2, almost no power). Using CRD in a long narrow field where one end is consistently more productive.

What StatVeda gives you for each design

For all three designs, the output table follows the same structure: an ANOVA table with df, SS, MS, F and p, treatment means with CLD letters, CD at 5 percent and 1 percent, CV percent and SE(m). The only thing that changes is which sources of variation appear in the ANOVA: Treatment + Error for CRD, Treatment + Block + Error for RBD, Treatment + Row + Column + Error for Latin Square. The same input grid lets you switch between designs and compare what blocking actually buys you on your data.

When you suspect a gradient but cannot confirm it

Run an RBD anyway. If the Block MS is small relative to Error MS, the blocking did not help much, and you have only spent two df. If the Block MS is large, the blocking absorbed real variance and the treatment F is correctly inflated. The cost of being wrong about a gradient is small. The cost of ignoring a real gradient is large.