CRD, RBD, LSD essentials

In one sentence

Each design removes one more source of variation from the error term, paying for it in degrees of freedom — and the degrees of freedom are where the arithmetic is tested.

Why the exam asks it

Because the three designs form a ladder with a clear logic, and because every question reduces to an ANOVA table whose entries can be checked against each other. A df count that does not add up is a wrong answer that announces itself.

The idea

Completely randomised design

$t$ treatments assigned entirely at random to $n$ units. Model $y_{ij}=\mu+\tau_i+\varepsilon_{ij}$.

source	df
treatment	$t-1$
error	$n-t$
total	$n-1$

Simplest, needs no structure, and gives the most error degrees of freedom — which is its advantage. Use it when the units are genuinely homogeneous, since any real heterogeneity goes straight into the error term and inflates it.

Randomised block design

Group units into $b$ blocks of $t$ homogeneous units, and randomise treatments within each block. Model $y_{ij}=\mu+\tau_i+\beta_j+\varepsilon_{ij}$.

source	df
treatment	$t-1$
block	$b-1$
error	$(t-1)(b-1)$
total	$tb-1$

Blocking removes between-block variation from the error. It pays when blocks genuinely differ — and costs $b-1$ error degrees of freedom, so blocking on something irrelevant makes the design worse by shrinking the error df without shrinking the error.

The principle is the same as stratification in sampling: remove a known source of variation rather than let it become noise.

Latin square design

Two blocking directions at once — rows and columns — with $t$ treatments in a $t\times t$ array, each appearing exactly once in every row and every column.

source	df
treatment	$t-1$
row	$t-1$
column	$t-1$
error	$(t-1)(t-2)$
total	$t^2-1$

Removes two sources for the price of one design, but requires rows $=$ columns $=$ treatments. That rigidity is the cost, and it also means small $t$ leaves very few error degrees of freedom — a $3\times3$ square has only $2$, which is too few to test anything reliably. Latin squares are worth using from about $t=5$ upwards.

The three principles

Randomisation makes the error estimate valid by ensuring assignment is independent of unit characteristics. Replication provides an error estimate at all — with one unit per treatment there is nothing to compare against. Local control (blocking) reduces the error by removing identified variation.

Randomisation is the one that cannot be traded away: without it the error term estimates the wrong thing, and no amount of replication repairs it.

Efficiency

Relative efficiency of RBD to CRD compares error mean squares with a df adjustment. If blocking removed nothing, the RBD is less efficient, because it spent degrees of freedom for no reduction. That is the trade the question usually turns on.

Where intuition breaks

"Blocking always improves the design." Only if the blocks differ. Otherwise it costs error df and gains nothing.

"A Latin square needs a square array of any size." Rows, columns and treatments must all equal $t$. That is a real constraint on when it can be used.

"More blocking is always better." Each blocking factor costs degrees of freedom. A $3\times3$ Latin square retains only $2$ for error.

"Randomisation is a formality." It is what makes the error estimate unbiased. Without it, systematic differences between units are confounded with treatments and nothing can be separated.

"CRD is always inferior." With homogeneous units it is best — most error df and no wasted structure.

"Replication and repetition are the same." Replication means independent experimental units; repeated measurements on one unit do not provide an independent error estimate, and treating them as if they did is pseudoreplication.

"The error df is always $n-t$." Only for CRD. Each blocking factor removes more.

The exam's angle

1. Write the ANOVA table and check the df sum: the parts must total $n-1$ or $t^2-1$. A mismatch is an immediate error.
2. Identify the design from the structure described — one grouping means RBD, two crossed groupings mean Latin square.
3. For efficiency comparisons, look at whether the blocks actually differ.
4. Watch the Latin square constraint — if treatments do not equal rows, the design named is impossible.
5. Count error df before trusting a test — a design with $2$ error df cannot support a conclusion.

The night before

• CRD: treatment $t-1$, error $n-t$, total $n-1$. Most error df; use with homogeneous units.
• RBD: treatment $t-1$, block $b-1$, error $(t-1)(b-1)$, total $tb-1$. Blocking pays only if blocks differ.
• LSD: treatment, row, column each $t-1$; error $(t-1)(t-2)$; total $t^2-1$. Requires rows $=$ columns $=$ treatments.
• Three principles: randomisation (valid error), replication (any error estimate at all), local control (smaller error).
• Blocking on something irrelevant makes the design worse.
• A $3\times3$ Latin square has $2$ error df — too few to be useful.