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Missing Value in LSD Bias Correction Efficiency RBD vs CRD Efficiency LSD vs RBD Efficiency LSD vs CRD
On this page
  1. 1. Missing Value in LSD
  2. 2. Efficiency of RBD Relative to CRD
  3. 3. Efficiency of LSD Relative to RBD
  4. 4. Efficiency of LSD Relative to CRD
  5. 5. Summary of Design Comparison
  6. Key Take-aways

1. Missing Value in LSD

DEFINITION

If one observation is missing in a Latin square (say at row \(i\), column \(j\), bearing treatment \(k\)), estimate it by minimizing error sum of squares; this gives a closed-form formula.

YATES' FORMULA — Missing value in LSD \[ \hat y_{ijk} \;=\; \dfrac{t\,(R'_i + C'_j + T'_k) - 2\,G'}{(t-1)(t-2)}, \]

where \(R'_i, C'_j, T'_k\) are sums of the available values in the same row, column, treatment respectively; \(G'\) is the grand total of all available values; \(t\) = number of rows = columns = treatments.

Adjustments after substitution

  1. Insert \(\hat y_{ijk}\) and analyse using normal LSD ANOVA.
  2. Reduce error df by 1: dfE = \((t-1)(t-2) - 1\).
  3. SSTr is biased upward; correct by subtracting: \[ B \;=\; \dfrac{\bigl[ R'_i + C'_j - (t-1) T'_k - G' / t \bigr]^2}{(t-1)^2(t-2)^2}\cdot t. \] (For practical work, an approximate correction \(B \approx \dfrac{[(t-1) \hat y_{ijk} - T'_k]^2}{(t-1)^2(t-2)}\) is often used.)
EXAMPLE 1

4 × 4 LSD with one missing value. Row total \(R'_i = 65\), Column total \(C'_j = 70\), Treatment total \(T'_k = 60\), Grand total \(G' = 240\), \(t = 4\).

\(\hat y_{ijk} = \dfrac{4(65 + 70 + 60) - 2(240)}{3 \cdot 2} = \dfrac{780 - 480}{6} = \dfrac{300}{6} = 50\).

EXAMPLE 2

5 × 5 LSD: \(R' = 80, C' = 88, T' = 92, G' = 432, t = 5\).

\(\hat y = \dfrac{5(80 + 88 + 92) - 2(432)}{4 \cdot 3} = \dfrac{1300 - 864}{12} = \dfrac{436}{12} = 36.33\).

2. Efficiency of RBD Relative to CRD

DEFINITION

The relative efficiency (RE) of design B over design A measures how many times more replications design A would need to achieve the same precision as design B.

Formula

\[ \text{RE}(\text{RBD over CRD}) \;=\; \dfrac{\text{Estimated MS}_E\text{ if CRD had been used}}{\text{MS}_E\text{ in RBD}}. \]

Estimating the CRD-MSE from the RBD analysis (Federer's formula):

\[ \text{MS}_E^{CRD} \;=\; \dfrac{(b-1)\, \text{MS}_{Bl} + b(t-1)\, \text{MS}_E}{bt - 1} \;=\; \dfrac{\text{SS}_{Bl} + \text{SS}_E}{(b-1) + (t-1)(b-1)}. \]

Adjusted RE (Fisher's correction)

\[ \text{RE}^{*} \;=\; \dfrac{(f_1 + 1)(f_2 + 3)}{(f_1 + 3)(f_2 + 1)}\cdot \dfrac{\text{MS}_E^{CRD}}{\text{MS}_E^{RBD}}, \]

where \(f_1\) = error df in RBD, \(f_2\) = error df in CRD. The factor in front adjusts for the loss of df due to blocking.

EXAMPLE 1

RBD with 4 treatments, 5 blocks. ANOVA gives \(\text{MS}_{Bl} = 12,\; \text{MS}_E = 4\). Compute RE.

\(\text{MS}_E^{CRD} = (4 \cdot 12 + 5 \cdot 3 \cdot 4)/(4 \cdot 5 - 1) = (48 + 60)/19 = 108/19 = 5.68\).

RE = 5.68/4 = 1.42 → RBD is 42 % more efficient than CRD.

EXAMPLE 2

RBD with 3 treatments, 4 blocks: \(\text{MS}_{Bl} = 8,\; \text{MS}_E = 2\).

\(\text{MS}_E^{CRD} = (3 \cdot 8 + 4 \cdot 2 \cdot 2)/11 = (24 + 16)/11 = 40/11 = 3.64\).

RE = 3.64/2 = 1.82 → RBD is 82 % more efficient.

3. Efficiency of LSD Relative to RBD

Two cases — taking either the rows or the columns of the LSD as blocks of an equivalent RBD.

RE of LSD over RBD (rows as blocks) \[ \text{MS}_E^{RBD\,(rows)} \;=\; \dfrac{(t-1)\, \text{MS}_C + (t-1)(t-2)\, \text{MS}_E}{(t-1) + (t-1)(t-2)} \;=\; \dfrac{\text{SS}_C + \text{SS}_E}{(t-1)(t-1)}. \]

Then RE = \(\text{MS}_E^{RBD}/\text{MS}_E^{LSD}\).

RE of LSD over RBD (columns as blocks) \[ \text{MS}_E^{RBD\,(cols)} \;=\; \dfrac{\text{SS}_R + \text{SS}_E}{(t-1)(t-1)}. \]
EXAMPLE 1

5 × 5 LSD with \(\text{SS}_R = 30, \text{SS}_C = 25, \text{SS}_E = 60\). dfE = 12.

\(\text{MS}_E^{LSD} = 60/12 = 5\).

\(\text{MS}_E^{RBD\,(cols)} = (30 + 60)/(4 \cdot 4) = 90/16 = 5.625\). RE = 5.625/5 = 1.125 (12.5 % gain).

\(\text{MS}_E^{RBD\,(rows)} = (25 + 60)/16 = 5.31\). RE = 1.06 (6 % gain).

EXAMPLE 2

4 × 4 LSD: \(\text{SS}_R = 20, \text{SS}_C = 30, \text{SS}_E = 18\). dfE = 6.

\(\text{MS}_E^{LSD} = 3\). \(\text{MS}_E^{RBD\,(cols)} = (20 + 18)/9 = 4.22\). RE = 4.22/3 = 1.41 (41 % gain).

4. Efficiency of LSD Relative to CRD

Pool all blocking SS into the error of an equivalent CRD.

\[ \text{MS}_E^{CRD} \;=\; \dfrac{\text{SS}_R + \text{SS}_C + \text{SS}_E}{t^2 - t} \;=\; \dfrac{\text{SS}_R + \text{SS}_C + \text{SS}_E}{t(t - 1)}. \]

RE of LSD over CRD = MSECRD / MSELSD.

EXAMPLE 1

From LSD Example 1 above (5 × 5): \(\text{MS}_E^{CRD} = (30 + 25 + 60)/(5 \cdot 4) = 115/20 = 5.75\).

RE = 5.75/5 = 1.15 (15 % gain over CRD).

EXAMPLE 2

From LSD Example 2 (4 × 4): \(\text{MS}_E^{CRD} = (20 + 30 + 18)/12 = 5.67\). RE = 5.67/3 = 1.89 → LSD 89 % more efficient than CRD.

5. Summary of Design Comparison

DesignPrinciples UseddfEBest when
CRDReplication, Randomization\(N - k\)Units are homogeneous
RBD+ Local control (one direction)\((t-1)(b-1)\)One blocking variable
LSD+ Local control (two directions)\((t-1)(t-2)\)Two blocking variables

General Rule

Decision Flow

For an experiment with \(t\) treatments:

  1. Are the units homogeneous? → use CRD.
  2. Is there one obvious source of heterogeneity (rows or columns)? → use RBD.
  3. Are there two perpendicular sources of heterogeneity? → use LSD (with \(t \times t\) plots).

Key Take-aways

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