• When the experimental material is divided into rows and columns and the treatments are allocated such that each treatment occurs only once in a row and once in a column, the design is known as latin square design. In this design eliminating fertility variations consists in an experimental layout which will control variation in two perpendicular direction.
• Latin square designs are normally used in experiments where it is required to remove the heterogeneity of experimental material in two directions.
• This design requires that the number of replications (rows) equal the number of treatments. In LSD the number of rows and number of columns are equal. Hence the arrangement will form a square. ^{NABARD Mains-2020}

Layout of LSD

• In this design the number of rows is equal to the number of columns and it is equal to the number of treatments. Thus, in case of ‘m’ treatments, there have to be m x m = m² experimental units (plots) arranged in a square so that each row as well as each column contain ‘m’ plots.
• The ‘m’ treatments are then allocated at random to these rows and columns in such a way that every treatment occurs once and only once in each row and each column such a layout is known as m x m L.S.D. and is extensively used in agricultural experiments.
• The minimum and maximum number of treatments required for layout of LSD is 5 to 12. Because the minimum error degree of freedom should be 12. Shouldn’t be used for less than 5 treatments.
• In LSD the treatments are usually denoted by alphabets like A, B, C…etc. For a latin square with five treatments the arrangement may be as follows:

Mathematical Model

y_ijk = μ + α_i + β_j + γ_k + ξ_ijk

• i = j = k = 1,2,……
• Where
  • Y_ijk denote the response from the unit (plot) in the i^th row, j^th column and receiving the k^th treatment.
  • μ = general mean effect
  • α_i = i^th row effect
  • β_j = j^th column effect
  • γ_k = k^th treatment effect
  • ξ_ijk = error effect
• We know that total variation =

  Variation due to rows + variation due to columns + Variation due to treatments + variation due to error

  • Null hypothesis (H₀) = There is no significant difference between Rows, Columns and Treatment effects.
• i.e.
  • H₀₁: α₁ = α₂ = ………. α_m
  • H₀₂: β₁ = β₂ = ……… = β_m and
  • H₀₃: γ₁ = γ₂ = ……………….. = γ_m
• The steps in the analysis of the data for verifying the null hypothesis are: Different component variations can be calculated as follows:

ANOVA

• If calculate value of F(Tr) < table value of Fat 5% LOS, H₀ is accepted and hence we may conclude that there is no significance difference between treatment effects.
• If calculate value of F(Tr) > table value of F at 5% LOS, H₀ is rejected and hence we may conclude that there is significance difference between treatments effects.
• If the treatments are significantly different, the comparison of the treatments is carried out on the basis of Critical Difference (C.D.).

Advantages of Latin Square Design

• With two way grouping or stratification LSD controls more of the variation than C.R.D. or R.B.D.
• L.S.D. is an incomplete 3-way layout. Its advantage over complete 3-way layout is that instead of m³ experimental units only m² units are needed.
• Thus a 4 x 4 L.S.D. results in saving of 64 - 16 = 48 observations over a complete 3-way layout.
• The statistical analysis is simple though slightly complicated than for R.B.D. Even with missing data the analysis remains relatively simple.
• More than one factor can be investigated simultaneously.
• The missing observations can be analysed by using missing plot technique.
• Three way classification two way control of error.
• This design used when fertility gradient is in two directions.

Number of replications = Number of treatments.

Number of rows = Number of columns = Number of treatments.

• Randomization of treatments is done in such a way that each treatments occurs once and only once in each row and each column.
• Error degree of freedom in LSD: (n - 1) x (n - 2)

• When the experimental material is divided into rows and columns and the treatments are allocated such that each treatment occurs only once in a row and once in a column, the design is known as latin square design. In this design eliminating fertility variations consists in an experimental layout which will control variation in two perpendicular direction.
• Latin square designs are normally used in experiments where it is required to remove the heterogeneity of experimental material in two directions.
• This design requires that the number of replications (rows) equal the number of treatments. In LSD the number of rows and number of columns are equal. Hence the arrangement will form a square. ^{NABARD Mains-2020}

Layout of LSD

• In this design the number of rows is equal to the number of columns …

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