For $N>3$, there are many possible Latin Squares. Tables of standard squares are available, and non-standard squares can be generated by permuting rows/columns/treatments.

4. Analysis of Variance (ANOVA)

Hypotheses:
- $H_0: \tau_1 = \tau_2 = ... = \tau_N = 0$ (No treatment effect)
- $H_1:$ At least one $\tau_k \ne 0$ (Treatment effects exist)
- Similar hypotheses can be tested for row and column effects.

ANOVA Table Structure:

Source of Variation	Degrees of Freedom (df)	Sum of Squares (SS)	Mean Square (MS)	F-statistic
Rows	$N-1$	$SSR = \sum_{i=1}^N \frac{R_i^2}{N} - CF$	$MSR = \frac{SSR}{N-1}$	$F_{Rows} = \frac{MSR}{MSE}$
Columns	$N-1$	$SSC = \sum_{j=1}^N \frac{C_j^2}{N} - CF$	$MSC = \frac{SSC}{N-1}$	$F_{Cols} = \frac{MSC}{MSE}$
Treatments	$N-1$	$SSTr = \sum_{k=1}^N \frac{T_k^2}{N} - CF$	$MSTr = \frac{SSTr}{N-1}$	$F_{Tr} = \frac{MSTr}{MSE}$
Error	$(N-1)(N-2)$	$SSE = SST - SSR - SSC - SSTr$	$MSE = \frac{SSE}{(N-1)(N-2)}$
Total	$N^2-1$	$SST = \sum_{i,j} Y_{ij}^2 - CF$

Calculations:
- $CF = \frac{(\sum Y_{ijk})^2}{N^2}$ (Correction Factor)
- $R_i$: Sum of observations in row $i$.
- $C_j$: Sum of observations in column $j$.
- $T_k$: Sum of observations for treatment $k$.
F-test: Compare $F_{Tr}$ to $F_{\alpha, (N-1), (N-1)(N-2)}$ to test for treatment effects.

5. Multiple Comparisons (Post-Hoc Tests)

If $H_0$ for treatments is rejected, multiple comparison procedures (e.g., Tukey's HSD) can be used to determine which specific treatment means differ.
Tukey's Honestly Significant Difference (HSD): $$HSD = q_{\alpha, N, (N-1)(N-2)} \sqrt{\frac{MSE}{N}}$$ Compare absolute differences between treatment means $|\bar{Y}_{T_k} - \bar{Y}_{T_l}|$ to HSD.

6. Replicated Latin Squares

When $(N-1)(N-2)$ is small (e.g., $N=3$, df error = 2), the power of the test is low.
To increase degrees of freedom for error and improve precision, replicate the Latin Square.
Replication can be done in two ways:
1. Several independent Latin Squares: Each square is a complete experiment. Each square has its own set of rows and columns.
2. One Latin Square with multiple observations per cell: Not a true Latin Square. This is more like a factorial design with blocking.

7. Example Scenario

An experiment to test the effect of 4 different fertilizers (A, B, C, D) on crop yield.
Two blocking factors: soil type (rows 1-4) and plot orientation (columns 1-4).
A $4 \times 4$ Latin Square ensure