ALATN

Analyzes a Latin square design.

Required Arguments

NTRT — Number of treatments. (Input)
NTRT must also be the number of rows and the number of columns.

NRESP — Number of repeated responses within each row-column position. (Input)

Y — Vector of length NTRT * NTRT * NRESP containing the responses. (Input)
The first NRESP elements of Y contain the responses for row 1, column 1; the second NRESP elements of Y contain the responses for row 1, column 2. The last NRESP elements of Y contain the responses for row NTRT, column NTRT.

ITRT — Vector of length NTRT * NTRT containing the treatment numbers for the responses in Y. (Input)
The treatment numbers must be from the set 1, 2, …, NTRT. For I = 1, 2, …, NTRT**2, element numbers (I ‑ 1) * NRESP + 1 through (I ‑ 1) * NRESP + NRESP of Y correspond to treatment number ITRT(I).

AOV — Vector of length 15 containing statistics relating to the analysis of variance. (Output)

 

I	AOV(I)
1	Degrees of freedom for the model (blocks and treatments)
2	Degrees of freedom for error (interaction is pooled with the within-cell error)
3	Total (corrected) degrees of freedom
4	Sum of squares for the model (blocks and treatments)
5	Sum of squares for error (experimental error pooled with the within-cell error)
6	Total (corrected) sum of squares
7	Model mean square
8	Error mean square
9	F -statistic
10	p-value
11	R2 (in percent)
12	Adjusted R2 (in percent)
13	Estimated standard deviation of the model error
14	Overall mean of Y
15	Coefficient of variation (in percent)

Optional Arguments

IPRINT — Printing option. (Input)
Default: IPRINT = 0.

 

IPRINT	Action
0	No printing is performed.
1	Print AOV, EFSS, and TESTLF (if NRESP > 1) only.
2	Print YMEANS only.
3	All print is performed.

EFSS — Vector of length 12 containing statistics relating to the sums of squares for the effects in the model. (Output)
Elements of EFSS are described as follows:

 

Elem	Description
1, 2, 3	Degrees of freedom for rows, columns, and treatments, respectively.
4, 5, 6	Sum of squares for rows, columns, and treatments, respectively.
7, 8, 9	F-statistics for rows, columns, and treatments, respectively. F-statistics are computed using AOV(8) as the estimated error variance.
10‑12	p-values associated with the F-statistics.

TESTLF — Vector of length 10 containing statistics relating to the test for lack of fit of the model.(Output if NRESP > 1)
If NRESP = 1, TESTLF is not referenced and can be a vector of length one. Elements of TESTLF are described as follows:

 

Elem.	Description
1	Degrees of freedom for experimental error
2	Degrees of freedom for within-cell error
3	Degrees of freedom for error (TESTLF(1) + TESTLF(2))
4	Sum of squares for experimental error
5	Sum of squares for within-cell error
6	Sum of squares for error
7	Mean square for experimental error
8	Mean square for within-cell error
9	F -statistic
10	p-value

YMEANS — Vector of length 3 * NTRT + NTRT * NTRT containing the row means, column means, treatment means, and the row-column means, respectively. (Output)

FORTRAN 90 Interface

Generic: CALL ALATN (NTRT, NRESP, Y, ITRT, AOV [, …])

Specific: The specific interface names are S_ALATN and D_ALATN.

FORTRAN 77 Interface

Single: CALL ALATN (NTRT, NRESP, Y, ITRT, IPRINT, AOV, EFSS, TESTLF, YMEANS)

Double: The double precision name is DALATN.

Description

Routine ALATN performs an analysis for a Latin square design. The model is

yijkm = μ + α i + βj + δk + ɛijkm      i, j, k = 1, 2, …, p; m = 1, 2, …, n

where the observed value of yijkm constitutes the m-th response on the k-th treatment in row i column j of the Latin square design; μ + αi + βj + δk is the population mean for the response, and the ɛ ijkm’s are identically and independently distributed normal errors with mean zero and variance σ2. This model assumes the row effects (αi), column effects (βj), and treatment effects (δk) are additive. Often in practice, there are interactions between two or more of these factors. For this reason, ALATN computes a test for nonadditivity (lack of fit), in addition to summary statistics for the additive model. This test requires at least two responses in each cell. A test for nonadditivity with one response per cell in a Latin square design is discussed by Snedecor and Cochran (1967, pages 334‑337).

Routine ALATN requires yijk’s to be entered in single vector Y with the data for each cell occupying contiguous elements. The cells must be in standard order, i.e., (1, 1), (1, 2), …, (1, p), (2, 1), (2, 2), …, (2, p), …, (p, 1), (p, 2), …, (p, p). A discussion of formulas and interpretations for the analysis of a Latin square design appears in many elementary statistics texts, e.g., Snedecor and Cochran (1967, pages 312‑317).

Example

This example performs an analysis for a Latin square design using data discussed by Kirk (1982, Table 7.3-2, pages 312‑317). The responses are thickness of tread remaining on each of 32 tires after 10,000 miles of driving. The tires are divided equally among four different types, labeled A, B, C, and D. Four cars are used in the study. The experiment is performed twice, sixteen tires are used in each experiment. Each of the sixteen tires occupies one of the four wheel positions on one of the cars. The data are given in the following table:

Wheel Position	Car 1	Car 2	Car 3	Car 4
Right Front	A: 1, 2	B: 2, 3	C: 5, 6	D: 9, 8
Left Front	B: 3, 4	C: 8, 6	D: 9, 8	A: 2, 3
Right Rear	C: 5, 7	D: 10, 11	A: 3, 2	B: 5, 4
Left Rear	D: 7, 10	A: 6, 3	B: 3, 4	C: 6, 6

 

USE ALATN_INT

 

IMPLICIT NONE

INTEGER NRESP, NTRT

PARAMETER (NRESP=2, NTRT=4)

!

INTEGER IPRINT, ITRT(NTRT*NTRT)

REAL AOV(15), Y(NTRT*NTRT*NRESP)

!

DATA Y/1.0, 2.0, 2.0, 3.0, 5.0, 6.0, 9.0, 8.0, 3.0, 4.0, 8.0, &

6.0, 9.0, 8.0, 2.0, 3.0, 5.0, 7.0, 10.0, 11.0, 3.0, 2.0, &

5.0, 4.0, 7.0, 10.0, 6.0, 3.0, 3.0, 4.0, 6.0, 7.0/

DATA ITRT/1, 2, 3, 4, 2, 3, 4, 1, 3, 4, 1, 2, 4, 1, 2, 3/

DATA IPRINT/3/

!

CALL ALATN (NTRT, NRESP, Y, ITRT, AOV, IPRINT=IPRINT)

END

Output

 

Dependent R-squared Adjusted Est. Std. Dev. Coefficient of

Variable (percent) R-squared of Model Error Mean Var. (percent)

Y 89.809 85.640 1.044 5.375 19.43

 

* * * Analysis of Variance * * *

Sum of Mean Prob. of

Source DF Squares Square Overall F Larger F

Model 9 211.5 23.50 21.542 0.0000

Error 22 24.0 1.09

Corrected Total 31 235.5

 

* * * Decomposition of Variation Attributable to the Model * * *

Source Sum of Prob. of

DF Squares F Larger F

Row 3 9.2 2.826 0.0622

Column 3 7.8 2.368 0.0983

Treatment 3 194.5 59.431 0.0000

 

Test for Lack of Fit

Source Sum of Mean Prob. of

DF Squares Square F Larger F

Experimental Error 6 5 0.833 0.702 0.6525

Within Cell 16 19 1.188

Error 22 24

 

* * * Row Means * * *

Row Mean (N=4)

1 4.500

2 5.375

3 5.875

4 5.750

 

* * * Column Means * * *

Column Mean (N=4)

1 4.875

2 6.125

3 5.000

4 5.500

 

* * * Treatment Means * * *

Treatment Mean (N=4)

1 2.8

2 3.5

3 6.2

4 9.0

 

* * * Cell Means * * *

Row Column Mean (N=2)

1 1 1.500

1 2 2.500

1 3 5.500

1 4 8.500

2 1 3.500

2 2 7.000

2 3 8.500

2 4 2.500

3 1 6.000

3 2 10.500

3 3 2.500

3 4 4.500

4 1 8.500

4 2 4.500

4 3 3.500

4 4 6.500