Analyzes a balanced incomplete block design or a balanced lattice design.
NTRT Number of treatments. (Input)
NREP Number of replications. (Input)
NBLK Number of blocks. (Input)
NTBLK Number of treatments within each block. (Input)
NRESP Number of responses within each treatment-block combination. (Input)
Y Vector of
length NBLK
* NTBLK * NRESP containing the
responses. (Input)
The first NRESP elements of
Y
contain the responses for the first treatment in the first block in the first
replicate. The second NRESP elements of
Y
contain the responses for the second treatment in the first block in the first
replicate. The NTBLK-th NRESP elements of
Y
contain the responses for the NTBLK-th treatment in
the first block in the first replicate. The last NRESP elements of
Y
contain the responses for the NTBLK-th treatment in
the NBLK-th
block in the NREP-th replicate.
ITRT Vector of
length NBLK
* NTBLK 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,
,
NBLK * NTBLK, element numbers
(I
− 1) * NRESP + 1 thru
(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 regression |
2 |
Degrees of freedom for error |
3 |
Total degrees of freedom |
4 |
Sum of squares for regression |
5 |
Sum of squares for error |
6 |
Total sum of squares |
7 |
Regression 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 |
Mean of the response (dependent) variable |
15 |
Coefficient of variation (in percent) |
INTER
Interblock analysis option. (Input)
Default: INTER
= 0.
INTER |
Means |
|
0 |
Intrablock analysis is requested. (Blocks are fixed effects.) | |
1 |
Interblock analysis is requested. (Blocks are random effects.) |
IPRINT Printing
option. (Input)
Default: IPRINT
= 0.
INTER |
Means |
|
0 |
No printing is performed. | |
1 |
Print AOV, SQSS, and TESTLF (if NRESP > 1). | |
2 |
Print YMEANS only. | |
3 |
All printing is performed. |
SQSS Vector of length 12 containing statistics relating to the sequential sum of squares for the model. (Output)
Elem. Description
1, 2, 3 Degrees of freedom for replicates, blocks within replicates, and treatments (adjusted), respectively
4, 5, 6 Sum of squares for replicates, blocks within replicates, and treatments (adjusted), respectively
7, 8, 9 F -statistics for replicates, blocks, and treatments, respectively, computed using AOV(8) as the estimated error variance
10−12 p-values associated with the F -statistics
SSALT Vector of
length 2 containing an alternative partitioning of the model sum of
squares. (Output)
SSALT(1) is the
treatment sum of squares (unadjusted) and SSALT(2) is the block
sum of squares (adjusted).
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 NREP +
NBLK + NTRT + NTBLK * NBLK containing the
replicate means, block by replicate means, treatment means (adjusted), and
treatment by block means, respectively. (Output)
The treatment
means (adjusted) in YMEANS are used for
estimating treatment differences.
SETRTD Estimated standard error of a treatment difference. (Output)
EFNCY Estimated
efficiency of this design relative to a randomized complete block
design. (Output)
The randomized complete block design has NBLK * NTBLK/NTRT complete
blocks.
Generic: CALL ABIBD (NTRT, NREP, NBLK, NTBLK, NRESP, Y, ITRT, AOV [, ])
Specific: The specific interface names are S_ABIBD and D_ABIBD.
Single: CALL ABIBD (NTRT, NREP, NBLK, NTBLK, NRESP, Y, ITRT, INTER, IPRINT, AOV, SQSS, SSALT, TESTLF, YMEANS, SETRTD, EFNCY)
Double: The double precision name is DABIBD.
Routine ABIBD performs analyses for balanced incomplete block designs. The basic model used is the randomized block design with the source of variation for blocks subdivided into replications and blocks within replications. For INTER = 0, the model is
yijtm = μ + α i + βjj + δt + ɛ ijkm i = 1, , r; j = 1, , k; t = 1, , p; m = 1, , n
where the observed value of yijtm constitutes the m-th response with treatment t in block j within the i replicate, μ + α i + βij + δt is the population mean for the response, and the ɛ ijtms are independently distributed normal errors with mean zero and variance σ2. This model assumes the block effects and treatment effects are additive. Often in practice, there are interactions between the blocks and treatments. For this reason, ABIBD 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.
The analysis performed with the βijs regarded as fixed effects in the model (INTER = 0) is called an intrablock analysis. For INTER = 1, the βijs are assumed to be random effects in the model, the analysis performed for this mixed model is called an interblock analysis.
Routine ABIBD requires the yijtms to be entered in a single vector Y ordered lexicographically, so that the i subscript varies least rapidly, the j subscript the next most rapidly, and so forth. Formulas and interpretations for the analysis of balanced incomplete block designs are discussed by Anderson and Bancroft (1952, Chapters 19 and 24) and Kempthorne (1975, pages 532−539).
Workspace may be explicitly provided, if desired, by use of A2IBD/DA2IBD. The reference is:
CALL A2IBD (NTRT, NREP, NBLK, NTBLK, NRESP, Y, ITRT, INTER, IPRINT, AOV, SQSS, SSALT, TESTLF, YMEANS, SETRTD, EFNCY, WK)
The additional argument is:
WK Work vector of length NTRT or 2 * NTRT.
This example performs an intrablock analysis for a balanced incomplete block design using data discussed by Anderson and Bancroft (1952, pages 254−256). The responses are weight gains of rats fed p = 9 different rations. There are four replications with k = 3 blocks within each replicate. (Since p = k2, this balanced incomplete block design is a balanced lattice design.) The data with the treatment numbers in parentheses are given in the following table:
Replicate |
Block |
(Treatment): Weight Gain | ||
1 |
1 |
(1): 20 |
(4): 15 |
(7): 11 |
1 |
2 |
(3): 8 |
(6): 18 |
(9): 26 |
1 |
3 |
(2): 18 |
(5): 16 |
(8): 2 |
2 |
1 |
(7): 8 |
(8): 12 |
(9): 16 |
2 |
2 |
(1): 20 |
(2): 2 |
(3): 2 |
2 |
3 |
(4): 20 |
(5): 6 |
(6): 2 |
3 |
1 |
(1): 13 |
(9): 19 |
(5): 14 |
3 |
2 |
(8): 14 |
(4): 34 |
(3): 2 |
3 |
3 |
(6): 14 |
(2): 20 |
(7): 14 |
4 |
1 |
(5): 19 |
(7): 23 |
(3): 6 |
4 |
2 |
(1): 22 |
(6): 12 |
(8): 2 |
4 |
3 |
(9): 27 |
(2): 7 |
(4): 20 |
USE ABIBD_INT
IMPLICIT NONE
INTEGER NBLK, NREP, NRESP, NTBLK, NTRT
PARAMETER (NBLK=12, NREP=4, NRESP=1, NTBLK=3, NTRT=9)
!
INTEGER IPRINT, ITRT(NBLK*NTBLK)
REAL AOV(15), Y(NBLK*NTBLK*NRESP)
!
DATA Y/20.0, 15.0, 11.0, 8.0, 18.0, 26.0, 18.0, 16.0, 2.0, 8.0, &
12.0, 16.0, 20.0, 2.0, 2.0, 20.0, 6.0, 2.0, 13.0, 19.0, &
14.0, 14.0, 34.0, 2.0, 14.0, 20.0, 14.0, 19.0, 23.0, 6.0, &
22.0, 12.0, 2.0, 27.0, 7.0, 20.0/
DATA ITRT/1, 4, 7, 3, 6, 9, 2, 5, 8, 7, 8, 9, 1, 2, 3, 4, 5, 6, &
1, 9, 5, 8, 4, 3, 6, 2, 7, 5, 7, 3, 1, 6, 8, 9, 2, 4/
!
IPRINT = 3
CALL ABIBD (NTRT, NREP, NBLK, NTBLK, NRESP, Y, ITRT, &
AOV, IPRINT=IPRINT)
END
Dependent R-squared Adjusted Est. Std.
Dev.
Coefficient of
Variable (percent) R-squared of Model
Error Mean Var.
(percent)
Y
79.771
55.748
5.345
14 38.18
* * * Analysis of Variance * * *
Sum of Mean Prob. of
Source DF Squares Square Overall F Larger F
Model 19 1802.8 94.88 3.321 0.0095
Error 16 457.2 28.57
Corrected Total 35 2260.0
* * * Decomposition of Variation Attributable to the
Model * * *
Sum of Prob. of
Source DF Squares F Larger F
Replicates 3 219.6 2.561 0.0913
Blocks within
Replicates 8 127.1 0.556 0.7980
Treatments
(adjusted) 8 1456.1 6.370 0.0009
* * * Replicate means * * *
Replicate Mean (N=4)
1 14.8889
2 9.7778
3 16.0000
4 15.3333
* * * Block by Replicate Means * * *
Replicate Block Mean (N=3)
1 1 15.3333
1 2 17.3333
1 3 12.0000
2 1 12.0000
2 2 8.0000
2 3 9.3333
3 1 15.3333
3 2 16.6667
3 3 16.0000
4 1 16.0000
4 2 12.0000
4 3 18.0000
* * * Adjusted Treatment Means * * *
Treatment Mean (N=1)
1 22.11
2 11.67
3 0.67
4 23.89
5 14.78
6 11.11
7 12.89
8 6.44
9 22.44
* * * Treatment by Block
Means * * *
Replicate Block Treatment Mean (N=1)
1 1 1 20.0000
1 1 4 15.0000
1 1 7 11.0000
1 2 3 8.0000
1 2 6 18.0000
1 2 9 26.0000
1 3 2 18.0000
1 3 5 16.0000
1 3 8 2.0000
2 1 7 8.0000
2 1 8 12.0000
2 1 9 16.0000
2 2 1 20.0000
2 2 2 2.0000
2 2 3 2.0000
2 3 4 20.0000
2 3 5 6.0000
2 3 6 2.0000
3 1 1 13.0000
3 1 9 19.0000
3 1 5 14.0000
3 2 8 14.0000
3 2 4 34.0000
3 2 3 2.0000
3 3 6 14.0000
3 3 2 20.0000
3 3 7 14.0000
4 1 5 19.0000
4 1 7 23.0000
4 1 3 6.0000
4 2 1 22.0000
4 2 6 12.0000
4 2 8 2.0000
4 3 9 27.0000
4 3 2 7.0000
4 3 4 20.0000
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