Chapter 4: Analysis of Variance

ATWOB

Analyzes a randomized block design or a two-way balanced design.

Required Arguments

NBLK — Number of blocks.   (Input)

NTRT — Number of treatments.   (Input)

NRESP — Number of repeated responses within each block-treatment combination.   (Input)

Y — Vector of length NBLK * NTRT * NRESP containing the responses.   (Input)
The first NRESP elements of Y contain the responses for block one, treatment one, the second NRESP elements of Y contain the responses for block one, treatment two; ; the last NRESP elements of Y contain the responses for block NBLK, treatment NTRT.

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 (interaction is 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

AOV(I)

0

No printing is performed.

1

Print AOV, EFSS, and TESTLF (if NRESP > 1).

2

Print YMEANS only.

3

All printing is performed.

EFSS — Vector of length 8 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

Degrees of freedom for blocks and treatments, respectively

3, 4

Sum of squares for blocks and treatments, respectively

5, 6

F-statistics for blocks and treatments, respectively. F-statistics are computed using AOV(8) as the estimated error variance.

7, 8

p-values associated with the F -statistics

TESTLF — Vector of length 10 containing statistics relating to the test for lack of fit of the two-way model without interaction.   (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 interaction

2

Degrees of freedom for within-cell error

3

Degrees of freedom for error (TESTLF(1) + TESTLF(2))

4

Sum of squares for interaction

5

Sum of squares for interaction

6

Sum of squares for within-cell error

7

Mean square for interaction

8

Mean square for within-cell error

9

F-statistic

10

p-value

YMEANS — Vector of length NBLK + NTRT + NBLK * NTRT containing the block means, treatment means and block-by-treatment means, respectively.   (Output)

FORTRAN 90 Interface

Generic:          CALL ATWOB (NBLK, NTRT, NRESP, Y, AOV [,…])

Specific:                             The specific interface names are S_ATWOB and D_ATWOB.

FORTRAN 77 Interface

Single:            CALL ATWOB (NBLK, NTRT, NRESP, Y, IPRINT, AOV, EFSS, TESTLF, YMEANS)

Double:                              The double precision name is DATWOB.

Description

Routine ATWOB performs an analysis for a two-way classification design with balanced data. For balanced data, there must be an equal number of responses in each cell of the two-way layout. The basic model is the same as for the randomized block design. The block and treatment effects are additive, i.e., there are no interactions. The model is

yijk = μ + α i + βj + ɛ ij     i = 1, 2, , n1; j = 1, 2, , n2; k = 1, 2, , n3

where the observed value of yijk constitutes the k-th response in the ij-th cell of the two-way layout, μ + α i + βj is the population mean for the ij-th cell, and the ɛ ijk’s are identically and independently distributed normal errors with mean zero and variance σ2. This model assumes that the effects for the two factors are additive. Often in practice, there are interactions between the two factors. For this reason, in addition to summary statistics for the additive model, ATWOB computes a test for nonadditivity (lack of fit). The test used here requires at least two responses in each cell. Tests for nonadditivity with one response per cell are given by Tukey (1949) and Mandel (1961). Tukey’s test is discussed by Snedecor and Cochran (1967, pages 331334).

The routine ATWOB requires yijk’s as input into a 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, n2), (2, 1), (2, 2), , (2, n2), , (n1, 1), (n1, 2), , (n1n2):

Example 1

This example performs an analysis for a randomized block design using data discussed by Neter and Wasserman (1974,Table 23.2, pages 725730). Fifteen businessmen were shown one of three methods for quantifying the maximum risk premium they would be willing to pay to avoid uncertainty. The responses are a stated degree of confidence, on a scale of 0 (no confidence) to 20 (highest confidence). The fifteen businessmen were grouped into five blocks by age. The three businessmen in each block were randomly assigned to a rating method. The data are given in the following table:

 

 

Confidence Rating

Block

Method 1

Method 2

Method 3

1

1

5

8

2

2

8

14

3

7

9

16

4

6

13

18

5

12

14

17

 

      USE ATWOB_INT

 

      IMPLICIT   NONE

      INTEGER    NBLK, NRESP, NTRT

      PARAMETER  (NBLK=5, NRESP=1, NTRT=3)

!

      INTEGER    IPRINT

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

!

      DATA Y/1.0, 5.0, 8.0, 2.0, 8.0, 14.0, 7.0, 9.0, 16.0, 6.0, 13.0, &

          18.0, 12.0, 14.0, 17.0/

!

      IPRINT = 3

      CALL ATWOB (NBLK, NTRT, NRESP, Y, 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             94.003     89.506           1.727          10           17.27


                   * * * Analysis of Variance * * *

                               Sum of        Mean             Prob. of

 Source                DF     Squares      Square  Overall F  Larger F

 Model                  6       374.1       62.36     20.901    0.0002

 Error                  8        23.9        2.98

 Corrected Total       14       398.0


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

                                Sum of             Prob. of

        Source          DF     Squares          F  Larger F

        Blocks           4       171.3     14.358    0.0010

        Treatment        2       202.8     33.989    0.0001


 * * * Block Means * * *

    Block  Mean (N=3)

        1      4.6667

        2      8.0000

        3     10.6667

        4     12.3333

        5     14.3333

 

 

 


 * * * Treatment Means * * *

    Treatment  Mean (N=5)

            1      5.6000

            2      9.8000

            3     14.6000


    * * * Cell Means * * *

 Block  Treatment  Mean (N=1)

     1          1      1.0000

     1          2      5.0000

     1          3      8.0000

     2          1      2.0000

     2          2      8.0000

     2          3     14.0000

     3          1      7.0000

     3          2      9.0000

     3          3     16.0000

     4          1      6.0000

     4          2     13.0000

     4          3     18.0000

     5          1     12.0000

     5          2     14.0000

     5          3     17.0000

Additional Example

Example 2

This example fits an additive two-way analysis of variance model and performs a test for nonadditivity (lack of fit) using data discussed by Kirk (1982,Table 8.3-1, pages 354359). The data for the two-way layout is given in the following table:

 

 

BLOCK

TREATMENT

1

2

3

1

24, 33, 37, 29, 42

44, 36, 25, 27, 43

38, 29, 28, 47, 48

2

30, 21, 39, 26, 34

35, 40, 27, 31, 22

26, 27, 36, 46, 45

3

21, 18, 10, 31, 20

41, 39, 50, 36, 34

42, 52, 53, 49, 64

 

      USE ATWOB_INT

 

      IMPLICIT   NONE

      INTEGER    NBLK, NRESP, NTRT

      PARAMETER  (NBLK=3, NRESP=5, NTRT=3)

!

      INTEGER    IPRINT

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

!

      DATA Y/24.0, 33.0, 37.0, 29.0, 42.0, 30.0, 21.0, 39.0, 26.0, &

          34.0, 21.0, 18.0, 10.0, 31.0, 20.0, 44.0, 36.0, 25.0, 27.0, &

          43.0, 35.0, 40.0, 27.0, 31.0, 22.0, 41.0, 39.0, 50.0, 36.0, &

          34.0, 38.0, 29.0, 28.0, 47.0, 48.0, 26.0, 27.0, 36.0, 46.0, &

          45.0, 42.0, 52.0, 53.0, 49.0, 64.0/

!

      IPRINT = 3

      CALL ATWOB (NBLK, NTRT, NRESP, Y, 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             33.206     26.526           9.336          35           26.68


                   * * * Analysis of Variance * * *

                               Sum of        Mean             Prob. of

 Source                DF     Squares      Square  Overall F  Larger F

 Model                  4      1733.3       433.3      4.971    0.0024

 Error                 40      3486.7        87.2

 Corrected Total       44      5220.0


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

                                Sum of             Prob. of

        Source          DF     Squares          F  Larger F

        Blocks           2      1543.3      8.853    0.0007

        Treatment        2       190.0      1.090    0.3460


                  * * * Test for Lack of Fit * * *

                           Sum of        Mean              Prob. of

 Source            DF     Squares      Square          F   Larger F

 Interaction        4      1236.7       309.2      4.947     0.0028

 Within cell       36      2250.0        62.5

 Error             40      3486.7


 * * * Block Means * * *

    Block  Mean (N=3)

        1     27.6667

        2     35.3333

        3     42.0000


 * * * Treatment Means * * *

    Treatment  Mean (N=3)

            1     35.3333

            2     32.3333

            3     37.3333


    * * * Cell Means * * *

 Block  Treatment  Mean (N=5)

     1          1     33.0000

     1          2     30.0000

     1          3     20.0000

     2          1     35.0000

     2          2     31.0000

     2          3     40.0000

     3          1     38.0000

     3          2     36.0000

     3          3     52.0000



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