Chapter 4: Analysis of Variance

AONEW

Analyzes a one-way classification model.

Required Arguments

NI — Vector of length NGROUP containing the number of responses for each group.   (Input)

Y — Vector of length NI(1) + NI(2) + L + NI(NGROUP) containing the responses for each group.   (Input)

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

I

AOV(I)

1

Degrees of freedom for among groups

2

Degrees of freedom for within groups

3

Total (corrected) degrees of freedom

4

Sum of squares for among groups

5

Sum of squares for within groups

6

Total (corrected) sum of squares

7

Among-groups mean square

8

Within-groups mean square

9

F -statistic

10

p-value

11

R2 (in percent)

12

Adjusted R2 (in percent)

13

Estimated standard deviation of the error within groups

14

Overall mean of Y

15

Coefficient of variation (in percent)

Optional Arguments

NGROUP — Number of groups.   (Input)
Default: NGROUP = size (NI,1).

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

IPRINT                  Action

0                              No printing is performed.

1                              AOV is printed only.

2                              STAT is printed only.

3                              All printing is performed.

STATNGROUP by 4 matrix containing information concerning the groups.   (Output)
Row I contains information pertaining to the I-th group. The information in the columns is as follows:

Col.     Description

1          Group number

2          Number of nonmissing observations

3          Group mean

4          Group standard deviation

LDSTAT — Leading dimension of STAT exactly as specified in the dimension statement in the calling program.   (Input)
Default: LDsTAT= size (STAT , 1)

NMISS — Number of missing values.   (Output)
Elements of Y containing NaN (not a number) are omitted from the computations.

FORTRAN 90 Interface

Generic:                              CALL AONEW (NI, Y, AOV [,…])

Specific:                             The specific interface names are S_AONEW and D_AONEW.

FORTRAN 77 Interface

Single:            CALL AONEW (NGROUP, NI, Y, IPRINT, AOV, STAT, LDSTAT, NMISS)

Double:                              The double precision name is DAONEW.

Description

Routine AONEW performs an analysis of variance of responses from a one-way classification design. The model is

yij = μi + ɛ ij     i = 1, 2, , k; j = 1, 2, , ni

where the observed value of yij constitutes the j-th response in the i-th group, μi denotes the population mean for the i-th group, and the ɛ ij’s are errors that are identically and independently distributed normal with mean zero and variance σ2. AONEW requires the yij’s as input into a single vector Y with responses in each group occupying contiguous locations. The analysis of variance table is computed along with the group sample means and standard deviations. A discussion of formulas and interpretations for the one-way analysis of variance problem appears in most elementary statistics texts, e.g., Snedecor and Cochran (1967, Chapter 10).

Example

This example computes a one-way analysis of variance for data discussed by Searle (1971, Table 5.1, pages 165179). The responses are plant weights for 6 plants of 3 different types3 normal, 2 off-types, and 1 aberrant. The responses are given by type of plant in the following table:

Type of Plant

Normal

Off-Type

Aberrant

101

84

32

105

88

 

94

 

 

Note that for the group with only one response, the standard deviation is undefined and is set to NaN (not a number).

 

      USE AONEW_INT

 

      IMPLICIT   NONE

      INTEGER    NGROUP, NOBS

      PARAMETER  (NGROUP=3, NOBS=6)

!

      INTEGER    IPRINT, NI(NGROUP)

      REAL       AOV(15), Y(NOBS)

!

      DATA NI/3, 2, 1/

      DATA Y/101.0, 105.0, 94.0, 84.0, 88.0, 32.0/

!

      IPRINT = 3

      CALL AONEW (NI, 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             98.028     96.714            4.83          84           5.751

                  * * * Analysis of Variance * * *
                              Sum of        Mean             Prob. of
Source                DF     Squares      Square  Overall F  Larger F
Among Groups           2        3480      1740.0     74.571    0.0028
Within Groups          3          70        23.3
Corrected Total        5        3550

             Group Statistics
                                  Standard
 Group           N        Mean   Deviation
     1           3         100       5.568
     2           2          86       2.828
     3           1          32         NaN



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