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### IMSL_ANOVANESTED

IMSL_ANOVANESTED

The IMSL_ANOVANESTED function analyzes a completely nested random model with possibly unequal numbers in the subgroups.

## Syntax

Result = IMSL_ANOVANESTED(n_factors, eq_option, n_levels, y [, ANOVA_TABLE=variable] [, CONFIDENCE=value] [, /DOUBLE] [, EMS=array] [, VAR_COMP=variable] [, Y_MEANS=array])

## Return Value

The p-value for the F-statistic.

## Arguments

#### eq_option

Equal numbers option.

• 0—Unequal numbers in the subgroups
•

• 1—Equal numbers in the subgroups

#### n_factors

Number of factors (number of subscripts) in the model, including error.

#### n_levels

One-dimensional array with the number of levels.

If eq_option = 1, n_levels is of length n_factors and contains the number of levels for each of the factors. In this case, the additional variables listed in Table 16-5 are referred to in the description of IMSL_ANOVANESTED:

Variable
Description
LNL
n_levels(1) +
... + n_levels(0) * n_levels(1) *
... * n_levels(n_factors – 2)
LNLNF
n_levels(0) * n_levels(1) * ...*
n_levels(n_factors – 2)
NOBS
The number of observations. NOBS equals
n_levels(0) * n_levels(1) * ... *
n_levels(n_factors-1)

If eq_option = 0, n_levels contains the number of levels of each factor at each level of the factor in which it is nested. In this case, the following additional variables are referred to in the description of IMSL_ANOVANESTED:

• LNL—Length of n_levels.
•

• LNLNF—Length of the subvector of n_levels for the last factor.
•

• NOBS—Number of observations. NOBS equals the sum of the last LNLNF elements of n_levels. n_levels(n_factors-1).

For example, a random one-way model with two groups, five responses in the first group and ten in the second group, would have LNL = 3, LNLNF = 2, NOBS = 15, n_levels(0) = 2, n_levels(1) = 5, and n_levels(2) = 10.

#### y

One-dimensional array of length NOBS containing the responses.

## Keywords

#### ANOVA_TABLE

Named variable which stores the size 15 array containing the analysis of variance table. Analysis of variance statistics are as follows:

• 0—Degrees of freedom for the model
•

• 1—Degrees of freedom for error
•

• 2—Total (corrected) degrees of freedom
•

• 3—Sum of squares for the model
•

• 4—Sum of squares for error
•

• 5—Total (corrected) sum of squares
•

• 6—Model mean square
•

• 7—Error mean square
•

• 8—Overall F-statistic
•

• 9—p-value
•

• 10—R2 (in percent)
•

•

• 12—Estimate of the standard deviation
•

• 13—Overall mean of y
•

• 14—Coefficient of variation (in percent)

#### CONFIDENCE

Confidence level for two-sided interval estimates on the variance components, in percent. Confidence percent confidence intervals are computed, hence, Confidence must be in the interval [0.0, 100.0). Confidence often will be 90.0, 95.0, or 99.0. For one-sided intervals with confidence level ONECL, ONECL in the interval [50.0, 100.0), set Confidence = 100.0 – 2.0 * (100.0 - ONECL). Default: Confidence = 95.0

#### DOUBLE

If present and nonzero, then double precision is used.

#### EMS

One-dimensional array of length n_factors * ((n_factors + 1)/2) with expected mean square coefficients.

#### VAR_COMP

Named variable into which an array of size n_factors by 9 containing statistics relating to the particular variance components in the model is stored. Rows of Var_Comp correspond to the n_factors factors. Columns of Var_Comp are as follows:

• 1—Degrees of freedom
•

• 2—Sum of squares
•

• 3—Mean squares
•

• 4—F -statistic
•

• 5—p-value for F test
•

• 6—Variance component estimate
•

• 7—Percent of variance explained by variance component
•

• 8—Lower endpoint for confidence interval on the variance component
•

• 9—Upper endpoint for confidence interval on the variance component

If a test for error variance equal to zero cannot be performed, Var_Comp(n_factors, 4) and Var_Comp(n_factors, 5) are set to NaN.

#### Y_MEANS

One-dimensional array containing the subgroup means.

Table 16-6: eq_option for Y_Means

eq_option
Length of y means
0
1 + n_levels(0) + n_levels(1) + ... n_levels((LNL - LNLNF)-1)
(See description of argument n_levels for definitions of LNL and LNLNF.)
1
1 + n_levels(0) + n_levels(0) * n_levels(1) + ... + n_levels(0)* n_levels(1) * ... * n_levels (n_factors – 2)

If the factors are labeled A, B, C, and error, the ordering of the means is grand mean, A means, AB means, and then ABC means.

## Discussion

The IMSL_ANOVANESTED function analyzes a nested random model with equal or unequal numbers in the subgroups. The analysis includes an analysis of variance table and computation of subgroup means and variance component estimates. Anderson and Bancroft (1952, pages 325-330) discuss the methodology. The analysis of variance method is used for estimating the variance components. This method solves a linear system in which the mean squares are set to the expected mean squares. A problem that Hocking (1985, pages 324-330) discusses is that this method can yield negative variance component estimates. Hocking suggests a diagnostic procedure for locating the cause of a negative estimate. It may be necessary to reexamine the assumptions of the model.

## Example

An analysis of a three-factor nested random model with equal numbers in the subgroups is performed using data discussed by Snedecor and Cochran (1967, Table 10.16.1, pages 285-288). The responses are calcium concentrations (in percent, dry basis) as measured in the leaves of turnip greens. Four plants are taken at random, then three leaves are randomly selected from each plant. Finally, from each selected leaf two samples are taken to determine calcium concentration. The model is:

yijk = m + ai + bij + eijk i = 1, 2, 3, 4; j = 1, 2, 3; k = 1, 2

where yijk is the calcium concentration for the k-th sample of the j-th leaf of the i-th plant, the ai's are the plant effects and are taken to be independently distributed:

the bij's are leaf effects each independently distributed:

and the eijk's are errors each independently distributed N(0, s2). The effects are all assumed to be independently distributed. The data is given in Table 16-7:

Table 16-7: Calcium Concentrations

Plant
Leaf
Samples
1
1
2
3
3.28
3.52
2.88
3.09
3.48
2.80
2
1
2
3
2.46
1.87
2.19
2.44
1.92
2.19
3
1
2
3
2.77
3.74
2.55
2.66
3.44
2.55
4
1
2
3
3.78
4.07
3.31
3.87
4.12
3.31

`.RUN  PRO print_results, p, at, ems, y_means, var_comp      anova_labels = [';degrees of freedom for model', \$        ';degrees of freedom for error', \$        ';total (corrected) degrees of freedom', \$        ';sum of squares for model', 'sum of squares for error', \$        ';total (corrected) sum of squares', 'model mean square', \$        ';error mean square', 'F-statistic', 'p-value', \$        ';R-squared (in percent)', \$        ';adjusted R-squared (in percent)', \$        ';est. standard deviation of within error', \$        ';overall mean of y', \$        ';coefficient of variation (in percent)']     ems_labels  = [';Effect A and Error', 'Effect A and Effect B', \$        ';Effect A and Effect A', 'Effect B and Error', \$        ';Effect B and Effect B', 'Error and Error']      components_labels  =  [';degrees of freedom for A', \$        ';sum of squares for A', 'mean square of A', \$        ';F-statistic for A', 'p-value for A', \$        ';Estimate of A', 'Percent Variation Explained by A', \$        ';95% Confidence Interval Lower Limit for A', \$        ';95% Confidence Interval Upper Limit for A', \$        ';degrees of freedom for B', 'sum of squares for B', \$        ';mean square of B', 'F-statistic for B', 'p-value for B', \$        ';Estimate of B', 'Percent Variation Explained by B', \$        ';95% Confidence Interval Lower Limit for B', \$        ';95% Confidence Interval Upper Limit for B', \$        ';degrees of freedom for Error', \$        ';sum of squares for Error', 'mean square of Error', \$        ';F-statistic for Error', 'p-value for Error', \$        ';Estimate of Error', 'Percent Explained by Error', \$        ';95% Confidence Interval Lower Limit for Error', \$        ';95% Confidence Interval Upper Limit for Error']     means_labels = [';Grand mean', \$        '; A means 1', \$        '; A means 2', \$        '; A means 3', \$        '; A means 4', \$        ';AB means 1 1', \$        ';AB means 1 2', \$        ';AB means 1 3', \$        ';AB means 2 1', \$        ';AB means 2 2', \$        ';AB means 2 3', \$        ';AB means 3 1', \$        ';AB means 3 2', \$        ';AB means 3 3', \$        ';AB means 4 1', \$        ';AB means 4 2', \$        ';AB means 4 3']     PRINT, ';p value of F statistic =', p          PRINT                   PRINT, ';               * * * Analysis of Variance * * *'     FOR i  =  0, 14 DO \$        PM, anova_labels(i), at(i), FORMAT = ';(A40, F20.5)'                        PRINT           PRINT, ';          * * * Expected Mean Square Coefficients * * *'                  FOR i  =  0, 5 DO \$        PM, ems_labels(i), ems(i), FORMAT = ';(A40, F20.2)'     PRINT     PRINT, ';      * * Analysis of Variance / Variance Components * *';     k = 0     FOR i  =  0, 2 DO BEGIN        FOR j  =  0, 8 DO BEGIN           PM, components_labels(k), var_comp(i, j), \$           FORMAT = ';(A45, F20.5)'            k = k + 1        ENDFOR     ENDFOR     PRINT     PRINT, ';means', FORMAT = '(A20)'     FOR i  =  0, 16 DO \$        PM, means_labels(i), y_means(i), FORMAT =';(A20, F20.2)'  END    y = [3.28, 3.09, 3.52, 3.48, 2.88, 2.80, 2.46, 2.44, 1.87, \$     1.92, 2.19, 2.19, 2.77, 2.66, 3.74, 3.44, 2.55, 2.55, \$     3.78, 3.87, 4.07, 4.12, 3.31, 3.31]  n_levels  =  [4, 3, 2]  p = IMSL_ANOVANESTED(3, 1, n_levels, y, Anova_Table = at, \$     Ems=ems, Y_Means = y_means, Var_Comp = var_comp)  print_results, p, at, ems, y_means, var_comp    p value of F statistic =      0.00000                    * * * Analysis of Variance * * *                 degrees of freedom for model            11.00000                 degrees of freedom for error            12.00000         total (corrected) degrees of freedom            23.00000                     sum of squares for model            10.19054                     sum of squares for error             0.07985             total (corrected) sum of squares            10.27040                            model mean square             0.92641                            error mean square             0.00665                                  F-statistic           139.21599                                      p-value             0.00000                       R-squared (in percent)            99.22248              adjusted R-squared (in percent)            98.50976      est. standard deviation of within error             0.08158                            overall mean of y             3.01208        coefficient of variation (in percent)             2.70826               * * * Expected Mean Square Coefficients * * *                           Effect A and Error                1.00                        Effect A and Effect B                2.00                        Effect A and Effect A                6.00                           Effect B and Error                1.00                        Effect B and Effect B                2.00                              Error and Error                1.00              * * Analysis of Variance / Variance Components * *                          degrees of freedom for A             3.00000                              sum of squares for A             7.56034                                  mean square of A             2.52011                                 F-statistic for A             7.66516                                     p-value for A             0.00973                                     Estimate of A             0.36522                  Percent Variation Explained by A            68.53015         95% Confidence Interval Lower Limit for A             0.03955         95% Confidence Interval Upper Limit for A             5.78674                          degrees of freedom for B             8.00000                              sum of squares for B             2.63020                                  mean square of B             0.32878                                 F-statistic for B            49.40642                                     p-value for B             0.00000                                     Estimate of B             0.16106      Percent Variation Explained by B                   30.22121      95% Confidence Interval Lower Limit for B           0.06967      95% Confidence Interval Upper Limit for B           0.60042      degrees of freedom for Error                        12.00000      sum of squares for Error                             0.07985      mean square of Error                                 0.00665                   F-statistic for Error                    NaN                   p-value for Error                        NaN                   Estimate of Error                       0.00665     Percent Explained by Error                            1.24864     95% Confidence Interval Lower Limit for Error        0.00342     95% Confidence Interval Upper Limit for Error        0.01813                    means               Grand mean                3.01                A means 1                3.17                A means 2                2.18                A means 3                2.95                A means 4                3.74             AB means 1 1                3.18             AB means 1 2                3.50             AB means 1 3                2.84             AB means 2 1                2.45             AB means 2 2                1.89             AB means 2 3                2.19             AB means 3 1                2.72             AB means 3 2                3.59             AB means 3 3                2.55             AB means 4 1                3.82             AB means 4 2                4.10             AB means 4 3                3.31  `

## Version History

 6.4 Introduced

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