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

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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:

y_ijk = μ + a_i + β_ij + e_ijk i = 1, 2, 3, 4; j = 1, 2, 3; k = 1, 2

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

the β_ij’s are leaf effects each independently distributed:

and the ε_ijk’s are errors each independently distributed N(0, σ2). The effects are all assumed to be independently distributed. The data is given below.

.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

IDL prints:

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  1               3.59

  AB	means 3  3               2.55

  AB	means 4  1               3.82

  AB	means 4  1               4.10

  AB	means 4  3               3.31

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 overall 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 the following table 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

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 (optional)

Named variable into which the analysis of variance table is stored. The 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)
• 11: Adjusted R² (in percent)
• 12: Estimate of the standard deviation
• 13: Overall mean of y
• 14: Coefficient of variation (in percent)

CONFIDENCE (optional)

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: 95.0

DOUBLE (optional)

If present and nonzero, then double precision is used.

EMS (optional)

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

VAR_COMP (optional)

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 (optional)

One-dimensional array containing the subgroup means.

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.

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