IMSL_ANOVANESTED
Syntax | Return Value | Arguments | Keywords | Discussion | Example | Version History
The IMSL_ANOVANESTED function analyzes a completely nested random model with possibly unequal numbers in the subgroups.
Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.
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.
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:
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:
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)
- 11—Adjusted 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.
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 = μ + αi + βij + 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 αi's are the plant effects and are taken to be independently distributed:
the βij's are leaf effects each independently distributed:
and the εijk's are errors each independently distributed N(0, σ2). The effects are all assumed to be independently distributed. The data is given in Table 16-7:
.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
