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IDL Analyst Reference Guide: Analysis of Variance |
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The IMSL_ANOVANESTED function analyzes a completely nested random model with possibly unequal numbers in the subgroups.
| Note This routine requires an IDL Analyst license. For more information, contact your ITT Visual Information Solutions sales or technical support representative. |
Result = IMSL_ANOVANESTED(n_factors, eq_option, n_levels, y [, ANOVA_TABLE=variable] [, CONFIDENCE=value] [, /DOUBLE] [, EMS=array] [, VAR_COMP=variable] [, Y_MEANS=array])
The p-value for the F-statistic.
Equal numbers option.
Number of factors (number of subscripts) in the model, including error.
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.
One-dimensional array of length NOBS containing the responses.
Named variable which stores the size 15 array containing the analysis of variance table. Analysis of variance statistics are as follows:
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
If present and nonzero, then double precision is used.
One-dimensional array of length n_factors * ((n_factors + 1)/2) with expected mean square coefficients.
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:
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.
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.
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.
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:
.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
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