anova_nested

Analyzes a completely nested random model with possibly unequal numbers in the subgroups.

Synopsis

#include <imsls.h>

float *imsls_f_anova_nested (int n_factors, int equal_option, int n_levels[], float y[], ..., 0)

The type double function is imsls_d_anova_nested.

Required Arguments

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

int equal_option (Input)
Equal numbers option.

equal_option

Description

0

Unequal numbers in the subgroups.

1

Equal numbers in the subgroups.

int n_levels[] (Input)
Array with the number of levels for each factor.

If equal_option = 1, n_levels is of length n_factors and contains the number of levels for each of the factors.

Example: Suppose there are 3 factors, A, B, and C. A has two levels (A1, A2), B has 3 levels at each level of A, and C has 2 levels at each level of B. n_levels = {2,3,2} and the number of observations is nobs = 2 × 3 × 2 = 12.

 

A levels

B levels

C levels

y indices

 

1

1

1

0

 

2

1

2

1

2

 

2

3

3

1

4

 

2

5

2

1

1

6

 

2

7

2

1

8

 

2

9

3

1

10

 

2

11

 

 

 

 

n_levels

2

3

2

 

nobs

2×

3×

2

=12

 

equal_option = 1 example

If equal_option = 0, n_levels contains the number of levels of each factor at each level of the factor in which it is nested.

Example: Suppose there are 3 factors, A, B, and C, with C nested in B and B nested in A. A has two levels (A1, A2), B has up to 3 levels, and C has up to 2 levels. In the equal_option = 0 case, the function needs to know explicitly how the number of levels varies throughout. As specified in the table, A has two levels (n_levels[0] = 2), B has 3 levels in level 1 of A (n_levels[1] = 3) and 2 levels in level 2 of A (n_levels[2] = 2). Similarly, factor C has 2 levels in the A1‑B1 and A1‑B2 combinations (n_levels[3] = 2, n_levels[4] = 2), but only 1 level in the A1‑B3 combination (n_levels[5] = 1). n_levels = {2,3,2,2,2,1,1,2} and the number of observations is the sum of the number of levels in the last factor, C, nobs = 2 + 2 + 1 + 1 + 2 = 8:

 

 

A levels

B levels

C levels

y indices

 

1

1

1

0

 

 

2

1

 

2

1

2

 

 

2

3

 

3

1

4

2

1

1

5

 

2

1

6

 

 

2

7

 

 

 

 

n_levels

2

3, 2

2 , 2, 1, 1, 2

 

nobs

 

 

2 + 2 + 1 + 1 + 2

= 8

 

equal_option = 0 example

float y[] (Input)
Array of length nobs containing the responses.

Return Value

The p-value for the F‑statistic, anova_table[9].

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_anova_nested (int n_factors, int equal_option, int n_levels[], float y[],

IMSLS_ANOVA_TABLE, float **anova_table,

IMSLS_ANOVA_TABLE_USER, float anova_table[],

IMSLS_CONFIDENCE, float confidence,

IMSLS_VARIANCE_COMPONENTS, float **variance_components,

IMSLS_VARIANCE_COMPONENTS_USER, float variance_components[],

IMSLS_EMS, float **expect_mean_sq,

IMSLS_EMS_USER, float expect_mean_sq[],

IMSLS_Y_MEANS, float **y_means,

IMSLS_Y_MEANS_USER, float y_means[],

0)

Optional Arguments

IMSLS_ANOVA_TABLE, float **anova_table, (Output)
Address of a pointer to an internally allocated array of size 15 containing the analysis of variance table. The analysis of variance statistics are as follows:

Element

Analysis of Variance Statistics

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).

Note that the p‑value is returned as 0.0 when the value is so small that all significant digits have been lost.

IMSLS_ANOVA_TABLE_USER, float anova_table[] (Output)
Storage for array anova_table is provided by the user. See IMSLS_ANOVA_TABLE

IMSLS_CONFIDENCE, float confidence (Input)
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.

IMSLS_VARIANCE_COMPONENTS, float **variance_components (Output)
Address to a pointer to an internally allocated array. variance_components is an n_factors by 9 matrix containing statistics relating to the particular variance components in the model. Rows of variance_components correspond to the n_factors factors. Columns of variance_components are as follows:

Column

Description

0

Degrees of freedom.

1

Sum of squares.

2

Mean squares.

3

F–statistic.

4

p-value for F test.

5

Variance component estimate.

6

Percent of variance of variance explained by variance component.

7

Lower endpoint for a confidence interval on the variance component.

8

Upper endpoint for a confidence interval on the variance component.

A test for the error variance equal to zero cannot be performed. variance_components [(n_factors-1)*9+3] and variance_components [(n_factors-1)*9+4] are set to NaN (not a number). Note that the p-value for the F test is returned as 0.0 when the value is so small that all significant digits have been lost.

IMSLS_VARIANCE_COMPONENTS_USER, float variance_components[] (Output)
Storage for array variance_components is provided by the user. See IMSLS_VARIANCE_COMPONENTS.

IMSLS_EMS, float **expect_mean_sq (Output)
Address to a pointer to an internally allocated array of length n_factors * (n_factors +1) / 2 with expected mean square coefficients.

IMSLS_EMS_USER, float expect_mean_sq[] (Output)
Storage for array expect_mean_sq is provided by the user. See IMSLS_EMS.

IMSLS_Y_MEANS, float **y_means (Output)
Address to a pointer to an internally allocated array containing the subgroup means.

Equal options

Length of y_means

0

1 + sum of values in n_levels for the first (n_factors-1) factors

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.

IMSLS_Y_MEANS_USER, float y_means[] (Output)
Storage for array y_means is provided by the user. See IMSLS_Y_MEANS

Description

Function imsls_f_anova_nested 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.

Examples

Example 1

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 + eijki = 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 ɛijk’s are errors each independently distributed N(0, σ2). The effects are all assumed to be independently distributed. The data are given in the following table:

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

 

#include <imsls.h>

#include <stdio.h>

 

int main()

{

float pvalue, *aov, *varc, *ymeans, *ems;

float 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

};

int n_levels[] = {4, 3, 2};

char *aov_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)"

};

char *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"

};

char *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"

};

char *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"

};

 

pvalue = imsls_f_anova_nested(3, 1, n_levels, y,

IMSLS_ANOVA_TABLE, &aov,

IMSLS_Y_MEANS, &ymeans,

IMSLS_VARIANCE_COMPONENTS, &varc,

IMSLS_EMS, &ems,

0);

 

printf("pvalue = %f\n", pvalue);

 

imsls_f_write_matrix("* * * Analysis of Variance * * *", 15, 1, aov,

IMSLS_ROW_LABELS, aov_labels,

IMSLS_WRITE_FORMAT, "%11.4g",

0);

 

imsls_f_write_matrix(

"* * * Expected Mean Square Coefficients * * *",

6, 1, ems,

IMSLS_ROW_LABELS, ems_labels,

IMSLS_WRITE_FORMAT, "%6.2f",

0);

 

imsls_f_write_matrix("* * * Means * * *", 17, 1, ymeans,

IMSLS_ROW_LABELS, means_labels,

IMSLS_WRITE_FORMAT, "%6.2f",

0);

 

imsls_f_write_matrix(

"* * Analysis of Variance / Variance Components * *",

27, 1, varc,

IMSLS_ROW_LABELS, components_labels,

IMSLS_WRITE_FORMAT, "%11.4g",

0);

}

Output

 

pvalue = 0.000000

 

          * * * Analysis of Variance * * *

degrees of freedom for model                      11

degrees of freedom for error                      12

total (corrected) degrees of freedom              23

sum of squares for model                       10.19

sum of squares for error                     0.07985

total (corrected) sum of squares               10.27

model mean square                             0.9264

error mean square                           0.006655

F-statistic                                    139.2

p-value                                   6.769e-011

R-squared (in percent)                         99.22

adjusted R-squared (in percent)                98.51

est. standard deviation of within error      0.08158

overall mean of y                              3.012

coefficient of variation (in percent)          2.708

 

* * * 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

 

  * * * 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

 

    * * Analysis of Variance / Variance Components * *

degrees of freedom for A                                 3

sum of squares for A                                  7.56

mean square of A                                      2.52

F-statistic for A                                    7.665

p-value for A                                     0.009725

Estimate of A                                       0.3652

Percent Variation Explained by A                     68.53

95% Confidence Interval Lower Limit for A          0.03955

95% Confidence Interval Upper Limit for A            5.787

degrees of freedom for B                                 8

sum of squares for B                                  2.63

mean square of B                                    0.3288

F-statistic for B                                    49.41

p-value for B                                   5.092e-008

Estimate of B                                       0.1611

Percent Variation Explained by B                     30.22

95% Confidence Interval Lower Limit for B          0.06967

95% Confidence Interval Upper Limit for B           0.6004

degrees of freedom for Error                            12

sum of squares for Error                           0.07985

mean square of Error                              0.006655

F-statistic for Error                          ...........

p-value for Error                              ...........

Estimate of Error                                 0.006655

Percent Explained by Error                           1.249

95% Confidence Interval Lower Limit for Error     0.003422

95% Confidence Interval Upper Limit for Error      0.01813

Example 2

An analysis of a three-factor nested random model with unequal numbers in the subgroups is performed. The data are given in the following table:

A

B

C

1

1

2

23.0

31.0

19.0

37.0

 

2

1

2

33.0

29.0

29.0

 

3

1

36.0

29.0

33.0

4

1

2

3

4

5

6

7

8

9

11.0

23.0

33.0

23.0

26.0

39.0

20.0

24.0

36.0

21.0

18.0

 

5

1

25.0

33.0

 

6

1

2

3

4

5

6

7

8

9

10

28.0

25.0

32.0

41.0

35.0

16.0

30.0

40.0

32.0

44.0

31.0

42.0

36.0

 

 

#include <imsls.h>

 

int main()

{

float *aov, *ems, *vc, *ymeans;

float y[36] = {23.0, 19.0, 31.0, 37.0,

33.0, 29.0, 29.0,

36.0, 29.0, 33.0,

11.0, 21.0,

23.0, 18.0,

33.0, 23.0, 26.0, 39.0, 20.0, 24.0, 36.0,

25.0, 33.0,

28.0, 31.0,

25.0, 42.0,

32.0, 36.0,

41.0, 35.0, 16.0, 30.0, 40.0, 32.0, 44.0

};

int nl[32] = {

6, /* Factor A */

2, 2, 1, 9, 1, 10, /* Factor B */

2, 2, /* Factor C */

2, 1,

3,

2, 2, 1, 1, 1, 1, 1, 1, 1,

2,

2, 2, 2, 1, 1, 1, 1, 1, 1, 1

};

int i, ymeans_length;

 

char *aov_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)"

};

char *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"

};

char *means_labels[] = {

"Grand mean", " A means 1", " A means 2",

" A means 3", " A means 4", " A means 5",

" A means 6", "AB means 1 1", "AB means 1 2",

"AB means 2 1", "AB means 2 2", "AB means 3 1",

"AB means 4 1", "AB means 4 2", "AB means 4 3",

"AB means 4 4", "AB means 4 5", "AB means 4 6",

"AB means 4 7", "AB means 4 8", "AB means 4 9",

"AB means 5 1", "AB means 6 1", "AB means 6 2",

"AB means 6 3", "AB means 6 4", "AB means 6 5",

"AB means 6 6", "AB means 6 7", "AB means 6 8",

"AB means 6 9", "AB means 6 10"

};

char *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"};

 

imsls_f_anova_nested (3, 0, nl, y,

IMSLS_ANOVA_TABLE, &aov,

IMSLS_EMS, &ems,

IMSLS_VARIANCE_COMPONENTS, &vc,

IMSLS_Y_MEANS, &ymeans,

0);

 

imsls_f_write_matrix("***AnalysisofVariance ***", 15, 1, aov,

IMSLS_ROW_LABELS, aov_labels,

IMSLS_WRITE_FORMAT, "%10.5f",

0);

imsls_f_write_matrix("***ExpectedMeanSquare Coefficients ***",

6, 1, ems,

IMSLS_ROW_LABELS, ems_labels,

IMSLS_WRITE_FORMAT, "%6.2f",

0);

 

/* sum level count for factors 1 and 2 */

ymeans_length = 1;

for (i=0; i<=6;i++) ymeans_length += nl[i];

 

imsls_f_write_matrix("* * * Means ***", ymeans_length, 1, ymeans,

IMSLS_ROW_LABELS, means_labels,

IMSLS_WRITE_FORMAT, "%6.2f",

0);

imsls_f_write_matrix(

"** Analysis of Variance / Variance Components **", 27, 1, vc,

IMSLS_ROW_LABELS, components_labels,

IMSLS_WRITE_FORMAT, "%10.5f",

0);

}

Output

 

***AnalysisofVariance ***

degrees of freedom for model 24.00000

degrees of freedom for error 11.00000

total (corrected) degrees of freedom 35.00000

sum of squares for model 1810.80591

sum of squares for error 310.16650

total (corrected) sum of squares 2120.97241

model mean square 75.45025

error mean square 28.19695

F-statistic 2.67583

p-value 0.04587

R-squared (in percent) 85.37621

adjusted R-squared (in percent) 53.46977

est. standard deviation of within error 5.31008

overall mean of y 29.52778

coefficient of variation (in percent) 17.98334

 

***ExpectedMeanSquare Coefficients ***

Effect A and Error 1.00

Effect A and Effect B 1.97

Effect A and Effect A 5.38

Effect B and Error 1.00

Effect B and Effect B 1.29

Error and Error 1.00

 

* * * Means ***

Grand mean 29.53

A means 1 27.50

A means 2 30.33

A means 3 32.67

A means 4 24.91

A means 5 29.00

A means 6 33.23

AB means 1 1 21.00

AB means 1 2 34.00

AB means 2 1 31.00

AB means 2 2 29.00

AB means 3 1 32.67

AB means 4 1 16.00

AB means 4 2 20.50

AB means 4 3 33.00

AB means 4 4 23.00

AB means 4 5 26.00

AB means 4 6 39.00

AB means 4 7 20.00

AB means 4 8 24.00

AB means 4 9 36.00

AB means 5 1 29.00

AB means 6 1 29.50

AB means 6 2 33.50

AB means 6 3 34.00

AB means 6 4 41.00

AB means 6 5 35.00

AB means 6 6 16.00

AB means 6 7 30.00

AB means 6 8 40.00

AB means 6 9 32.00

AB means 6 10 44.00

 

* * Analysis of Variance / Variance Components * *

degrees of freedom for A 5.00000

sum of squares for A 461.42230

mean square of A 92.28446

F-statistic for A 0.98770

p-value for A 0.46007

Estimate of A -0.21371

Percent Variation Explained by A ..........

95% Confidence Interval Lower Limit for A ..........

95% Confidence Interval Upper Limit for A ..........

degrees of freedom for B 19.00000

sum of squares for B 1349.38354

mean square of B 71.02019

F-statistic for B 2.51872

p-value for B 0.05965

Estimate of B 33.19880

Percent Variation Explained by B 54.07344

95% Confidence Interval Lower Limit for B 0.00000

95% Confidence Interval Upper Limit for B 100.58640

degrees of freedom for Error 11.00000

sum of squares for Error 310.16650

mean square of Error 28.19695

F-statistic for Error ..........

p-value for Error ..........

Estimate of Error 28.19695

Percent Explained by Error 45.92656

95% Confidence Interval Lower Limit for Error 14.14990

95% Confidence Interval Upper Limit for Error 81.28591