crd_factorial

Analyzes data from balanced and unbalanced completely randomized experiments. Funtion crd_factorial does permit a factorial treatment structure. However, unlike anova_factorial, function crd_factorial allows for missing data and one or more locations.

Synopsis

#include <imsls.h>

float *imsls_f_crd_factorial (int n_obs, int n_locations, int n_factors, int n_levels[], int model[], float y[], …, 0)

The type double function is imsls_d_crd_factorial.

Required Arguments

int n_obs (Input)
Number of missing and non-missing experimental observations.

int n_locations (Input)
Number of locations n_locations must be one or greater.

int n_factors (Input)
Number of factors in the model.

int n_levels[] (Input)
Array of length n_factors+1. The n_levels[0] through n_levels[n_factors-1] contain the number of levels for each factor. The last element, n_levels[n_factors], contains the number of replicates for each treatment combination within a location.

int model[] (Input)
A n_obs by (n_factors+1) array identifying the location and factor levels associated with each observation in y. The first column must contain the location identifier and the remaining columns the factor level identifiers in the same order used in n_levels. If n_locations = 1, the first column is still required, but its contents are ignored.

float y[] (Input)
An array of length n_obs containing the experimental observations and any missing values. Missing values are indicated by placing a NaN (not a number) in y. The NaN value can be set using either the function imsls_f_machine(6) or imsls_d_machine(6), depending upon whether single or double precision is being used, respectively.

Return Value

A pointer to the memory location of a two dimensional, n_anova by 6 array containing the ANOVA table, where:

 

where

 

and m = model_order.

Each row in this array contains values for one of the effects in the ANOVA table. The first value in each row, anova_tablei,0 = anova_table[i*6], is the source identifier which identifies the type of effect associated with values in that row. The remaining values in a row contain the ANOVA table values using the following convention:

j	anova_tablei,j = anova_table[i*6+j]
0	Source Identifier (values described below)
1	Degrees of freedom
2	Sum of squares
3	Mean squares
4	F-statistic
5	p-value for this F-statistic

 

Note: Note that the p-value for the F-statistic is returned as 0.0 when the value is so small that all significant digits have been lost.

The values for the mean squares, F-statistic and p-value are set to NaN for the residual and corrected total effects.

The Source Identifiers in the first column of anova_tablei,j are the only negative values in anova_table. The absolute value of the source identifier is equal to the order of the effect in that row. Main effects, for example, have a source identifier of –1. Two-way interactions use a source identifier of –2, and so on.

Source Identifier	ANOVA Source
-1	Main Effects †
-2	Two-Way Interactions ‡
-3	Three-Way Interactions ‡
.	.
.	.
.	.
-n_factors	(n_factors)-way Interactions ‡
-n_factors-1	Effects Error Term
-n_factors-2	Residual ⇑
-n_factors-3	Corrected Total
-1	Main Effects †

Notes: By default, model_order = n_factors when treatments are replicated, or n_locations >1. However, if treatments are not replicated and n_locations =1, model_order = n_factors -1.

† The number of main effects is equal to n_factors+1 if n_locations >1, and n_factors if n_locations =1. The first row of values, anova_table[0] through anova_table[5] contain the location effect if n_locations >1. If n_locations=1, then these values are the effects for factor 1.

⇑The residual term is only provided when treatments are replicated, i.e., n_levels[n_factors]>1.

‡ The number of interaction effects for the nth-way interactions is equal to

 

The order of these terms is in ascending order by treatment subscript. The interactions for factor 1 appear first, followed by factor 2, factor 3, and so on.

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_crd_factorial (int n_obs, int n_locations, int n_factors, int n_levels[], int model[], float y[],

IMSLS_RETURN_USER, float anova_table[],

IMSLS_N_MISSING, int *n_missing,

IMSLS_CV, float *cv,

IMSLS_GRAND_MEAN, float *grand_mean,

IMSLS_FACTOR_MEANS, float **factor_means,

IMSLS_FACTOR_MEANS_USER, float factor_means[],

IMSLS_FACTOR_STD_ERRORS, float **factor_std_err,

IMSLS_FACTOR_STD_ERRORS_USER, float factor_std_err[],

IMSLS_TWO_WAY_MEANS, float **two_way_means,

IMSLS_TWO_WAY_MEANS_USER, float two_way_means[],

IMSLS_TWO_WAY_STD_ERRORS, float **two_way_std_err,

IMSLS_TWO_WAY_STD_ERRORS_USER, float two_way_std_err[],

IMSLS_TREATMENT_MEANS, float **treatment_means,

IMSLS_TREATMENT_MEANS_USER, float treatment_means[],

IMSLS_TREATMENT_STD_ERROR, float **treatment_std_err,

IMSLS_TREATMENT_STD_ERROR_USER, float treatment_std_err[],

IMSLS_ANOVA_ROW_LABELS, char ***anova_row_labels,

IMSLS_ANOVA_ROW_LABELS_USER, char *anova_row_labels[],

0)

Optional Arguments

IMSLS_RETURN_USER, float anova_table[] (Output)
User defined n_anova by 6 array for the anova_table.

IMSLS_N_MISSING, int *n_missing (Output)
Number of missing values, if any, found in y. Missing values are denoted with a NaN (Not a Number) value.

IMSLS_CV, float *cv (Output)
Coefficient of Variation computed by:

 

IMSLS_GRAND_MEAN, float *grand_mean (Output)
Mean of all the data across every location.

IMSLS_FACTOR_MEANS, float **factor_means (Output)
Address of a pointer to an internally allocated array of length n_levels[0] + n_levels[1] + … + n_levels[n_factors-1] containing the factor means.

IMSLS_FACTOR_MEANS_USER, float factor_means[] (Output)
Storage for the array factor_means, provided by the user.

IMSLS_FACTOR_STD_ERRORS, float **factor_std_err (Output)
Address of a pointer to an internally allocated n_factors by 2 array containing factor standard errors and their associated degrees of freedom. The first column contains the standard errors for comparing two factor means and the second its associated degrees of freedom.

IMSLS_FACTOR_STD_ERRORS_USER, float factor_std_err[] (Output)
Storage for the array factor_std_err, provided by the user.

IMSLS_TWO_WAY_MEANS, float **two_way_means (Output)
Address of a pointer to an internally allocated one-dimensional array containing the two-way means for all two by two combinations of the factors. The total length of this array when n_factors > 1 is equal to:

 

If n_factors = 1, NULL is returned. If n_factors>1, the means would first be produced for all combinations of the first two factors followed by all combinations of the remaining factors using the subscript order suggested by the above formula. For example, if the experiment is a 2x2x2 factorial, the 12 two-way means would appear in the following order: A1B1, A1B2, A2B1, A2B2, A1C1, A1C2, A2C1, A2C2, B1C1, B1C2, B2C1, and B2C2.

IMSLS_TWO_WAY_MEANS_USER, float two_way_means[] (Output)
Storage for the array two_way_means, provided by the user.

IMSLS_TWO_WAY_STD_ERRORS, float **two_way_std_err (Output)
Address of a pointer to an internally allocated n_two_way by 2 array containing factor standard errors and their associated degrees of freedom, where

 

The first column contains the standard errors for comparing two 2-way interaction means and the second its associated degrees of freedom. The ordering of the rows in this array is similar to that used in IMSLS_TWO_WAY_MEANS. For example, if n_factors = 4, then n_two_way = 6 with the order AB, AC, AD, BC, BD, CD.

IMSLS_TWO_WAY_STD_ERRORS_USER, float two_way_std_err[] (Output)
Storage for the array two_way_std_err, provided by the user.

IMSLS_TREATMENT_MEANS, float **treatment_means (Output)
Address of a pointer to an internally allocated array of size

 

containing the treatment means. The order of the means is organized in ascending order by the value of the factor identifier. For example, if the experiment is a 2x2x2 factorial, the 8 means would appear in the following order: A1B1C1, A1B1C2, A1B2C1, A1B1C2, A2B1C1, A2B1C2, A2B2C1, and A2B2C2.

IMSLS_TREATMENT_MEANS_USER, float treatment_means[] (Output)
Storage for the array treatment_means, provided by the user.

IMSLS_TREATMENT_STD_ERROR, float **treatment_std_err (Output)
The array of length 2 containing standard error for comparing treatments based upon the average number of replicates per treatment and its associated degrees of freedom.

IMSLS_TREATMENT_STD_ERROR_USER, float treatment_std_err[] (Output)
Storage for the array treatment_std_err, provided by the user.

IMSLS_ANOVA_ROW_LABELS, char ***anova_row_labels (Output)
Address of a pointer to a pointer to an internally allocated array containing the labels for each of the n_anova rows of the returned ANOVA table. The label for the i-th row of the ANOVA table can be printed with printf("%s", anova_row_labels[i]);

The memory associated with anova_row_labels can be freed with a single call to imsls_free(anova_row_labels).

IMSLS_ANOVA_ROW_LABELS_USER, char *anova_row_labels[] (Output)
Storage for the anova_row_labels, provided by the user. The amount of space required will vary depending upon the number of factors and n_anova. An upperbound on the required memory is char *anova_row_labels[n_anova × 60].

Description

The function imsls_f_crd_factorial analyzes factorial experiments replicated in different locations. Missing observations for each treatment are allowed. All factors are regarded as fixed effects in the analysis. However, if multiple locations appear in the data, i.e., n_locations > 1, then all effects involving locations are treated as random effects.

If n_locations = 1, then the residual mean square is used as the error mean square in calculating the F-tests for all other effects. That is

 

when n_locations = 1.

If n_locations > 1 then the error mean squares for all factor F-tests is the pooled location interaction. For example, if n_factors = 2 then the error sum of squares, degrees of freedom and mean squares are calculated by:

 

Example

The following example is based upon data from a 3x2x2 completely randomized design conducted at one location. For demonstration purposes, observation 9 is set to missing.

 

#include <imsls.h>

#include <stdio.h>

 

int main(){

int n_obs = 12;

int n_locations = 1;

int n_factors = 3;

int n_levels[4] ={3, 2, 2, 1};

int page_width = 132;

 

/* model information */

int model[]={

1, 1, 1, 1,

1, 1, 1, 2,

1, 1, 2, 1,

1, 1, 2, 2,

1, 2, 1, 1,

1, 2, 1, 2,

1, 2, 2, 1,

1, 2, 2, 2,

1, 3, 1, 1,

1, 3, 1, 2,

1, 3, 2, 1,

1, 3, 2, 2

};

 

/* response data */

float y[] ={

4.42725419998168950,

2.12795543670654300,

2.55254390835762020,

1.21479606628417970,

2.47588264942169190,

5.01306104660034180,

4.73502767086029050,

4.58392113447189330,

5.01421167794615030,

4.11972457170486450,

6.51671624183654790,

4.73365202546119690

};

 

int model_order;

int i, j, k, l, m, n_missing, i2, j2;

int n_factor_levels=0, n_treatments=1;

int n_two_way_means=0, n_two_way_std_err=0;

int n_two_way_interactions=0;

int n_subscripts, n_anova_table=2;

float cv, grand_mean;

float *anova_table;

float *two_way_means, *two_way_std_err;

float *treatment_means, *treatment_std_err;

float *factor_means;

float *factor_std_err;

float aNaN = imsls_f_machine(6);

char **anova_row_labels;

char *col_labels[] = {" ", "\nID", "\nDF", "\nSSQ ",

"Mean \nsquares", "\nF-Test", "\np-Value"};

 

/* Compute the length of some of the output arrays. */

model_order = n_factors-1;

for (i=0; i < n_factors; i++){

n_factor_levels = n_factor_levels + n_levels[i];

n_treatments = n_treatments*n_levels[i];

for (j=i+1; j < n_factors; j++){

n_two_way_interactions++;

}

}

n_two_way_std_err = n_two_way_interactions;

for (i=0; i < n_factors-1; i++){

for (j=i+1; j < n_factors; j++){

n_two_way_means = n_two_way_means + n_levels[i]*n_levels[j];

}

}

 

n_subscripts = n_factors;

n_anova_table = 2;

 

for (i=1; i <= model_order; i++){

n_anova_table +=

(int)imsls_f_binomial_coefficient(n_subscripts, i);

}

 

/* Set observation 9 to missing. */

y[8] = aNaN;

anova_table = imsls_f_crd_factorial(n_obs, n_locations, n_factors,

n_levels, model, y,

IMSLS_N_MISSING, &n_missing,

IMSLS_CV, &cv,

IMSLS_GRAND_MEAN, &grand_mean,

IMSLS_FACTOR_MEANS, &factor_means,

IMSLS_FACTOR_STD_ERRORS, &factor_std_err,

IMSLS_TWO_WAY_MEANS, &two_way_means,

IMSLS_TWO_WAY_STD_ERRORS, &two_way_std_err,

IMSLS_TREATMENT_MEANS, &treatment_means,

IMSLS_TREATMENT_STD_ERROR, &treatment_std_err,

IMSLS_ANOVA_ROW_LABELS, &anova_row_labels,

0) ;

 

/* Output results. */

imsls_page(IMSLS_SET_PAGE_WIDTH, &page_width);

 

/* Print ANOVA table. */

imsls_f_write_matrix(" *** ANALYSIS OF VARIANCE TABLE ***",

n_anova_table, 6, anova_table,

IMSLS_WRITE_FORMAT, "%3.0f%3.0f%8.3f%8.3f%8.3f%8.3f",

IMSLS_ROW_LABELS, anova_row_labels,

IMSLS_COL_LABELS, col_labels,

0);

 

printf("\n\nNumber of Missing Values Estimated: %d", n_missing);

printf("\nGrand Mean: %7.3f", grand_mean);

printf("\nCoefficient of Variation: %7.3f", cv);

 

m=0;

/* Print Factor Means. */

printf("\n\nFactor Means\n");

for(i=0; i < n_factors; i++){

printf(" Factor %d: ", i+1);

for(j=0; j < n_levels[i]; j++){

printf(" %f ", factor_means[m]);

m++;

}

k = (int)factor_std_err[2*i+1];

printf("\n std. err.(df): %f(%d) \n",

factor_std_err[2*i], k);

}

 

/* Print Two-Way Means. */

printf("\n\nTwo-Way Means");

m = 0;

l=0;

 

for(i=0; i < n_factors-1; i++){

for(j=i+1; j < n_factors; j++){

printf("\n Factor %d by Factor %d: \n", i+1, j+1);

for(i2=0; i2 < n_levels[i]; i2++){

for(j2=0; j2 < n_levels[j]; j2++){

printf(" %f ",two_way_means[m]);

m++;

}

printf("\n");

}

k = (int)two_way_std_err[l+1];

printf(" std. err.(df): = %f(%d) \n", two_way_std_err[l], k);

l+=2;

}

}

 

/* Print Treatment Means. */

printf("\n\nTreatment Means\n");

m = 0;

for(i=0; i < n_levels[0]; i++){

for(j=0; j < n_levels[1]; j++){

for(k=0; k < n_levels[2]; k++){

printf(" Treatment[%d][%d][%d] Mean: %f \n",

i+1, j+1, k+1, treatment_means[m]);

m++;

}

}

}

k = (int)treatment_std_err[1];

printf("\n Treatment Std. Err (df) %f(%d) \n",

treatment_std_err[0], k);

}

Output

 

*** ANALYSIS OF VARIANCE TABLE ***

Mean

ID DF SSQ squares F-Test p-Value

[1] -1 2 13.061 6.530 7.844 0.245

[2] -1 1 0.107 0.107 0.129 0.781

[3] -1 1 1.302 1.302 1.563 0.429

[1]x[2] -2 2 3.768 1.884 2.263 0.425

[1]x[3] -2 2 5.253 2.626 3.154 0.370

[2]x[3] -2 1 0.560 0.560 0.672 0.563

Error -4 1 1.665 1.665 ........ ........

Total -5 10 25.715 ........ ........ ........

 

 

Number of Missing Values Estimated: 1

Grand Mean: 3.962

Coefficient of Variation: 32.574

 

Factor Means

Factor 1: 2.580637 4.201973 5.101940

std. err.(df): 0.912459(1)

Factor 2: 3.866924 4.056109

std. err.(df): 0.745020(1)

Factor 3: 4.290849 3.632185

std. err.(df): 0.745020(1)

 

 

Two-Way Means

Factor 1 by Factor 2:

3.277605 1.883670

3.744472 4.659474

4.578696 5.625184

std. err.(df): = 1.290412(1)

 

Factor 1 by Factor 3:

3.489899 1.671376

3.605455 4.798491

5.777192 4.426688

std. err.(df): = 1.290412(1)

 

Factor 2 by Factor 3:

3.980268 3.753580

4.601429 3.510790

std. err.(df): = 1.053617(1)

 

 

Treatment Means

Treatment[1][1][1] Mean: 4.427254

Treatment[1][1][2] Mean: 2.127955

Treatment[1][2][1] Mean: 2.552544

Treatment[1][2][2] Mean: 1.214796

Treatment[2][1][1] Mean: 2.475883

Treatment[2][1][2] Mean: 5.013061

Treatment[2][2][1] Mean: 4.735028

Treatment[2][2][2] Mean: 4.583921

Treatment[3][1][1] Mean: 5.037668

Treatment[3][1][2] Mean: 4.119725

Treatment[3][2][1] Mean: 6.516716

Treatment[3][2][2] Mean: 4.733652

 

Treatment Std. Err (df) 1.824918(1)