latin_square

Chapter 4: Analysis of Variance and Designed Experiments

latin_square

Analyzes data from latin-square experiments. Function latin_square also analyzes latin-square experiments replicated at several locations.

Synopsis

#include <imsls.h>

float * imsls_f_latin_square (int n, int n_locations,
int n_treatments, int row[], int col[], int treatment[],
float y[], …, 0)

The type double function is imsls_d_latin_square.

Required Arguments

int n (Input)
Number of missing and non-missing experimental observations. imsls_f_latin_square verifies that:

hint n_locations (Input)
Number of locations. n_locations must be one or greater. If n_locations>1 then the optional array locations[] must be included as input to imsls_f_latin_square.

int n_treatments (Input)
Number of treatments. n_treatments must be greater than one. In addition the number of rows and columns must be equal to n_treatments.

int row[] (Input)
An array of length n containing the row identifiers for each observation in y. Each row must be assigned values from 1 to n_treatments. imsls_f_latin_square verifies that the number of unique factor A identifiers is equal to n_treatments.

int col[] (Input)
An array of length n containing the column identifiers for each observation in y. Each column must be assigned values from 1 to n_treatments. imsls_f_latin_square verifies that the number of unique column identifiers is equal to n_treatments.

int treatment[] (Input)
An array of length n containing the treatment identifiers for each observation in y. Each treatment must be assigned values from 1 to n_treatments. imsls_f_latin_square verifies that the number of unique treatment identifiers is equal to n_treatments.

float y[] (Input)
An array of length n containing the experimental observations and any missing values. Missing values cannot be omitted. They 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. The location, row, column, and treatment number for each observation in y are identified by the corresponding values in the arguments locations, row, col, and treatment.

Return Value

Address of a pointer to the memory location of a two dimensional, 7 by 6 array containing the ANOVA table. Each row in this array contains values for one of the effects in the ANOVA table. The first value in each row,
anova_table_i,₀= anova_table[i*6], identifies the source for the effect associated with values in that row. The remaining values in a row contain the ANOVA table values using the following convention:

J	*anova_table_i,j= anova_table[i6+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

The Source Identifiers in the first column of anova_table_i,jare the only negative values in anova_table[]. Assignments of identifiers to ANOVA sources use the following coding:

Source Identifier	ANOVA Source
-1	LOCATIONS †
-2	ROWS
-3	COLUMNS
-4	TREATMENTS
-5	LOCATIONS × TREATMENTS †
-6	ERROR WITHIN LOCATIONS
-7	CORRECTED TOTAL

Notes: † If n_locations=1 rows involving location are set to missing (NaN).

Synopsis with Optional Arguments

#include <imsl.h>

float * imsls_f_latin_square (int n, int n_locations, int n_treatments, int row[], int col[], int treatment[], float y[],
IMSLS_RETURN_USER, float anova_table[],
IMSLS_LOCATIONS, int locations[],
IMSLS_N_MISSING, int *n_missing,
IMSLS_CV, float *cv,
IMSLS_GRAND_MEAN, float *grand_mean,
IMSLS_TREATMENT_MEANS, float **treatment_means,
IMSLS_TREATMENT_MEANS_USER, float treatment_means[],
IMSLS_STD_ERRORS, float **std_err,
IMSLS_STD_ERRORS_USER, float std_err[],
IMSLS_LOCATION_ANOVA_TABLE float **location_anova_table,
IMSLS_LOCATION_ANOVA_TABLE_USER,
float location_anova_table[],
IMSLS_ANOVA_ROW_LABELS, char ***anova_row_labels,
IMSLS_ANOVA_ROW_LABELS_USER, char *anova_row_labels[],

Optional Arguments

IMSLS_RETURN_USER, float anova_table[] (Output)
User defined array of length 42 for storage of the 7 by 6 anova table described as the return argument for this routine. For a detailed description of the format for this table, see the previous description of the return arguments for imsls_f_latin_square.

IMSLS_LOCATIONS, int locations[] (Input)
An array of length n containing the location identifiers for each observation in y. Unique integers must be assigned to each location in the study. This argument is required when n_locations>1.

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)
The coefficient of variation computed by using the within location standard deviation.

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

IMSLS_TREATMENT_MEANS, float **treatment_means (Output)
Address of a pointer to an internally allocated array of size n_treatments containing the treatment means.

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

IMSLS_STD_ERRORS, float **std_err (Output)
Address of a pointer to an internally allocated array of length 2 containing the standard error and associated degrees of freedom for comparing two treatment means. std_err[0] contains the standard error and its degrees of freedom are returned in std_err[1].

IMSLS_STD_ERRORS_USER, float std_err[] (Output)
Storage for the array std_err, provided by the user.

IMSLS_LOCATION_ANOVA_TABLE, float **location_anova_table (Output)
Address of a pointer to an internally allocated 3-dimensional array of size n_locations by 7 by 6 containing the anova tables associated with each location. For each location, the 7 by 6 dimensional array corresponds to the anova table for that location. For example, location_anova_table[(i-1)×42+(j-1)×6 + (k-1)] contains the value in the kth column and jth row of the anova-table for the ith location.

IMSLS_LOCATION_ANOVA_TABLE_USER, float anova_table[] (Output)

Storage for the array location_anova_table, 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 ith 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 free(anova_row_labels).

IMSLS_ANOVA_ROW_LABELS_USER, char *anova_row_labels[] (Output)
Storage for the array 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[600].

Description

The function imsls_f_latin_square analyzes latin-square experiments, possibly replicated at multiple locations. Latin-square experiments block treatments using two factors: rows and columns. The number of levels associated with rows and columns must equal the number of treatments. Treatments are blocked by rows and columns in a balanced arrangement to ensure that every row contain one replicate of every treatment. The same balance is required for every column, see Table 1. Notice that the four treatments, T1, T2, T3, and T4, appear exactly once in every column and every row.

		Columns
		C1	C2	C3	C4
Rows	R1	T1	T2	T3	T4
	R2	T2	T3	T4	T1
	R3	T3	T4	T1	T2
	R4	T4	T1	T2	T3

Table 1 Latin-Square Experiment with Four Treatments

A necessary assumption in Latin-Square experiments is that there are no interactions between treatments and the row and column blocking factors. For data collected at a single location, the Anova table for a Latin-Square experiment is usually organized into five rows, see Table 2.

SOURCE	DF	Sum of Squares	Mean Squares
ROWS		SSR=	MSR
COLUMNS		SSC=	MSC
TREATMENTS		SST=	MST
ERROR		SSE=SSTot-SSR-SSC-SST	MSE
TOTAL		SSTot=

Table 2 – The ANOVA Table for a Latin-Square Experiment at one Location

The statistical model used to represent data is from a single location:

where

is the observation for the kth treatment in the ith row and jth column of the Latin Square, and,is the effect associated with the kth treatment.and are the ith row and jth column effects, respectively, and is the noise associated with this observation.

If multiple locations are involved, imsls_f_latin_square assumes that treatments are crossed with locations, but that row and column effects are nested within locations, see Table 3. The statistical model used to represent these data is:

where

is the effect associated with the kth treatment, and

is the interaction effect between location l and treatment k.

SOURCE	DF	Sum of Squares	Mean Squares
LOCATIONS		SSL=	MSL
ROWS		SSR=	MSR
COLUMNS		SSC=	MSC
TREATMENTS		SST=	MST
LOCATIONS X TREATMENTS		SSLT by difference	MSLT
ERROR		SSE=	MSE
TOTAL		SSTot=

Table 3 – The ANOVA Table for a Latin-Square Experiment at Multiple Locations

Example

This example uses four treatments organized into a latin square. This example also uses the function l_print_LSD(), which is defined in the first example for imsls_f_lattice().

#include <stdio.h>

#include <math.h>

#include "imsls.h"

void l_print_LSD(int n1, int* equalMeans, float *means);

void main()

{

char **anova_row_labels;

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

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

float alpha = 0.05;

int i, l, page_width = 132;

int n = 16; /* Total number of observations */

int n_locations = 1; /* Number of locations */

int n_treatments = 4; /* Number of rows, columns and treatments */

int n_aov_rows = 7; /* Number of rows in the latin-square anova table */

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

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

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

float y[]={

1.167, 1.185, 1.655, 1.345, 1.64, 1.29, 1.665, 1.29,

1.475, 0.71, 1.425, 0.66, 1.565, 1.29, 1.4, 1.18};

float grand_mean;

float cv;

float *aov;

float *treatment_means;

float *std_err;

int df;

int *equal_means;

printf("\n\n*** Experimental Design ***");