This is tamu_anova.info, produced by makeinfo version 4.8 from
tamu_anova.texi.
INFO-DIR-SECTION Statistics software
START-INFO-DIR-ENTRY
* tamu_anova: (tamu_anova). TAMU ANOVA extensions to GSL
END-INFO-DIR-ENTRY
File: tamu_anova.info, Node: Top, Next: Introduction, Prev: (dir), Up: (dir)
TAMU_ANOVA
**********
This file documents TAMU_ANOVA, a package for the GNU Scientific
Library (GSL) containing functions for performing one and two way NOVA
tests.
Information about GSL can be found at the project homepage,
`http://www.gnu.org/software/gsl/'.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License.
* Menu:
* Introduction::
* Motivation::
* Computational Method::
* Functions::
* Examples::
* References and Further Reading::
* GNU Free Documentation License::
* Function Index::
* Concept Index::
File: tamu_anova.info, Node: Introduction, Next: Motivation, Prev: Top, Up: Top
1 Introduction
**************
ANOVA, or Analysis of Variance, is a method for comparing levels of
some continuous response variable between different groups. The main
idea is to compare variation within each group to variation between the
groups; if the groups vary considerably from one group to another in
comparison to the within group variation, we can reject the null
hypothesis that all the groups have similar levels of the response
variable.
TAMU ANOVA contains both single and two factor ANOVA. Use of the
package can be facilitated through linking to the compiled library
tamuanova. The package function definitions are accessible through
`tamu_anova.h'. Another option for use of the package is to include
the original function definitions in the files `anova_1.h' and
`anova_2.h'. With the use of these files the program must still be
linked to the GSL (see GSL documentation on linking and compiling.)
File: tamu_anova.info, Node: Motivation, Next: Computational Method, Prev: Introduction, Up: Top
2 Motivation
************
The first reason for creating this package was that, to the best of the
programmers' knowledge, there was not an open source package in
existence that computed sums of squares correctly in the case of
unequal cell sizes. The model for the two-way ANOVA is commonly y_ijk
= \alpha_i + \beta_j + \gammaij + \epsilon_ijk, but the sums of squares
are computed in different ways according to the hypotheses desired to
test. This package computes Type III sums of squares, as described in
Searle (1987). The Type III Sums of Squares are the only correct sums
of squares in the case of unequal cell sizes (but no cells are empty)
when the desired hypotheses are as follows: H_o: \alpha_1 = \alpha_2 =
... = \alpha_k = 0 vs. H_a: at least one \alpha_i \neq 0, H_o: \beta_1
= \beta_2 = ... = \beta_m = 0 vs. H_a: at least one \beta_i \neq 0,
and H_o: \gamma_1,1 = \gamma_1,2 = ... = \gamma_k,m vs. H_a: at least
one \gamma_i,j = 0.
The second reason for creating this package was to calculate
F-statistics for models with random and mixed effects as described in
Section 4.10 of Sahai and Ageel's book, The Analysis of Variance
(2000). Some packages seem superficially to calculate F-statistics
correctly, but upon further investigation, do not actually find the
F-statistics recommended by Sahai and Ageel(2000).
File: tamu_anova.info, Node: Computational Method, Next: Functions, Prev: Motivation, Up: Top
3 Computational Method
**********************
The one-way ANOVA tables were computed by the methods described in
Devore (2004). First, sums of squares are computed in the usual way (n
is the total number of observations, and k is the number of populations
in the study): SST = \sum_i=1^k \sum_j=1^n_i (x_ij -
\barx_..)^2, SSM = \sum_i=1^k \sum_j=1^n_i (\barx_i. -
\barx_..)^2, and SSE = \sum_i=1^k \sum_j=1^n_i (x_ij -
\barx_i.)^2. Then the package constructs the ANOVA table as follows:
Source: SS: df: MS: F-stat: p-value:
Model SSM k-1 SSM/k-1 SSM/SSE P(F > F-stat)
Error SSE n-k SSE/n-k
Total SST n-1 SST/n-1
The two-way models require further computation when the desired
hypotheses to be tested are not conditional on the previous factors
entered into the model but are the most common (where the model is:
y_ijk = \mu + \alpha_i + \beta_j + \gamma_ij + \epsilon_ijk): H_o:
\alpha_1 = \alpha_2 = ... = \alpha_k = 0, H_o: \beta_1 = \beta_2 =
... = \beta_m = 0, and H_o: \gamma_11 = \gamma_12 = ... =
\gamma_km = 0. Note that the hypotheses for the individual factors
only hold in the absence of significant interaction.
Following Searle (1987), then, the package computes Type III Sums of
Squares: SST = \sum_i n_i. \bary_i..^2 + \sum_j=1^b-1 \mathaccent
94 \tau_j r_j, where for c_jj = n_.j - \sum_i=1^a n_ij^2/n_i.,
c_jj' = -\sum_i=1^a n_ijn_ij'/n_i., and r_j = y_.j. - \sum_i=1^a
n_ij \bary_i.., we solve the b-1 linear equations c_jj \mathaccent
94 \tau_j + \sum_j'=1, j' \neq j^b-1 c_jj' \mathaccent 94 \tau_j' = r_j
for \mathaccent 94 \tau_1, \mathaccent 94 \tau_2,..., \mathaccent 94
\tau_b-1. These computations are only correct in the case that no
cells are empty. In the case that some cells are completely empty,
the experimenter must use other methods to compute sums of squares,
depending on the desired hypotheses.
Those methods only are appropriate in the case that the effects are
fixed. In the case that the effects are random, the expected values of
the mean squares are not the same as for the fixed case, so one uses a
different denominator for finding the F-statistics in the ANOVA table.
The computation follows the recommendation of Sahai and Ageel (2000):
For the random effects model, F_\alpha = MS_\alpha / MS_\gamma
and F_\beta = MS_\beta / MS_\gamma.
For the mixed effects model, F_\alpha = MS_\alpha / MS_\gamma and
F_\beta = MS_\beta / MSE.
Thus we see that for the mixed effects model, the data must be
entered in such a way that the first effect is the fixed effect, while
the second effect is the random effect.
File: tamu_anova.info, Node: Functions, Next: Structures, Prev: Computational Method, Up: Top
4 Functions
***********
* Menu:
* Structures::
* Function Definitions::
File: tamu_anova.info, Node: Structures, Next: Function Definitions, Prev: Functions, Up: Functions
4.1 Structures
==============
One way table
-- struct tamu_anova_table:
`struct tamu_anova_table{ long df_tr, df_err, df_tot; double
SSTr, SSE, SST, MSTr, MSE, F, p;}; '
Two way table
-- struct tamu_anova_table_twoway:
`struct tamu_anova_table_twoway { long dfA, dfB, dfAB, dfT,
dfE; double SSA, MSA, FA, pA, SSB, MSB, FA, pB,
SSAB, MSAB, FAB, pAB, SSE, MSE, SST; }; '
File: tamu_anova.info, Node: Function Definitions, Next: Examples, Prev: Structures, Up: Functions
4.2 Function Definitions
========================
-- Function: tamu_anova_table tamu_anova (double DATA[],long FACTOR[],
long I, long J);
This performs the one way ANOVA and returns a oneway table struct. The
arguments are as follows:
1. data - an array of all data values of type double.
2. factor - an array of factor codings for the data. Acceptable
values are 1..J with no breaks (I.E. the function does not allow
for empty cells).
3. I - the length of the data and factor arrays (yes they must be the
same length).
4. J - The number or factors in the experiment.
-- Function: tamu_anova_table_twoway tamu_anova_twoway (double
DATA[],long FACTOR[][2], long I, long J[2], enum
gsl_anova_twoway_types TYPE )
This performs the two way ANOVA and returns a twoway table struct.
The arguments are as follows:
1. data - an array of all data values of type double.
2. factor - a I x 2 matrix of type long for factor codings.
Essentially an array of ordered pair of the type {factor A, Factor
B}. Acceptable values for are 1..J_A for factor A and 1..J_B for
factor B. With no skipped numbers (I.E. the function does not
allow for empty cells)
3. I - the length of the data and factor arrays (yes they must be the
same length).
4. J - an array of {J_A,J_B} which tells the function the number of
groups for factor A and factor B.
5. type - an enumerated variable to tell the function what kind of
model to use. This only affects the F and P values.
* anova_fixed = 0
* anova_random = 1
* anova_mixed = 2
File: tamu_anova.info, Node: Examples, Next: References and Further Reading, Prev: Function Definitions, Up: Top
5 Examples
**********
Here are three examples of how to use the program. They all assume
that you have the library installed and are able to link to it without
any extra declarations. If this is not the case extra compiler
directive may be needed to compile and link properly.
WARNING: The TAMU ANOVA library must be linked with the GSl libraries,
as shown in the examples. The functions in TAMU ANOVA use the gsl and
cannot be ran without it.
Example 1: One-way balanced ANOVA
#include
//Data set from Devore(2004).
double data[20] = {
88.60,73.20,91.40,68.00,75.20,63.00,53.90,69.20,
50.10,71.50,44.90,59.50,40.20,56.30,38.70,31.00,
39.60,45.30,25.20,22.70 };
long factor[20]={
1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4 };
int main()
{
struct tamu_anova_table tbl1 = tamu_anova(data,factor,20,4);
tamu_anova_printtable(tbl1);
return 0;
};
This file would be compiled by the following:
% gcc -c testfile.c
% gcc testfile.o -ltamuanova -lgsl -lgslcbls -lm
Example(2): One-way unbalanced ANOVA
#include
//Data from Devore(2004)
double data[22] = {
45.50,45.30,45.40,44.40,44.60,43.90,44.60,44.00,44.20,
43.90,44.70,44.20,44.00,43.80,44.60,43.10,46.00,45.90,
44.80,46.20,45.10,45.50
};
long factor[22]={
1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3
};
int main()
{
struct tamu_anova_table tbl1 = tamu_anova(data,factor,22,3);
tamu_anova_printtable(tbl1);
return 0;};
this example is compiled the same as above
An example of two way unbalanced ANOVA with data from Searle(1987).
#include
double d[15]={
6,10,11,13,15,14,22,12,15,19,18,31,18,9,12};
long f[15][2]={
{1,1},
{1,1},
{1,1},
{1,2},
{1,2},
{1,3},{1,3},
{2,1},{2,1},{2,1},{2,1}
,{2,2},
{2,3},{2,3},{2,3}};
int main()
{
struct tamu_anova_table_twoway r;
long J[2]={2,3};
r=tamu_anova_twoway(d,f,15,J,0);
tamu_anova_printtable_twoway(r);
};
This is compiled the same as above for the one way library linked
example. The output is (slightly reformatted from what is printed on
the screen):
Source: SS: df: MS: F-stat: p-value:
Factor-A123.771429 1 123.771429 9.282857 0.013865
Factor-B192.127660 2 96.063830 7.204787 0.013546
Interact222.765957 2 111.382979 8.353723 0.008888
Error 120.000000 9 13.333333
Total 520.000000 14
File: tamu_anova.info, Node: References and Further Reading, Next: GNU Free Documentation License, Prev: Examples, Up: Top
6 References and Further Reading
********************************
Devore, Jay L. (2004), Probability and Statistics for Engineering and
the Sciences (6th ed.), Canada, Brooks/Cole.
Milliken and Johnson (1992), Analysis of Messy Data: Volume I:
Designed Experiments, New York, NY, Van Nostrand Reinhold.
Sahai, Hardeo and Ageel, Mohammed I. (2000), The Analysis of Variance:
Fixed, Random, and Mixed Models, Ann Arbor, MI, Sheridan Books, Inc.
Searle, Shayle R. (1987), Linear Models for Unbalanced Data, New York,
NY, John Wiley and Sons, Inc.
Yandell, Brian S. (1997), Practical Data Analysis for Designed
Experiments, Cornwall, Great Britian, Chapman & Hall.
File: tamu_anova.info, Node: GNU Free Documentation License, Next: Function Index, Prev: References and Further Reading, Up: Top
GNU Free Documentation License
******************************
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File: tamu_anova.info, Node: Function Index, Next: Concept Index, Prev: GNU Free Documentation License, Up: Top
Function Index
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[index]
* Menu:
* : Structures. (line 9)
* tamu_anova: Function Definitions. (line 8)
* tamu_anova_twoway: Function Definitions. (line 24)
File: tamu_anova.info, Node: Concept Index, Prev: Function Index, Up: Top
Concept Index
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[index]
* Menu:
* FDL, GNU Free Documentation License: GNU Free Documentation License.
(line 6)
Tag Table:
Node: Top234
Node: Introduction915
Node: Motivation1928
Node: Computational Method3374
Node: Functions6250
Node: Structures6426
Node: Function Definitions6949
Node: Examples8696
Node: References and Further Reading11504
Node: GNU Free Documentation License12298
Node: Function Index34736
Node: Concept Index35127
End Tag Table