National Bureau of Standards

I'i^iiry, N.J^_B1

.anaaras

8

'1965"^'^ Reference book not to be

' - »,^'. from the library.

^ecknlcai ^yiote 259

A FORTRAN PROGRAM FOR DETERMINING AN EMPIRICAL

EXPRESSION FOR A QUANTITY MEASURED AT COMBINATIONS

OF SEVERAL LEVELS OF EACH OF TWO VARIABLES

MARY NAN STEEL, FRANK L. McCRACKIN, AND JOHN MANDEL

^i^^t i^f^/'f'^ ' ,>-

i

v^ '.i/i^ '&'4A' 'i^.Md&H- i-' i^--'-'

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

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listing of its four Institutes and their organizational units. Institute for Basic Standards. Electricity. Metrology. Heat. Radiation Physics. Mechanics. Applied Mathematics. Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radio Standards Laboratory: Radio Standards Physics; Radio Standards Engineering.** OfiBce of Standard Ref- erence Data. Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Materials. Reactor Radiations. Cryogenics.** Office of Standard Reference Materials.

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* NBS Group, Joint Institute for Laboratory Astrophysics at the University of Colorado. ** Located at Boulder, Colorado.
I

NATIONAL BUREAU OF STANDARDS

Technical Note 259

ISSUED APRIL 12, 1965

A FORTRAN PROGRAM FOR DETERMINING AN EMPIRICAL

EXPRESSION FOR A QUANTITY MEASURED AT COMBINATIONS

OF SEVERAL LEVELS OF EACH OF TWO VARIABLES

Mary Nan Steel, Frank L. McCrackin, and John Mandel

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 40 cents
Contents

Page

1. Introduction 1

2. Computer Input 2

2.1 An Example With A single Value In Each Cell 3

2.2 An Example With Replicate Values In Each Cell 5

2.3 Analysis of the Data 6

2.31 The Linear Model- LMODEL 6

2.32 The Weighted Linear Model- LMl*JCOL 6

2.33 The Quadratic Model- QMODEL 7

2.4 Modification of the Data ~ 8

2.41 Omit Data- OMIT 8

2.42 Transform Data- TRAFOR ~ 8

2.43 Correct Data- CORECT 9

2.44 Reverse Data- REVERS 10

2.45 Combine Data- COMBIN 10

2.46 New Data- NEWDAT 11

2.47 Stop- STOP 11

3. Interpretation of Computer Output 11

3.1 The Table With A single Value In Each Cell 11

3.2 The Table With Replicate Values In Each Cell l4

4. References 18

Appendix A. Tables of Computer Input and Output 19

Appendix B. Listing of Fortran Program 32

Appendix C. Functions and Subroutines

Used by the Program ' 50

Appendix D. Variables Used in Program 51

Appendix E. Flow Diagrams of Program and Subprograms 53

31
A FORTRAN Program for Determining an Empirical Expression for

a Quantity Measured at Combinations of Several Levels of Each

of Two Variables

Mary Nan Steel, Frank L. McCrackln, and John Mandel

This note relates to a general procedure, reported

elsewhere, for finding an empirical expression for

the functional dependence of a measured quantity on

two variables. The procedure is applicable when

measurements of the dependent variable are available at m.n combinations of m levels of one variable and n levels of the othe"rj it leads to an

algebraic expression relating the three variables.

A Fortran program to perform the various steps of

the general procedure is presented along with

detailed instructions for its use. The program

also Includes the calculation of an analysis of

variance, of components of variance, of significance tests, tables of residuals, and a number

of other quantities.

1. Introduction

In many scientific studies, a quantity is measured at

several levels for each of two variables. The data are often

presented in a table with the rows of the table representing

one variable and the columns representing the other variable.

When both variables are continuous, the data can be considered as representing either a surface in three-dimensional

space or a family of plane curves [1]^. In the latter

representation each curve represents the relation between the

measured quantity and one of the variables, for a selected

fixed value of the second variable. For different fixed values

of this second variable, different curves are obtained. In

this fashion a family of curves is generated. An example of

this type is the relationship between the volume, the pressure

and the temperature of a gas. The objective is to find a mathematical representation of such a relationship.

One or both of the variables may represent categories

rather than continuous variables. For example, in interlaboratory studies of test methods [2], the data are often

classified in accordance with laboratories (rows) and

materials (columns). In such cases, the main objective is to

Figures in brackets Indicate the literature references at

the end of this paper.
determine the effect of the variables (laboratories and

materials) on the variability of the data. Methods to determine the relationsTTIp between the measured quantity and the

variables have been developed by Mandel, et al [1,2,3,4,5].

These methods, which have been referred to as the linear

model and the quadratic model , constitute essentially a

general procedure for the empirical fitting of the data by

means of algebraic expressions. The fitting procedure

comprises several steps

1) the selection of a particular algebraic expression from

among a class of such expressions for the most appropriate

representation of the data;

2) the calculations of numerical values for the parameters

and coefficients occurring in the chosen expression;

5) the estimation of the precision of the calculated

parameters and coefficients, tests of significance related to

them, and the calculation of confidence intervals for their

true values;

4) the derivation of components of variance related to the

variability of some of the parameters and coefficients;

5) further statistical interpretations of the parameters,

coefficients, and components of variance in terms of the

general objectives of the study.

A computer program was developed to perform these analyses

and this note presents the program. Section 2 describes how

the data are organized to be read into the computer and the

instructions required to perform the calculations. Possible

modifications to be made to the data by the computer are also

described. Section 3 presents two examples with interpretations of the computer output. The tables of computer output

examined in Section 3 are given in Appendix A, Appendixes B

through D give the FORTRAN program and information required

for understanding the program. Appendix E consists of Figures

1 through 9 J the flow diagrams for the computer program and

the principal subprograms

2. Computer Input

Two examples are presented as a guide for the practical

use of the program. Section 2.1 describes a setup in which

each cell contains a single measurement. Section 2.2 presents

the case in which replicate measurements in each cell are to

be analyzed. It will be noted that the data are set up in a

different way in each case (Examples 1 and 2). Each line of

data in the lower portion of both examples represents a single

card of computer input and the first card is the title card

in both cases. The nominal number of replicates of a particular test is put in columjis 1 and 2 of this card, followed by

any descriptive title.

2
Example 1. A Single Value In Each Cell

The Data

Round Robin on Dielectric Strength

Material

Lab 1 2345 6 7 8

LOH .80 1.15 .5^ .47 .22 .34 .17 .20

LMG .96 1.26 .32 .61 .23 .32 .17 .23

LST .98 1.24 .34 .52 .31 .45 .20 .24

LUM .88 1.33 .32 .51 .22 .31 .15 .18

Setup for Computer Input

Coliimn NiAinber

1 7 15 25 35 45 55 65

01 ROUND ROBIN ON DIELECTRIC STRENGTH

MATl MAT2 MAT3 MAT4 MAT5 MAT6 MAT7

LOH .80 1.13 .3^ Aj .22 .34 .17

LMG .96 1.26 .32 .61 .23 .32 .17

LST .98 1.24 .34 .52 .31 .45 .20

LUM .88 1.33 .32 .51 .22 .31 .15

MORKDA

MAT8

LOH .20

LMG .23

LST .24

LUM .18

ENDATA

LMWCOLCAL

STOP

2.1 An Example With A Single Value In Each Cell

Example 1 gives experimental data with a single value in

each cell. The top table gives the original data. The lower

table gives the setup for computer input. In the latter^ the

first card has 01 in columns 1 and 2 and the title that

identifies the data follows. The second card contains the

column headings from the table and may have as many as seven

per card, beginning in card columns 7, 15, 25, 35, 45, 55, 65

as' shown in the example. Next follow the data cards, each of^

which has a row heading in columns 1-6 followed b^ data

corresponding in number and order to the column headings,i .e. the data are placed in card columns 7 through l4, 15 through

24, 25 through 3^, 35 through 44, 45 through 5^, 55 through 64

3
Example 2. Replicate Values In Each Cell

The Data

Adhesion of Plastic Patches, Method A

Condition

Material 1385 l4l5 2385

AA 18.2

18.6

17.9

20.0

19.7

20.0

22.1

22.0

20.8

AB

AC

AD

16.8

15.2

15.6

14.6

13.1

14.2

19.5

18.7

16.5

24.5

25.1

23.4

l4.7

14.7

l4.7

15.9

16.5

17.7

16.7

17.3

17.7

17.7

14.6

14.2

14.9

18.0

20.1

20.3

Setup for Computer Input

Column Number

1 7 15 25 35

T5j> ADHESION UJf FLA^TIU PATCHES, METHOD

AA 1385 18.2 18.6 17.9

AB 1385 16.8 15.2 15.6

AC 1385 14.6 13.1 14.2

AD 1385 19.5 18.7 16.5

AA l4l5 20.0 19.7 20.0

AB l4l5 24.5 25.1 23.4

AC l4l5 14.7 14.7 14.7

AD l4l5 0415.9 16.5 17.7

AA 2385 22.1 22.0 20.8

AB 2385 17.3 17.7 17.7

AC 2385 14.6 14.2 14.9

AD 2385 18.0 20.1 20.3

ENDATA

QMODEL

REVERS

LMODEL

STOP

45

16.7
and 65 through 72. The column and row headings must all be

different and each heading may consist of from 1 to 6

characters. All data must contain decimal points. If more

than 7 columns of data are to be read, a card with MOREDA In

columns 1-6 Is placed after the last row of data of the first

group of columjis. That Is, all rows for each set of columns

are read before additional coliomn headings with their respective rows of data are Introduced. After the MOREDA card, the

next set of coliunn headings, again beginning In column 7 are

read, followed by rows of data corresponding to the new column

headings . At this point It Is Important that the rows are In

the exact order and the same In ni;imber as before, for the row

headings are Ignored after the first reading, the Identity of

the order Is assumed, and no provision Is made for a difference

In the number of rows per column. The MOREDA card occurs only

once, regardless of the number of columns over 7 that are to

be read.

The reading of the data Is concluded by placing a card

with ENDATA In columns 1-6 after the last row of data.

2.2 An Example With Replicate Values In Each Cell

In Example 2, the case In which replicate values occur

In each cell, we note that 05 Is placed In columns 1 and. 2,

followed by the title on the first card. On the second card,

a row and column heading appear, for In the case of replicate

measurements row and column headings and replicates for the

cell thus designated, all appear on the same data card. The

row heading Is put In columns 1-6 and the coliomn heading In

card columns 7-12. Columns 13-1^ are blank If the number of

replicates In the cell Is the same as the number appearing on

the title card - In our example 03. If the number of

replicates for any one cell differs from that on the title

card, that number Is placed In columns 1^-1^ (note 04 In line

AD l4l5 In Example 2). The data then follow on the same line

beginning In coluraiis 15, 25, 35, ^5, 55, 65. If more than

six replicates occur per cell, the additional data are placed

In groups of six on successive cards, making certain that the

number of replicates Is either the same number Indicated on

the title card or that placed In columns 13-1^ of the Initial

card of the cell listing. There may be as many as 99

replicate values per cell.

It Is Important that there be the same number of rows

with the same row headings per column and the same number of

columns with the same column headings per row, otherwise, the

program will not run. The data cards may be placed In any

order. The program will handle a table of data containing as

many as 120 rows and 50 columns. A card with ENDATA In

columns 1-6 Is always placed after the final data card.
2.3 Analysis of the Data

After the data are read into the computer some basic

calculations are made. In the case of replicate measurements

per cell J the cell averages ^ standard deviations , and the

coefficients of variation are computed.

Row and column averages are calculated for either the

cell averages (in the case of replicate measurements per cell)

or for the cell entries (in the case of single measurements

in the cells). Then the data of the table are rearranged in

such a way that the column-averages are in ascending order

from left to right and the row-averages are in ascending order

from top to bottom. A table of averages and, in the case of

replicates, a table of standard deviations with the coefficients of variation are printed. Analysis of the data is

accomplished by placing instruction cards for the desired

analysis at the end of the data deck after the card ENDATA.

The instruction cards are the same for data consisting of

either single or replicate measurements per cell and the

choice of instructions determines the analysis that will be

performed on the data.

Examples 1 and 2 show the way in which the instruction

cards appear in an input setup. Details for making these

cards and notes on what the instruction accomplishes are

given below.

2.31 The Linear Model - LMODEL

By placing LMODEL in columns 1-6 of a card following the

data, the equation y. .=A +B.C., the linear model, is fitted

to the cell averages y. .. The residuals and their sums of

squares are calculated and printed and the analysis of variance

is computed. The analysis is then printed along with the

values calculated for A, B and C. Interpretations of the

computer output for this and other calculations are given in

Section 3.

2.32 The Weighted Linear Model - LMWCOL

If one wishes to weight the columns before analyzing the

data, LMWCOL is placed in columns 1-6 on a card. A linear

model analysis is then performed, in which the columns are

weighted. Weights may be given or calculated. When the

weights are given, they are placed on cards immediately

following the LMWCOL card. One weight is given for each

column and six weights may be read from a single card beginning in card columns 15, 25, 35, 45, 55, and 65, as in

Example 3. If a column is not to be weighted, 1. is given as

its weight and of course the weights are given in the same

6
order as the columns of data being weighted, i.e., in the

ascending order of the column averages, from left to right,

as ordered by the computer.

Example 3

Column Number

1 7 15 23 35 ^5 53 63

KNDATA

' LMWCOL

1.35 1. .5 1.5 .75 1.^5

STOP

If the user wishes the weights to be calculated he places

CAL in columns 7-9 following the L]vn//COL, thus LMl^COLCAL

appears on the instruction card. The calcul-'tion of weights

is an iterative process discussed in reference [5]. When the

weights are calculated, the analysis is not alid unless the

data consist of a minimum of eight columns. The weights are

derived from the original data themselves. It is well to note

that if the user has sufficient information to provide his

own weights, it is at this point in the program that they

should be introduced.

After the weights have been read or computed, the square

root of the weights, referred to as the weighting factor, is

determined and both weights and weighting factors are printed.

In addition to these, the weighted values for A, B, and C,

the point of concurrence-^ and the analysis of variance are

calculated and the results printed. The residuals with the

sums of squares of both rows and columns are computed and the

Table of Deviations, preceded by the number of rows and

columns, is printed.

2.33 The Quadratic Model - QMODEL

The quadratic model fits the data to the equation

where Ej Is a quadratic function of Cj. The instruction

QMODEL is placed on a card in columns 1-6 when this model is

desired. All the variables for the equation are calculated

and printed as in the analysis of variance. In addition to

the Table of Deviations with the row and column sums of

squares for the quadratic model, this instruction also

calculates and prints the linear model Table of Deviations

with sums of squares.

* See Section 3.1

7
2.4 Modification of the Data

It is frequently desirable to analyze a set of data,

then modify it in some fashion and reanalyze the new set.

Instructions are included to accomplish a number of these

modifications

2.4l Omit Data - OMIT

The OMIT instruction placed in columns 1-6 of a card

causes the designated rows and/or columns to be omitted in

the analyses following it, or until another OMIT card is

placed in the sequence of instructions.

On the same card with OMIT, in columns 13-1^^ appears

the number of rows and columns to be omitted followed by their

labels beginning in columns 15, 25, 35, 45, 55, 65. These

may be given in any order. If there are more than 6, the

additional labels in groups of six appear on cards immediately

following the first OMIT instruction.

An OMIT card cancels all previous omit cards . In other

words a single set of data may be analyzed using different

rows and columns within the same run by first placing the

OMIT card followed by the card designating the analysis

desired, followed by another OMIT card and another analysis

card, etc.

Example 4

Column Number

1 7 13 25

HiNDATA

QMODEL

OMIT 02A

B2

QMODEL

OMIT 02DUP0NT Al

LMODEL

STOP

In Example 4, the first QMODEL analysis will include all

laboratories and materials, then materials Al and B2 will be

omitted in the second QMODEL analysis. LMODEL will be

computed omitting laboratory DUPONT and material Al but will

include material B2 previously omitted.

2.42 Transform Data - TRAFOR

This instruction may be on a card in columns 1-6 placed

before or after the data. Placed before the data, af'ter the

title card, the function TRAFOR causes each measurement to

be transformed before any other calculations are made. Any

8
transformation may be employed. This program does not include

a transformation, in order to use the instruction a special

subprogram called TRAFOR must be written and added by the

user.

If one wishes to transform the averages after all of the

cell averages have been computed, TRAFOR is placed on a card

in columns 1-6 after the ENDATA card. The subroutine TRAFOR

will then transform the averages. As in the case of the

function TRAFOR, any transformation may be used, and again

no subroutine TRAFOR is included and it is left to the user

to write his own.

It is important to note that the user must choose either

one or the other of these two transformations and include

only the one desired in using the program. Of course neither

is necessary to the running of the program; the decision

depends entirely on the nature of the data and the wishes of

the user.

2.43 Correct Data - CORECT

The averages and standard deviations may be replaced by

values corrected in the form AX + B and ko respectively, where

A is a multiplicative correction, B is an additive correction,

Y is the original average and _a is the original standard

"3'eviation.

The data may be corrected before the initial printing

of the Table of Averages and Standard Deviations, in which

case CORECT should appear in columns 7-12 on the same card

with ENDATA. (Example 5a) If one wishes the original data

to be printed before correction is made, the CORECT in

columns 1-6 should appear on a separate card after ENDATA.

(Example 5b)

In either case the row and column designations of those

values to be corrected are placed on a card immediately

following the CORECT card in columns 1-6 and 7-12, respectively, along with the multiplicative term (columns 15-24)

and additive term (columns 25-^4). These designations may be

for single values or, by placing ALL in place of a single row

or column heading, may include all row values for a single

column or all column values for a single row, as shown in

Example 5. After these instructions an END card (columns 1-

3) is placed.

If a purely multiplicative correction is desired, it is

not necessary to put 0. in column 25 but if a purely additive

correction is desired, a 1. must be placed in column 15.
Example 5

(a)

Column Number

1 7 15 25

ENDATACORECT

ALL MAT02 1. 2.

LAB03 MATH 2.

LAB07 ALL 2. 5.

END

(b)

Column Number

1 7 15 25

ENDATA

CORECT

R0W3 C0L7 1. 5.

ALL C0L2 2. 1.5

R0W6 ALL 1. 2.

END

2.44 Reverse Data - REVERS

This instruction card with REVERS in columns 1-6 causes

the rows and columns to be interchanged in the analyses that

follow.

2.45 Combine Data - COMBIN

Sometimes data previously treated as two or more sets

are found not to be significantly different and the experimenter wishes to treat them as one set. The instruction

COMBIN serves the purpose of combining such sets into new

averages. After the COMBIN in columns 1-6, the designation

"ROWS"^ or "cols" is placed on the card in columjis 7-10 and

the labels of the sets to be combined are placed in columns

beginning at 15, 25, 35, 45, 55. This causes the cell

averages in the rows and/or columns thus designated to be

averaged together into a single row or column. The label for

the newly formed row or column is placed on the card in the

position immediately following the listing of the rows and

columns to be combined. Thus it is possible to combine as

many as 5, their new label being placed on the card beginning

in position 65.

If more than one combination is to be computed,

additional lines are added as shown in Example 6. COMBIN is

placed on the first card only, its place being left blank on

the following lines. An instruction card with END in columns

7-9 completes the setup for this subroutine,

10
iiNDATA

COMBINROWS Al A2 A3

COLS CI C3 C6

ROWS L7 l8 L9

END

It should also be noted that the new table of averages

is treated as though it were a table of single values and no

standard deviations are calculated.

Example 6

Column Number

1 7 15 25 35 45 55 65

AlO

LIO Ll4 LI5

These instructions combine rows Al, A2 and A3 in a single

row labeled AlO; combine columns CI and C3 in a single column

called C6; combine rows L7, l8, L9, LIO, Ll4 in a single row

called LI5.

2.46 New Data - NEIaIDAT

If the user wishes to analyze more than one set of data

in a single run, a card with NEWDAT in columns 1-6 is placed

after the last analysis card of the preceding set of data.

This is followed by the title card for the new set of data.

The rest of the setup for the set follows the same rules as

for the initial set. Again, the number of such sets to be

run at one time is the user's choice.

2.47 Stop - STOP

The very last card in any analysis is the card with STOP

in columns 1-4. If several sets of data are to be analyzed

the STOP card is placed only at the end of the last set of

data.

3. Interpretation of Computer Output

3.1 The Table With A Single Value In Each Cell

The output from the computer is illustrated by two

examples. The data shown in table 1 will give the output

shown in table 2. Each line of table 1 represents one card

of computer input. The first line is the title preceded by

01 designating a table with a single value in each cell. The

second line gives the column headings which, in this particular

example, are numbers. The data follow and the instructions

come after the data.

The first page of output, table 2a, gives the original

data with row and column averages. The data have been

rearranged so that the row and column averages are in

11
ascending order of magnitude, from top to bottom and from

left to right, respectively. The data were then fitted by

the linear model with the results shown In table 2b. The

Instruction LMODEL Is printed on the second line to Identify

the source of the print out; the -0 following LMODEL Indicates

that no further Information was on the instruction card. The

parameters A, B, and C fit the original data by the model

equations

A

Ya A = y^ 4+ residual error Ij Ij

^11 = '^i+^i'^j

(1)

where y . . is the fitted value for the observation y^ ^. A and

B values*^are given for each row; C is given for eacn*^column.

The analysis of variance for the fitting is given next.

The total sum of square, SS, and degrees of freedom, DF, are

partitioned into the effect of the mean (MEAN), rows (ROWS]

columns (COLS) and interaction (RxC). The mean squares (MS)

for the row, column and interaction effects are also given.

This part of the analysis of variance corresponds to the

analysis of the data by the additive model given by

,A

where kJ^^Jq^ Is the fitted value for the observation y^^^.

The interaction (RxC) is next partitioned into two

components according to the following equation (which is just

a different way of writing equation (l)):

A

y. . = y^ 4 + residual error

«= A . + C . + (B -l)C . + residual error /^x

interaction (RxC)

The component denoted linear (LIN) corresponds to the

term (B. -l)C., while the component denoted residual (RESID)

corresponds to^ the residual error term in equation {3),

Mandel and McCrackin [1] have shown that if B^ is a

linear function of A^, the linear model, equation Xl) , may be

reduced to the simpler concurrent model given by

(^ij)J'o = ^o + ^iS (^)

12
In particular. If

B^ = 1 + a (A..J) (5)

then y^ = TT-l/a (6)

Pi = h'^o (7)

and Q. = 1-KxC (8)

To aid in determining the applicability of the concurrent

model, the sum of squares for the linear part has been

further divided into a concurrent (C) and non- concurrent (NC)

part. The concurrent model, equation (4), applies when the

non- concurrent part is sufficiently small.

The ratios, F, of mean squares are given to be compared

with tabulated values of the F-statistic to determine the

significance level of the various effects. The ratios given

are P

ROWS MS mows ) /MS (RES II!))

COLS MS(C0LS)/MS(RESID)

RXC MS ( RXC )/MS ( RESID

)

LIN MS (LIN VMS (RESID)

C MS(C)/MS(NC)

NC MS (NC )/MS (RESID

)

For the example in table 2b, the F value for rows (lOOOO)

is larger than F^q ^q for the 0.005 significance level

(5.;544y, so the eff'ect of rows is significant at better than

the 0.005 level. Likewise, the columns, interaction, and

linear parts are significant. Since the interaction and

linear parts are significant, the data cannot be adequately

represented by the additive model.

The F values for both the concurrent and non- concurrent

parts are also highly significant. Therefore, the data can

be fitted approximately by equation (4), the concurrent

model, but the fit by equation (4) is not as good as that by

equation (l), the linear model.

The next page, table 2c, shows the deviations of the

data from the linear model, i.e., YA^-yA^* The sums of

squares of the deviations over both the rows and columns are

also given. This table should be carefully scrutinized for

trends, either in the rows or in the columns, and for

excessive individual residuals. Trends can be detected by

systematic patterns in the + and - signs. If a row or column

contains excessive residuals, the analysis should be run

13
omitting the suspect row or column^ or else a weighted

analysis should be run. Other systematic patterns or

residuals may indicate that the linear model is inadequate to

represent the data to within their precision. In this case,

the data should be fitted by a more general model, such as

the quadratic model. In order that conclusions drawn from

the analysis of variance be valid, it is essential that the

residuals be free of marked systematic effects or serious

outliers

The linear model may also be applied with rows and

columns interchanged. For this purpose the data were reversed

(i.e., the rows and columns were interchanged) by the

Instruction REVERS, and the data were reanalyzed by the linear

model with the results shown in table 2d. The residual mean

square (0.233x10-5) is seen to be larger than the residual

mean square (0.879x10"^) in table 2b, before the data were

reversed, so for this case the first analysis gave the better

fit and is to be preferred.

3.2 The Table With Replicate Values In Each Cell

Table 3 is an example of a set of data with 2 replicate

values in each cell. The computer output for these data are

given in table 4. The title for the data appears as the first

line of table 3 and the 02 preceding the title indicates that

there are two values in each cell to be used in calculating

the cell averages and standard deviations. The second line

gives a row heading, followed by a column heading and the two

replicate values of the designated cell.

The data are from an inter-laboratory test on natural

rubber. The rows labeled LAB12 through LAB59 In table 4a

represent different laboratories that performed the test and

the columns labeled MAT345 through MAT245 represent the

materials on which the test of stress at 100 percent elongation was made. In table 4a the average of the two replicate

measurements is given for each cell. The row and column

averages are also given. The data have been arranged so that

their row and column averages appear in ascending order of

magnitude. Table 4b gives the standard deviations per cell

and the percent coefficient of variation xf or each column

(material).

Table 4c is a print out of the COMBIN instructions which

appear after the data cards In table 3. It will be noted that

in each line the instruction Is given to combine four rows

into a single row with a new label. (See Section 2.45 - COMBIN) The four rows in each case are really four tests

within a single laboratory but with different labels. In a

previous computer run of these data In which each of the four

14
tests was analyzed as a separate laboratory, it was found that

there was no significant difference within the set of four.

Therefore in this run the computer has been instructed to

combine each set of four - averaging the cell averages into a

new "table of averages", table 4d. The result is that the

cell averages from table 4a for the rows labeled LABO9, LAB 10,

LABll, LAB12 are averaged in a single row of averages labeled

LABOl in table 4d, etc. This new table is treated as a set

of data with single measurements per cell and therefore no

new standard deviations are computed. In table 4e the first

line after the heading, LMICOLCAL, indicates that the computer

was instructed to analyze the data using the linear model

weighted by columns, with the weights to be calculated.

Printed in this table are the computed weights (WEIGHT) and

their square roots (SQRTW) for the various columns of combined

data. A, B, and C are the values obtained for the parameters

in the model equation (l) discussed on page 12. B*(B STAR)

is the estimate of B based on the assumption that Ba is a

linear function of A.. If Bf- = B. , equation (5) holds and

the model is concurrent. To what extent equation (5) holds

(i.e., to what extent the model is concurrent) can be Judged

by comparing B and B STAR: a close agreement indicates a high

degree of concurrence. In table 4e they do not agree, so

these data do not fit the concurrent model.

The calculated value of a is also given in table 4e.

The last line of table 4e gives the "Point of Concurrence"

which is discussed in reference [4].

Table 4f gives the analysis of variance of the weighted

data. The usual row and coliAmn SS are given and the row x

column interaction. The latter is partitioned into "Fit" and

"Slope". The value of the ratio

MS T F=MS-^^^ (9)

^^fit

is compared with the tabulated value of the F-statistic with

the degrees of freedom given in the column DF (in this case

5 and 30). A significant value for this F indicates that the

model is not additive; it may then be either concurrent or

non-concurrent . To decide on this point, the sum of squares

for "slope" is partitioned into "concurrence" and "non-concurrence". If the mean square for"non-concurrence' is not

appreciably larger than that for "fit" and the mean square for

"concurrence" is significantly larger than that for "nonconcurrence", the model is probably a concurrent one. This

must be verified by:

(a) comparing BSTAR with B in table 4e

(b) plotting B versus A: a straight line indicates the

15
concurrent model

(c) examining R SQUARED V7hich is the square of the correlation

coefficient between A and B. A value of R squared close to

unity indicates a tendency for concurrence. Conclusions

drawn from the analysis of variance are always tentative

subject to an examination of the residuals _, as described below.

The Table of Deviations for this example is given in

table 4g. These are the residuals in the weighted scale^ i.e.

deviation ij = tT" (y^f^i^) (1°)

As noted in the preceding example, the table should be

carefully scrutinized for trends, either across the rows or

down the columns, and for excessive individual residuals.

Trend can be detected by systematic patterns in the + and - signs. The total sum of squares in this table is identical

with the sum of squares for "fit" in the analysis of variance.

Again it should be emphasized that in order that conclusions

drawn from the analysis of variance be valid, it is essential

that the residuals be free of marked systematic effects or

serious outliers.

In table 4h the line below the title "qmODEl" indicates

that the data were analyzed by the quadratic model as

requested in the instructions at the end of Table 5.

Analysis by the quadratic model is in accordance with

the relation

y. . = A.+B.C .+D.E. (ll)

or the equivalent form:

y. , = a! +B.'C .+D,C^ (12)

In equation (ll). A., B. , and C .. are the estimates corresponding to a fit by the-^lin^ar modes!. The set E. is orthogonal

to C ., i.e. : 5;c .E. = 0. ^

J J J J E . is a quadratic function of C . If terms of the same

order are collected in equation (11), the expression given

by equation (12) is obtained, where C . and D. are the same

as in equation (11). "^ "^

The analysis of variance for the quadratic model is

given in table 4i. The total sum of squares, SS, and the

total degrees of freedom, DF, are partitioned into the effect

of the mean (MEAN), rows (ROWS), columns (COLS) and

16
interaction (RxC), The row-column Interaction is further

partitioned into three parts: linear (LINK quadratic (QUAD),

and residual (RES). The mean squares (MS) = (SS)/(DF) are

also computed.

By means of the F-test^ both the mean squares of the

"linear" and "quadratic" can be tested against "residual".

Comparing the resulting quotients with the values in the F

table, a conclusion may be reached as to whether the model is

essentially additive (both "linear" and "quadratic" nonsignificant against "residual"), linear ("linear" significant,

"quadratic" non-signlfleant )

, or quadratic ("quadratic"

significant against "residual"). As pointed out in the

previous examples of analysis of variance, the conclusions

drawn here are not valid unless substantiated by a table of

residuals free of large systematic effects.

The lower part of table 4l contains the table of

residuals from the linear fit. On careful examination for

systematic patterns or excessive residuals in this example,

it is obvious that material 7^5 gives rise to residuals that

are substantially larger than those corresponding to all

other materials. Therefore the analysis of variance is not

usable for the drawing of valid inferences from the data.

Table 4j contains the quadratic model residuals. Again

a careful scrutiny is necessary and for the present example,

the residuals for material 7^5 are abnormally large as

compared to all others. The analysis of variance can therefore not be trusted and the data are reanalyzed omitting the

abnormal- material 7^5.

Table 5a shows the instruction OMIT with 1 before the

label MAT7'^5 on the second line. Thus the computer is

instructed to omit 1 column. The Table of Averages is the

same as that in table 4d except that material 7^5 is omitted

and the row averages are therefore different.

Table 5b gives the values determined by the QMODEL

analysis on the data without material 7^5, and table 5c gives

the analysis of variance from the quadratic model and the

table of residuals from the linear model both excluding

material 7^5. Note the marked decrease in error variance

resulting from the omission of the abnormal material. There

also appears evidence in the new analysis of a quadratic

effect.

The sums of squares of residuals from the linear model

for the materials are now all of the same order of magnitude.

And in table 5d the sums of squares of the residuals from the

quadratic model are also all of the same order of magnitude.

Therefore the quadratic effect observed in the analysis of

17
variance appears to be real.

This program was written as a research tool. The first

program "multiple curve fitting" was written by Alfred E.

Beam. Necessary changes were made as the research progressed

and the present FORTRAN program was written as a result of

the experiences gained in the use of Mr. Beam's program.

4, References

[1] Mandel, John and McCrackin, F, L., Analysis of Families

of Curves J J, Res. Natl. Bur. Standards 67A (Phys. and

Chem.), No. 5, 259-26? (May-June 1963).

[2] Mandel, John and Lashof, T. W., The Interlaboratory

Evaluation of Testing Methods ^ ASTM Bull. 239, 55-61

(July 1959).

[3] Mandelj John, The Measuring Process, Technometrics,

1, 251-267 (Aug. 1959).

[4] Mandel, John, Non-Additivity in Two-Way Analysis of

Variance, J. Am. Stat. Assoc. 56, 878-888 (Dec. 196I).

[5] Mandel, John, Estimation of Weighting Factors in Linear

Regression and Analysis of Variance, Technometrics 6,

1-25 (Feb. 1964).

""

18
Appendix A. Tables of Gorrroijiter Input and Outuut

Table 1

COMPUTER INPUT WITH A SINGLE VALUE IN EACH CELL

01 SPECIFIC VOLUME OF RUBBER - WEIR

81.5 64.0 50.2 38.5 21.0

.96677 .95639 .94826 .94143 .93397

1 .9407 7 .93344 .92673 .92204 .91678

2 .91360 .90953 .90464 .90189 .89941

3 .89436 .89231 .88880 ,88645 .88447

4 .87937 ,87864 .87572 .87336 •87214

5 .86680 .86697 .86424 .86188 .86056

6 .85636 .85654 .85410 .85134 •85038

7 .84708 .84717 .84480 .84201 •84113

8 .83828 .83828 .83608 .83345 •83263

9 .83074 .82996 •82859 .82563 •82647

10 .82360 .82249 .82129 .81829 ,81834

ENDATA

LMODEL

REVERS

LMODEL

STOP

19
Table 2. Computer Output For Data In Table 1

5 COLUf'NS

SPECIFIC VOLUME OF RUBBER - HEIR TABLE OF AVERAGES

Table 2a

10

987

6

.81833<59q6 826459995

832629994

841129996

850379996

.818289995 .825629994

.833449997

.842C09999

.851339996

.821289994

.828589998

.836080000

.844799995

.854099996

.822489999

.829959995

.838280000

.347169995

.856539994

.823599994 830739997

.838280000 84708000C

856359996

.820801981 .828277975

.835743986

.844437987

.853743978

54

32

1

0.0.0.

860560000

872139998

884469993

899409994

916779995 0.

861879997

873359993 886449993

901839995

922039993

864239998

875719994

888799995 904639997

925729999

.866969995

.878639995

.392309994 909529999 933440000

0.0.0.

0.

856799995

879369996 894359998

913599998

940759993

C

C

.864089988 875845976

889277980 905813977

927951992

AVG

0.

0. 933969997 865934516

0.0. 941429995

868888147

0. 948259994

872113600 956389993

875610881

0.

0. 966769993

87797S419 0, 949363977 872304499

SPECIFIC VOLUME OF RUBBER - WEIR LMCOEL -0

Table 2ti

10

9

87

6

5

0.820801981 0.828277975

0.835743986

0.844437987 0.853743978

0.864089988

0.51668030

0.45763269

0.56575614 0.59856942

0.60197268

0.51487932

0.875845976

0.839277980

0.905813977 0.927951992

0.949363977

0.68988606

0.89047448

1.24970494 2.04841876 2.77372935

21.0

38.5 50.2

-0.005369984

-0.003415352 -0.000190899

64.0 81.5 0.003306381 0.00557C920

ANALYSIS OF VARIANCE

TOTAL 55 41.9396214

MEAN 1 41.8503318

ROMS 10 0.087371805

COLS 4 0.000919990

RXC 40 0.000494681

LIN 10 0.000468358

C 1 0.000391498

NC 9 0.768659E-04

RESIO 30 0.2635CI3E-04

0.00B787177

0.000229992

C. 123572E-04 C.468365E-04

0.000391498 C.854066E-05

C.878677E-06

10000.4635 251.756298

14.0747378 53.3034325 45.8394866

9.71989989

SPECIFIC VOLUME OF RUBBER Table 2o TABLE OF DEVIATIONS

10

9

8

7

SUM SU

21.0 0.312582E-03

0.549504E-03 -0.758842E-04

-0.935761E-04 -0.131398E-03

-0.228092E-03 -0.129540E-05

-0.261515E-04

0.305919E-03 -0.172019E-03

-0.499092E-C3

0.976957E-06

38.5 0.746824E-03

•0. 108454E-02

-C.361167E-03 -0.383057E-03

-0.347428E-03

-C.109345E-03

-C. 129089E-03 n.214189E-03

C.345454E-03

0. 10e612E-02 0.154205E-02

0.588329E-05

50.2 0.586651E-03

0.399388E-03 0.444017E-03 0.475278e-03

0.470936E-03

0.267394fc-C3 0.572205E-05 0.307992E-03

-0.935413E-03

-0.830948E-03 -0.574477E-03 0.321133E-05

64.0 -0.203177E-04 0.168920E-03

0.665411E-03 0.752911E-03

0.805669E-03

0.846982E-03

0.512995E-03 0.877678E-04

-C.415973E-03

-0.128485E-02

-0.214498E-C2 0.910076E-05

81.5 -0.132039E-03

-0.133172E-03

-0.672340E-03

-0.752419E-03

-0.797719E-03

-0.776924E-03

-0.388265E-03 0.322163E-04

0.699051E-03

0.120158E-02

167643E-02 718798E-05

SUM SQ C.101746E-0S 0.180387E-05

0.122817E-05

0.151536E-05 C.164521E-05

0.145647E-05

0.430611E-06 0.150161E-C6

0.175024E-05

0.499436E-05 C.103684E-04

0.263603E-04

20
SPECIFIC VOLUME OF RUBBER - WEIR REVERS -0

SPECIFIC VOLUME OF RUBBER - WEIR LMOOEL -0

A B 0.90821072

0.96198194

0.98454904

21.0

38.5

50.2

0.866934516

0.868888147

0.872113600

10

9

87

6

5

C -0.051502518 -0.044026524

-0.036550513

-0.027866513

-0.018560521

-0.008214511

TOTAL

MEAN ROMS

COLS

RXCLIN

CNC RESID

ANALYSIS OF VARIANCE

OF

55

14

1040

4

13 36

SS 41.9396214

41.8503323 0.000919990

0.087871805

0.000494581 0.000410840

0.000391461 0.193813E-04

0.838441E-04

64.0

81.5

0.003541477

0.016973481 0.033509478

0-055647492

0.077059478

Table 2d

0.875510881 0.877975419

1.03570965

1.10952595

0.000229992 0.008787177 0.123672E-04

0.000102706

0.000391451

C. 645044 E-05

C. 232900 E-05

98.7544155

3772.93661 5.31006300 44.1007395 60.5937004

2.77390614

5 ROWS 11 COLUMNS

SPECIFIC VOLUME OF RUBBER TABLE OF DEVIATIONS

10 9 8 21.0 -0.181938E-02 -C .479154E-03 -0. 109987E-02 -0.495851E-03

38.5 -0.105365E-02 -0. 905432E-03 -0.267595E-03 -0. 710636E-04

50.2 -0.111599E-03 -0. 172928E-03 -0. 343248E-04 0.125133E-03

64.0 0.220776E-03 -0.521839E-04 0.535198E-03 0.420734E-03

81.5 0.276796E-02 0.161315E-02 0.869423E-03 0.232011E-04

0.302352E-03 0.108600E-02

0.305740E-03 0.894062E-03

0.262000E-03 0.214815E-03

0.152431E-03 -0. 133030E-03 -0. 538820E-03 0.102204E-02 -0. 2061 21E-02 -0. 253478E-02

0.198908E-02

0.105502E-02

0. 119284E-03

0.211999E-02 0.123357E-02

-0.265241E-04

-0.8804B0E-03

-0.244793E-02

SUM SQ 0.121431E-04 0.358429E-05 0.232483E-05 0.444132E-05 0.132195E-05 0.629117E-05 0.119381E-04 0.127845E-04

SPECIFIC VOLUME OF RUBBER - WEIR TABLE OF DEVIATIONS

1 21.0 0.204182E-02 -0. 694163E-03 -0. 295076E-02

38.5 0.766337E-03 -0. 380032E-03 -0. 158797E-02

50.2 -0.468579E-03 -0. 176847E-03 0.269860E-03

64.0 -0.785958E-03 0.194475E-03

81.5 -0.155506E-02 0.105224E-02

0.967875E-03

0.329509t-02

SUM SQ 0.28C748E-04

0.880953E-05

0.512598E-05 0.333311E-05

0.431139E-04

SUM SO 0.801346E-05 0.180259E-05 0.230958E-04 0.838441E-04

SPECIFIC VOLUME OF RUBBER - WEIR STOP -0

21
Table 3- Computer Input With Repllcate '

02 ISO NATURAL RUBBFR TEST STR

LAB09 MAT145 6.93 6.78

LAB09 MAT245 8.86 8.76

LABOg MAT345 4.26 4.24

LAB09 MAT4'!f5 6.48 6.46

LAB09 MAT545 5.17 5.82

LAB09 MAT645 4.58 4.56

LAB09 MAT745 4.37 4.38

LAB09 MAT845 5.50 5.47

LABIO MAT145 6.96 6,81

LABIO MAT245 9.13 9,09

LABin MAT345 4.39 4,40

LABIO MAT445 6.32 6,42

LABIO MAT545 5.13 5,12

LABIO MAT645 4.58 4,57

LABIO MAT745 4,48 4,51

LABIO MAT845 5.55 5,55

LABU MAT145 6.81 6,94

LABll MAT245 9.12 9,29

LABll MAT345 4.32 4,37

LABll MAT445 6.29 6,30

LABll MAT545 5.15 5,12

LABll MAT645 4.50 4,55

LABll MAT745 4.44 4,41

LABll MAT845 5.51 5,54

LAB12 MAT145 6.82 6.82

LAB12 MAT245 8.57 8,61

LA612 MAT345 4.41 4.37

LAB12 MAT445 6.46 6,41

LAB12 MAT545 5.13 5,11

LAB12 MAT645 4.48 4.46

LAB12 MAT745 4.20 4.16

LAB12 MAT845 5.52 5.48

LAB21 MAT145 6.99 7.14

LAB21 MAT245 8.03 8.80

LAB21 MAT345 4.29 4,37

LAB21 MAT445 6,44 6,43

LAB21 MAT545 4.95 5,12

LAB21 MAT645 4.57 4,59

LAB21 MAT745 6.61 6,21

LAB21 MAT845 5.58 5.54

LAB22 MAT145 6.91 7.10

LAB22 MAT245 8.75 8.82

LAB22 MAT345 4.45 4.49

LAB22 MAT445 6.46 6.47

LAB22 MAT545 5.02 5.21

LAB22 MAT645 4.53 4.50

LAB22 MAT745 4.89 4.72

LAB22 MAT845 5.41 5.48

ESS AT 100 40

22
Table ^. Coiiiputer Input w/ith Replicate Values in Each Cell (Cent.)

LAB23 MAT145 7.09 7.14

LAB23 MAT245 8.40 8.46

LAB23 MAT345 4.34 4.35

LAB23 MAT445 6.35 6.19

LAB23 MAT545 5.16 5.18

LAB23 MAT645 4.45 4.40

LAB23 MAT745 4.83 4.82

LAB23 MAT845 5.52 5.47

LAB24 MAT145 7.18 7.81

LAB24 MAT245 8.83 8.87

LAB24 MAT345 4.45 4.45

LAB24 MAT445 6.51 6.46

LAB24 MAT545 5.44 5.26

LAB24 MAT645 4.61 4.64

LAB24 MAT745 6.82 6.62

LAB24 MAT845 5.56 5.67

LAB57 MAT145 6.97 6.97

LAB57 MAT245 9.29 9.25

LAB57 MAT345 4.51 4.54

LAB57 MAT445 6.61 6.67

LAB57 MAT545 5.24 5.29

LAB57 MAT645 4.77 4.79

LAB57 MAT745 7.60 7.51

LAB57 MAT845 5.85 5.88

LAB58 MAT145 7.14 7.14

LAB58 MAT245 9.37 9.28

LAB58 MAT345 4.60 4.57

LAB58 MAT445 6.72 6.64

LAB58 MAT545 5.31 5.37

LAB58 MAT645 4.86 4.85

LAB58 MAT745 8.41 8.34

LAB58 MAT845 6.02 5.86

LAB59 MAT145 7.11 7. ]4

LAB59 MAT245 9.70 9.68

LAB59 MAT345 4.67 4.61

LAB59 MAT445 6.76 6.71

LAB59 MAT545 5.41 5.37

LAB59 MAT645 4.94 4.84

LAB59 MAT745 8.32 8.32

LAB59 MAT845 5.90 5.99

LAB60 MAT145 7.04 7.06

LAB60 MAT245 9.16 9.21

LAB60 MAT345 4.63 4.59

LAB60 MAT445 6.74 6.76

LAB60 MAT545 5.43 5.39

LAB60 MAT645 4.78 4.78

LAB60 MAT745 8.28 8.22

LAB60 MAT845 5.88 5.99

LAB61 MAT145 6.90 6.95

LAB61 MAT245 9i59 9.90

LAB61 MAT345 4.97 5.06

LAB61 MAT445 6.91 6.88

23
Computer Input Witti Replicate Values in iiach Cell (Cont.)

LAB61 MAT545 5.42 5.37

LAB61 MAT645 4.92 5.12

LAB61 MAT745 4.47 4.42

LAB61 MAT845 5.73 5.65

LAB62 MAT145 7.14 7.07

LAB62 MAT245 9.33 9.65

LAB62 MAT345 4.92 5.13

LAB62 MAT445 6.51 6.50

LAB62 MAT545 5.44 5.37

LAB62 MAT645 5.07 5.12

LAB62 MAT745 7.34 7.25

LAB62 MAT845 6.01 6.13

LAB63 MAT145 7.20 7.16

LAB63 MAT245 Q.77 10.07

LAB63 MAT345 4.59 4.57

LAB63 MAT445 6.02 6.20

LAB63 MAT545 5.27 5.20

LAB63 MAT645 4.99 5.17

LAB63 MAT745 6.53 6.46

LAB63 MAT845 5.80 5.90

LAB64 MAT145 7.09 7.01

LAB64 MAT245 9.20 9.26

LAB64 MAT345 4.16 4.35

LAB64 MAT445 6.28 6.29

LAB64 MAT545 5.26 5.16

LAB64 MAT645 4.84 4.88

LAB64 MAT745 7.54 7.72

LAB64 MAT845 5.91 5.94

LAB69 MAT145 7.16 7.14

LAB69 MAT245 8.85 8.89

LAB69 MAT345 4.5 4.42

LAB69 MAT445 6.56 6.64

LAB69 MAT545 5.34 5.39

LAB69 MAT645 4.77 4.73

LAB69 MAT745 7.39 7.43

LAB69 MAT845 5.72 5.83

LAB70 MAT145 7.13 7. 15

LAB70 MAT245 9.16 9.81

LAB70 MAT345 4.73 4.68

LAB70 MAT445 6.82 6.87

LAB70 MAT545 5.58 5.48

LAB70 MAT645 5.05 5.05

LAB70 MAT745 7.64 7.59

LAB70 MAT845 5.81 5.77

LAB71 MAT145 7.03 7.03

LAB71 MAT245 9.34 9.29

LAB71 MAT345 4.46 4.47

LAB71 MAT445 6.61 6.66

LAB71 MAT545 5.28 5.23

LAB71 MAT645 4.83 4.83

LAB71 MAT745 7.51 7.52

LAB71 MAT845 5.84 5.79

Jl

24
Table 3. Computer Input With Replicate Values in Each Cell (Cent.)

LAB72 MATINS 7.17 7.10

LAB72 MAT245 9.22 9.14

LAB72 MAT345 4.47 4.47

LAB72 MAT'^45 6.62 6.67

LAB72 MAT545 5.48 5.5

LAB72 MAT645 4.76 4.86

LAB72 MAT745 6.94 7.05

LAB72 MAT845 5.86 5.72

LAB85 MAT145 7.0 7.0

LAB85 MAT245 8.8 8.7

LAB85 MAT345 4.5 4.5

LAB85 MAT445 6.6 6.7

LAB85 MAT545 4.9 5.3

LAB85 MAT645 4.5 4.5

LAB85 MAT745 4.8 5.0

LAB85 MAT845 5.7 5.7

LAB86 MAT145 7.0 7.0

LAB86 MAT245 8.7 8.9

LAB86 MAT345 4.4 4.5

LAB86 MAT445 6.6 6.6

LAB86 MAT545 5.3 5.3

LAB86 MAT645 4.6 4.7

LAB86 MAT745 4.5 4.7

LAB86 MAT845 5.9 5.9

LAB87 MAT145 7.0 7.1

LAB87 MAT245 9.0 9.1

LAB87 MAT345 4.4 4.5

LAB87 MAT445 6.5 6.6

LAB87 MAT545 4.9 5.0

LAB87 MAT645 4.5 4.6

LAB87 MAT745 5.0 5.2

LAB87 MAT845 5.6 5.7

LAB88 MAT145 7.0 7.0

LAB88 MAT245 9.2 9.2

LAB88 MAT345 4.5 4.5

LAB88 MAT445 6.7 6.8

LAB88 MAT545 5.1 5.2

LAB88 MAT645 4.8 4.8

LAB88 MAT745 5.1 5.3

LAB88 MAT845 5.7 5.8

ENDATA

COMBINROWS LAB09 LABIO LABll LAB12 LABOl

COMBINROWS LAB21 LAB22 LAB23 LAB24 LAB02

COMBINROWS LAB57 LAB58 LAB59 LAB60 LAB03

COMBINROWS LAB61 LAB62 LAB63 LAB64 LAB04

COMBINROWS LAB69 LAB70 LAB71 LAB72 LAB05

COMBINROWS LAB85 LAB86 LAB87 LAB88 LAB06

END

LMWCOLCAL

QMODEL

OMIT 01MAT745

OMODEL

STOP

25
Table 4. Computer Output For Data In Table 3

ISO NATURAL RUBBER TEST

TABLE OF AVERAGES

STRESS AT 100 40

HAT345 MAT645 MAT545 MAT845 HAT745 KAT445 HAT145 HAT245 LAB12 LAB23

LAB09

LABll LABIO

4.38999993

4.34499997

4.24999994

4.34499997

4.39499998

4.46999997

4.42499995 4.56999993

4.52499998 4.57499999

5.11999995

5.16999996

5.49499995

5.13499993 5.12499994

5.49999994

5.49499995 5.48499995

5.52499998

5.54999995

4.17999995 4.82499999

4.37499994 4.42499995

4.49499995

6,6

6.

6.

6.

43499994

26999998 46999997

29499996

36999995

6.

7.

6.

6.

6.

81999999

11499995 85499996

87499994

88499993

8.58999991 8.42999995

8.80999994 9.20499992

9.10999990

LAB22

LAB85 LAB86

LAB87

LAB21

4.46999997

4.50000000

4.44999999

4.44999999

4.32999992

4.51499999 4.50000000

4.64999998

4.54999995

4.57999998

5.11499995 5.09999996

5.29999995 4.94999999

5.03499997

5.44499993

5.69999999

5.89999998

5.64999998 5.55999994

4.80499995 4.89999998

4.59999996

5.09999996 6.40999997

6

6.

6.66

46499997

64999998

59999996

54999995 43499994

7.

7.

7.

7.7.

00499994 00000000

00000000

04999995 06499994

8.78499997

8.74999988

8.79999995

9.04999995 8.41499996

LAB88

LAB61 LAB24

LAB69 LAB64

4.50000000

5.01499993 4.44999999

4.45999998 4.25499994

4.79999995

5.01999992 4.62499994

4.74999994

4.85999995

5.14999998

5.39499992

5.34999996

5.36499995

5.20999998

5.74999994

5.68999994 5.61499995

5.77499998

5.92499995

5.19999993 4.44499993

6.71999997

7.40999997 7.62999994

666

6

6.

74999994

89499998 48499995

59999996 28499997

7,

6.

7,

7.

7.

00000000

92499995 49499995

14999998 04999995

9.19999993

9.74499989

8.84999990 8.86999989

9.22999990

LAB63

LAB72

LAB71 LAB57

LAB60

4.57999998

4.46999997

4.46499997

4.52499996

4.60999995

5.07999992 4.80999994

4.82999998

4.77999997

4.77999997

5.23499995

5.48999995 5.25499994

5.26499993

5.40999997

5.84999996

5.78999996 5.81499994

5.86499995

5.93499994

6.49499995 6.99499995

7.51499999 7.55499995

8.24999988

66

6.6

6

10999995 64499992

63499993

63999993

74999994

7.7.

7.6

7.

17999995

13499993 02999997

96999997

04999995

9.91999996

9.17999995 9.31499994

9.26999998

9.18499994 LAB62

LAB70

LAB5e LAB59

AVG

5.02499998 4.70499992

4.58499998 4.63999993

4.50874972

5.09499997

5.04999995 4.85499996

4.88999999

4.73270798

5.40499997

5.52999997

5.33999991 5.38999993

5.26395798

6.06999999

5.78999996 5.93999994

5.94499993

5.73187459

7.29499996

7.61499995

8,37499988 8.31999993

6.16395783

6

66

6

6

50500000 84499997

67999995

73499995

54583275

7

7.7.7

7.

10499996

13999999 13999999

12499994 04854119

9.48999989

9.48499990

9.32499993

9.68999994 9.U249936

ISO NATURAL RUBBER TEST TABLE OF AVERAGES

STRESS AT 100 40

LAB12 5.68812484

LAB23 5.75937486

LAB09 5.78874981

LABll 5.79124975

LABIO 5.81312484

LAB22 5.82562482

LAB85 5.88749987

LAB86 5.91249985

LAB87 5.91874981

LAB21 5.97874987

LAB88 6.04374981

LA861 6.14124978

LAB24 6.19874978

LAB69 6.29749978

LAB64 6.30562478

LAB63 6.30624992

LAB72 6.31437486

LAB71 6.35749984

LAB57 6.35874981

LAB60 6,49624979

LAB62 6,49874979

LAB70 6.51999986

LAB58 6.52999973

LAB59 6.59187484

AVG 6.13851511

26

I
Table 4b

ISO NATURAL RUBBER TEST STRESS AT 100 40 TABLE OF STANDARD DEVIATIONS

MAT345 fAT645 MAT545 HAT845 MAT745 MAT44S MAT145 MAT245

LAB12 LAB23

LAB09

LABll LABIO

0.

0.0.

0.

028284281 007071055

014142163

035355300 007071055

014142118 035355344

014142163 035355300

0C7071O55

0.014142163

0.014142118

0.459619425

0.021213219 0.007071100

0.028284281 0.035355344

0.021213219

0.021213181

0.

0.028284281 0.007071055

0.00707110C 0.021213219

0.021213219

0.035355344

0.113137059 0.014142118

0.007071055

0.070710652

0.0.035355344

0.106065996 0.091923915

0.106066041

0.028284244

0.042426363 0.070710607

0.120208204

0.028284326

LAB22

LAB85 LAB86 LAB87

LAB21

0.

0.

0.

0.

0.

028284244 070710689

070710689

056568526 0.

021213181 070710689

070710652

014142118

0.134350285

0.282842688

0.0.070710689

0.120208122

0.049497463

0.

0.0.070710689

0.028284281

0.120208159 0.141421385

0.14142134C 0.14142134C

0.282842688

0.007071055 0.070710689

0.0.070710652

C. 007071100

0.134350285

0.

0.0.070710652

0.106066041

0.049497426

0.070710689

0.141421385

0.070710607

0.544472203

LAB88

LAB61 LAB24

LAB69

LAB64

0.

0.

0.

0.

0636395 89 056568570

134350285

141421340 021213219

028284281

028284281

0.070710689

0.035355344

0.127279222

0.035355344

0.070710689

0.070710652 0.056568526 0.077781752

0.077781752

0.021213219

0.14142134C

0.035355344 0.141421385

0.028284244

0.127279222

0.070710652

0.021213181 0.035355344 0.056568570 0.007071055

0.0.035355344

0.445477262 0.014142118

0.056568526

0.0.219203144 0.028284244

0.028284326

0.042426445

LAB63

LAB72

LAB71 LAB57

LAB60

0.

0.

0.

014142118 007071055

021213181

028284281

127279222 070710652

014142118

0.049497463

0.014142163

0.035355344

0.035355344

0.028284244

0.070710689 0.098994933

0.035355344 0.021213219

0.077781707

0.049497463

0.077781707

0.007071055 0.063639589

0.042426445

C. 127279222 0.035355344 0.035355344

0.042426407 0.014142163

0.028284281 0.049497463

0.0.0.014142U8

0.212131999

0.056568488

0.035355300 0.028284244

0.035355300

LAB62 LAB70

LAB58

LABS9

AVG

148492448 035355344

021213181 042426407

037123099

035355300

007071055 070710689

034471445

0.049497508

0.070710689

0.042426407

0.028284281 0.075719349

0.084852815 0.028284244

0.113137096

0.063639589

0.048024334

0.063639589

0.035355344

0.049497508

0.0.07395158C

0.007071055

0.035355300 0.056568526

0.035355344

0.040658630

0.049497463

0.014142118

0.0.021213219

0.057452418

0.226274200 0,459619388 0.063639544

0.014142118

0.109012276

ISO NATURAL RUBBER TEST TABLE OF STANDARD DEVIATIONSSTRESS AT 100 40

LAB12 0.022097088

LAB23 0.036239207

LAB09 0.088388346

LABll 0.044194177

LABIO 0.030935928

LAB22 0.058059012

LAB85 0.070710681

LAB86 0.053033009

LAB87 0.079549491

LAB21 0.144956879

LAB88 0.044194162

LAB61 0.076013975

LAB24 0.109501550

LA669 0.040658653

LAB64 0.060987954

LAB63 0.084852807

LAB72 0.050381340

LAB71 0.019445427

LAB57 0.0282842 59 LAB60 0.030052029

LAB62 0.083085045

LAB70 0.084852800

LAB58 0.044194162

LAB59 0.034471452

AVG 0.059551641

PERCENT CV MAT345 0.81497547

HAT645 0.71768180

MAT545 1.43588310 HAT845 0.83216553

PERCENT CV MAT745 1.27107066

HAT445 0.62585288

HAT145 0.80329586

MAT245 1.19633342

27
Table ho

ISO NATURAL RUBBER TEST STRESS AT 100 40 COHBINROHS - 0LAB09 LABIC LABll LAB12 LABOl ROWS LAB21 LAB22 LAB23 LAB24 LAB02 ROWS LAB57 LAB58 LAB59 LAB60 LABO? ROWS LAB61 LAB62 LAB63 LAB64 LAB04

ROWS LAB69 LAB70 LAB71 LAB72 LAB05 ROWS LAB85 LAB86 LAB87 LAB88 LAB06

END

Table Hd

ISO NATURAL RUBBER TEST

TABLE OF AVERAGES

STRESS AT 100 40

MAT345 MAT645 HAT545 MAT845 MAT745 MAT445 MAT14S MAT245

LABOI LAB02 LAB06 LAB04

LAB05

4

44

4

4

34499991

.39874989 47499996

.71874988 52499992

4.53499991 4.53624994

4.62499994

5.01374990 4.85999990

5

5

55

5

21874988

16749990 12499994

31124985

40999985

5.51499987

5.52874988

5.74999988 5.88374990

5.79249996

4.36874992

5.68999988 4.94999993

6.46624988 7.3837499C

6.39249992 6.41374993

6.63749987 6.44874990

6.68124986

6.

7.

77.

7.

85874993

16999990 01249999

06499994

11374992

8.92874992 8.61999989

8.94999981 9.59624982 9.21249986

LAB03

AVG

4

4 .58999991 508749 84 4.826249964.73270822

5

5 3512498726395816 5.921249935.73187476

8.12499988

6.16395813

6.70124984 6.54583317

77 07124990

04854143 9.367499839.11249971

ISO NATURAL RUBBER TEST

TABLE OF AVERAGES

STRESS AT 100 40

LABOl 5 .77031231 LAB02 5 .94062483 LAB06 5 94062483

LA804 6 .31296849 LAB05 6 37234348 LAB03 6 .49421853 AVG 6 .13851535

Table 4e

ISO NATURAL RUBBER TEST STRESS AT 100 40

LMWCOLCAL -0 WEIGHT SORT W MAT345 0.130694E 04 36.1516733

MAT645 0.I78694E 03 13.3676336

MAT545 0.845907E 02 9.1973194

MAT845 0.439850E 03 20.9726107

LABOl LAB02 LAB064.77612406

4.81412500

4.92038530

0.98501815

0.97C05575

1.02875741

WEIGHT

MAT745 0.575277E 00 MAT445 0.904790E 02 MAT145 0.294502E 02 MAT245 0.885176E 01

B STAR 1.00000301 1.00000304

1.00000310

LAB04

LAB05 LAB03

SORT M 0.7584704 9.5120463

5.4268070

2.9751900

5.13244158 4.99554825

5.05768508

0.93052941

1.03660026

1.04903851

B STAR 1.00000323

1.00000314 1.00000319

MAT345 -0.44063497

MAT645 -0.21667659

MAT545 0.31457335

HAT845 0.78248995

ALPHA=

POINT OF CONCURRENCE=

MAT745 MAT445

MAT145

MAT245

1.21457332

1.59644830 2.09915662 4.163114910.632179E-06 -1581831.50
Table 4f

ANALYSIS OF WEIGHTED DATA SOURCE OF SS MS ROMS 5 206.509474 41.3018947

COLS 7 6326.78949 903.827065

RXC 35 40.9357047 1.16959156

FIT 30 29.8441887 0.994806290

SLOPE 5 11.0915161 2.21830320

CONC 1 0.939660E-04 0.939660E-04

NONC 4 11.0914221 2.77285552

R SQUARED^ 0.847188E-05

R= 0.002910644

Table 4g

ISO NATURAL RUBBER TEST TABLE OF DEVIATIONS

STRESS AT 100 40

MAT 345 MAT645 MAT545 MAT845 MAT745 MAT445 HAT145 MAT245

LABOl

LAB02

LAB06- LAa04 LAB05

0.105176158

0.436184779

0.286363162 -0.132600397

-0.4983854 59

-0.37019988 -0.90481069

-0.96885531

1.10860850 1.19050403

1.22108589 0.44350605 -1.09452666

-1.04767998

0.81271965

-0.66883728

-0.93189596

0.51639310

0.48611278 -0.29734046

-1-0

-0

.21639760

.22930755 92524794

15443379 85644557

0.41705905

0.48493462

0.71108788 -1.60975917

0.29318916

0.08095985

1.73427707 -0.36581131 -0.11270709

-0.31358596

-0

-0

1 -0

15435924 69196562

75337961 75508659 29315839

LAB03

SUM SO -0.196740396

0.588000752 -0.05524824

4.54377592 -0.33510166

4.75617659

0.89558908 2.70923194

14 36007446 99544007 -0.29649965

3.67994228 -1.02312286

4.30590874

-0.

4. 17093369 26571304

ISO NATURAL RUBBER TEST STRESS AT 100 40 TABLE OF DEVIATIONS

SUN SO LABOl 3.7704466

LAB02 5.8483440

LAB06 4.5484633 LAB04 8.2887406

LAB05 3.4183493

LAB03 3.9598453

SUM SO 29.8441887

Table Uh

ISO NATURAL RUBBER TEST QMODEL -0STRESS AT 100 40

LABOl LAB02

LAB06

LAB04 LAB05

A 5.77031231 5.94062483

5.94062483

6.31296849

5.37234348

B 0.98995132 0.94821443

0.98772637

1.03774434

1.00552733

-0

-0

D086043470

027748495

038834378

.055445835 063444003

AP 5.60465795 5.99404734

5.86585927

5.20622188

6.49448842

BP 0.88223404

0.98295260 0.93910985

0.95833204

1.08495247 LAB03 6.49421853 1.03083800 -0 089132071 6.66581917 1.14242187

MAT345

MAT645

MAT545 MAT845

C-1.62975551 -1.40580714

-0.87455720

-0.40564059

E 2.77118859

1.81C97450 -0.06553725

-1.25081283

HAT745

MAT445 MAT145

MAT245

C 0.02544278

0.40731782

0.91002607

2.97398436

E-1.95644450

-2.26925C99 -2.23634863

3.19523C44

29
ANALYSIS OF VARIANCE

TOTAL 48 1912.73088

MEAN 1 1808.7C580

ROWS 5 3.40423223 0.680846445

COLS 7 92.4115286 13.2016469

RXC 3b 8.20929074 0.234551162

LIN 5 0.082239047 0.016447805

QUAD 5 0.907975405 0.181595080

RES 25 7.21907640 0.288763054

Table 4i

ISO NATURAL RUBBER TEST STRESS AT 100 40 TABLE OF DEVIATIONS LINEAR fODEL

MAT 345 f'AT645 HAT545 HAT845 MAT745 HAT445 MATI45 HAT245

LABOl 0.188076138 0.156368256

LAB02 0.003492236 -0.071368277

LAflC6 0.144137502 0.072927892

LAB04 0.097061336 0.159649849

LABC5 -0.208569765 -0.098766088

0.314206660

0.056142867

0.048198342

-0.094151855

0.147241950

-0.026292443 0.211024702

-0.007229567 -0.082952440 -0.170955241

-1.42674947

-0.27475011 -1.01575536 0.12687832

0.98582304

0.218962848

C. 086900473 C.294556499

-0.286910295

-0.100662768

0.187556148 0.366475224

0.173018456

-0.192342937

-0.173649609

0.214337945

-0.140599728

0.071892261 0.197045922

-0.150266170

LAB03 -0.224194407

SUM SQ 0.159345828

-0.218809128

0. 117983185 -0.241441846 0.178240769

-0.153788030 0.119831622

1.6045540C 6.70539916

-0.212847352

C. 280014448

-0.361058056 0.396929681-0.192414761 0.169307984

ISO NATURAL RUBBER TEST STRESS AT 100 40

TABLE OF DEVIATIONS LINEAR HOOEL

LABOl LAB02

LABa6 LAB04

LABOS

2

1

1

34490645 24606062 22657490

21806434

12407701 LAB03

SUM SQ

28 96736929

12705255

Table H

ISO NATURAL RUBBER TEST STRESS AT 100 40 TABLE OF DEVIATIONS

MAT345 FAT645 MAT545 MAT845 HAT745 MAT445 MAT145 MAT245

LABOl LA602 LA806

LAB04

LABOS

-0

-0

-0

050366543

080388561 036520094

0565895 36 032754466

-0 000545710

021116443

002599806

059238844

016129389

-0-0

319845706

054324299

050743438

090518080 087110385

0.254866228 -0.061000615

0.259599246

0.062122792 -0.250311807

-1-0

-0

.25841019

.32903851 .93977804

.23535502

.86169836

-0

-0

414217077

023932159

382681459

161089770 244633131

-0-0

379979342

304419905

259865671 068346709

.315532513

-0

-0

-0

.060676813

.051909126

.052231379 .019828245

.052515484 LAB03

SUM SQ 022807382

015128233

-0 057393201 007516332

-0 247283317

184758082 -0.265275568

0.272956006

15 .43017203

.41835326

-0

c 415110402576705560

-0 560388446

722853191 092471890020806432

ISO NATURAL RUBBER TEST TABLE OF DEVIATIONS

STRESS AT 100 40

LABOl LAB02

LAB 06 LAB04

LABC5

2

1

07303306

21778555

17119324

10517071 97626529

LAB03

SUM SO

2

7 67562920

21907705

3i0
Table 5. Computer Output For Data In Table 3 Omitting Column MAT745

ISO NATURAL RUBBER TEST STRESS AT 100 40 OMIT 1MAT745

6 ROWS AND 7 COLUMNS

ISO NATURAL .RUBBER TEST TABLE OF AVERAGES

STRESS AT 100 40 Table 5a

MAT345 HAT645 MAT545 MAT84S MAT445 HAT145 HAT245 AVG LA601 LAB02

LAB06 LAB05 LAB03

4.34499991 4.39874989

4.47499996 4.52499992

4.58999991

4.53499991 4.53624994

4.62499994

4.85999990 4.82624996

5.21874988

5.16749990 5.12499994

5.40999985

5.35124987

5

5.5

5.5.

51499987

52874988

74999988 79249996 92124993

6

6

66

6

.39249992 41374993

63749987

68124986

70124984

6

7

7

7

7

.85874993

.16999990 01249999

11374992 07124990

8.92874992 8.61999989

8.94999981 9.21249986

9.36749983

5

566

6

97053552

.97642833 08214265

22785687

26124978 LAB04

AVG 4.71874988

4.50874984

5.01374990 4.73270822

5.31124985

5.26395816

5.

5. 88374990 73187476

66 4487499C

54583311

7

7 06499994

04854143

9.59624982

9.11249971

6

6 29107118

13488066

ISO NATURAL RUBBER TEST STRESS AT 100 40 Table 5b OHOOEL -0

A B D AP BP LABOl 5.97053552 0.99264501 -0 .000944868 5.97261435 0.99383824

LAB02 5.97642833 0.94873317 -0 .050493434 6.08752239 1.01249915 LAB06 6.08214265 0.98964411 -0 .026128277 6.13962924 1.02264038 LABOS 6.22785687 1.00366606 -0 .003878571 6.23639035 1.00856414 LAB03 6.26124978 1.02780862 .009729356 6.23984349 1.01552182 LAB04 6.29107118 1.03750479 .071714893 6.13328630 0.94693914

C E C E MAT34S -1.62613082 2.49770308 PAT445 0.41095245 -2.55026C96

MAT645 -1.40217245 1.53666186 MAT145 0.91366076 -2.51921552

HAT545 -0.87092251 -0.34181234 MAT245 2.97761905 2.90573967

ANALYSIS OF VARIANCE Table 5c

TOTAL 42 1674.41855

MEAN 1 1580.74394

ROWS 5 0.727348790 0.145469755

COLS 6 92.4070892 15.4011815

RXC 30 0.540115476 0.018003844

LIN 5 0.076744035 0.015348807

QUAD 5 0.274498112 0.054899618

RES 20 0.18B873298 0.009443663

ISO NATURAL RUBBER TEST STRESS AT 100 40 TABLE OF DEVIATIONS LINEAR PODEL

HAT345 ^'AT645 MAT545 MAT845 MAT445 »^AT145 MAT245 SUM SO LABOl LA802 LAB06

LABOS

LAB03

-0

-0

-0

011364937 034914196

002148092

070764661 000101447

-0.

-0.

-0.

0.

043676138

109890878 069490969

039455950 006165147

-0

-0

112731278 017344654

095239341

.056258380

.014858246

-0-0

-0

.055493832 .065333366

.066689670

.030873537

.074213088

01403451C 047437429

14866060C 040933967

01761961C

-0

-0

-0

.018726349 326751292 026158392

031117201

129068255

-0

-0-0

002495766

181394339 078925967 003892183

045827627

:c

.018378541 159784913 047365196

013341606

024835601

LAB04

SUM SO 114797235 019538827 0. 177439392 051892117

-0 076235056

.031277262 .010799289.018372908

-0 268686354 098725647

-0 173998654

155703768 215884686 087860957

199665628 463371493

6 ROWS 7 COLUMNS 'i aole 5u,

ISO NATURAL RUBBER TEST TABLE OF DEVIATIONS

STRESS AT 100 40

MAT345 MAT645 MAT545 HAT845 MAT445 NAT145 MAT245 SUM SG LABOl LAB02

LAB06

LAB05 LAB03

-0

-0

-0

009004921

091203421 067408770

061077125

024199605

-0-0

-0

-0

042224184

032299533

029340632 045416005

008785583

0. -0

-0

112408E-00 853722E-04

1041706-00

549326E-01

115326E-01

-0.056938365 -0.142528556

0.026744328 -0.036803164

0.089087486

-0 011624835 081334017

082026668

031042576 04243201C

-C-0

-0

-c

021106683

199547432 039664373

040888168

104557894

-0-0

.005241312 034673542

003003985 007377952

017556638

c

01834967U

077312373

025282182

.012855016 021773450

LAB04

SUM SO -0 064325280

021396868 067237847

010347739

-0 517220E-01

293129E-01 0.120438181

0.048068032

-0 085794650

023603573

c 006666623

054486565

007499866

001657665

033300601

188873313

ISO NATURAL RUBBER TEST

STOP -0

STRESS AT 100 40

31
Appendix B. Listing of Fortran Program

C A PROGRAM FOR FITTING FUNCTIONS OF TOO VARIABLES

DIMENSION AV(120.50)»Z(120»50),XR(120)»XC(^0).A(120),B(120).C(50),

1 TITLE! 12) tXNS(26) .R ( 99 ) »D ( 120

»WT ( 120

»WFAC(12 0) ,XL( 120)

COMMON AV,Z.XR,XC»A.B»C»M»N»WT»WFAC»MT.NT»NREP»NR,NX,XNS.TITLE»XY,

1XL,D,R

C BOOLEAN STATEMENTS FOR INSTRUCTIONS AND GETTING TO SUBROUTINES

B XNSd )=626346476060 STOP

B XNS(2 )=452566242163 ' NEWDAT

B XNS( 3 ) =4344^46242543 LMODEL

B XNS(4)=504446242543 QMODEL

B XNS(5 )=234446242543 CMODEL

B XNS(6)=444446242543 MMODEL

B XNS(7 )=434466514666 LMWROW

B XNS( 8

) =434466234643 LMWCOL

B XNS(9)=626422016060 SUBl

B XNS( 10)=234651252363 CORECT

B XNS( 11 )=464431636060 OMIT

B XNS( 12)=314523436424 INCLUD

B XNS( 13)=635121264651 TRAFOR

B XNS( 14)=474666255160 POWER

B XNS( 15)=234644223145 COMBIN

B XNS( 16)=626422026060 SUB2

B XNS( 17)=512565255162 REVERS

B XNS( 18)=314563432122 INTLAB

B XNS(19)=254524216321 ENDATA

B XNS(20)=232143606060 CAL

B XNS(21 )=606060606060 BLANK

B XNS(22)=626444606250 SUM SO

B XNS(23)=444651252421 MOREDA

B XNS(24)=606060606060 BLANK

B XNS(25)=216527606060 AVG

100 READ 110»NR, TITLE

C PROGRAM RETURNS TO THIS STATEMENT FOR EACH NEW SET OF DATA

C NR - NUMBER OF REPLICATES TITLE - ANY SUITABLE IDENTIFICATION

110 FORMAT! I2»11A6,A4)

PRINT 120, NR. TITLE

120 F0RMATI1H1I2.11A6.A4)

M = N = TRANS=0.

DO 130 1=1.120

130 XR! I )=0«

DO 140 J=l,50

140 XC!J)=0.

C IF NR IS GREATER THAN 1. REPLICATE VALUES FOR EACH CELL ARE READ

C AT STATEMENT 250

C IF NR EQUALS 1 PROGRAM CONTINUES AND READS IN SINGLE VALUES / CELL

IFINR-1)200.200.250

200 READ 210.XRX, (XC! J)

J=l»7)

C READ COLUMN HEADINGS XC

210 FORMAT!2A6,2X,5!A6»4X) »A6)

DO 220 J=2.7

IFIXCIJ)-XNS(21) ) 220, 22 1.2 20 BLANK

220 CONTINUE

N»7

GO TO 222

221 N«J-1

222 DO 227 I=l»120

C READ ROW HEADINGS XR AND DATA AV

READ 225»XR(n»<AV(I«J) .J-l.N)

225 FOR«AT(A6»F8.0»5F10.0.F8.0)

IF(XR( I)-XNS(19) )226,228.225 ENDATA

C ENDATA - END OF DATA

226 IFfXR(n-XNS(23))227,230.227 MOREDA

C MOREDA - MORE DATA* WHEN MORE THAN 7 COLUMNS

52
227 CONTINUE

CALL ENDJOB

228 M=I-1

229 MT=M

NT = N MT1=MT+1

NT1=NT+1

GO TO 425

JUMPS TO 425 WHERE ROW AND COLUMN AVERAGES ARE COMPUTED

230 M=I-1

DO 244 JX=8.50»7

JXX=JX+6

READ 210,XRX, (XC( J) ,J=JX,JXX)

AT THIS POINT MORE DATA ARE READ

IF(XRX-XNS( 19) ) 240*2 29. 240 ENDATA

240 DO 241 J=JX»JXX

IF(XC( J)-XNS(21 ) ) 241 242*241 BLANK

241 CONTINUE

N = JXX

GO TO 243

242 N=J-1

243 DO 244 I=1*M

244 READ 245 » ( AV( I , J ) . J= JX » N

245 FORMAT(6X*F8.0»5F10.0*F8.0)

CALL ENDJOB

250 DO 300 1=1,120

DO 300 J=l*50

300 AV( I* J)=-1.0E37

USED NEAR STATEMENT 402 TO CHECK FOR MISSING VALUES

301 READ 305,XRX»XCX*NRX,(R(K) ,K=1,6)

DATACONSISTING OF REPLICATE MEASUREMENTS ARE READ

305 FORMAT(2A6,I2,5F10.0,F8.0)

IF(XRX-XNS( 19) ) 3 10*400, 310 ENDATA

310 IF(XRX-XNS(13) )312.311,312 TRAFOR

311 TRANS=1.

AX=R(1)

BX=R(2)

CX=R(3)

FOR TRANSFORMING DATA BEFORE AVERAGING OR PRINTING

GO TO 301

312 DO 313 1 = 1,

IF(XRX-XR( I))313,314,313

313 CONTINUE

M=M+1

XR(M)=XRX

I=M 314 DO 315 J=1»N

IF(XCX-XC(J) )315, 320,315

315 CONTINUE

N = N+1

XC{N)=XCX

J = N 320 IF(NRX)331, 330,331

330 NRX=NR

331 IF(NRX-6)336,336,332

332 READ 335, ( R ( K ) ,K=7 »NRX

AS MANY AS 99 REPLICATES PER CELL MAY BE READ

335 FORMAT(14X*5F10.0*F8,0)

336 IF(TRANS)337,339,337

337 DO 338 K=1,NRX

338 R(K)=TRAFOR{R(K),AX,BX,CX)

CALLS FUNCTION TRANSFOR NOTE** USERWRITES OWN TRANSFORMATION

339 FNRX=NRX

AV( I , J)t=SUMl (R,NRX)/FNRX

IF(NRX-1)350, 301,350

350 Z(I,J)=0.

DO 351 K=1,NRX

351 Z(I,J)=(R(K)-AV( I, J) )**2+Z( I, J)

Z(I,J)=SORTF(Z(I ,J)/(FNRX-1. >)
60 TO 301

400 MT=M

NT = N MT1=MT+1

NT1=NT+1

MEMORY=0

401 DO 408 1 = 1.

DO 408 J=1.N

IF(AV( I ,J)+1.0E3 7)402,402,408

C CHECK FOR MISSING VALUES

402 PRINT 405.XR( I ) »XC( J)

405 FORMAT(16H0MISSING VALUE A6.2X.A6)

MEM0RY=1

408 CONTINUE

409 IF(MEMORY)420»421.420

420 CALL ENDJOB

C CUTS OFF BAD PROGRAM WITH MISSING VALUES

421 IF(XCX-XNS( 10) )425.422,425 CORECT

C CORECT IN COLUMNS 7-12 AFTER ENDATA - CORRECTS DATA BEFORE

C PRINTING TABLES OF AVERAGES AND STANDARD DEVIATIONS

422 CALL CORECT

425 MP1=M+1

NP1=N+1

FM = M FN = N C CALCULATES ROW AND COLUMN AVERAGES FOR TABLE OF AVERAGES AND IF

C REPLICATE MEASUREMENTS* FOR TABLE OF STANDARD DEVIATIONS

450 DO 451 J=1»N

451 AV(M+1»J)=SUM1 (AV( 1»J) tMj/FM

452 DO 453 1=1, MPl

453 AV( I ,N+1 )=SUM2 (AV,I ,N)/FN

454 IF(NR-1 1500,500,455

455 DO 456 J=1»N

456 Z(M+1,J)=SUM1(Z(1,J) ,M)/FM

457 DO 458 1=1, MPl

458 Z( I,N+1)=SUM2(Z,I,N)/FN

C ORDER AV,Z,XR,XC IN ASCENDING ORDER OF AVERAGES OF AVERAGE

500 DO 505 1=1,

IS=I

DO 502 IX=I,M

IF(AV< IX,N + 1 )-AV( IS,N + 1 ) )501 ,502,502

501 IS=IX

502 CONTINUE

HOLD=XR( I

XR( I )=XR( IS)

XR( IS)=HOLD

DO 505 J=1,NT1

HOLD=AV( I, J)

AV( I ,J)=AV( IS, J)

AV( IS,J)=HOLD

IF(NR-1)505,505,503

503 HOLD=Z(I ,J)

Z(I,J)=Z( IS, J)

504 Z(IS,J)=HOLD

505 CONTINUE

DO 520 J=1,N

JS = J DO 507 JX=J,N

IF(AV(M+1,JX)-AV(M+1,JS) ) 5 06,507,507

506 JS=JX

507 CONTINUE

HOLD=XC( J)

XC(J)=XC( JS)

XC(JS)=HOLD

DO 520 1=1, MTl

HOLD=AV(I,J)

AV( I ,J)=AV( I ,JS)

AV( I,JS)=HOLD

IF(NR-1)520,520,508

54

I
508 HOLD=Z( I»J)

Z( I»J)=Z( I.JS)

509 Z( I »JS)=HOL0

520 CONTINUE

XR(M+1)^XNS(25) AVG

XC(N + 1 )=XNS(25 ) AVG

C PRINTING OF TABLE OF AVERAGES AND STANDARD DEVIATIONS

C CALCULATE AND PRINT PER CENT COEFFICIENT OF VARIATION

PRINT 521»M,N

521 FORMAT (1H1I3.5H R0WSI9.8H COLUMNS)

CALL PRMXLT ( AV »MP1 ,NPl ,XR »XC , T I TLE , 17HTABLE OF AVERAGES)

IF(NR-1)522.540.522

522 PRINT 521, M,N

CALL PRMXLT(Z,MP1,NP1,XR»XC.TITLE»28HTABLE OF STANDARD DEVIATIONS)

DO 531 J=1.N

C( J)=0.

DO 530 1=1.

530 C(J)=C(J)+Z(I»J)/AV(I,J)*100.

531 C(J)=C(J)/FM

CALL PRLKXCCN.IOHPERCENT CV)

C INSTRUCTIONS TO GET TO VARIOUS SUBROUTINES AND FOR CALCULATING SS

C OF DEVIATIONS

540 READ 541,XX,XY,NX»(XL(K) .K=l»6)

541 FORMAT(2A6,I2»5(A6,4X) .Ae)

PRINT 542,TITLE»XX.XY.NX»(XL(K) ,K=1»6)

542 FORMAT(1H112A6/1H02A6.I2.5(A6,4X) .A6)

C CALCULATED GO TO STATEMENT

DO 550 K=l»18

IF(XX-XNS(K) ) 550 552, 550

550 CONTINUE

551 CALL ENDJOB

C THESE CAN BE READILY IDENTIFIED FROM LIST OF BOOLEAN STATEMENTS

C AT BEGINNING OF PROGRAM. EXAMPLE - XNS(l) = STOP CAUSES PROGRAM

C TO GO TO STATEMENT 551 WHERE ENDJOB IS CALLED.

552 GO TO (551,100,1100,1200,1300,1400,1500,1600,1700,422,1900,2000,

1 2100,2200,23 00,2400,2500,2600) ,K 1100 CALL LWDDEL

GO TO 3000

1200 CALL OMODEL

GO TO 3000

1300 CALL CMODEL(PTCONC)

GO TO 3000

1400 CALL MMODEL

GO TO 3000

C CALCULATES ROW WEIGHTS

1500 IF(XY-XNS(20) ) 1501,1515,1501 CAL

1501 READ 1502,(WT( I) ,I=1,M)

1502 FORMAT(14X,5F10.0,F8.0)

1503 DO 1504 1 = 1,

1504 WFAC( I )=SQRTF(WT(I )

CALL PRL2(XR,WT,WFAC,N,6HWEIGHT»6HSQRT W)

1505 CALL LMWROW

GO TO 3000

1515 CALL CALWR

60 TO 1503

C CALCULATES COLUMN WEIGHTS

1600 IF{XY-XNS(20) ) 1601,1615,1601 CAL

1601 READ 1502,(WT< J) ,J=1,N)

1602 DO 1603 J=1,N

1603 WFAC( J)=SQRTF(WT(J)

CALL PRL2(XC,WT,WFAC,N,6HWEIGHT,6HSORT W)

1605 CALL LMWCOL

GO TO 3000

1615 CALL CALWC

GO TO 1602

1700 CALL SUBl

GO TO 3000

1900 CALL OMIT

60 TO 425

2000 CALL INCLUD

55
GO TO 425

2100 CALL TRAFOR

GO TO 425

2200 CALL POWER

GO TO 425

2300 CALL COMBIN

GO TO 425

2400 CALL SUB2

GO TO 540

2500 CALL REVERS

GO TO 540

2600 CALL INTLAB

GO TO 540

C CALCULATES ROW AND COLUMN SUM OF SQUARES FOR TABLE OF DEVIATIONS

3000 DO 3001 1=1.

3001 Z( I»N+1)=SUM5(Z.I.N)

DO 3002 J=1.N

3002 Z(M+1.J)=SUM3(Z(1.J) »Z(1«J) .M)

Z(M+1»N+1)=SUM2(Z.M+1,N)

XR(M+1)=XNS(22

SUM SO

XC(N+1)=XNS(22) SUM SQ

PRINT 3003. M,N

3003 FORMAT (1H1I3.5H R0WSI9,8H COLUMNS)

CALL PRMXLT(Z.M+1,N+1,XR.XC. TITLE. 19HTABLE OF DEVIATIONS)

GO TO 540

END

SUBROUTINE LMODEL

C FITS AV( I.J)=A{ I)+B{ I)*C( J) Z( I.J)=AV( I.J)-A( I)-B( I)*C( J)

C ASSUMES AVERAGES FOR AV ARE IN AV DIMENSION AV(120.50).Z( 120.50) .XR(120) .XCCSO) . A ( 120 ) .B ( 120 ) .C(50).

1 TITLE (12) .XNS(26) »R ( 99 ) .D ( 120 ) .WT(120) .WFAC(120) .XL ( 120)

COMMON AV.Z.XR.XC.A.B.C.M.N.WT.WFAC.MT.NT.NREP.NR.NX.XNS.TITLE.XY.

IXL.D.R

FM=M

FN=N

CALL LINM

CALL PRL2(XR.A.B.M.1HA.1HB)

CALL PRLKXC.C.N.IHC)

PRINT 101

101 FORMAT(1H010X,20HANALYSIS OF VARIANCE/1H08X.2HOF8X. 2HSS12X.2HMS 12X

1 .IHF/IH ) S2=SUM3(C.C.N)

SSR=0.

SST=0.

SSRXC=0.

RSO=0.

SSL=0.

SSRES=0.

AVDD=AV(M+1.N+1)

DO 10 I=1.M

SST=SST+SUM5(AV»I»N)

SSL=SSL+(B( I )-l. )*»2

RSO=RSQ+(A( I )-AVDD)*(B( I )-l.

10 SSR=SSR+(A( I )-AVDD)**2

RS0=RSQ**2/SSR/SSL

SSM=FM*FN»AVDD**2

SSR=FN«SSR

SSC=FM*S2

SSL=SSL»S2

DO 12 1=1,

DO 12 J=1.N

SSRXC = SSRXC+(AV(I.J)-A( I)-C( J) )**2

Z( I.J)=AV(I.J)-A( I )-B(I )*C( J) 12 SSRES=SSRES+Z( I ,J)**2

SRES=SSRES/(FM-1. )/(FN-2.

CALL PRLI1(5HT0TAL»M*N,SST)

CALL PRLI1(4HMEAN.1,SSM)

SR=SSR/(FM-1,

^6
CALL PRLI3(4HROWS»M-l,SSR» SR. SR/ SRES)

SC = SSC/(FN-1. ) CALL PRLI3(4HC0LS,N-1,SSC» SC » SC/ SRES)

SRXC=SSRXC/(FM-1, ) /(FN-l.

CALL PRLI3(3HRXC.(M-1)*(N-1

tSSRXC* SRXC , SRXC/SRES

SL = SSL/(FM-1. ) CALL PRLI3(5H L IN , M-1 , SSL ,SL ,SL/SRES

SSCN=SSL*RSO

SSNC=SSL-SSCN

SNC=SSNC/(FM-2.)

CALL PRLI3(5H C» 1 SSCN SSCN , SSCN/SNC

CALL PRLI3(6H NC ,M-2 »SSNC ,SNC ,SNC/SRES

CALL PRLI2(5HRESID. (M-l)*(N-2) .SSRES, SRES)

RETURN

END SUBROUTINE LINM

TO DO BASIC CALCULATIONS FOR LINEAR AND QUADRATIC MODELS

DIMENSION AV(120.50)»Z(120»50)»XR(120).XC(50)»A(120)»B(120).C(50).

1 TITLE(12).XNS(26).R(99).D(120).WT(120).WFAC(120)»XL{120)

COMMON AV»Z.XR,XC.A.8.C»M.N»WT.WFAC»MT,NT,NREP»NR,NX,XNS,TITLE»XY,

1XL,D»R

DO 1 J=1,N

1 C( J)=AV(M+1 ,J)-AV(M+1,N+1

S2=SUM3(C»C»N)

DO 2 I=1»M

A()=AV{IfN+l)

2 B( I )=SUMA(AV.C.I N)/S2

RETURN

END SUBROUTINE LMWCOL

WEIGHTED WITH WFAC. Z ( I . J ) =WFAC ( J ) * ( Y ( I » J )-A ( I

) -B ( I )*C { J ) ) DIMENSION AV(12O.5O)»Z(12O.5O),XR(120)»XC(5O)»A(120)»B(12O)»C{5O)»

1 TITLE(12).XNS(26).R(99)»D(120).WT{120) .WFACt 12 0)»XL(120)»U(50).BE

2STAR( 120)

COMMON AV,Z»XR,XC»A»B.C»M»N,WT,WFAC.MT,NT»NREP»NR»NX,XNS»TITLE»XY,

IXL.D.R

CALL LMWCL

FM=M

SBIM1=0.

SAIMAV=0.

DO 106 1=1.

105 SBIM1=SBIM1+(B(I)-1.)*(A(I )-AVA)

106 SAIMAV=SAIMAV+(A( I )-AVA)**2

ALPHA=SBIM1/SAIMAV

PTC0NC=AVA-1. /ALPHA

DO 107 1=1,

107 BESTAR( I ) =1 . +ALPHA* ( A ( I )-AVA)

DO 109 J=1»N

109 Z(M+1 .J)=SUM3{Z(1,J) .Zd »J) »M)

Z(M+1»N+1)=SUM2(Z»M+1 .N)

CALL PRL3 (XR.A,B»BESTAR,M,1HA,1HB,6HB STAR)

CALL PRLl (XCC.N.IHC)

CALL PLINEl (6HALPHA=. ALPHA)

CALL PLINE1(21HP0INT OF CONCURRENCE= ,PTC0NC)

AS=SUM1(A,M)

W=SUM3(WFAC,WFAC»N)

ABAR=AS/FM

AZA=0.

DO 1 I=1,M

1 AZA=AZA+(A( I )-ABAR)**2

SSR=W*AZA

SWC=0.

DO 2 J=1,N

2 SWC=SWC+WFAC( J)**2*C( J)»*2

SSC=FM»SWC

SSF = Z(M+1,N+1)

BZB=0.

37
DO 3 1=1,

3 BZB=BZB+(B( I )-l. )**2

SSS = SWC*BZB

RSO = (SUM3(A,B»M)-AS)**2/(AZA*BZB)

SSCO= SSS*RSO

SSNC= SSS*(1.-RSQ)

SSRC= SSF+SSS

PRINT 110

110 FORMAT (26H1ANALYSIS OF WEIGHTED DATA/7H0S0URCEAX ,2H0F 8X,2HSS12X,

12HMS)

CALL PRLI2 (4HR0WS»M-1,SSR,SSR/FL0ATF(M-1 )

CALL PRLI2 (4HC0LS»N-1,SSC,SSC/FL0ATF(N-1 ) CALL PRLI2 (3HRXC,(M-i)*(N-l ) ,SSRC,SSRC/FLOATF( (M-l)*(N-l) )

CALL PRLI2 (5H FIT. ( M-1 ) * ( N-2 ) ,SSF ,SSF/FLOATF ( ( M-l ) *{ N-2 ) )

CALL PRLI2 (5HSL0PE* M-1 , SSS SSS/FLOATF ( M-1)

CALL PRLI2 (6H CONC » 1 SSCO ,SSCO

CALL PRLI2 (6H NONC »M-2 .SSNC SSNC/FLOATF ( M-2 ) ) CALL PLINEKIOHR SQUARED= » RSO ) CALL PLINE1(2HR=.S0RTF(RS0) ) RETURN

END

SUBROUTINE LMWCL

DOES BASIC CALCULATIONS FOR SUBROUTINES LMwCOL AND CALWC

DIMENSION AV(12O.50),Z{12O»5O),XR(12O),XC(5O).A{120),B(120),C(50),

1 TITLE (12) .XNS<26) .R ( 99 ) ,D (120 ) » WT ( 120 ) ,WFAC(120) .XL (120)

COMMON AV.Z.XR.XC.A.B.C.M.N.WT.WFAC.MT.NT.NREP.NR.NX.XNS.TITLE.XY,

IXL.D.R

FM = M DO 100 1=1.

DO 100 J=1»N

100 Z( I .J)=AV( I .J)*WFAC( J)

DO 101 J=1.N

101 D( J)=SUM1{Z(1.J) ,M)/FM

SW=SUM3(WFAC.WFAC.N)

DO 102 1=1.

102 A( I)=SUM4(Z.WFAC,I.N)/SW

AVA=SUM1 (A.M)/FM

DO 103 I=1»M

103 B(I )=(SUM4{ Z.D.I .N)-SW»A( I

) *AVA ) / ( SUM3 ( D .D.N ) -SW*AVA**2

DO 104 J=1.N

104 C( J)=SUM1(AV(1 .J) .M)/FM-AVA

DO 108 1=1.

DO 108 J=1,N

108 Z( I.J)=(AV( I.J)-A(I )-B( I )»C( J) )*WFAC{ J)

RETURN

END

SUBROUTINE CALWC

ITERATIVE F-PROCEDURE WITH ADJUSTMENT FOR HIGH WEIGHTS

DIMENSION AV(120.50),Z(120,50).XR(120).XC(50).A(120).B(120),C(50).

1 TITLE (12) .XNS(26) .R(99),D(120).WT(120) .WFAC( 120).XL(120).XM(50)»L

2Y(50)

COMMON AV.Z.XR.XC.A.B.C.M.N.WT.WFAC.MT.NT.NREP.NR.NX.XNS.TITLE.XY,

IXL.D.R

FM=M

NUM=25

NUMX=20

DO 1 J=1,N

1 WT(J)=1.

DO 50 LX=1.NUM

DO 17 J=1.N

17 WFAC( J)=SORTF{WT(J)

CALL LMWCL

D02 J=1.N

2 XM( J)=SUM3{Z(1 .J).Z(l.J) .M)/(FM-1,)

W=SQRTF(SUM1 (WT.N)

SUM=0.

58
DO 3 J=1,N

3 SUM=SUM+WT( J)*C( J)**2

SUM=SORTF(SUM)

DO 4 J=1,N

A( J)=WFAC( J)/W

4 B{ J)=WFAC( J)»C( J)/SUM

DO 5 J=1,N

A( J)=l.-A( J)»*2-B( J)*»2

5 WT(J) =WT( J)*A( J) /XM( J)

IF(LX-NUMX)50,50»18

18 DO 19 J=1,N

19 LY(J)=J

N1=N-1

DO 8 J=l Nl

SMALL=WT(J)

JX = J DO 7 K=J»N

IF(SMALL-WT(K) )7»7»6

6 JX = < SMALL=WT(K)

7 CONTINUE

WT( JX)=WT( J)

WT( J) =SMALL

IHOLD=LY(JX)

LY{ JX)=LY( J)

8 LY(J)=IHOLD

FD=N-5

R=L06F(WT(N-2) /WT ( 3 )

NX=(N+l)/2

NY=(N+2)/2

S=0.5»LOGF(WT(NX)*WT(NY)

SF1=EXPF(R* (.5 501+1. 0553 /FD)+S)

SF2=EXPF(R*(.6 5 29+2.5405/FD)+S)

rF(WT(N-l)-SFl)ll»ll,9

9 PRINT lO.LY(N-l) ,WT(N-1) ,SF1

10 F0RMAT(14H0REPLACED WT I2,E15.6,5H BYE15.6)

WT(N-1)=SF1

11 IF(WT(N)-SF2)13»13.12

12 PRINT 10,LY(N) WT(N) SF2

WT(N)=SF2

13 DO 16 J=1»N1

DO 16 K=J»N

IF( J-LY(IC) )16.15.14

14 CALL ENDJOB

15 LY(tC)=LY(J)

HOLO=WT(K)

WT(K) =WT( J)

WT(J)=HOLD

16 CONTINUE

50 CALL PRN1(WT»N.6HWEIGHT)

RETURN

END

SUBROUTINE OMODEL

C SUBROUTINE TO FIT AV ( I , J ) =A ( I )+6 ( I )*C ( J ) +D ( I )*E ( J )

C AV( I,J)=AP( I )+BP( I )*C( J)+D( I )*C( J)**2

C Z(I ,J)=AV( I,J)-A{ n-B( I )»C(J)-D( I )»E(J)

DIMENSION AV (120. 50) »Z( 120*50) »XR( 120). XC(50).A( 120) BC 120 )»C(50).

1 TITLE(12).XNS(26) .R ( 99 ) .D ( 120 ) .WT ( 120 ) ,WFAC( 120 ) .XL { 120 ) . E ( 50 ) .AP 2(120) »BP(120)

COMMON AV.Z.XR.XC.A.B.C.M.N.WT.WFAC.MT.NT.NREP.NR.NX.XNS.TITLE.XY.

IXL.D.R

FM=M

FN=N

CALL LINM

S2=SUM3(C.C.N)

S3=0.

DOl J=1.N

39
1 S3=C( J)**3+S3

DO 2 J=1»N

2 E(J)=C(J)**2-S3/S2*C( J)-S2/FN

Q=SUM3(E»E,N)

DO 4 1=1,

R = 0. DO 3 J=1,N

3 R=AV( I,J)*C(J)**2+R

4 D( n=(R-B( I)*S3-A( I )*S2) /O

DO 5 I=1»M

AP( I )=A(I )-D(I )*S2/FN

5 BP( I )=B( I )-D( I )*S3/S2

AVD0=AV{M+1,N+1

CALL PRL5(XR.A,B,D»AP,BP,M»1HA,1HB,1HD»2HAP.2HBP)

CALL PRL2(XC,C,E.N,1HC,1HE)

PRINT 101

101 FORMAT(1H110X,21H ANALYSIS OF VAR I ANCE/ 1H08X, 2HDF8X .2HSS 12X

1 2HMS/1H ) SST=0,

DO 10 1=1,

10 SST=SST+SUM5(AV,I ,N)

SSM=FM*FN*AVDD»*2

SSR=0.

DO 11 1=1,

11 SSR=SSR+(A( I )-AVDD)*»2

SSR=FN*SSR

SSC=FM»S2

SSRXC=0.

DO 12 1=1,

DO 12 J=1,N

12 SSRXC=SSRXC+(AV( I , J

) -A ( I )-C ( J ) )*»2

SSL=0.

DO 13 1=1,

13 SSL=SSL+(B( I )-l. )»*2

SSL=S2*SSL

SSQ=Q*SUM3(D»D»M)

SSRES=0.

DO 14 1=1,

DO 14 J=1,N

14 SSRES=SSRES+(AV< I , J )-A (I )-B ( I

) C ( J

) -D ( I

) *E ( J ) )**2

CALL PRLU (5HT0TAL,M*N,SST)

CALL PRLIl (4HMEAN,1,SSM)

CALL PRLI2 (4HR0WS,M-i,SSR,SSR/FL0ATF(M-l )

CALL PRLI2 (4HC0LS,N-1,SSC,SSC/FL0ATF(N-1 ) CALL PRLI2 {3HRXC,(M-1)*(N-1 ) ,SSRXC,SSRXC/FL0ATF(<M-1 )*(N-1)

CALL PRLI2 (6H L I N ,M-1 ,SSL ,SSL/FL0ATF ( M-1 ) ) CALL PRLI2 (6H QUAD , M-1 ,SS0,SSQ/FL0ATF ( M-1 )

CALL PRLI2 (6H RES , ( M-1 ) * ( N-3 ) ,SSRES«SSRES/FLOATF (( M-1 ) ( N-3 ) )

DO 20 1=1,

DO 20 J=1,N

20 Z( I,J)=AV( I,J)-A(I )-B(I)*C( J)

DO 21 1=1,

21 Z( I,N+1)=SUM5(Z,I,N)

DO 22 J=1,N

22 Z(M+1,J)=SUM3(Z(1»J) ,Zt 1»J) ,M>

Z(M+1 ,N+1)=SUM2(Z,M+1,N)

XR(M+1)=XNS(22)

XC(N + 1)=XNS(22 ) CALL PRMXLT(Z,M+1,N + 1,XR,XC, TITLE, 33H TABLE OF DEVIATIONS LINEAR M lODEL)

DO 23 1=1,

DO 23 J=1,N

23 Z( I,J)=Z(I,J)-D( I)*E( J)

RETURN

END

SUBROUTINE OMIT

C TO OMIT COLS OR ROWS OR BOTH - READ NX AND XL(K)

C EXAMPLE

40

4l
CCI COLUMN NUMBER

15 25 35 45

C OMIT NXROWl R0W5 C0L3 C0L7

C ANY NUMBER OF ROWS AND COLUMNS IN ANY ORDER MAY BE OMITTED

DIMENSION AV(120.50).Z(120.50)»XR(12O).XC(50)»A(12O)«B(120)»C(50)»

1 TITLE(12).XNS(26).R(99).D(120).WT(120) WFAC( 120) »XL( 120)

COMMON AV»Z.XR,XC»A,B.C>M,N»WT.WFAC.MT»NT,NREP»NR,NX,XNS.TITLE»XY.

1XL,D»R

MT1=MT+1

NT1=NT+1

NR = 1 IF(XL(1)-XNS(21) ) 10. 180. 10

10 XR(M+1 )=XR(MT1 ) XC{N+1 )=XC(NT1 ) DO 20 I=1»MT

AV( I .N + 1 )=AV( I .NTl

20 Z( I.N+1)=Z( I .NTl)

DO 30 J=1.NT

AV(M+1,J)=AV(MT1.J)

30 Z(M+1,J)=Z(MT1,J)

M = MT

N = NT DO 40 K=l,6

40 A(K)=XL(K)

IF(NX-6)65.65.50

50 READ 60. (A(K) ,K=7.NX)

60 FORMAT! 14X»A6.4X.A6»4X,A6»4X.A6.4X.A6»4X.A6)

PRINT 61,(A(K) ,K = 7,NX)

61 FORMAT! 1H014X , A6 .4X . A6 , 4X , A6 . 4X » A6 4X . A6 . 4X » A6

65 DO 150 K=1,NX

DO 70 1 = 1,

IF(A(K)-XR( I ) ) 70.80,70

70 CONTINUE

GO TO 100

80 DO 90 J=1,NT1

AV(M+1,J)=AV( I ,J)

AV( I . J)=AV(M,J)

Z(M+1.J)=Z( I.J)

90 Z( I .J)=Z(M,J)

XR(M+1)=XR( I

XR( I )=XR(M)

M=M-1

GO TO 150

100 DO 110 J=1,N

IF(A(IC)-XC( J) ) 110.130,110

110 CONTINUE

PRINT 120, A(K)

120 FORMAT(1HOA6,13H IS INCORRECT)

CALL ENDJOB

130 DO 140 1 = 1, MTl

AV( I ,N + 1 )=AV( I ,J)

AV( I , J)=AV( I ,N)

Z( I.N+1)=Z( I .J)

140 Z( I.J)=Z( I,N)

XC(N + 1 )=XC( J)

XC(J)=XC(N)

N = N-1

150 CONTINUE

160 PRINT 170. M,N

170 FORMAT(lH0I6,9H ROWS ANDI6,8H COLUMNS)

180 RETURN

END

SUBROUTINE CORECT

C CORRECTS DATA(AV) AND STANDARD ERRORS ( Z ) ,AV=AX*AV+BX Z=Z»AV

C EXAMPLE

C COLUMN NUMBER

C 1 7 15 25

C CORECT

kl
C ALL COL02 1. 2.

C R0W02 COLll 2. C COL03 ALL 2. 5. C END

DIMENSION AV(120»50)»Z(120.50)»XR(120)«XC(50)»A(120)»B(120).C(50)»

1 TITLE (12) ,XNS(26) .R(99).D{120)»WT(120) .WFAC( 120) .XL (120)

COMMON AV»Z»XR.XC»A,B.C.M,N»WT,WFAC.MT»NT»NREP.NR,NX,XNS.TITLE»XY,

IXL.D.R

B ALL=2143A3606060

1 READ 100,XX»XY,AX.BX

100 FORMAT(2A6»2X,2F10.0)

PRINTlOl* XX.XY.AX.BX

101 FORMAT(lH02A6.2X.7HFACTOR=E15.8,5X.9HCONSTANT=E15.8)

IF(XX-ALL)20»2.20

2 IF(XY-ALL)7,3,7

3 DO 4 1=1,

DO 4 J=1,N

AV( I , J)=AX*AV( I,J)+BX

4 Z( I .J)=AX*Z( I .J)

GO TO 1 7 DO 8 J=1,N

IF(XY-XC( J) )8.10.8

8 CONTINUE

9 RETURN

10 DO 11 I=1»M

AV( I J)=AX*AV( I,J)+BX

11 Z( I J)=AX*Z(I»J)

GO TO 1 20 DO 21 1 = 1,

IF(XX-XR( I) )21, 23,21

21 CONTINUE

GO TO 9 23 IF(XY-ALL)27,24,27

24 DO 25 J=1,N

AV(I , J)=AX» AV(I,J)+BX

25 Z( I,J)=AX»Z(I ,J)

60 TO 1 27 DO 28 J=1,N

IF(XY-XC( J) )28,29,28

28 CONTINUE

GO TO 9 29 AV( I,J)=AX*AV( I,J)+BX

Z(I,J)=AX*Z(I ,J)

GO TO 1 END

SUBROUTINE COMBIN

c COMBINES ROWS OR COLUMNS OF AV

c EXAMPLE

C COLUMN NUMBER

C 1 7 15 25 35 45

C COMBINROWS LABOl LAB02 LABIO

c ROWS LAB03 LAB34 LABll

c END

c LABOl AND LAB02 WILL BE COMBINED AND LABELED LABIO

c LAB03 AND LAB04 WILL BE COMBINED AND LABELED LABll

DIMENSION AV(120,50),Z(120,50),XR(120),XC(50),A(120),B(120)»C(50)»

1 TITLE(12) »XNS(26) ,R(99) ,D( 120) ,WT{120) ,WFAC ( 120 ) »XL( 120

COMMON AV,Z,XR,XC,A,B,C»M,N,WT,WFAC,MT,NT,NREP,NR,NX,XNS»TITLE,XY.

1XL,D,R

B ROWS=514666626060

B COLS=234643626060

1 IF{XY-ROWS)2,3»2

2 IF(XY-C0LS)24, 14,24

3 D04 1=3,6

IF(XL( I )-XNS(21) )4»5,4

4 CONTINUE

NR = 5 GO TO 6

42
5 NR=I-2

6 DO 30 J=1,N

30 A(J)=0.

DO 11 K=1»NR

DO 7 1=1,

IF(XL(K.)-XR( I ) )7.8.7

7 CONTINUE

CALL ENDJOB

8 DO 10 J=1,N

9 A(J)=A(J)+AV( I.J)

10 AV( I

,

J)=AV(M,J)

XR(I )=XR(M)

11 M=M-1

M = M+1

FNR=NR

DO 12 J=1.N

12 AV(M,J)=A( J)/FNR

XR(M)=XL(NR+1)

13 READ lOO.XY. (XL(K) .K=l,6)

100 FORMAT(6X,A6»2X,5(A6»4X) .AS)

PRINT 10l,XY,(XL(K) .K=l,6)

101 FORMATdH 6X A6 » 2X 5 ( A6 .4X ) , A6 ) GO TO 1 14 DO 15 I=3»6

IF(XL(I)-XNS(21 ) )15,16,15

15 CONTINUE

NR = 5 GO TO 17 16 NR=I-2

17 DO 31 1=1.

31 A( I)=0.

DO 22 K=1.NR

DO 18 J=1,N

IF(XL(K)-XC(J) )18.19.18

18 CONTINUE

CALL ENDJOB

19 DO 21 I=1.M

20 A( I)=A(n+AV(I ,J) 21 AVI I.J>=AV(I.N)

XC( J)=XC(N)

22 N=N-1

N=N + 1 DO 23 1=1.

23 AV(I.N)=A(I)/FNR

XC(N)=XL(NR+1)

GO TO 13 24 MT=M

NT = N NR = 1 RETURN

END

SUBROUTINE REVERS

REVERSES ROWS AND COLUMNS OF DATA AV FOR MATRIX CURVE FITTING

DIMENSION AV(120,50) ,Z(120»50) .XR(120) .XC(50) . A ( 120 ) .B( 120 ) .C(50)

,

1 TITLE(12).XNS<26).R(99).D(120).WT(120) .WFAC(120) .XL (120)

COMMON AV.Z.XR.XC.A.B.C.M.N.WT.WFAC.MT.NT.NREP.NR.NX.XNS.TITLE.XY,

IXL.D.R

MX=XMAX0F(M.N)

MXP1=MX+1

DO 9 1 = 1. MX HOLD = XR(n

XR(I)=XC(I)

XC(I)=HOLD

M2=I+1

DO 9 J=M2.MXP1

HOLD=AV( I.J)

AV{ I.J)=AV(J.I )

45
9 AV(J,I)=HOLD

IOLD=M

M=N

N=IOLD

RETURN

END

FUNCTION SUMKA.M)

C SUMS OVER ROWS. MAY BE IN THE FORM SUMKA.M) OR SUMl ( A ( I , J ) ,M)

DIMENSION A(lOl)

SUM1=0,

DO 1 1=1,

1 SUM1 = A( n+SUMl

RETURN

END

FUNCTION SUM2(A.I,N)

C SUMS ROW I OF A TWO DIMENSIONAL VARIABLE A OVER COLUMNS

DIMENSION A(120»50)

SUM2=0.

DO 1 J=1,N

1 SUM2=A{I,J)+SUM2

RETURN

END

FUNCTION SUM3(A,B»M)

C SUM OF A PRODUCT OVER ROWS. MAY BE IN THE FORM SUM3(A»B»M) OR

C SUM3(A( I »J) .BtM) OR SUM3 ( A ( I . J ) »B ( I J ) .M

DIMENSION AtlOl) »B( 101)

SUM3 =0.

DOl I=1«M

1 SUM3 =A( I )»B( I )+SUM3

RETURN

END

FUNCTION SUM4(A, B.I.N)

C SUM OF A PRODUCT CONSISTING OF A VECTOR TIMES THE VALUES IN A ROW

C OF A MATRIX - SUMMED OVER COLUMNS

DIMENSION A( 120.50) .6(120)

SUM4 =0,

DO 1 J=1.N

1 SUM4 =A( I,J)»B(J)+SUM4 ~—

RETURN

END

FUNCTION SUM5(A,I.N)

C SUM OF SQUARES OVER COLUMNS

DIMENSION A{120.50)

SUMS =0.

DO 1 J=1,N

1 SUM5 =A(I.J)*»2+SUM5

RETURN

END

SUBROUTINE PLINEKL.X)

C PRINTS 1 LINE OF 1 TO 48 CHARACTERS (1 TO 8 CELLS) AND 1 NUMBER

DIMENSION NFMT(^) ,L(8) »LX(8 ) »FMT(4)

EQUIVALENCE (NFMT.FMT)

C CALCULATE FORMAT (NFMT OR FMT

B FMT( 1 )=740130006010

B FMT( 2)=210673606060

NFMT( 3)=NFRMT( X.l

B FMT(4 )=0167346n6060

CALL 0RDH0L(L,LX,8

PRINT FMT , (LX( J) ,J = 1.8) .X RETURN

END

^4
SUBROUTINE PRMXLT ( A tN »M »XR ,XC .TITLE»NT

C PRINT MATRIX A WITH N ROWS AND M COLUMNS

C MATRIX LABELED WITH TITLE

C DIMENSION STATEMENT MAY BE CHANGED

C ROWS LABELED WITH XR AND COLS LABELED WITH XC

C TITLE(12) IS PRINTED BEFORE NT, THE SUBTITLE

DIMENSION FMT(11)»NT(10),NU(10),A(120.50),XR(120),XC(50).TITLE(12]

1,NFMT( 11

EQUIVALENCE ( FMT .NFMT ) , ( XND . ND

B FMT( 1 1=7^0130606060

B FMT(2)=210673606060

B FMT( 11 )=016734606060

B XL4= 606060606060

B XND= 000000000100

B XNE= 250104330673

B XNF9= 260104331173

CALLORDHOL (NT,NU.10)

PRINT 102

102 FORMATdH ) DO 500 1=1, M,

PRINT 104,TITLE, (NU( J) ,J=1 ,10)

104 FORMAT{ 1H012A6/1H010A6)

J=XMINOF( 1+7, M)

PRINT 106, (XC(K) ,K=I ,J)

106 FORMAT(1H06X,8(6X,A6,2X)

DO 120 <=3,10

120 FMT(K)=XL4

DO 190 K=I ,J

KK=K-I+3

FMT(KK )=XNF9

Al=.0001

A2 = l. DO 170 JX=1 ,10

DO 150 IX=1,N

IF (A(IX,K)) 130,150,130

130 IF (ABSF(A( IX,K) )-Al ) 180,140,140

140 IF (ABSF(A( IX,K) )-A2) 150,150,160

150 CONTINUE

GO TO 190

160 A1=A1*10.

A2=A2*10.

170 NFMT( KK)=NFMT(KK)-ND

180 FMT(KK)=XNE

190 CONTINUE

DO 500 L=1,N,5

LL=XMIN0F(N,L+4)

PRINT 200

200 FORMATdH ) 500 PRINT FMT,(XR(KK) ,(A(KK,K) ,K=I,J) ,KK=L,LL)

RETURN

END SUBROUTINE PRL 1 { LABEL , A ,N , X ) C PRINTS 1 COLUMN HEADED BY X CONSISTING OF N VALUES A WITH

C RESPECTIVE LABELS

DIMENSION LABEL (100) ,A(100) ,NFMT(7) ,FMT(7) X(2)

EQUIVALENCE (NFMT, FMT)

C FMT(1)=(1H FMT(2)=A6, NFMT(4)=10X, NFMT(5)=A6, NFMT(7)=1X)

B FMT( 1 )=740130606060

B FMT(2)=210673606060

B FMT(4)=010067736060

B FMT(5)=210673606060

B FMT(7)=016734606060

B IF(-X(0)) 1,2,1

1 Y=X(0)

GO TO 3 B 2 Y=606060606060

3 PRINT 4, Xd) ,Y,Xd) ,Y 4 FORMAT dH09X,2A6,18X,2A6)

45
NFMT( 3)=NFRMT( A,N)

NFMT( 6)=NFMT(3)

ND2=N/2

J=N-ND2

DO 5 K=1,ND2

KX=K+J

5 PRINT FMT,LABEL(K) ,A(K) ,LABEL(KX) ,A(KX)

IF(J-ND2) 6»7,6

6 PRINT FMT,LABEL( J) »A( J)

7 RETURN

END SUBROUTINE PRL2(LABEL,A.B»N.XA.XB)

C PRINTS 2 COLUMNS HEADED BY XA AND XB CONSISTING OF N VALUES EACH

C OF A AND B WITH THEIR RESPECTIVE LABELS

DIMENSION LABEL (100 )»A( 100 )»B( 100) ,NFMT(9) .FMT(9) »XA(2) »XB(2)

EQUIVALENCE (NFMT»FMT)

B IF{-XA(0)) 1.2.1

1 X=XA(0)

GO TO ^ R 2 X=60606n606060

B 3 IF(-XB(0)) 4,5.4

4 Y=XB( 0)

GO TO 6

B 5 Y=606060606060

6 PRINT 7. XA( 1 ) ,X.XB( 1 ) ,Y.XA ( 1 ) »X.XB(1 ) .Y

7 FORMAT(lH09X.2A6»2X,2A6?18X,2A6,2X,2A6)

B FMT( 1 )=740130606060

B FMT(2)=210673606060

NFMT(3)=NFRMT(A,N)

NFMT(4)=NFRMT(-B.N)

B FMT(5)=010067736060

FMT(6)=FMT(2)

NFMT(7)=NFMT(3)

NFMT(8)=NFMT(4)

B FMT(9)=016734606060

ND2=N/2

J=N-ND2

DO 8 IC = 1.ND2

KX = IC+J

8 PRINTNFMT.LABEL(IC).A(K) ,B(K) .LABEL ( KX ) »A ( KX ) »B(KX)

IF(J-ND2)9»10»9

9 PRINTNFMT.LABEL(J) .A( J) .B( J)

10 RETURN

END

SUBROUTINE PRL3 ( LABEL.A .B.C »N,LA»LB,LC)

C PRINTS 3 COLUMNS HEADED BY LA, LB AND LC CONSISTING OF N VALUES

C EACH OF A, B AND C WITH THEIR RESPECTIVE LABELS

C NFMT(1)=(1H (2)=A6» (6)=10X. (7)=A6. {11)=1X)

DIMENSION LABEL (100) ,A(100) .B(IOO) .CdOO) .NFMT(ll) FMT(ll)

EQUIVALENCE (NFMT.FMT)

B FMT{ 1 )=740130606060

B FMt(2)=210673606060

NFMT(3)=NFRMT(A»N)

NFMT(4)=NFRMT(B.N)

NFMT(5)=NFRMT(C»N)

B FMT(6)«010067736060

B FMT(7)=210673606060

NFMT(8)=NFMT(3)

NFMT(9)trNFMT(4)

NFMT(10)=NFMT(5)

B FMT{11)=016734606060

ND2=N/2

I=N-2»ND2+1

J=N-ND2

PRINT l.LA.LB.LC.LA.LB.LC

1 F0RMAT(1H04X.3(8X,A6) . 16X . 3 ( 8X .Ae )

DO 2 IC='1.ND2

46
I<;X = K +

2 PRINT NFMT,LABEL(K) .A(K),B(K),C(K) .LABEL (KX) . A ( KX ) ,B ( KX ) .C ( <X

GO TO (^.S) .1

3 PRINT NFMT.LABEL( J) .A( J) »B( J) ,C( J) W RETURN

END

SUBROUTINE PRL 5 ( LABEL ,A ,B ,C »D,E»N,LA .LB LC ,LD, LE

C PRINTS 5 COLUMNS HEADED BY LA, LB, LC, LD AND LE CONSISTING OF N C VALUES EACH OF A, B, C, D AND E WITH THEIR RESPECTIVE LABELS

C NFMT(1)=(1H (2)=A6. (8)=1X)

DIMENSION LABEL (99) , A ( 99 ) ,B ( 99 ) ,C ( 99 ) , E ( 99 ) ,NFMT(8) ,FMT(8)

EQUIVALENCE (NFMT.FMT)

B FMT( 1 )=7^0130606060

B FMT(2 )=210673606060

NFMT(3)=NFRMT( A.N)

NFMT(4)=NFRMT(B,N)

NFMT{ 5)=NFRMT(C.N)

NFMT(6)=NFRMT(D.N)

NFMT(7)=NFRMT(E»N)

B FMT(8)=016734606060

PRINT l.LA.LB.LC.LD.LE

1 FORMAT(1H04X,5 (8X.A6)

DO 2 1=1, N.

J=XMINOF( 1+4, N)

PRINT NFMT,(LABEL(K) ,A(IC),B(K),C(K),D(K),E(K).K=I,J)

2 PRINT NFMT

RETURN

END SUBROUTINE PRL 1 1 ( L , INT , A

C PRINTS 1 LINE OF LABEL, INTEGER AND 1 FLOATING POINT NUMBER

C NFMT(1)=(1H {2)=A6,I6, (4)=1X)

DIMENSION NFMT (4) .FMT(4)

EQUIVALENCE (NFMT.FMT)

B FMT( 1 )=740130606060

B FMT(2)=210673310673

NFMT(3)=NFRMT(A.l)

B FMT(4)=016734606060

PRINT NFMT. L. INT.

RETURN

END

SUBROUTINE PRL 12 ( L , INT. A.B

C PRINTS 1 LINE OF LABEL. INTEGER AND 2 FLOATING POINT NUMBERS

DIMENSION NFMT(5) .FMT(5)

EQUIVALENCE (NFMT.FMT)

C NFMT(1)=(1H {2)=A6.I6, (5)=1X)

B FMT( 1 )=740130606060

B FMT(2)=210673310673

NFMT(3)=NFRMT(A.l)

NFMT(4)=NFRMT(B,1)

8 FMT(5 )=016734606060

PRINT NFMT. L.INT. A.B

RETURN

END

SUBROUTINE PRL 13 (L. INT , A.B .C

C PRINTS 1 LINE OF LABEL, INTEGER AND 3 FLOATING POINT NUMBERS

DIMENSION NFMT (6) .FMT(6)

EQUIVALENCE (NFMT.FMT)

B FMT( 1)=740130606060

B FMT(2)=210673310673

B FMT(6)=016734606060

NFMT(3)=NFRMT( A,l)

NFMT(4)=NFRMT( B.l)

NFMT(5)=NFRMT(C,1)

PRINT NFMT.L,INT,A.B,C

RETURN

END

47
SUBROUTINE PRN1(A»M»LA)

C PRINTS COLUMN OF DATA A LABELED WITH NUMBERS M - 1.2.ETC IN

C CONSECUTIVE ORDER AND BY HEADING LA DIMENSION A (100) »FMT(7) .NFMT(7) .LA(2) .MA(2)

EQUIVALENCE (NFMT.FMT)

B FMTd 1=740130606060

B FMT(2)=310673606060

NFMT(3)=NFRMT(A»M)

CALL ORDLAe(LA,MA)

PRINT 1»MA»MA

1 F0RMAT(1H07X.2A6»18X.2A6)

B FMT(4)=010067606060

IF(M-5)2.2.3

2 NFMT( 5)=NFMT(7)

PRINT NFMT,(I»A(I).I-1»M)

GO TO 6

3 NFMT(5)=NFMT(2)

NFMT(6)=NFMT(3)

B FMT(7)=016734606060

MD2=M/2

J=M-MD2

DO 4 IC = 1«MD2

KX=K+J

4 PRINT NFMT,K,A(K),KX»A(KX)

IF(J-M02)6»6»5

5 PRINT NFMT.J»A(J»

6 RETURN

END

FUNCTION NFRMT(A»N)

C CALCULATES A FORMAT TO PRINT A COLUMN OF N NUMBERS WITHIN

C ANOTHER PRINT SUBROUTINE

DIMENSION A(500)

EQUIVALENCE (XNE.NE) .(XNF9»NF9) .(XND»ND)

B XNE=250104330673

B XNF9=260104331173

B XND=000000000100

NFRMT»NF9

Ale.OOOl

A2 = l. DO 10 J=l,10

DO 1 1=1,

IF(ABSF(A( I) )-Al )4,2,2

2 IF(ABSF(A( I) )-A2)l»l,3

4 IF(A( in 11.1.11

1 CONTINUE

GO TO 12

3 A1=A1»10.

A2=A2»10.

10 NFRMT=NFRMT-ND

11 NFRMT=NE

12 RETURN

END

.48
SUBROUTINE 0RDHOL( G, CM)

C ORDERS HOLLERITH CHARACTERS G FROM CALL STATEMENT TO C FOR

C PRINTING, M IS DIMENSION OF C. IF DIMENSION OF G IS LESS THAN

C C IS FILLED WITH BLANKS

DIMENSION G(10) , C(10)

B XMASK=777777777777

K = l DO 1 1=1,

C(I )= G(K)

IF(XMASK-G(K) ) 1,2.1

C INCREMENT K BY -1

1 K = IC + 32767

GO TO 4

2 DO 3 J=I,M

B 3 C( J)=606060606060

4 RETURN

END

SUBROUTINE ORDLAB( G, C) C CALCULATES A COLUMN HEADING FOR

C CHARACTERS ( 1 OR 2 CELLS) GIVEN

DIMENSION G(2), C(2)

B IF(- G(0) ) 1,2,1

1 C(l)= G(l)

C(2)= G(0)

GO TO 3 B 2 C(l) =606060606060

C(2)= G(l)

3 RETURN

END

A TABLE FROM HEADING OF 1 TO 12

IN A CALL STATEMENT

49
ABSP

CALWC

COMBIN

CORECT

ENDJOB

EXPF

LINM

LMODEL

LMWCL

LMWCOL

LOGF

NFRMT

OMIT

ORDHOL

ORDLAB

PLINEl

PRLl

PRL2

PRL3

PRL5

PRLIl

PRLT2

PRLI3

PRMXLT

PRNl

QMODEL

REVERS

SQRTP

APPENDIX C

Functions and Subroutines Used By Program

Absolute Value

Calculates weights by columns for LMWCOL, Uses

LMWCL

Combines into one average 2 or more rows or columns

Corrects averages in the form AX + B and standard

deviations by factor C

End of computation

Exponential

Calculates A, B and C for both LMODEL and QMODEL

Fits y. . = A + BC to averages

Calculates weighted values for A, B and C and a

table of residuals for LMWCOL and CALWC

Weights the columns then fits y. . = A + BC to

averages '^ Logarithm to base e Calculates part of a format statement. Used with

printing subroutines

Omits rows and/or columns, as designated, from

analysis

Orders labels for printing. Used with printing

subroutines

Orders column labels for printing. Used with

printing subroutines

Prints 1 line 1 to 48 characters (l to 8 cells)

and 1 number

Prints 1 column of values with heading and labels

Prints 2 columns of values with their respective

headings and labels

Prints 3 columns of values with their respective

headings and labels

Prints 5 columns of values with their respective

headings and labels

Prints 1 line consisting of label. Integer and a

floating point value

Prints 1 line consisting of

2 floating point values

Prints 1 line consisting of

3 floating point values

Prints a table of values with row and column

headings and a title and subtitle

Prints 1 column of values with heading and

labeled by numbers in consecutive order

Fits y^ . = A^ + B^Cj + D.E^ = a' + B^C^

to averages

Interchanges rows and columns

Square root

label, integer and

label, integer and

+ D^C

J

50
SUMl

SUM2

SUM5

SUM4

SUM5

TRAFOR

XMINOF

Sums over rows. Z^ A. or.Y.^k. . 1=1 1 1=1 1

Sums over columns .Z^ A. . J=l IJ

Sum over rows.ZTA.B. or.r-,A. .B. or.r-,A^ .B. 1=1 i 1 1=1 ij 1 1=1 1,1 1 Sum over columns ^l^ A. .B. j=l iJ J

Sum of squares over columns ^il- A. . J=l ij

See section 2.42

Chooses the smallest value

J iJ

The following subroutines are called in the program to

facilitate its use as a research tool. They are not

included in this note for they are not necessary to the

program's use as outlined in the preceding paragraphs.

CALWR Calculates weights by rows

CMODEL Research model

INCLUD To include data previously omitted

INTLAB A complex subroutine for interlab oratory test

analysis. A note will be written explaining

this process [Reference 2]

LMWROW Weights the rows then fits y. = A+BC to averages

MMODEL Research model -^^ POWER Research tool

SUBl Research tool

SUB2 Research tool

A(l20)

AV(120,50)

B(l20)

C 50)

d(i20)

R(99)

TITLE (12

V/FAC(120

WT(120)

XC(50)

APPENDIX D

Variables Used in Main Program

a. Dimensional Variables

(All are included in COMT/ION)

Row averages

Averages of original data or original single

values per cell

Slopes

Column main effect

Variable in calculations

Replicates per cell; with TRAFOR, transformation

constants

Title to Identify run

Weighting factor = square of v/eight

Weight

Column labels

51
XL(120

XNS(26'

XR(120'

Z(120,50)

M

MT

N

NT

NR

NREP

NX

XY

AX

BX

CX

FM

FN

FNRX

HOLD

I

IS

IX

J

JS

JX

JXX

K

MEMORY

MPl

MTl

NPl

NRX

NTl

PTCONC

TRANS

xcx

XRX

XX

Storage variables

Instructions In Boolean statements

Row labels

Standard deviations for cells with replicate

values; residuals from fitting various models

b. Non-dlmenslonal Variables

In COMMON

Number of rows

Total number of rows

Number of columns

Total number of columns

Original number of replicates per cell

Number of replicates

Variable Integer

Variable label in card columns 7-12

Other

For transforming data

For transforming data

For transforming data

Floating point M

Floating point N

Floating point NRX

Storage variable for ordering

Index for rows

Index for ordering rows

Index for ordering rows -,__ Index for columns

Index for ordering columns

Index for column count; for ordering

Index for column count

Index for replicates; for computed go to

Check for missing values

M + 1

MT + 1

N + 1

Integer in card columns 13-1^; number of

replicates; number of omissions , etc.

NT + 1

Point of concurrence

For transformation instruction

Label in card columns 7-12

Label in card columns 1-6

Label in card columns 1-6

52
Appendix E. Flow Diagrams of Pro^;!- and Subprogi'ams

CT?eai i Print ^

Nff, TITLE J

^ WT(DjwFAOft))

LMWRHiW

t 05? 4

Ca.ll LMW(14>l

Figure 1. Plow Diagram of Computer Program. Numbers

in ovals refer to statement numbers in

Fortran program

53
SUBR(|)UTINE m<pDEL

Start P Call LIM

PRINT A,B,C

and heading

ANALYSIS (pF VARIANCE

Calculate Elements of

Analysis of Variance

and Deviations

f PRINT elements ^

of

\Analysis of Varianc/

> RETURN

SUBR(j)UTINE LINM ®

Start k

D(}) J s 1,N

Cj « AVjj4]_ j - Grand Average

Sura Cj^

Put Row Averages

in Ai^ ®

at) I = i,M

Bl =

J #-1 AVi jC

J

RETURN

Figure 2. A. Flow Diagram for Subroutine LMODEL

B. Flow Diagram for Subroutine LINM

54
SUBR<?UTINE LMIfCL

(To3>

Start

Calculate Table ^ of Weighted Averages

Zij ^ AVi jWFACj

Calculate

Colunin Averages

for Z in Dj

AVA -

M2 A i^A

j^

Calculate Weighted

Ro»r Averages

Ai= ^^ ZijWFACj/SW

I

Calculate

Sum of (WFAC)2

in SW

^l

Calculate Weighted Slopes

N

2ZijDj-(SW)(Ai)(AVA)

M

-2 D^2-(SW)(AVA)

i-1 ^

(Tgg)

Calculate Weighted

Column Main Effect

M

ifl^^iJ

Ca = - AVA

M

RETURN

Calculate Weighted Residuals

2ij=(AVij-Ai-RiCj)(WFACj)

Figure 3. Flow Diagram for Subroutine LMWCL

55
SUBR^UTIIE LMWC(J>L

dog)

start

Call

BIWCL

Calculate - the slope of B on A

& P<|)INT (t)F CONCURRENCE

PRINT A,B,B*,C,

& P0INT (fJF CdJNCURRENCE>

Calculate Elements of

Weighted Analysis of Variance

& Correlation Coefficient

(jo^

Calculate

B'*«i*l 4.0c (Ai - AVA)

^Tg

PRINT ANALYSIS ^r

WEIGHTED DATA -

J heading & elements - R and r2

RETURN

Figure 4. Plow Diagram for Subroutine LMWCOL

56
SUBROUTINE CALWC

®

Start mm = 2^

NUMX^20

Set all

wr(j)^i.

T

DO LX = 1,NUM — —;f (Q) DO J=1,N

1 Call LIMCL 1 WFAc(j)= ^wr(j) ^! DO J=1,N M

M « XM(J) = ^ Zfj/ }^1

M.

DO J = 1,N

Aj = 1.-

-.2 WFAC^

ni

- BCalculate new WTj

by "WT^AV XM.

4-

(B

DO J=1,N

CjWFACj

!zFrjCj2

O)

Set all

LY(J) = J

Order Wts from

Small to Large

Calciilate Estimated

Values for 2 Largest

Weights - SFl ^ SF2

PRINT - REPLACED

Wr(N - 1) BY SFlD r<

wrCN-1) = sFi

PRINT - REPIACED

Wr(N) BY SF2

M'(N) =SF2

Check for Order of

^

r

I Weights & ERROR

Call ENDJOB

* Any numbers may be used

hereo These gave optimum

RETURN

results in repeated experiments.

Figure 5. Flow Diagram for Subroutine CALV/C

57
SUBR0UTI1E QMJ)DEL

Start

Call

LIM

®

rxt) i = i,M

Di:.

jAVikCk2-Ai2Ck2-Bi2Ck3

D(|) I = 1,M

Ati n Ai -

B'i= B^-

k

N

DiZCk^

k

/ PRINT Elements of j^

V Analysis of Variance

f^Z.O) Calculate

Table of Deviations

for Linear Model

ZijcAVij-Aj-BjCj

I

D(t) J=1,N

E -i — C -i — \J n J J ^ 2 "^ k

^

5^1c^ C-l

^¥ N

PRINT

.A,B,D,A',B',C,E

Cioi)

PRINT

heading

ANALYSIS 0F VARIANCE

Cfo)

Calculate Elements

of Analysis of Variance

C^g;

Calculate

Table of Deviations

for Quadratic Model

Zij = AVij-Aj,-BiCj-DiEj

(PRINT

Table of Deviations

for Linear Model

I

RETURN

Figure 6. Flov; Diagram for Subroutine QMODEL

58
SUBR(|)UTINE (1)MIT

Start

ri[Tl = MT-l

READ NX Rows

and Columns to

be omitted. Store

in A(K)

Replace Stored Row

& Column Headings

& Respective Data

in RIplfMTl

and/or C^imi

Dlj) K = l, NX

D0 J=1,NT1

AV(M+1,J) sAV(I,J)

AV(I,J) = A(M,J)

XR(M+1) =.XR(I)

XR(I)=XR(M)

M=M-1

dB)

D(J) IrrljMTl

AV(I,I*fl) = AV(I,J)

AV(I,J)=AV(I,M)

PRINT A(K)

IS INC(1)RRECT

XC(N+1)=XC(J)

XC(J)sXC(N)

N = N-1

I

Call

ENDJ^B

@) i

PRINT M R^i/?S

N C(|)LUMNS

(2B> i

RETURN

Figure 7. Flow Diagram for Subroutine OMIT

59
SUBROUTINE TRAF(t)R

Start

READ Factors

& Constants

PRINT Formula

SUBR(|JUTINE C^RECT

Start

Correct ALL

AVs & SDs

@

Correct AVs & SDs

for ALL WflS in

Designated COLUMN

&

Correct AVs & SDs

for ALL COLUMNS in

Designated R(^W

®

Correct AV & SD

for Designated

Cell

READ & PRINT

XX, XY, Factor AX,

Constant BX

Figure 3. A. Plow Diagram for Subroutine TRAFOR

B. F10V7 Diagram for Subroutine CORECT

60
sdbrjIjutine revers

start 1^ Selects

Larger

of M or N

Reverses

Averages

yr

iL

'f Exchanges XC(I)

for XT?!'!') arsH Reverses ^ RETURK

visa versa

M & N P

SUBR^'UTINE C^JffilN

Start

Combine Rows

Listed in XL(K)

Regroup & Relabel

Rows

(0)

f KERAD & PRINTA

Combine Columns

Listed in XL(K)

Regroup & Relabel

Columns

I Next Card

J

RETURN

Figure 9. A. Flow Diagram for Subroutine REVERS

B. Flovi Diagram for Subroutine COMBIN

61


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