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]]>This is a function to find the associated probability of a value (FR) of an

F Ratio

With DA representing Degrees of Freedom Above

And DB representing Degrees of Freedom Below

It returns -999 if any input parameter is less than or equal to zero.

It can also be used to evaluate

Chi-Squared

With DA representing the degrees of freedom

DB set to as large number as possible (999 is probably sufficient)

FR set to the chi-squared value divided by the degrees of freedom DA

Student’s “t”

With DA set to unity (1)

DB set to the degrees of freedom

Fr set to the value of t squared

A normal variate

With DA set to Unity

DB set to as large a number as possible

FR set to the square of the normal variate

(The function GAUS, which is called by PRBF, gives this probability directly.)

It is based on

Jaspen, N., “The calculation of Probabilities corresponding to Values of Z, t, F, and Chi-squared” Education and Psychological Measurement, 1965 Vol 25, pp 887-880

This function is written in ASA FORTRAN IV – but it should be fairly easy to translate into a more modern language.

Enjoy

FUNCTION PRBF(DA,DB,FR)

C PRBF (SEE 11.8) STATISTICAL FINDS PROBABILITY OF F-RATIO

C DA = DF ABOVE,DB=DF BELOW,FR=F RATIO – PRBF=-999 =OUT OF RANGE

PRBF = -999.

IF(DA.LE.0.0.OR.DB.LE.0.0.OR.FR.LT.0.0) RETURN

PRBF = 1.

IF(FR.EQ.0.) RETURN

LI = 500

IF(DA.EQ.1.0.AND.DB.GT.LI) X = SQRT(FR)

IF(DA.EQ.1.0.AND.DB.GT.LI) GO TO 15

FT = 0.

PRBF = 0.

X = DB/(DB + DA*FR)

NA = DA + 0.5

NB = DB + 0.5

VP = NA + NB – 2

FA = NA

FB = NB

IF(NA.EQ.NA/2*2.AND.NA.LT.LI) GO TO 1

IF(NB.EQ.NB/2*2.AND.NB.LT.LI) GO TO 4

IF(NA+NB.LT.LI) GO TO 6

GO TO 14

C FA EVEN AND LESS THAN LI (THE LIMIT FOR ITERATION)

1 XX = 1.-X

2 FA = FA – 2.

IF(FA.LT.1.) GO TO 3

VP = VP – 2.

FT = XX*VP/FA*(FT+1.)

GO TO 2

3 FT = X**(.5*FB)*(FT+1.)

IF(FT.GT.0.)PRBF = FT

RETURN

C FB EVEN AND LESS THAN LI (THE LIMIT FOR ITERATION)

4 FB = FB – 2.

IF(FB.LT.1.) GO TO 5

VP = VP – 2.

FT = X*VP/FB*(FT+1.)

GO TO 4

5 FT = 1. – (1.-X)**(0.5*FA)*(FT+1.)

IF(FT.GT.0.)PRBF = FT

RETURN

C FA AND FB ODD – FA+FB LESS THAN LIMIT FOR ITERATION

6 TH = ATAN(SQRT(FA*FR/FB))

ST = SIN(TH)

CT = COS(TH)

C2 = CT*CT

S2 = ST*ST

A = 0

B = 0

IF(DB.LE.1.) GO TO 9

7 FB = FB – 2.

IF(FB.LT.2.) GO TO 8

A = C2*(FB-1.)/FB*(1.+A)

GO TO 7

8 A = ST*CT*(1.+A)

9 A = TH + A

IF (DA.LE.1.) GO TO 13

10 FA = FA – 2.

IF (FA.LT.2.) GO TO 11

VP = VP – 2.

B = S2*VP/FA*(1.+B)

GO TO 10

11 GF = 1.

K = DB*0.5

DO 12 I = 1,K

X1 = I

12 GF = X1*GF/(X1-0.5)

B = GF*ST*CT**DB*(1.+B)

13 FT = 1. + 0.636619772368*(B-A)

IF(FT.GT.0.)PRBF = FT

RETURN

C FA AND/OR FB EXCEED LIMITS

14 FA = 2./9./FA

FB = 2./9./FB

CR =FR**0.333333333333

X = (1.-FA +(FB-1.)*CR )/SQRT(FB*CR*CR + FA)

15 FT = GAUS(X)

IF(FT.GT.0.) PRBF = FT

RETURN

END

]]>

Function to derive the associated probability of a value on a Normal distribution with Arithmetic mean 0 and Standard Deviation 1

From my book STATCAT – no home should be without one.

This function is based on based on a numerical approximation, since the normal distribution cannot be integrated. (Don’t ask me why, I only work here.)

The numerical approximation was based on an ALGOL program by D. Ibbotsen (ACM Collected Al;gorithms No. 209 , 1963). The original ALGOL formulation has been modified to reduce the rounding errors associated with a negative value of X.(If you estimate the size of the tail of the distribution as, say, 3.141 x 10-9 and subtract this from 1.0 you will get a row of 9s or even 1.0, which doesn’t help much.)

Most computers will not cope with all twelve significant figures, but someone might need them.

This function is written in ASA FORTRAN IV – originally punched on cards, the lines beginning 1,2,3,4,5 are extensions of the previous line, if anyone cares.

FUNCTION GAUS(X)

C GAUS (SEE 11.8 ) STATISTICAL FINDS ASS. PROB. OF N VALUE

IF(X.EQ.0.) GAUS = 0.5

IF(X.LT.-6.)GAUS = 0.0

IF(X.GT.6.) GAUS = 1.0

Y =ABS(X)*0.5

IF(X.EQ.0..OR.Y.GT.3.)RETURN

IF(Y.GT.1.) GO TO 15

W = Y*Y

Z =((((((((0.000124818987 *W – 0.001075204047)*W + 0.005198775019)

1 * W – 0.0191982914)*W + 0.059054035642)*W – 0.151968751364)

2 * W + 0.319152932694)*W – 0.53192171)*W + 0.797884560593)

3 * Y

IF(X.LE.0.) GAUS = 0.5 – Z

IF(X.GT.0.) GAUS = 0.5 + Z

RETURN

15 Y = Y – 2.

Z = (((((((((((((-0.000045255659*Y + 0.000152529290)*Y

1 – 0.000019538132)*Y – 0.000676904986)*Y + 0.001390604284)*Y

2 – 0.0007948820)*Y – 0.002034254874)*Y + 0.006549791214)*Y

3 – 0.010557625006)*Y + 0.0119447319)*Y – 0.009279453341)*Y

4 + 0.005353579108)*Y – 0.002141268741)*Y + 0.0005353849)*Y

5 – 0.000063342476

IF(X.LE.0.) GAUS = – Z * 0.5

IF(X.GT.0.) GAUS = 1.+ Z*0.5

RETURN

END

]]>

The sixty-four groups of six digits following form a Hyper-Greco-Latin Cube. You can consider the last three columns to represent the X, Y and Z coordinates of the cube. In an experimental design, each of the six columns can be used to denote an experimental variable at four levels. Each level of each variate occurs sixteen times with each level of each of the others, and (I think) four times with each pair of values of two other variates.

You can then analyse whatever variate you are measuring with a classic Analysis of Variance, providing six factors with three degrees of freedom, and 45 degrees of freedom for the residual.

(I don’t need to say that you would, of course, assign the four levels of each variate at random, using a suitable random number source)

1 1 1 1 1 1 3 4 4 1 2 1 4 2 2 1 3 1 2 3 3 1 4 1

4 3 4 2 1 1 2 2 1 2 2 1 1 4 3 2 3 1 3 1 2 2 4 1

2 4 2 3 1 1 4 1 3 3 2 1 3 3 1 3 3 1 1 2 4 3 4 1

3 2 3 4 1 1 1 3 2 4 2 1 2 1 4 4 3 1 4 4 1 4 4 1

4 4 3 1 1 2 2 1 2 1 2 2 1 3 4 1 3 2 3 2 1 1 4 2

1 2 2 2 1 2 3 3 3 2 2 2 4 1 1 2 3 2 2 4 4 2 4 2

3 1 4 3 1 2 1 4 2 3 2 2 2 2 3 3 3 2 4 3 2 3 4 2

2 3 1 4 1 2 4 2 4 4 2 2 3 4 2 4 3 2 1 1 3 4 4 2

2 2 4 1 1 3 4 3 1 1 2 3 3 1 3 1 3 3 1 4 2 1 4 3

3 4 1 2 1 3 1 1 4 2 2 3 1 3 2 2 3 3 4 2 3 2 4 3

1 3 3 3 1 3 3 2 2 3 2 3 4 4 4 3 3 3 2 1 1 3 4 3

4 1 2 4 1 3 2 4 3 4 2 3 1 2 1 4 3 3 3 3 4 4 4 3

3 3 2 1 1 4 1 2 3 1 2 4 2 4 1 1 3 4 4 1 4 1 4 4

2 1 3 2 1 4 4 4 2 2 2 4 3 2 4 2 3 4 1 3 1 2 4 4

4 2 1 3 1 4 2 3 4 4 2 4 1 1 2 3 3 4 3 4 3 3 4 4

1 4 4 4 1 4 3 1 1 4 2 4 4 3 3 4 3 4 2 2 2 4 4 4

]]>

Published in Contemporary Ergonomics and Human Factors 2016

Eds Waterson, P. Sims, R., Hubbard E-M ISBN 978-0-9554225-9-1

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]]>Originally appeared in “Contemporary Ergonomics and Human Factors 2016 ” Eds Waterson, A., Sims, R., and Hubbard, E.M.

ISBN 978-0-9554225-9-1

For the full text in PDF format click HERE

]]>

It presents a synthesis of work carried out in France on decision making in the activities of air traffic controllers, considered as an archetype of an expert task. These experimental studies illustrate a synthetic presentation of the principles of the theoretical background to representation. Examples are also given of practical applications in training and in the ergonomics of computer assisted training.

This work is intended for researchers, instructors and students of cognitive psychology as well as practising trainers or ergonomists. It will also interest other disciplines in the cognitive sciences which use ideas of representation in other senses. Finally, the staff of the air traffic control system, and particularly the controllers, will find, or rediscover, a different point of view on the much misunderstood skill of the air traffic controller.

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]]>

Potential conflicts, conflict resolution orders and differences in subsequent traffic generated are investigated.

Samples begin to differ about three minutes after the start, but are not completely different until about 30 minutes after the start.

Published in Contemporary Ergonomics 2004, Taylor and Francis, Ed. P.T. McCabe ISBN 0-8493-2342-8 pp.298-302

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]]>Appeared in CREATE2007 – Proceedings of the Conference on Creative Inventions, innovations and Everyday Designs in HCI 13-14 June 2007 London UK Edited David Golightly, Tony Rose, B.L.William Wong and Ann Light Pages 97-102

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]]>Published in Contemporary Ergonomics 1998, Taylor and Francis,1998, ISBN 0-7484-0811-8 p429-433

To read the complete article in PDF format click HERE

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