STATISTICS
A. THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS
Head of Department: Professor M. H . Belz STAFF
Full-time Officers
Professor MAURICE HENRY BELZ, Chevalier dc la Legion d'Honneur, B.Sc. (Svd.), M.Sc. (Cantab.), M . A , F.A.S.A, F.I.S.I.
Senior Lecturer, RUPERT THOMAS LESLIE, B.Sc, M . A , Ph.D.
Lecturer, BRIAN LINDSAY ADKINS, B.Sc. (Q'land) Senior Tutor, GEORGE W I L L I A M ROCERSON, B . A , B.Sc.
Tutor, WARREN JOHN EWENS, B.A.
Research Assistant, BETTY LABY
The acute shortage of staff, mentioned in the last Annual Report, has imposed on the Department a burden of teaching, consulting and administrative activities so heavy that little research into the mathematical theory of Statistics was possible. A considerable amount of consultative advice on statistical aspects of research problems was given to other University Departments, as well as to other educa- tional organizations and to industrial concerns.
Interest in Operational Research continues to grow, the newly- formed Operational Research Society of Victoria being closely linked with the Department. I n July 1958 a seminar on the use of statistical methods in management was again offered to thc Assistant Secre- taries' Conference, Canberra, by invitation of the Commonwealth Public Service Board. Under the same auspices, a week-end residential appreciation course in Operational Research was offered in August 1958 to senior executives in the various sections of the Commonwealth Public Service. In October a visit was paid to Kalgoorlie to advise one of the mining companies on aspects of its sampling procedures.
A. THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS
(1) Dual Interpretation of Random Variables Workers: R. T. Leslie and F. E. Binet
It is well known that the negative binomial distribution is sus- ceptible of interpretation as a compound Poisson distribution or as a contagious distribution. Work has been proceeding on further dual interpretations of random variables with respect to the underlying stochastic process; for example, recurrent and compound, recurrent and contagious, contagious and compound variates. A paper dealing with a portion of this problem was presented at the Australian Mathematical Society meeting in Adelaide, 1958, under the title 'Recurrent events generated hy clustering of rhe events of a primary stochastic process'.
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(2) Distribution of waiting times
Workers: R. T. Leslie and F. E. Binet
It has been shown that in non-trivial compound processes con- secutive waiting times cannot bc both independently and identically distributed.
(3) Some applications of Monte Carlo methods using C S I R A C Worker: B. L. Ad kins
The solution of problems by Monte Carlo methods requires con- sideration of three points.
(i) Choice of the probability model to bc used.
(ii) Generation of random variables from given probability dis- tributions.
(iii) Methods of increasing the efficiency of thc estimates obtained from thc sampling process.
Thc use of high-speed computers has tended to make variance- reducing techniques relatively less interesting than hitherto, but has by no means eliminated their usefulness. However, the development of the Monte Carlo method has depended on access of the worker to a computer, and thc present project is designed to test initially the suitability of CSIRAC for work of this type.
Thc starting point in this investigation is the classical Gambler's Ruin problem, in which the probability of ruin is, in Feller's notation, the solution of the difference equation
q. = p-q,+i + q-q^t, i<z<«-i,
with boundary conditions q0=\, qu = o, where the combined wealth of the two gamblers is a and one starts with wealth z.
Under certain limiting conditions, but with thc same boundary conditions, this difference equation goes formally to the differential
equation d2u du _
dx' + ~a dx ~ °"
A t the same time, the solution of the difference equation rigorously approaches that of thc differential equation.
The development from the probability problem to the functional equation is now reversed, i.e. it is now considered that a solution"of the differential equation is obtainable from that of a random walk in one dimension with absorbing barriers.
A programme has been devised, and tested, to do this using CSIRAC, and the practicability of the Monte Carlo method for the solution of problems of this type is being investigated. Use is made of a standard programme for thc generation of random numbers.
The work is being extended to the investigation of thc efficiency of the two-dimensional random walk in solving thc Laplace equation.
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(4) Factor analysis
Worker: G. W. Rogerson
A n assessment of the more common methods of calculating factors, Thurston's Centroid method and the method of Principal Com- ponents, has been followed by use of CSIRAC i n thc extraction of principal components for some industrial data. A programme has been developed for extracting cigen values and vectors from a sym- metric matrix, resulting in the reduction of the associated arith- metical labour to manageable proportions. The method has been applied to data with four measured characteristics, two of thc four principal components seeming to account for most of thc variation.
One of the chief problems seems to be that of deciding on the number of factors that should be taken into account before allotting the remaining variation to residuals. Some approximate significance tests have been suggested and it is hoped to develop a satisfactory test along sequential lines.
(5) Applications of the Logarithmic-Normal Distribution Workers: M. E. Joseph and M. H. Belz
A n investigation of thc logarithmic-normal distribution for esti- mation purposes in connection with gold-mine sampling has been commenced.
B. APPLICATIONS OF STATISTICAL METHODS . Computers: Gwenda E. Macdonald and Johanna H. Ronayne (6) Measurements on Bite Plate Treatment
Referred by R. C. Case (Department of Conservative Dentistry) Workers: G. W. Rogerson and Betty Laby
Children suffering from overbite had been divided into matched pairs. One member of each pair was treated, the other studied as a control.
Measurements of overbite were analysed for temporary and for permanent improvement in the treated children. Changes in various other facial measurements were analysed in an attempt to explain the effects of the treatment. Thc difference in permanent change between treated and control children was significant for mandibular plane angle for both male and female children, for upper incisor height for females, and for anterior face height for males, but not significant for any other measurements.
(7) The Effect of Physical Training on Amino-acids in Rat Muscle Referred by M. K. Goidd (Department of Biochemistry) Worker: G. W. Rogerson
Experiments were carried out to study thc effect of swimming on
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the levels of amino-acids in rat muscles, for tiained and untrained rats. (The data were analysed on a factorial basis.)
Early experiments gave inconclusive results. A later experiment using 80 rats from 10 different litters showed that swimming increased the level of amino-acid by 5'8 per cent, and this increase was the same for trained and untrained rats. Trained rats gave results 5-2 per cent higher than untrained rats.
(8) Carbon Translocation in the Soybean
Referred by (Mrs) Stella L. Ovenden (Department of Botany) Workers: R. T. Leslie and G. W. Rogerson
The effects of masking or removing leaves from soybean plants, and of masking the apices of the plants, on rate of movement of carbon compounds were measured using a Balanced Incomplete Block experiment. Analysis of the results showed that masking the apex or the side leaves of a plant did not affect the rate of carbon movement to the apex. However, cutting off the side leaves increased the rate of movement by about 60 per cent. Differences between blocks containing the same treatments were significant at the 5 per cent level. Thus some reduction in the error mean square was obtained by carrying out the work in blocks.
(9) Relation of Vital Capacity and Time Taken to Run a Mile Referred by A. W. Willee (Department of Physical Education) Worker: G. W. Rogerson
Analysis of measurements of vital capacity and time taken to run one mile, on first-year and second-year Physical Education students, showed no evidence of any correlation for either year.
(10) Comparison of Isotonic and Isometric exercises of Physical Education Students
Referred by A. W. Willee (Department of Physical Education) Worker: G. W. Rogerson
Strengths of leg muscles in thc inner and outer ranges were measured for a number of Physical Education students. These were divided into three groups, one being subjected to isometric exercises, another to isotonic exercises, and thc third to no special exercises so that it served as a control. Analysis of increase in strength failed to reveal any differences between the two types of exercise.
A regression of length of kick on leg-strength, using total strength of the preferred leg, was significant for second-year students but not for first-year students. Regression of High Jump and Broad Jump performance on total inner range and total outer range muscle strengths showed a significant relationship between inner range strengths and both jumps.
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(11) Intermediate Physical Education Examination Standards Referred by A. W. Willee (Department of Physical Education)
Worker: G. W. Rogerson
Standards for Intermediate Physical Education performances were calculated from data provided by various schools. These were to bc used for future examination-marking.
(12) Casual Arterial Blood Pressure in the People of Fiji
Referred by R. R. ti. Lovell and I. Maddocks (Department of Medicine)
Workers: G. W. Rogerson and Betty Laby
Blood pressure measurements on Fijians and Indians were analysed to establish relationships between systolic and diastolic pressures and age, race and sex. The data were also compared with published data for Londoners.
Rates of increase of systolic and diastolic pressures with age were higher for Indians than for Fijians. Thesc rates of increase were less than that derived from published data on Londoners. The pressures of male subjects increased less steeply than those of females. This difference was consistent for all races.
(13) Activity of the Thyroid Gland
Referred by R. E. Loughlin (Department of Biochemistry) Worker: G. W. Rogerson
The effect of Thyrotrophic hormone and Methyl-Thiouracil on 'Cysteinyltyrosinase' activity of thyroid glands in rats was measured.
A n analysis of variance showed that thc activity was higher for treated rats than for controls. This difference was highly significant for both treatments.
(14) Tutorial Aid to First-year University Students
Referred by D. S. Anderson (Educational Research Officer) Worker: R. T. Leslie
Further examination was made of thc performance of test and control groups in thc tutorial experiment of 1957 in the light of examination results in the Matriculation and University first-year examinations. This work is continuing.
(15) Pregnancy and Hypertension
Referred by S. L. Townsend (Department of Obstetrics and Gynae- cology )
Workers: R. T. Leslie and M. H. Belz
Thc influence of hypertension and of its treatment during preg- nancy was investigated with respect to foetal outcome. Thc methods
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of treatment used in 1956 were compared with those used in 1946-47 by considering various aspects of pregnancy, labour and foetal out- come. Treatment i n 1956 was characterized by more regular ante- natal visits, control of weight gain, restriction of salt intake when necessary, hypotensive drugs when necessary and adequate facilities for in-patient treatment.
The significant results included a reduction in the number of thera- peutic abortions, in the blood-pressure at the end of pregnancy, in the incidence of recurrent proteinuria, in thc incidence of prema- turity, in foetal wastage, in peri-natal mortality, in the number of small babies and in the incidence of proteinuria.
(16) Ventilatory Capacity Studies on Children
Referred by B. H. Gandevia (Royal Children's Hospital) Worker: R. T. Leslie
The relationship between several factors associated with ventilatory capacity and thc age and stature of the child has been investigated.
The following equations were obtained:
FEV 1%= 113-9 —0-561 Standing height ^1 for FEV 1%= 96-2-0-104 Age J females FEV 1%= 87-1-0-098 Standing height "1 for FEV 1%= 83-9-0-017 Age J males (17) Germination Time of Algae
Referred by Mary M. Ross (Department of Botany) Workers: B. L. Adkins and R. T. Leslie
Experiments have been designed and analysed to determine the effect of sterilization and anaerobic conditions on germination times of spores of Chara. Two different techniques for obtaining anaerobic conditions (evacuation and nitrogen) were tested and the difference between them was not significant. The effect of sterilization of spores was highly significant, as was the effect of evacuation.
(18) Studies in Asbestos
Referred by B. H. Gandevia (Department of Medicine) Worker: R. T. Leslie
The following factors likely to affect pulmonary function of two groups of workers exposed to varying amounts of asbestos dust and a control group were investigated: severity of exposure, smoking habits, and age. Presence of asbestos bodies in sputum, X-ray diagnosis, and measures of ventilatory capacity were taken as in- dicators of pulmonary function. The investigation involved analysis of covariance in a two-way classification with disproportionate cell frequencies. The dependence of pulmonary function on age was
S T A T I S T I C S 63
found to be significant; there was no difference between the three groups in the measures of ventilatory capacity employed.
(19) Body Hair Distribution Related to Age
Referred by R. A. Melick (Department of Medicine) Worker: R. T. Leslie
In order to establish a norm in body hair distribution in male and female subjects in age groups above 60, studies have been carried out relating hair density to age. This has involved the investigation of linear trends in percentages, which showed significant results in 5- out of 7 area categories for both sexes.
(20) Effectiveness of Hydrocortisone in the Treatment of Allergic Rhinitis
Referred by R. IT. O. Donald (Royal Melbourne Hospital) Worker: R. T. Leslie
An experiment involving two preparations of hydrocortisone snuff and controls has been designed and the results analysed. The experi- ment has been reported and will bc published in the Australian Medical Journal.
(21) Measurement of Change of Cardiac Output Induced by Cold Stimulation
Referred by J. R. E. Fraser (Department of Medicine) Worker: R. T. Leslie
The rise in B.P. following exposure of a limb to intense cold has been attributed in the past to local constriction of the blood vessels.
A n alternative explanation now advanced is increase in cardiac out- put. Apparently not all subjects react with such an increase; there is little doubt that a category of high reactors does exist.
Tolerance limits were established for control observations and the significance of the changes observed during the exposure to cold were assessed.
(22) Pulping of Pine
Referred by N. E. Norman (Australian Paper Manufacturers) Worker: B. D. Craven in association with members of the Depart-
ment
An analysis was made of a scries of pilot mill trials, in which the factors time, temperature, and chemical concentration were varied, and the effects measured in several properties of thc resulting pulp.
The combination of conditions chosen was such as to allow an exploration of the response surface by means of an orthogonal composite design. The analysis led to a prediction of the pulping conditions for optimum pulp quality.
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(23) Statistical Problems in Gold-mine Sampling Workers: M. E. Joseph and M. H. Belz
A visit was made to the Kalgoorlie gold-fields to investigate sta- tistical problems associated with the current procedures in grab, chip, channel and diamond-drill sampling. Experiments were de- signed and carried through for determining the sources of varia- bility introduced by crushing and splitting during the preparation of samples for assaying, thc effects of human error in weighing, and the effect and extent of bias in the selection of small-sized particles in grab samples. The investigation is continuing.
THESES PASSED FOR HIGHER DEGREES
DOCTOR OF PHILOSOPHY
i. LESLIE, R. T. Statistical Treatment of Observer Effects in Scientific Experiments
The intrinsic accuracy of various types of measuring instruments has been studied widely, and limitations to this accuracy are usually taken into account when design- ing or selecting an instrument for a specific purpose. Tt is usually assumed that these limits will bc realized in actual use. However, unless precautions are taken, personal idiosyncrasies and aberrations may introduce bias and variability of high order into a set of observations. Examples of such phenomena arising in thc use of a wide range of instruments and in a variety of observing situations are considered, and remedies arc discussed. In particular the problem of graticules and reference marks for optical instruments is considered and the accuracy of various designs is investigated experimentally. It is shown that a new combination, referred to as the "split-grating"
design, is superior to conventional designs and is, moreover, applicable to automatic machines and servo-mechanisms.
Intensive investigation of the visual criteria employed by observers in making settings with optical instruments shows that different observers employ different criteria, with varying degrees of success. These criteria bear little relation to the apparent graticule-reference mark configuration but depend essentially on artefacts produced bv the intimate structure of local regions of thc field of view. Systematic examination of thesc phenomena might yield useful results, but this line of research was dropped in favour of the more general approach of information theory applied to both psychological and physiological functions.
Applications of communication theory to psychological situations are critically reviewed, and a new interpretation is given of recent work taking account of signal repetition as a means of coding to realize thc channel capacity of the system in error-free transmission. It is suggested that action potential discharges mav he under- stood in terms of this concept. Also, a statistical model is given to account for some of the phenomena of visual discrimination (joint work with R. L . Gregory, Cambridge Psychological Laboratory). Some interesting consequences concerning the nature of noise in thc central nervous system arc mentioned.
Finally, a probability model is suggested for thc discharge of retinal receptors.
Starting from thc fact that light quanta falling on a fixed receiving area are known to follow a Poisson process, it is postulated that a receptor has, for a given mean illumination, a fixed integration time (y). Furthermore, a discharge will occur provided a certain number (k) of quanta arrive, no two quanta being separated in time bv more than y. This mean discharge frequency is obtained for given light intensity oil this model. This is derived from the Laplace Transform of the distribution of time intervals between discharges, which is obtained in two ways—dircctlv bv the sum- mation of random variables, and indirectly as the limit of a Bernoulli process. The inversion of the Laplace Transform to yield the distribution is not given, as this is tedious and complex, and for physiological purposes the main interest centres in thc mean discharge frequency.
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MASTER OF ARTS
DOIG, ALISON G. Thc Elementary Theory of Linear Program- ming and its Application to the Integral Solution of an Industrial Problem
WATTERSON, G. A. Ordered and Censored Samples from a Multi- variate Normal Population
HEATHCOTE, C. R. Some Methods Applicable to thc Study of Birth and Death Processes, with Particular Refer- ence to the Logistic Process