Tractors in Spain, Further Logistic Analysis

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Tractors in Spain: A Further Logistic Analysis

Author(s): F. R. OliverSource: The Journal of the Operational Research Society, Vol. 32, No. 6 (Jun., 1981), pp. 499-502Published by: Palgrave Macmillan Journalson behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/2581538.

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J.OplRes. oc.Vol. 2.pp.499 o502.1981 0160-5682/81/060499-04S02.00/0Printedn Great ritain.llrightseserved Copyright 1981Operationalesearchociety td

Tractors in Spain: a Further Logistic AnalysisF. R. OLIVER

University of Exeter

The three parameter logistic growth function is known to give a good fit to annual statistics of thenumber of tractors in Spain. A parameterisation directly yielding the upper asymptote or saturationlevel is advanced, and the variance-covariance matrix of the parameter estimates is computed. Ageneralisation of the logistic, which proved useful in studying the growth of the number of tractors inGreat Britain is examined and gives appreciably different results.

INTRODUCTIONThe logistic growth curve has a long history in the analysis of time series. It is particu?larly useful for statistics which have a natural upper asymptote or saturation level. It canbe derived from a simple model of the growth process. Among recent studies is that ofMar-Molinero,1 who finds that the function fits annual data relating to the number oftractors in Spain between 1951 and 1976 extremely closely. The use of such a functionwith naturally bounded statistics allows one to make extrapolatory predictions and toanalyse observed residuals in the context of economic and other influences, as Mar-Molinero does, in a manner impossible with polynomial trends. An extensive comparisonbetween polynomials and a number of other functions, including the logistic, is given byGregg et al.2

The purpose of the present note is to supplement Mar-Molinero's logistic analysis intwo directions. First, a reparameterisation of the function admits of a ready geometricalinterpretation with a convenient asymptotic variance-covariance matrix; in particular ameasure of the accuracy of the estimate of the asymptote, or saturation level, is obtained.Secondly, a simple generalisation of the logistic, which has already proved useful inanalysing the growth of tractor numbers in Great Britain, is discussed. In neither case isthe problem of autocorrelated stochastic terms considered; the effect of such autocorrela?tion would be to cause underestimation of the sampling variability of the parameterestimates.

A REPARAMETERISATIONMar-Molinero writes the logistic as

?- Th? >

The geometrical interpretation of the function may be clarified by the following repara?meterisation. By writing a = l/tc, b = ~P/k and r = exp(-a) the logistic becomes

_ _^_ /y\h~ 1 + jSexp(-af)'The parameter k gives the upper asymptote directly, a is related to the maximum gradient,and j8 then determines the horizontal position of the function. In fact, d(y/K;)/d? has amaximum of a/4, and the abcissa of this point of inflexion and symmetry is at t = (log jg)/a.In addition,

dy/dt =ayk ? y k499

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Journal of the Operational Research Society Vol. 32, No. 6that is, the ratio of the rate of increase in y to the possible increase is proportional to theactual value of y. Thus the function arises naturally in the context of the diffusion of aninnovation such as tractors in that it accords with the hypothesis that the innovationspreads through an imitative process, but necessarily only among those yet to adopt it.In Oliver3 methods of fitting this non-linear function were reviewed and ordinary leastsquares advocated, despite its computational problems. This conclusion was recentlyreinforced by Stone.4 Under the assumption that the observations are generated by thelogistic process subject to an additive disturbance term with zero mean, variance a2 andwithout autocorrelation, the results are maximum-likelihood and hence unbiased in largesamples. The variance-covariance matrix of the least squares estimates of a2, a, /? and kis given in Oliver.5 The matrix is estimated using the fitted parameters, except that theestimate of a2 should take account of the degrees of freedom used.

Fitting (2) to Mar-Molinero's data, which are measured in tens of thousands of trac?tors, iteratively by least squares gives a = 0.1953, J3 = 50.1338 and k = 52.0804, with


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F. R. Oliver?Tractors in Spainwithin the country concerned. In this case, the quantity of tractors had been artificiallyrestricted in the early years of the post-war period, so the example of other countrieswould amount to such a stimulus, and one might therefore look for a negative value of y.This would yield a trend which rose somewhat faster in the early years of the period thanthe three parameter function.This generalisation was suggested by Oliver,6 who also applied it to tractors, in thiscase in Great Britain between 1950 and 1965. Here the estimate of y proved positive,suggesting a minimum quantity of tractors, perhaps employed in other functions, whoseexample did not lead to any expansion in tractor numbers.

Fitting this displaced function by direct least squares gives a = 0.1866, /? = 44.1952,y = -0.4316 and k = 54.1549, with b2 = {y - y)2/{n - 4) = 4.7249/22 = 0.2148 andR2 = 0.9989. It will be seen that y is negative, as expected, that the sum of residualssquared falls by 5.06%, and that the residual mean square falls, even when corrected forthe loss of the extra degree of freedom, although not by very much. The function fitsbetter than the simple function, by a margin sufficient to warrant the use of an extraparameter.

The variance-covariance matrix ofthe estimates of o2, a, /?, y and k is given in Oliver.6Evaluating it, using these estimated values of the parameters, gives standard errors andhence t-ratios as follows:

CoefficientaPyK

Standard error0.00985.67880.42772.6790

r-ratio19.11

7.78-1.0120.21

The correlation matrix of estimates of a, /?, y and k is1 0.9506 0.8757

1 0.93901

-0.9614-0.8412-0.8003

1

Evidently the correlation between sample estimates of y and of the other parametersincreases their standard errors.The upper asymptote is estimated as k + y tens of thousands, that is as 537,200. This is

appreciably larger than that given by the three parameter function. Its standard error isreduced by the negative correlation between k and y, but even so, at 23,500, is greaterthan before. The 95% confidence interval is 488,500-586,000, both wider and higher thanfor the estimate given by the three parameter function.The displaced function fits the more recent observations substantially better than doesthe simple function. Residual sums of squares are as follows:Improvement

Period Three parameter function Displaced function (%)1951-1976 4.9768 4.7249 5.061964-1976 2.3290 1.8680 19.791971-1976 0.4763 0.2652 44.32

This reflects that the displaced function is growing appreciably faster, to a higher asymp?tote, than the simple function, as are the original statistics. The simple function cannotfully reflect this, since the ordinary logistic is symmetric in the point of inflexion and isconstrained to have a zero lower asymptote.

Comparing the two functions, each gives the period of maximum growth of tractornumbers as about 1970 (t = 20.0404 for the logistic, t = 20.3079 for the displaced logis?tic). The logistic gives 260,402 tractors with a growth rate of 25,433 per year at this point;

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Journal of the Operational Research Society Vol. 32, No. 6the generalised function gives 266,459 with an annual growth rate of 25,258 per year. Aswould be expected, extrapolating the functions brings out their differences more mark-edly; for 1985, the logistic predicts 494,208 tractors increasing by 4930 per year, whereasthe generalisation predicts 504,416 tractors increasing by 6453 per year.The three parameter function is, effectively, the displaced function subject to the con?straint that the displacement is zero. In cases such as this, where there is no reason toimpose such a constraint, the more general function may be more realistic. In this case itfits the data better, although not by a great margin. However, the much better fit which itgives to the more recent data in particular strengthens the case for using it to makepredictions.

REFERENCES1C. Mar-Molinero (1980) Tractors in Spain: a logistic analysis. J. Opl Res. Soc. 31, 141-152.2J. V. Gregg, C. H. Hossell and J. T. Richardson (1964) Mathematical Trend Curves. Oliver & Boyd,Edinburgh.3F. R. Oliver (1964) Methods of estimating the logistic growth function, Appl. Statist. 13, 57-66.4R. Stone (1980) Sigmoids. Bull. Appl Statist. 7, 59-119.5F. R. Oliver (1966) Aspects of maximum likelihood estimation of the logistic growth function. J. Am. statist.Ass. 61, 697-705.6F. R. Oliver (1969) Another generalisation ofthe logistic growth function. Econometrica37, 144-147.

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Tractors in Spain, Further Logistic Analysis

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