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DOI: 10.1177/0010414068001001061968 1: 139Comparative Political Studies

Richard W. Chadwick and Karl W. DeutschRates of Change in Social Science Data

Doubling Time and Half Life: Two Suggested Conventions for Describing

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DOUBLING TIMEAND HALF LIFE

Two Suggested Conventions forDescribing Rates of Change

in Social Science Data

RICHARD W.

CHADWICKand

KARL W. DEUTSCHRICHARD W. CHADWTCK ~ on the staff of &dquo;..-Sydent Det)elo~ Corporations Beha0ford

CanzbV and SimuZation Program. KARLW, _ ,JM~~~CH,f~~MM~6~C~W~~M~~~~J~~- .

v~rd Unitvrsity, is the author of numerous books and journal CtttC~S in the ~Ms of international

&dquo;

~~~M~~~~~~~p~~M~h~K~~a~~~W~MM- J<

,.

tive government.&dquo;&dquo;..

.

l

...

a V

}- .., ..,...

-

-< t~ f

V ERYOFTEN IN THE STUDY of change in social phenomena, it is -.~

not the rate of gr~~th ~r decline per year, but rather the bnpliv~z-fiong of such hange-for sustained pmods of tine thai are #f th~~~ic~ interest. One is concerned not so much with whether a rate of growth is1 % or 37o per year, for example, but with the potential consequences fora system if that rate of gr~~vth were sustai~.e~.. We often ask: what kindsof difficulties would be generated for the system? for other systems?How long would it talce before small but sustained rates of change gen-erate important cbfferenoes in the condition of the system ?

~?Ue suggest that a simple, precise, and meaningfully descriptivemethod for denoting the potential implications of rates of change is

~.~~~ No=:A note of c~~~cic~t~n is due to Frr~ f~ssc,rrs Derek 1. doSoffa Price andAnatol Rapoport for their comments and s~~ge~t~s ~vari0u8 stages in this papees development.PuBmmWs NOTE: This paper is being published simultaneously in the.~~r~h-A~, 1968, ~s~ce of The tlmerican Behavioral Scientist (VolumeXl, Number 4), in an ~~ ~o bring it to the attention of a wider audienceof social sr;~nt~t~.

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to express them in terms of doubung titne, or, in the case of decline,half life. In either case, the appropriate figure is easily arrived at bytaking the number 70 (approximately) and dividing by the rate ofchange expressed as a percentage., Mathematieally, this may be writ-ten as

where DT is doubling time, HL is halt li~e, and r$ ~ an observedannual rate of change expressed as a decimal.A growth rate of 3%per year, for example, may be expressed as a potential doubling timeof approximately 23 years (ort in the cue of decline at the rate of 3%per year, at a half life of 23 years).

This is not, of course, to say that an observed rate of change willnecessarily-or even in all likelihood-be sustained constantly over along period of time. We are simply suggesting that the convention ofdenoting rates of change in terms of doubling times or half lives is intui-tively more immediately meaningful than the convention of denotingrates of change by percentages per year. ~r~ say, for example, that &dquo;a

population is growing at the rate of 3% per year&dquo; is not quite as cos-pelling as saying &dquo;a population has a potential doubling time of 23years.&dquo; Similarly, to say &dquo;the proportion of a nations gross nationalproduct engaged in trade is declining at the rate of 3% per year&dquo; doesmet bring home the intuitive sig1lifican of the statistic as quickly as thestatement that &dquo;the proportion of a nations gross national product emu-gaged in trade is currently declining at the rate of 50% every 23 years(has a potential half Me of 23 years).&dquo; Similarly, we could reformulatestatements concerning the relative proportion of world GNP engaged inthe international sector, rates of change in government budgets, alloca-tions to defense, and so forth.A useful extension of these conventions might be to project time series

forward(or backward) to assess the implications of differential rates ofchange. Suppose, for example, a countrys population grows at the rateof 2% per year, while its gross national product grows at the rate of3% per year. If these rates are sustained, it is evident that the popula-tions will double every thirty-five years and ~I~1P will double every 23

years. If we project these trends over a lQ year period, we see GNPwill have increased eightfold (doubling in 23 years, doubling again in43 years, and once again in ~ years), while population will only havequadrupled (doubling in 35 years, and once again in 70 years). Percapita income; therefore, will have doubled. Many similar tentative

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oompotatkm of this Mn4 can be carried out quickly, without recourseto computers. or even without pencil and paper. ,

The more widespread use of this technique among social scientistscould have some effects not limited to computational C01lveDence-though related to such convenience. Thwking about theories and pr0c-of social change is an operation involving many steps of thought orcalculations which must be passed fairly f8pid1y. if many possibilities andimplica dons are to be surveyed in a limited tune. Quantitative steps, inparticular, tend to be relatively slow and time-consuming; and yet theyare often essential if theories are to be improved and made morespecific.Any step that is relatively slow and laborious, or requires asearch for books and

tables,will slow up the entire How of thought,

much as a single bottleneck may slow up an extended stream of traffic.The method proposed here permits the quick and relatively accurateperception of potential consequences or implications of rates of change.Wherever the implications of rates of change form essential steps in thedevelopment of theories) this technique should make our now of thoughtfaster, more precise, and therefore more effective.

APPENDIX

1. A CoNvmqmNT Dr43p= oF APPRoxn&&noN~ The usual approach to computing changes in quantities given a rateof change (either increase or decrease) is to apply one of the followingtwo formulae:

where P = an original amount (such as &dquo;principal&dquo; in banks, or popu-lation size), F = a final amount after t-periods of time have elapsed,t = the number of elapsed time periods, r, == the observed rate ofchange per time period, r. = the &dquo;naturaf or &dquo;instantaneous&dquo; rate ofchange, and e = the irrational number 2.71828 .... generated by takingthe limit, x --+ 0 (1 + x) k . (This number, e, is used as the basis ofthe widely employed system of &dquo;natural logarithms,&dquo; for which convenienttables are

avai1able,and which can be converted to

logarithmsto the

base 10 by multiplying each natural logarithm by 0.4343, approximately.For the purpose of simplifying equations below, naturalareused and are denoted by the symbol ln(x), x being any number)

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,&dquo; --Mlheffier-,6ne employs equation (2) or (3) depeAd6~uwhedw one-has a~iIaMe, respectively, either r., say an annual rate of chtmge, Of t.,

4* 11attmd, iottantane6us rate -of changeA-3 it datMm.~ To convert ettherequation to an estiamtor of doubling time (DT) or half life (RL),one need oblyamun~e that by some time period, t, F ==: 2r or F = .5P,reSpeCtively, one then substitutes either 2P or .5P for F into eitherequaticia-~ 2) or (3), and solves for t, which. upon thb iubstituiloi3~ be-comes the DT (or HL) estimate. Going through this procedure forequation. (3), for e:Kamj?le, one obtains the fowwiug:

ta1dDg natural logariflmw on both sides, we obtain

but since the logarithm of any number to its awn base equals one,ln(e~ =1 1, and we obtain I ,

4ividing both sides by 1&dquo;11, we obtain

but since we set F = 2P to obtain the above, t is 01U estimate ofdoubling time; Owswe may write

-

-

,Ry a siniflw procedurei- through equation (2) we eventually obtain

Iteferrkag back to formula (4), we note that In( 2) is approximately,O.M (obtained from any tableof natural logaxidnns); ~ we uow sub-*~ate Oiis vouo into equation -(4), above, and multiply botb- theoumerator and denominator by 100, we obtain &dquo;

&dquo;

4 ~

Note that equation (41) it quite similar to the equation (1) given in the-text. Shft4Dff SOCia1scientists usually have annual rates of change as datit,1!quatou (5) would, strictly speaking, be,prefe-ral>le to (41) for accuratecomputation-, for substituting7 r. tor rl1 in eqU)1tion (41) intioduces, a

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downward bias In DT and HL estimates, making them slightjy smallerthan they should be. However, from a computational point of view,equation (4) is certainly more convonient. We therefore have cat-

cWqLted a series of ucorrection factors&dquo; to substitute for the value of 69.3in the numerator of equation (41), which will just offset the bias intro-duced by substituting r. for r. in the denominator, for given rates gfchange. These &dquo;correction factors&dquo; appear in Table It below.&dquo;

.

TABLEIPoTN~nAI. DOUBLING Ttux (HALF lavit) EsTm.&ns irost SBLECTED 1UTB8

OF CHANCE Pm YF4R

It can be seen from Table 1 why the value of 70 was chosen forequation (1) in the text: it eliminates the bias in equation (4)---fromwhich equation(1) was clearly derived-when r. = .01 or 1% peryear, and reduces bias somewhat for all values of r~ with absolute valuesgreater than .01. Moreover, note that, while the correction factornecessary to eliminate biases for r&dquo;s with larger absolute values in-creases rapidly, the actual amount of bias is not great-never becominglarger in absolute value than .3 years, albeit ~it does increase in &6-pdkn to doubling time rates. Given the purpose for which equation~ ~ ~ is put forth, however and given the amount of error in rates ofchange normally available to social scientists, and given the normal size

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4) or equation (5) would be preferable. We stress once again that ourpurpose here is to provide a heuristic for theorizing, not a computational

a~gor~ithm for accurate prediction in and of itself.s

U. THE CONCEPT 01 HALl L1n:

We should comment upon another aspect of Table 1, and the conceptof -half life,&dquo; in particular. The reader may be troubled to observe thatthe table is said to apply to estimating half life as well as doubling timewhen he observes a 100% or &dquo;A4~ rate of decline, only to discover thatat the end of a year the

originalamount has not disappeared

entirely.To explain this apparent anomaly, it will be helpful to recall that agrowth rate is conceived here as increases by a constant proportion ofthe amount that exists at each moment. ~f, for example, $100 weregrowing at the rate of 40% per year, then in one year we would have$140; in the next year if the 40% rate is maintained-it would benecessary to add $56, in the next approximately $78, and so on. In thiscase, the original $100 would have doubled shortly after the beamingof the thirdyear.,

Now, in the case of half life, $100 decreasing at the rate of 40%each year would drop to $60 by the end of the first year, by $24 thesecond year, and by approximately $14 the third year, and so on. Inthis case it would have shrunk to one half of its original value sometimesin the course of the second year.A constant rate of change, as,a percentage of the original and then

subsequent amounts, means increasing absolute additions in the -caseof growth, but it means declining absolute &M~ in the case of a rate ofdecline. These observations seem simple in where rates of

changeare srnall~-on the order of 3 to 5%-but they have to be carefully keptin mind to avoid confusion when rates of change are very large.

If $100 were dwindling at the rate of 100% per year, then the rateof loss during the first month would be { very roughly) 112th of $100or about $8; but in the second month it would be 1~ l.~h of $92 or 1.50;in the third month it would be 1/ 12+ of $84, which is only a $7 loss,Thus every month the residual amount dwindles by d!~~r~~n~ absoluteamounts.As a result, if $100 were shrinking at a 100% rate of decline,

there would be $50 left at the end of the year.Amounts that decline atconstant rates, therefore, shrink very quickly at first, but evermoreslowly thereafter. In theory, they can never fall to zero, though in prac>tice they will eventually fall below some threshold of indivisibility orhuignifi,cance. This fact makes it possible for rates of decline to be even

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higher than 100%. If $100 should decline at the rate of 300% per year,then it should decline by approximately 425 dol1ars during theAntmonth, by approximately $19 during the second month, by $14 the tbh-d,and so on. It would therefore have fit half life of somewhat less thanthree months, if we followed this crude calculation (using precisemethods, per equation (5), the half life is about .5 years or six months,and Rapoporfs adjustment yields about .6 years, as does the nesirestcorrection factor in Table 1, namely, the factor of 193. ; as stated earlier,for extreme rates of growth or decline, even single-digit accuracy re-quires fairly-precise calculation).

NOTES

1. See the attachedAppendix for derivation and general discussion.2. The relationship between the Natural, instantaneous rate of change, rn, and the ob-

served rate of change per some time period is given by the equation: rn = ln (l+ ra).3. The reader unfamiliar with logarithms may wonder why formula (4) satisfiesfor estimating both DT and HL, since we set F = 2P initially. It turns out that ifwe set F= .5P, we eventually obtain ln (.5) in the numerator instead of ln (2). Thesetwo logarithms, however, have identical absolute values, though opposite in sign. Butsince in such a case, we also have a change in sign of the ra value (being negative inthe case of half life) the sign changes cancel out in calculations. Thus we obtainexactly the same expression for estimating HL as we do for DT.

4. In personal communication to the authors,Anatol Rapoport has pointed out thatthe relationship between rn and ra given in note 2, above, implies the power expansionrn = lu(1 + ra)= ra -r2a/2 + r3a/3 - r4a/4 + .... If rn is small, terms of

degree higher than the first may be ignored, and, after some simplification, equation (4)becomes HL = DT &ap; 69.3/100ra (1+ ra/2). If one takes the trouble to employ this correctionfactor Rapoport suggests, one will find his modification yielding excellent estimates up to100% rates of change (at that rate, yielding a DT of 1.04 years; which compares quitefavorably with column 3 estimates in Table 1, above. Its only drawback is the labor ofcalculation.

5. Rates of change, particularly in the case of comparative economic and demo-graphic data, are seldom known within more than one digit of accuracy, if that. (Cf.Oskar Morgenstern, On theAccuracy of Economic Observations, 2nd ed., PrincetonUniv. Press, 1963, chap. 15, esp. pp. 298-301.) When we get in the neighborhood of50% rates of annual change, we begin to get a noticeable bias in the first digit, usingthe correction factor of 70. In applying this suggested convention, as with percentagevalues, great caution should be taken not to report more significant digits than estimated

error in the data or bias in calculations will allow.