riU.^h Journal of AdduU.n 75 (1980) I6<l-17:i. 00O7-0890/8O/0O780169S02.OO1980 Socifly iur ihr Study of Addifiion tu Alcohol and oihcr Drugs.

Is Alcohol Consumption Lognormally Distributed?

Ole-Jorgen SkogNational Institute for Alcohol Research, Oslo, Norway

SummaryThe empirical evidence for the kypotke.m of logriormalitj i.s reviewed. A theoretical argurtienl sugge.sting.•iyslemalic deviations from lognormality is outlined,and .'iome' new' data are presented.

IntroductionThe hypothesis that alcohol consumption is approximately lognormally distributedwas first advanced by Ledermann [1]. Ledermann failed to present a rationaltheoretical argument for his hypothesis, but he was obviously inspired by the work ofGibrat [2], who presented both theoretical arguments and empirical data to supportihe hypothesis for difierent economic variates.

The empirical data presented by Ledermann in support of his hypothesis wasnot convincing [3]. Some of the samples used were extremely small, some weredrawn from very special populations, and in some cases the variable was not alcoholconsumption at all.

Later studies, utilizing data from general population surveys, suggest that thelognormal model al best gives only a rough approximation to the distribution ofalcohol consumption. Makela [4], analysing P innish survey data, demonstrated asmall but systematic deviation from lognormality. Similar results were obtained bythe present author, when analysing survey data from a number of other countries [5]- significant deviations from lognormality were demonstrated in three out ol sixsurveys. The same result was obtained in a subsequent study [6].

The deviations from lognormality noted in these studies are not very dramatic,but they all go in the same direction. When plotted on logarithmic probability paper,a slightly curved pattern appears, indicating the empirical distributions to be some-what less skewed than the lognormal. The skewedness varies, however, and thegoodness-of-fit of the lognormal distribution seems to depend on the kind of popula-tion considered [7, p. II].

The Theoretical Foundation of the Hypothesis of LognormalityHow can this lack of agreement between empirical and theoretical distributions beexplained.^ One quite obvious possibility is that the samples considered may be seri-ously biased. Heavy drinkers and 'alcoholics' are in all probability under-represented and the deviations may therefore be an artefact. On the other hand,there are theoretical reasons to believe that the deviations are real, but before weconsider these reasons we should know why lognormality was expected in the firstplace.

170 Skog

A hypothesis of lognormality is in general based on the assumption that the traitin question is determined by a large number of'small' factors, which tend to combinemultiplicatively rather than additively. In longitudinal terms, multiplicativelyimplies that each individual tends to increase or decrease their intake by an amountproportional to the actual level of consumption - i.e. in accordance with the law ofproportionate effecls [ 1 ].

It is not unreasonable to assume that the law of proportionate effects is adequatefor alcohol consumption. This assumption can be derived from VVeber-Fechner'slaw, which leads us to expect that a person consuming 20 litres per year will perceivean increase of 5 litres as comparable to an increase of 1 litre by a person consuming 4litres. When a person is exposed to a stimulus which tends to influence his drinkingbehaviour, his response should in consequence be expected to depend on his initialconsumption level, and the change ought to be roughly proportional to this quantity.

Although the hypothesis of multiplicativety seems to be plausible, the secondpremise of the argument is not. This premise states that all factors should be 'small',i.e. contribute a negligible fraction to the total variance, but it is quite obvious thatsome factors contribute a very large amount. For example, the ratio of male to femalemean consumption is roughly one to three in a number of countries [8], and thisimplies that the sex factor contributes to the total logarithmic variance with a factorof 0.3. When the total variance is 2, the sex factor consequently explains 15 per centof the total variance.

Factors contributing a large fraction to the total variance obviously will have asubstantial impact on the distribution. The more they contribute, the stronger theimpact, and the more the distribution may deviate from the lognormal. Inhomogeneous substrata of the population, the distribution may well be approxi-mately lognormal, however, since no single factor contributes significantly to thevariance. If it did, the substratum would, by definition, not be homogeneous.

If the distributions within substrata are lognormal, the distribution in the totalpopulation may also be lognormal, on tbe condition that the mean consumptionvaries between substrata in accordance with a lognormal distribution, and that tbedispersion parameter is identical in all substrata [9, p. 110]. In consequence, log-normality in the total population may occur, in spite of the fact that some factors arecontributing a substantial fraction to the total variance.

In genera], lognormality will be the exception rather than the rule, however.Stability of the dispersion is very improbable and there are theoretical reasons forexpecting the dispersion to covary systematically (although not perfectly) with themean consumption level, the dispersion being high where the mean is low and viceversa [5, 6]. This relationship operates in the direction of making the distributionsomewhat less skewed than the lognormal [10], and as we have pointed out, thishypothesis is consistent with empirical findings.

The deviations noted in the above studies are not very dramatic. The reason forthis might be that, so far, mainly populations with low and modest/)«r capita con-sumption have been considered. In such populations the dispersion of the distribu-tion is likely to be fairly large, and socio-demographic factors such as sex, contributea smaller proportion to the total variance than would be the case in populationswhere the total variance is smaller. It has been noted above that the variance gener-ally ought to be small when the mean is high, and hence it is quite possible that thepotential deviations from lognormality are larger, the higher the mean.

Is Alcohol Consumption Lognormally Distributed? 171

Some' new' dataGoing back lo the data u.sed by Ledermann [11] in l%4, I have recently foundempirical distributions deviating sharply from lognormality. The data were pub-lished by Brezard [12-14] and were used by Ledermann to study the relationshipbetween mean consumption level and prevalence of heavy use. Oddly enough, hefailed to use these data to test the hypothesis of lognormality, however.

Brezard's data are exceptional, since under-reporting of consumption seems tobe very small [12, p. 272]. The data were obtained by random sampling from thegeneral population in seven regions of France (some of which have an extremely highper capita consumption) and each sample consisted of approximately 500 subjects.Table I reproduces the empirical distributions —abstainers being excluded.

Table 1 Distrihulion ofaduli consumers (male + female) according to consumption level measuredin litres of wine equivalents per day. (Source: Brezard [12, 14].)

Consumption

- .25.26- .75

.76-1.231.26-1.751.76-2.252.26-2.752.76-3.253.26-3.733.76-4.234.26 -1-

Mar-seille

17717456241021

444

St.Etienne

72241693323U51

433

Gironde

8714360674922121355

4<i5

Card

G7228943416321

443

Savoie

i;j398666351392117

488

Cotes-du-Nord

204117115327I2

478

Vendee

13811861513682

434

As can be seen from Figure 1, the distributions appear to be approximatelylognormal in three regions: Marseille, St. Etienne and Gard. In the remaining fourregions, strongly curved patterns appear. These deviations are highly significant, ascan be seen from Table 2.

The gamma distribution has also been fitted to Brezard's data. It has been sug-gested [6, 15] that this distribution by and large will give a better fit, since it issomewhat less skewed than the lognormal, and this hypothesis has gained someempirical support in previous studies [6, 16]. As can be seen from Table 2, however,the gamma distribution is not very successful in the present case. Although thegamma distribution appears to fit the data slighdy better than the lognormal (thechi-squares being smaller, on the average) large deviations remain, and the fit isindeed far from satisfying.

Concluding remarksEven though the distribution of alcohol consumption may be approximately log-normal in homogeneous substrata of a population, the approximation is in generalnot likely to be good when the total population is considered. The within-stratumvariances is likely to vary a great deal, and this will work in the direction of making

172 Skog

99-

10-

• Coies-du-Nord

• Savoie

0-5

99-

1 0 - •

• Vendfee

X Gironde

{)•!,

Marseille

SI, Elienne

(1-5

Figure 1 Log probability diagrams for Brezard's data

Table 2 Te.st of goodncss-of-fit of the lognormal distribution and [he gamma distribution on sur\'eydata from seven regions in France. The parameters have been estimated by the chi-square minimummethod.

Sample

MarseilleSt. EtienneGirondeVendeeSavoieGardCotes-du-Nord

I-ognormal distribution

A

-1.11- .60- .29- .78- .34- .61- .99

IT

.97

.801.081.241.29.73

1.09

X"

8.815.553.373.4

112.14.3

101.0

df

4574554

P

.066

.009<.OOOI<.(X)01<.OOO1

.511<.OOO1

Gamma distribution

a

2.052.20

.%1.00.67

2.811.56

1.011.641.09.77.81

1.97.88

x'1.7

34.420.733.843.66.7

55.5

df

4574554

P

.786<.OOO1

.004<.OOO1<.O0Ol

.244<.OOO1

Is Alcohol Consumption Lognormally Distributed? 173

the total distribution less skewed than the lognormai. Further, the distribution bet-ween substrata (i.e. distribution of substrata means) may easily deviate sharply fromlognormality, and the total result is difficult to predict.

It appears that when the within-stratum variances are small as compared to thetotal variance, i.e. when a large fraction of the total variance is explained by a smallnumber of factors, the potential deviations from a simple mathematical model suchas the lognormal or the gamma could be very large. This seems to be the case in thehigh consuming populations of Brezard, where the sex factor alone may explain asmuch as 40 per cent of the total variance.

When the within-stratum variances are large as compared to the total variance,i.e. when the between-strata variance is small in relative terms, the potential devia-tions may be smaller. Hence, in low consumption countries (where the total varianceought to be large), the deviations from lognormality may, as a rule, be less dramaticas compared to high consumption countries. Nevertheless, exceptional cases mayexist among low consumption countries as well.

References1 Ledermann, S. (1956). Alcoot, alcoolisme, alcoollmtion. Vol. I. Presses Universitaires de France,

Paris.2 Gibrat, R. (1931 ). Les inegalites economiques. Librairede Recueil Sirey, Paris.3 Smith, N.M.H.( ! 976). Research note on the Ledermann formulii and its recent applications. The

Drinking arid Dru^ Practices Surveyor., 12, \b~22.4 Makela, K. (1969). .Aikoholinkuiutuksen jakautuma. Kulutustulkiniuksen ennakkoraportti (Dis-

iributiun of the use of alcohol. A preliminary report of the consumption study). Reports from theSocial Research Institute for Alcohol Studies, Helsinki.

5 Skog, O.-J. (1971). Alkohotkon\umet.\fordetin_ii i befolkningen (The distribution of alcohol consumptioniii the population ). .Mimeographed. Natiotial Institute for .Mcobol Research, Oslo.

6 Skog, O.-J. (1974). Hidragtiten teoriom alkoholkomumelsfordeling,2. (Contribution to a theory of thedistrihution of alcohol consumption, 2). Mimeographed. National Institute for Alcohol Research,Oslo.

7 Skog, O.-J. (1973). Less aleohol - fewer alcoholics-'' The Drinking and Drug Practices Surveyor.,!, 7-14.8 Skog, O.-J. (1977). Does Ike same distributional model for alcohol comumption apply to both male and female

populatiom.' Mimeographed. National Institute for Alcohol Research, Oslo.9 Aitchison, J. and Brown, J. A. C. (1969). The lognormal distribution. The University Press, Cann-

10 Skog, O.-J. (1972). Noen reHeksjoner omkring alkoholkonsumets fordeling og den lognormaleinodetlen. (Some notes on the distribution of alcohol consumption and the lognormal model).Minieotjraphed. National Institute for Alcohol Research, Oslo.

11 Ledermann, S. (l%4). Atcool, alcoolisme, akoolisation. Vol. 2. Presses Universitaires de France,Paris.

12 Brezard, M. (1958). Presentation d'une enquete sur la consommation des boissons en France./W/f/mr/f/7..V./y.. 13,267-35(5.

13 Brezard, M. (1959). La consonimation des boissons en France. Deuxieme partie: Marseille./fw//f//«^f/7.;V.//., 14,95-163.

!4 Brezard, M. (l9(iO). La consommation des boissons en France. Quatriemc partie: quelques dh-ir'iclsrurnux.Hulletindel'L.WH., 15,229-263.

15 Skog, O.-J. (1977). On the distribution of alcohol consumption. In Tke Ledermann curve. Report of awmpo.uum. Alcohol Education Centre; London.

16 Skog, O.-J. (1979). Gamma vs. lognormal distributions - a reply toGuttorp and Song. The Drink-ing and Drug Practices .Surveyor, 14, 3—6.