583

PRINCIPLES OF MEDICAL STATISTICS

X-THE COEFFICIENT OF CORRELATION

A PROBLEM with which the statistician is frequentlyfaced is the measurement of the degree of relationshipbetween two, or more, characteristics of a population.For instance in a particular area the air temperatureis recorded at certain times and the mean air

temperature of each week is computed ; the numberof deaths registered as due to bronchitis and

pneumonia is put alongside it. Suppose the followingresults are reached :—

There is clearly some relationship between these twomeasurements. As the mean weekly temperaturerises there is a decrease in the average weekly numberof deaths from bronchitis and pneumonia, which fallfrom an average of 253 in the 5 coldest weeks to anaverage of 87 in the 4 warmest weeks. On theother hand there is at the same mean temperaturea considerable variability in the number of deathsregistered in each week, as shown in the ranges givenin the right-hand column. In individual weeksthere were sometimes, for instance, more deaths

registered when the temperature was 38°-41° than whenit was 35°-38°. In measuring the closeness of the

relationship between temperature and registereddeaths this variability must be taken into account.

35 37 39 41 43 45 47MEAN WEEKLY TEMPERATURE F°

FIG. 6.-Number of deaths registered in each week with varyingmean temperatures. A scatter diagram.

The declining number of deaths as the mean tempera-ture rises, and also the variability of this number inweeks of about the same temperature, are shownclearly in Fig. 6-known as a scatter diagram. Itappears from the distribution of the points (each ofwhich represents the mean temperature of and the

deaths registered in one week) that the relationshipbetween the temperature and the deaths could bereasonably described by a straight line, such as theline drawn through them on the diagram. The pointsare widely scattered round the line in this instancebut their downward trend follows the line and shows

35 37 39 41 43 45 47MEAN WEEKLY TEMPERATURE F 0

FIG. 7.-Number of deaths registered in each week with vary-ing mean temperatures. Hypothetical case of completecorrelation.

no tendency to be curved. The means in the tableabove very nearly fall on the line. In such instancesa satisfactory measure of the degree of relationshipbetween the two characteristics is the coefficient ofcorrelation, the advantage of which is that it givesin a single figure an assessment of the degree of therelationship which is more vaguely shown in the tableand diagram (both of which are, however, perfectlyvalid and valuable ways of showing associations).

Dependent and Independent Characteristics

There are various ways of considering this coefficient.The following is perhaps the simplest.

(i) Let us suppose first that deaths from pneumoniaand bronchitis are dependent upon the temperature ofthe week and upon no other factor and also follow a

straight line relationship-i.e., for each temperaturethere can be only one total for the deaths and thistotal falls by the same amount as the temperatureincreases each further degree. Then our scatter dia-

gram reduces to a series of points lying exactly upona straight line. For instance, in Fig. 7 the deathstotal 200 at 35°, 185 at 36°, 170 at 37°. For each

weekly temperature there is only a single value forthe deaths, the number of which falls by 15 as thetemperature rises one degree. If we know the

temperature we can state precisely the number ofdeaths. No error can be made for there is no scatterround the line.

(ii) Now let us suppose that the deaths are com-pletely independent of the temperature, but fluctuatefrom week to week for quite other reasons. Whenthe temperature is low there is then no reason

why we should observe a larger number of deathsthan when the temperature is high, or vice versa.If we had a very large number of weekly records weshould observe at each temperature all kinds of totalsof deaths. The scatter diagram would take the formshown in Fig. 8 (only roughly, of course, in practice).At 35° there were, for example, weeks in which wererecorded 50, 75, 100, 125, 150, 175, and 200 deaths ;at 36° we see the same totals, and similarly at each

584 PRINCIPLES OF MEDICAL STATISTICS

higher temperature. The average of the weeklytotals of deaths observed is the same at each

temperature and is equal to the average of allthe weekly observations put together, for the twocharacteristics being independent there is no tendencyfor these averages to move up or down as the

temperature changes. If we know the temperatureof the week we obviously cannot state the number ofdeaths in that week with any accuracy. We can,however, attempt to do so and we can measure theamount of our error. The best estimate of the

ncn _________________________________________

35 37 39 41 43 45 47MEAN WEEKLY TEMPERATURE F 0

FIG. 8.-Number of deaths registered in each week with varyingmean temperatures. Hypothetical case of complete absenceof correlation.

deaths we can make at any weekly temperature isthe average number of deaths taking place in all theweeks put together-for as pointed out we have noreason to suppose that the deaths recorded in a

specified week will be more or less numerous than theaverage, merely because the temperature was highor low. Our error in a specified week is, therefore, thedifference between the mean number of deaths in allweeks and the number of deaths that actually occurredin that particular week; e.g., the mean number ofdeaths in all the weeks recorded in Fig. 8 was

125 ; in one week with a mean temperature of 36°the number recorded was 50, and our error of estima-tion for this week is 125-50 = 75. We can computethis error for each week in turn and find the averagesize of our error, or, preferably, we can find the averageof the squared errors. This latter value will, in fact,be the square of the standard deviation of the weeklynumbers of deaths (which, as shown previously,is the mean squared deviation of the observationsfrom their average). Our weekly errors betweenestimation and observation can therefore be measured

by 2.(iii) Now let us suppose that neither of the

extreme cases is present-i.e., neither completedependence nor complete independence-but that wehave something between the two, as in Fig. 6,where the deaths certainly decline as the temperaturerises but show some variability at each temperature.How precisely can we now state the number of deathswhen we know the temperature Let us draw throughthe points a line which represents, broadly, theirtrend. From that line we can read off the expectednumber of deaths at each temperature and compareit with the observed number in that week. Thedifference will be the error we make in using this line.We can calculate this error for each week and theaverage of these squared errors we can call S2.

Are we any better off in our estimations of theactual weekly deaths by the use of this line than whenwe say that in each week we expect to see the averagenumber of deaths that took place in all the weeks ? ‘We can measure our relative success by comparinga2 with 82. If the two characteristics are entirelyindependent of one another (as in Fig. 8) the linewe draw can have no slope at all and will pass ateach temperature through the average number ofdeaths in all the weeks (there is nothing to make ithigher or lower than the average at different tempera.tures). S2 and o2 will then be the same. If thetwo characteristics are completely dependent, as inFig. 7, then there is no scatter round the line at alland S2 becomes 0. In practice we use as a measure

of the degree of association -B/1 - 2

which is

known as r or the correlation coefficient. If no

association at all exists S2 and c2 are, as pointed out,equal and r equals 0. If there is complete dependenceS2 is 0 and ’equals 1. For any other degree ofassociation r must lie between 0 and 1, being low asits value approaches 0 and high as it approaches 1.

The Use and Meaning of the CorrelationCoefficient

The actual mode of calculation of the correlationcoefficient is fully described in numerous statisticaltext-books, and attention will be confined here to itsuse and meaning. It is calculated in such a waythat its value may be either positive or negative,between z- 1 and - 1. Either plus or minus 1indicates complete dependence of one characteristicupon the other, the sign showing whether the associa-tion is direct or inverse; a positive value showsthat the two characteristics rise and fall together-e.g., age and height of school-children, a negativevalue that one falls as the other rises, as in our

example of deaths and temperature. In the latterinstance the value of the coefficient is, in fact, -0-90.This figure shows that there is in this short series ofobservations a very high degree of relationship betweenthe temperature and the deaths, but, as it and thegraph make obvious, not a complete relationship.Other factors are influencing the number of deathsas well as the temperature. If we knew the equationto the line we could certainly predict the numberof deaths that would take place in a particular weekwith a given mean temperature with more accuracythan would be possible without that information;but the diagram shows that in individual weeks wemight still be a long way out in our prediction. Itis the original variability in the number of deathsat each temperature than makes it impossible for theprediction to be accurate for an individual week,though we might be able to predict very closely theaverage number of deaths in a group of weeks of thesame temperature. The advantage of the coefficientis, as previously pointed out, that it gives in a singlefigure a measure of the amount of relationship.For instance, we might calculate two such coeffi-cients between, say, mean weekly temperatureand number of deaths from bronchitis and pneu-monia at ages 0-5, and between mean weeklytemperature and number of these deaths at

ages 65 and over, and thus determine in whichof the two age-groups are deaths from thesecauses more closely associated with temperaturelevel. We can also pass beyond the coefficient ofcorrelation and find the equation to the straight linethat we have drawn through the points. Readingfrom the diagram the straight line shows that at a

585PRINCIPLES OF MEDICAL STATISTICS

weekly temperature of 39° F. the estimated numberof deaths is about 205 ; at a weekly temperatureof 40° F. the deaths become about 185 ; at a

temperature of 41° F. they become 165. For eachxise of 1° F. in the mean weekly temperature thedeaths will fall, according to this line, by some20 deaths. In practice the method of calculatingthe coefficient of correlation ensures that this lineis drawn through the points in such a way as to makethe sum of the squares of the differences betweenthe actual observations at given temperatures and thecorresponding values predicted from the line for thosetemperatures, have the smallest possible value. Noother line drawn through the points could make thesum of the squared errors of the estimates have asmaller value, so that on this criterion our estimatesare the best possible.

THE REGRESSION EQUATION

The equation to the line can be found from thefollowing formula :-Deaths-Mean number of deaths=

Correlation Standard deviation of deathscoefficient Standard deviation of temperature

X (Temperature-Mean of the weekly temperatures)where the two means and standard deviations are thoseof all the weekly values taken together.Writing in the values we know from the data under

study this becomes-

Dea.ths-167-038= -0-9022 2.867 (Temperature-40-908)The fraction on the right hand side of the equation equals20.24 so that we have-

Deaths=-20-24 (Temperature-40-908)+167-038Removing the parentheses and multiplying the terms

within by —20’24 this becomes-

Deaths=-20.24 Temperature + 827-978 + 167-038or, finally,Deaths=-20-24 Temperature+995-02.The figure —20’24 shows, as we saw previously

from the diagram, that for each rise of 10 F. in thetemperature the deaths decline by about 20. (Forwhen the temperature is, say, 40 F., the deathsestimated from the line are 995-02-(20-24) 40= 185-4and when the temperature goes up 1° F. to 41’ F.,the estimated deaths are 995-02—(20-24) 41 = 165-2.)The figure - 20-24 is known as the regression

coefficient; as seen above it shows the change that,according to the line, takes place in one characteristicfor a unit change in the other. The equation isknown as the regression equation. As far as we havegone our conclusions from the example taken is thatdeaths from bronchitis and pneumonia are in a certainarea closely associated with the weekly air tempera-ture and that a rise of 10 F. in the latter leads, on theaverage, to a fall of 20 in the former.

Precautions in Use and InterpretationIn using and interpreting the correlation coefficient

certain points must be observed.

THE RELATIONSHIP MUST BE REPRESENTABLE

BY A STRAIGHT LINE

(1) In calculating this coefficient we are, as hasbeen shown, presuming that the relationship betweenthe two factors with which we are dealing is one whicha straight line adequately describes. If that is not

approximately true then this measure of associationis not an efficient one. For instance, we may supposethe absence of a vitamin affects some measurablecharacteristic of the body. As administration of the

vitamin increases a favourable effect on this bodymeasurement is observed, but this favourable effectmay continue only up to some optimum point.Further administration leads, let us suppose, to anunfavourable effect. We should then have a distinctcurve of relationship between vitamin administrationand the measurable characteristic of the body, thelatter first rising and then falling. The graph of thepoints would be shaped roughly like an inverted U andno straight line could possibly describe it. Efficientmethods of measuring that type of relationshiphave been devised-e.g., the correlation ratio-and thecorrelation coefficient should not be used. Plottingthe observations, as in Fig. 6 relating to temperatureand deaths from bronchitis and pneumonia, is a

rough but reasonably satisfactory way of determiningwhether a straight line will adequately describe theobservations. If the number of observations is

large it would be a very heavy test to plot theindividual records, but one may then plot the meansof columns in place of the individual observations-e.g., the mean height of children aged 6-7 years wasso many inches, of children aged 7-8 years so manyinches-and see whether those means lie approximatelyon a straight line.

THE LINE MUST NOT BE UNDULY EXTENDED

(2) If the straight line is drawn and the regressionequation found, it is dangerous to extend that linebeyond the range of the actual observations uponwhich it is based. For example, in school-childrenheight increases with age in such a way that a straightline describes the relationship reasonably well.But to use that line to predict the height of adultswould be ridiculous. If, for instance, at school agesheight increases each year by an inch and a half,that increase must cease as adult age is reached.The regression equation gives a measure of the

relationship between certain observations ; to presumethat the same relationship holds beyond the rangeof those observations would need justification onother grounds.

ASSOCIATION IS NOT NECESSARILY CAUSATION

(3) The correlation coefficient is a measure ofassociation and in interpreting its meaning one mustnot confuse association with causation. Proof thatA and B are associated is not proof that a changein A is directly responsible for a change in B or viceversa. There may be some common factor C whichis responsible for their associated movements. For

instance, in a series of towns it might be shown thatthe phthisis death-rate and overcrowding were

correlated with one another, the former being highwhere the latter was high and vice versa. This isnot necessarily evidence that phthisis is due to over-crowding. Possibly, and probably, towns with a

high degree of overcrowding are also those with alow standard of living and nutrition. This thirdfactor may be the one which is responsible for thelevel of the phthisis rate, and overcrowding is onlyindirectly associated with it. It follows that themeaning of correlation coefficients must always beconsidered with care, whether the relationship isa simple direct one or due to the interplay of othercommon factors. In statistics we are invariablytrying to disentangle a chain of causation and severalfactors are likely to be involved. Time correlationsare particularly difficult to interpret but are particu-larly frequent in use as evidence of causal relation-ships-e.g., the recorded increase in the death-ratefrom cancer is attributed to the increase in the

586 PRINCIPLES OF MEDICAL STATISTICS

consumption of tinned foods. Clearly such con-

comitant movement might result from quite unrelatedcauses and the two characteristics actually have norelationship whatever with one another except intime. Merely to presume that the relationship isone of cause and effect is fatally easy ; to secure

satisfactory proof or disproof, if it be possible at all,is often a task of very great complexity.

THE STANDARD ERROR

(4) As with all statistical values the correlationcoefficient must be regarded from the point of viewof sampling errors. In taking a sample of individualsfrom a universe it was shown that the mean and otherstatistical characteristics would vary from one sampleto another. Similarly if we have two measures foreach individual the correlation between those measureswill differ from one sample to another. For instance,if the correlation between the age and weight in allschool-children were 0-85, we should not alwaysobserve that value in samples of a few hundredchildren ; the observed values will fluctuate aroundit. We need a measure of that fluctuation, or, inother words, the standard error of the correlationcoefficient. Similarly if two characteristics are notcorrelated at all so that the coefficient would, if wecould measure all the individuals in the universe,be 0, we shall not necessarily reach a coefficientof exactly 0 in relatively small samples of thoseindividuals. The coefficient observed in such a

sample may have some positive or negative value.In practice we have to answer this question : couldthe value of the coefficient we have reached havearisen quite easily by chance in taking a sample, ofthe size observed, from a universe in which there is nocorrelation at all between the two characteristics For example in a sample of 145 individuals we findthe correlation between two characteristics to be+0-32. Is it likely that these two characteristicsare not really correlated at all, that if we had takena very much larger sample of observations thecoefficient would be 0 or approximately 0 ° It canbe proved that if the value of the correlationcoefficient in the universe is 0, then (a) the meanvalue of the coefficients that will be actually observedif we take a series of samples from that universe willbe 0, but (b) the separate coefficients will be scatteredround that mean, with a standard deviation, or

standard error, of 1/VjV—l, where N is the numberof individuals in each sample. In the example abovethe standard error will, therefore, be 1/V144 = 0 083.Values which deviate from the expected mean valueof 0 by more than twice the standard deviation are,we have previously seen, relatively rare. Hence ifwe observe a coefficient that is more than twiceits standard error we conclude that it is unlikelythat we are sampling a universe in which the twocharacters are really not correlated at all. In the

present case the coefficient of 0.32 is nearly four timesits standard error ; with a sample of 145 individualswe should only very rarely observe a coefficientof this magnitude if the two characters are notcorrelated at all in the universe. We may concludethat there is a "significant" correlation betweenthem-i.e., more than is likely to have arisen bychance due to sampling errors. If on the other handthe size of the sample had been only 26 the standarderror of the coefficient would have been 1/V25 = 0-2.As the coefficient is only 1-6 times its standard errorwe should conclude that a coefficient of this magnitudemight have arisen merely by chance in taking asample of this size, and that in fact the two charactersmay not be correlated at all. We should need more

evidence before drawing any but very tentativeconclusions. This test of " significance " should beapplied to the correlation coefficient before anyattempt is made to interpret it. Somewhat moreintricate methods are needed to test whether onecoefficient differs " significantly " from another-

e.g., whether deaths from bronchitis and pneumoniaare more closely correlated with air temperature atages 0-5 years than at ages 65 and over (see, forinstance, The Methods of Statistics. By L. H. C.Tippett. London : Williams and Norgate Ltd. 1931.15s.).

SummaryThe correlation coefficient is a useful measure of

the degree of association between two characteristics,but only when their relationship is adequatelydescribed by a straight line. The equation to thisline, the regression equation, allows the value ofone characteristic to be estimated when the valueof the other characteristic is known. The error ofthis estimation may be very large even when thecorrelation is very high. Evidence of associationis not necessarily evidence of causation, and thepossible influence of other common factors must beremembered in interpreting correlation coefficients.It is possible to bring a series of characteristics intothe equation, so that, for instance, we may estimatethe weight of a child from a knowledge of his age,height, and chest measurement, but the methodsare beyond the limited scope of these articles.

________

A. B. H.

CORRIGENDA.-In last week’s article the formula inline 17 of the second column of p. 527 should readn=(c-l) (f-1) and the numerator of the fourfold tableon p. 528 should read (ad-bcl2 (a+b+c-f-d).

KING’S COLLEGE HOSPITAL: NEW Wings onFeb. 23rd Viscount Wakefield, vice-president of the

hospital, opened a wing for private patients whichhas been presented to the hospital by the Stock ExchangeDramatic and Operatic Society and by other friends onthe Stock Exchange. The building is 266 feet long and80 feet wide and runs parallel with the main corridorsof the hospital. The offices are on the side nearest tothe hospital so that accommodation for the patients is

separated from sounds connected with its work, but thereis direct access to the X ray and other special departmentsin the main building. Above the entrance rises a tower70 feet high in which are situated the suite of rooms forthe resident medical officer and a muniment room for thesafe custody of records; it is a memorial to the flight ofMr. Giles Guthrie from England to Johannesburg erectedby his father, Sir Connop Guthrie. On the first floor thereare eighteen single rooms for which the charge will be8 guineas a week. The patient makes his own arrange-ment for the payment of fees for professional attendance,including the services of the pathologist and radiologist.In close proximity to the floor is an operating theatrespecially provided for visiting surgeons. On the floorabove is a complete maternity unit where accommodationis provided at varying charges ranging from 7 to 10 guineasin rooms with one, two, and four beds. The services ofthe resident medical officer will be available for antenatalattention as well as for the confinement, and a com-prehensive fee can be arranged to cover both. On thefloor above is a maternity isolation unit. In order to

provide the additional staff for the new wing it has beennecessary to extend the accommodation for residentmedical officers, nurses, and maids. On these extensionsthere is a debt outstanding of .El 5,000. The sum of E8000would enable the ground floor to be fitted up as a dentaldepartment readily accessible from the out-patientdepartment, which would release the general ward of thehospital where it is at present housed for occupation byordinary patients.