STATISTICAL METHODS FOR ASSESSING AGREEMENTBETWEEN TWO METHODS OF CLINICAL MEASUREMENT
J. Martin Bland, Douglas G. Altman
Department of Clinical Epidemiology and Social Medicine, St. George's Hospital Medical
School, London SW17 ORE; and Division of Medical Statistics, MRC Clinical ResearchCentre, Northwick Park Hospital, Harrow, Middlesex

SUMMARY
In clinical measurement comparison of a new measurement technique with an established one
is often needed to see whether they agree sufficiently for the new to replace the old. Suchinvestigations are often analysed inappropriately, notably by using correlation coefficients.
The use of correlation is misleading. An alternative approach, based on graphical techniquesand simple calculations, is described, together with the relation between this analysis and theassessment of repeatability.
(Lancet, 1986; i: 307-310)

INTRODUCTION
Clinicians often wish to have data on, for example, cardiac stroke volume or blood pressure
where direct measurement without adverse effects is difficult or impossible. The true values
remain unknown. Instead indirect methods are used, and a new method has to be evaluatedby comparison with an established technique rather than with the true quantity. If the new
method agrees sufficiently well with the old, the old may be replaced. This is very different
from calibration, where known quantities are measured by a new method and the resultcompared with the true value or with measurements made by a highly accurate method.
When two methods are compared neither provides an unequivocally correct measurement, so
we try to assess the degree of agreement. But how?
The correct statistical approach is not obvious. Many studies give the product-moment
correlation coefficient (r) between the results of the two measurement methods as an indicatorof agreement. It is no such thing. In a statistical journal we have proposed an alternative
analysis, 1 and clinical colleagues have suggested that we describe it for a medical readership.
Most of the analysis will be illustrated by a set of data (Table 1) collected to compare two
methods of measuring peak expiratory flow rate (PEFR).

SAMPLE DATA
The sample comprised colleagues and family of J.M.B. chosen to give a wide range of PEFR
but in no way representative of any defined population. Two measurements were made witha Wright peak flow meter and two with a mini Wright meter, in random order. All
measurements were taken by J.M.B., using the same two instruments. (These data werecollected to demonstrate the statistical method and provide no evidence on the comparabilityof these two instruments.) We did not repeat suspect readings and took a single reading as ourmeasurement of PEFR. Only the first measurement by each method is used to illustrate thecomparison of methods, the second measurement being used in the study of repeatability.

PLOTTING DATA
The first step is to plot the data and draw the line of equality on which all points would lie if
the two meters gave exactly the same reading every time (fig 1). This helps the eye in gauging
the degree of agreement between measurements, though, as we shall show, another type ofplot is more informative.

1

PEFR MEASURED WITH WRIGHT PEAK FLOW AND MINI WRIGHT PEAK FLOW METER
Wright peak flow meter
Mini Wright peak flow meter
First PEFR Second PEFRFirst PEFR Second PEFR
Subject(l/min)(l/mi)(l/min)(l/min)
1
494
490
512
525
2
395
397
430
415
3
516
512
520
508
4
434
401
428
444
5
476
470
500
500
6
7
8
9

557
413
442
650

611
415
431
638

600
364
380
658

625
460
390
642

10
11
12
13

433
417
656
267

429
420
633
275

445
432
626
260

432
420
605
227

14
15
16
17

478
178
423
427

492
165
372
421

477
259
350
451

467
268
370
443

Fig 1. PEFR measured with large Wright peak flow meter and mini Wright peak flow meter,with line of equality.

INAPPROPRIATE USE OF CORRELATION COEFFICIENT
The second step is usually to calculate the correlation coefficient (r) between the two
methods. For the data in fig 1, r = 0.94 (p < 0.001). The null hypothesis here is that themeasurements by the two methods are not linearly related. The probability is very small andwe can safely conclude that PEFR measurements by the mini and large meters are related.
However, this high correlation does not mean that the two methods agree:(1) r measures the strength of a relation between two variables, not the agreement between
them. We have perfect agreement only if the points in fig 1 lie along the line of equality, but
we will have perfect correlation if the points lie along any straight line.

2(2) A change in scale of measurement does not affect the correlation, but it certainly affects
the agreement. For example, we can measure subcutaneous fat by skinfold calipers. The
calipers will measure two thicknesses of fat. If we were to plot calipers measurement against
half-calipers measurement, in the style of fig 1, we should get a perfect straight line withslope 2.0. The correlation would be 1.0, but the two measurements would not agree — we
could not mix fat thicknesses obtained by the two methods, since one is twice the other.
(3) Correlation depends on the range of the true quantity in the sample. If this is wide, thecorrelation will be greater than if it is narrow. For those subjects whose PEFR (by peak flow
meter) is less than 500 l/min, r is 0.88 while for those with greater PEFRs r is 0.90. Bothare less than the overall correlation of 0.94, but it would be absurd to argue that agreement isworse below 500 l/min and worse above 500 l/min than it is for everybody. Since
investigators usually try to compare two methods over the whole range of values typicallyencountered, a high correlation is almost guaranteed.
(4) The test of significance may show that the two methods are related, but it would beamazing if two methods designed to measure the same quantity were not related. The test ofsignificance is irrelevant to the question of agreement.
(5) Data which seem to be in poor agreement can produce quite high correlations. For
example, Serfontein and Jaroszewicz 2 compared two methods of measuring gestational age.
Babies with a gestational age of 35 weeks by one method had gestations between 34 and 39.5
weeks by the other, but r was high (0.85). On the other hand, Oldham et al. 3 compared themini and large Wright peak flow meters and found a correlation of 0.992. They thenconnected the meters in series, so that both measured the same flow, and obtained a "material
improvement" (0.996). If a correlation coefficient of 0.99 can be materially improved upon,
we need to rethink our ideas of what a high correlation is in this context. As we show below,
the high correlation of 0.94 for our own data conceals considerable lack of agreement
between the two instruments.

MEASURING AGREEMENT
It is most unlikely that different methods will agree exactly, by giving the identical result for
all individuals. We want to know by how much the new method is likely to differ from theold: if this is not enough to cause problems in clinical interpretation we can replace the oldmethod by the new or use the two interchangeably. If the two PEFR meters were unlikely togive readings which differed by more than, say, 10 l/min, we could replace the large meter by
the mini meter because so small a difference would not affect decisions on patientmanagement. On the other hand, if the meters could differ by 100 l/min, the mini meter
would be unlikely to be satisfactory. How far apart measurements can be without causing
difficulties will be a question of judgment. Ideally, it should be defined in advance to help in
the interpretation of the method comparison and to choose the sample size.
The first step is to examine the data. A simple plot of the results of one method against thoseof the other (fig 1) though without a regression line is a useful start but usually the data points
will be clustered near the line and it will be difficult to assess between-method differences. A
plot of the difference between the methods against their mean may be more informative. Fig 2
displays considerable lack of agreement between the large and mini meters, withdiscrepancies of up to 80 l/min, these differences are not obvious from fig 1. The plot ofdifference against mean also allows us to investigate any possible relationship between themeasurement error and the true value. We do not know the true value, and the mean of thetwo measurements is the best estimate we have. It would be a mistake to plot the difference
against either value separately because the difference will be related to each, a well-known
statistical artefact. 4

3

Fig 2. Difference against mean for PEFR data.
For the PEFR data, there is no obvious relation between the difference and the mean. Under
these circumstances we can summarise the lack of agreement by calculating the bias,
estimated by the mean difference d and the standard deviation of the differences ( s ). If
there is a consistent bias we can adjust for it by subtracting d from the new method. For the
PEFR data the mean difference (large meter minus small meter) is − 2.1 l/min and s is 38.8
l/min. We would expect most of the differences to lie between d − 2 s and d + 2 s (fig 2).
If the differences are Normally distributed (Gaussian), 95% of differences will lie between
these limits (or, more precisely, between d − 1.96 s and d + 1.96 s ). Such differences arelikely to follow a Normal distribution because we have removed a lot of the variation between
subjects and are left with the measurement error. The measurements themselves do not haveto follow a Normal distribution, and often they will not. We can check the distribution of thedifferences by drawing a histogram. If this is skewed or has very long tails the assumption ofNormality may not be valid (see below).
Provided differences within d ± 2 s would not be clinically important, we could use the two
measurement methods interchangeably. We shall refer to these as the "limits of agreement".
For the PEFR data we get:d − 2 s = −2.1 − (2 × 38.8) = −79.7 l/min
d + 2s = −2.1 + (2 × 38.8) = −75.5 l/min
Thus, the mini meter may be 80 l/min below or 76 l/min above the large meter, which would
be unacceptable for clinical purposes. This lack of agreement is by no means obvious in fig
1.

PRECISION OF ESTIMATED LIMITS OF AGREEMENT
The limits of agreement are only estimates of the values which apply to the whole population.
A second sample would give different limits. We might sometimes wish to use standard
errors and confidence intervals to see how precise our estimates are, provided the differences
follow a distribution which is approximately Normal. The standard error of d is

s2 / n ,where n is the sample size, and the standard error of d − 2 s and d + 2 s is about 3s 2 / n .
95% confidence intervals can be calculated by finding the appropriate point of the tdistribution with n − 1 degrees of freedom, on most tables the columns marked 5% or 0.05,

4and then the confidence interval will be from the observed value minus t standard errors to
the observed value plus t standard errors.
For the PEFR data s = 38.8. The standard error of d is thus 9.4, For the 95% confidence
interval we have 16 degrees of freedom and t = 2.12. Hence the 95% confidence interval forthe bias is − 2.1 − (2.12 × 9.4) to − 2.1 + (2.12 × 9.4) , giving − 22.0 to 17.8 l/min. The
standard error of the limit d − 2 s is 16.3 l/min. The 95% confidence interval for the lower
limit of agreement is − 79.7 − (2.12 × 16.3) to − 79.7 + (2.12 × 16.3) , giving − 114.3 to− 45.1 l/min. For the upper limit of agreement the 95% confidence interval is 40.9 to 110.1
l/min. These intervals are wide, reflecting the small sample size and the great variation of thedifferences. They show, however, that even on the most optimistic interpretation there can beconsiderable discrepancies between the two meters and that the degree of agreement is not
acceptable.

EXAMPLE SHOWING GOOD AGREEMENT
Fig 3 shows a comparison of oxygen saturation measured by an oxygen saturation monitor
and pulsed oximeter saturation, a new non-invasive technique. 5 Here the mean difference is0.42 percentage points with 95% confidence interval 0.13 to 0.70. Thus pulsed oximetersaturation tends to give a lower reading, by between 0.13 and 0.70. Despite this, the limits ofagreement ( − 2.0 and 2.8) are small enough for us to be confident that the new method canbe used in place of the old for clinical purposes.

Fig 3. Oxygen saturation monitor and pulsed saturation oximeter

RELATION BETWEEN DIFFERENCE AND MEAN
In the preceding analysis it was assumed that the differences did not vary in any systematicway over the range of measurement. This may not be so. Fig 4 compares the measurement ofmean velocity of circumferential fibre shortening (VCF) by the long axis and short axis in Mmode echocardiography. 6 The scatter of the differences increases as the VCF increases. We
could ignore this, but the limits of agreement would be wider apart than necessary for small
VCF and narrower than they should be for large VCF. If the differences are proportional tothe mean, a logarithmic transformation should yield a picture more like that of figs 2 and 4,
and we can then apply the analysis described above to the transformed data.

5

Fig 4. Mean VCF by long and short axis measurements.
Fig 5 shows the log-transformed data of fig 4. This still shows a relation between the
difference and the mean VCF, but there is some improvement. The mean difference is 0.003 †on the log scale and the limits of agreement are − 0.098 † and 0.106.† However, although
there is only negligible bias, the limits of agreement have somehow to be related to theoriginal scale of measurement. If we take the antilogs of these limits, we get 0.80 and 1.27.
However, the antilog of the difference between two values on a log scale is a dimensionlessratio. The limits tell us that for about 95% of cases the short axis measurement of VCF will
be between 0.80 and 1.27 times the long axis VCF. Thus the short axis measurement may
differ from the long axis measurement by 20% below to 27% above. (The log transformationis the only transformation giving back-transformed differences which are easy to interpret,
and we do not recommend the use of any other in this context.)

Fig 5. Data of fig 4 after logarithmic transformation.

†

These numbers were incorrectly printed as 0.008, -0.226, and 0.243 in the Lancet. This mistake arose
because when revising the paper we dithered over whether to use natural logs or logs to base 10 and got
hopelessly confused.

6

Sometimes the relation between difference and mean is more complex than that shown in fig
4 and log transformation does not work. Here a plot in the style of fig 2 is very helpful incomparing the methods. Formal analysis, as described above, will tend to give limits ofagreement which are too far apart rather than too close, and so should not lead to theacceptance of poor methods of measurement.

REPEATABILITY
Repeatability is relevant to the study of method comparison because the repeatabilities of thetwo methods of measurement limit the amount of agreement which is possible. If one method
has poor repeatability — i.e. there is considerable variation in repeated measurements on thesame subject — the agreement between the two methods is bound to be poor too. When theold method is the more variable one, even a new method which is perfect will not agree with
it. If both methods have poor repeatability, the problem is even worse.
The best way to examine repeatability is to take repeated measurements on a series ofsubjects. The table shows paired data for PEFR. We can then plot a figure similar to fig 2,
showing differences against mean for each subject. If the differences are related to the mean,
we can apply a log transformation. We then calculate the mean and standard deviation of thedifferences as before. The mean difference should here be zero since the same method wasused. (If the mean difference is significantly different from zero, we will not be able to use
the data to assess repeatability because either knowledge of the first measurement is affecting
the second or the process of measurement is altering the quantity.) We expect 95% ofdifferences to be less than two standard deviations. This is the definition of a repeatability
coefficient adopted by the British Standards Institution. 7 If we can assume the maindifference to be zero this coefficient is very simple to estimate: we square all the differences,add them up, divide by n , and take the square root, to get the standard deviation of thedifferences.

Fig 6. Repeated measures of PEFR using mini Wright peak flow meter.
Fig 6 shows the plot for pairs of measurements made with the mini Wright peak flow meter.
There does not appear to be any relation between the difference and the size of the PEFR.
There is, however, a clear outlier. We have retained this measurement for the analysis,
although we suspect that it was technically unsatisfactory. (In practice, one could omit thissubject.) The sum of the differences squared is 13479 so the standard deviation of differences
between the 17 pairs of repeated measurements is 28.2 l/min. The coefficient of repeatability7

is twice this, or 56.4 l/min for the mini meter. For the large meter the coefficient is 43.2
l/min.
If we have more than two repeated measurements the calculations are more complex. We
plot the standard deviation of the several measurements for that subject against their mean
and then use one-way analysis of variance, 8 which is beyond the scope of this article.

MEASURING AGREEMENT USING REPEATED MEASUREMENTS
If we have repeated measurements by each of the two methods on the same subjects we cancalculate the mean for each method on each subject and use these pairs of means to compare
the two methods using the analysis for assessing agreement described above. The estimate ofbias will be unaffected, but the estimate of the standard deviation of the differences will be
too small, because some of the effect of repeated measurement error has been removed. We
can correct for this. Suppose we have two measurements obtained by each method, as in thetable. We find the standard deviation of differences between repeated measurements for eachmethod separately, s1 and s 2 , and the standard deviation of the differences between themeans for each method, s D . The corrected standard deviation of differences, sc , is1
1
s D2 + s12 + s 22 . This is approximately
44

2 s D2 , but if there are differences between thetwo methods not explicable by repeatability errors alone (i.e. interaction between subject andmeasurement method) this approximation may produce an overestimate. For the PEFR, wehave s D = 33.2, s1 = 21.6, s 2 = 28.2 ‡ l/min. sc is thus

33.2 2 + 1 × 21.6 2 + 1 × 28.2 2
4
4

or 37.7 l/min. Compare this with the estimate 38.8 l/min which was obtained using a single
measurement. On the other hand, the approximationl/min).

2 s D2 gives an overestimate (47.0

DISCUSSION
In the analysis of measurement method comparison data, neither the correlation coefficient(as we show here) nor techniques such as regression analysis 1 are appropriate. We suggest
replacing these misleading analyses by a method that is simple both to do and to interpret.
Further, the same method may be use to analyse the repeatability of a single measurement
method or to compare measurements by two observers.
Why has a totally inappropriate method, the correlation coefficient, become almost
universally used for this purpose?. Two processes may be at work here --- namely, patternrecognition and imitation. A likely first step in the analysis of such data is to plot a scatter
diagram (fig 1). A glance through almost any statistical textbook for a similar picture willlead to the correlation coefficient as a method of analysis of such a plot, together with a test of
the null hypothesis of no relationship. Some texts even use pairs of measurements by two
different methods to illustrate the calculation of r . Once the correlation approach has been
published, others will read of a statistical problem similar to their own being solved in thisway and will use the same technique with their own data. Medical statisticians who ask "why
did you use this statistical method?" will often be told "because this published paper used it".
Journals could help to rectify this error by returning for reanalysis papers which use incorrect
statistical techniques. This may be a slow process. Referees, inspecting papers in which two
methods of measurement have been compared, sometimes complain if no correlation
coefficients are provided, even when the reasons for not doing so are given.

‡

This was incorrectly printed as sc in the Lancet and in Biochimica Clinica.

8

We thank of our colleagues for their interest and assistance, including Dr David Robson who
first brought the problem to us, Dr P. D'Arbela and Dr H. Seeley for the use of their data; andMrs S Stevens for typing the manuscript.

REFERENCES
1. Altman DG, Bland JM. Measurement in medicine: the analysis of method comparison
studies. The Statistician 1983; 32, 307-317.
2. Serfontein GL, Jaroszewicz AM. Estimation of gestational age at birth: comparison of two
methods. Arch Dis Child 1978; 53: 509-11.
3. Oldham HG, Bevan MM, McDermott M. Comparison of the new miniature Wright peakflow meter with the standard Wright peak flow meter. Thorax 1979; 34: 807-08.
4. Gill JS, Zezulka AV, Beevers DG, Davies P. Relationship between initial blood pressure
and its fall with treatment. Lancet 1985; i: 567-69.
5. Tytler JA, Seeley HF. The Nellcor N-101 pulse oximeter - a clinical-evaluation inanesthesia and intensive-care. Anaesthesia 1986; 41: 302-305.
6. D'Arbela PG, Silayan ZM, Bland JM. Comparability of M-mode echocardiographic longaxis and short axis left ventricular function derivatives. British Heart Journal 1986; 56: 4459.
7. British Standards Institution. Precision of test methods 1: Guide for the determination andreproducibility for a standard test method (BS 597, Part 1). London: BSI (1975).
8. Armitage P. Statistical methods in medical research. Oxford: Blackwell Scientific
Publications, 1971: chap 7.

Reproduced by kind permission of the Lancet.

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