STATISTICS IN MEDICINE, VOL. 12,917-924 (1993)

CONSTRUCTION OF AGE-RELATED REFERENCE CENTILES USING ABSOLUTE RESIDUALS

DOUGLAS G. ALTMAN Medical Statislics Laboratory, Iniperial Cancer Research Fund, P.O. Box 123. Lincoln's Inn Fields, London WC2A 3 P X .

U .K .

SUMMARY

This paper proposes a simple approach to the parametric derivation of age-related reference ranges which avoids the creation of arbitrary age groups, copes easily with a non-linear relation between variability and age, and is computationally simple. After the mean is modelled as a function of age, the age-specific standard deviation is estimated by regressing the absolute residuals on age. The method assumes that the data are, or can be transformed to be, normal a t each age, and uses the properties of the half normal distribution. An example is given using 450 measurements of fetal foot length.

INTRODUCTION

For a normally distributed variable with standard deviation (SD) s, parametric reference centiles are calculated as mean ks, where k is often chosen as 1.645 or 1.96 to give a 90 or 95 per cent reference interval. When the variable changes with age it is common to use polynomial regression to model the mean as a function of age. However, ways of modelling the SD are not often considered, even though for many data sets the SD also changes with age. Indeed many published age-related reference ranges are calculated as mean f k x residual SD when the SD is clearly not constant.'-3 Goldstein4 suggested modelling the variance as a function of age by dividing age into several small groups and regressing the variance on age, and Isaacs et aL5 proposed the same procedure using the SD. Royston6 proposed a simpler approach based upon dividing age into three groups by tertiles and comparing the SDs of the observations in the lowest and highest thirds of age. This note describes a much simpler way of modelling variation in the SD which has several advantages over other methods when the data are normally distributed or can be transformed to be so.

METHOD

If the mean of the variable Y is modelled adequately by some function of age, most obviously a linear, quadratic or cubic curve, the information about the SD of the measurement is contained in the residuals around the fitted curve. The parametric method of deriving a reference range is based on the assumption that the variable has a normal distribution at all ages, so the residuals should have a normal distribution at each age and the absolute values of the residuals should have a half normal distribution. The mean of a half standard normal distribution is ,/(2/7c).' Thus the mean of the absolute residuals multiplied by J(z/2) is an estimate of the SD of the residuals. It

0277-671 5/93/100917-08$09.00 0 1993 by John Wiley & Sons, Ltd.

Received October 1991 Revised September 1992

918 D. G . ALTMAN

follows that, if the SD is not reasonably constant over age, the predicted values from regression of the absolute residuals on age multiplied by ,/(n/2) will be age-specific estimates of the SD of the signed residuals, and hence of Y. It is unlikely that a curve more complex than a quadratic will be required to get a satisfactory fit to the SD.

Because the absolute residuals have a skewed distribution, tests based on the standard errors of the regression coefficients are not strictly correct. This should not be a major worry as the choice of the best model should not be based solely on P-values but should also take account of the visual acceptability of the fitted reference range and the goodness of fit. In large samples some high-order terms may be statistically significant yet have negligible effect on the goodness of fit. Royston6 recommended plotting the residuals against age and testing them for normality. If the SD is not constant these procedures should be performed using standardized residuals. These are defined as

Yi - Ywed(Xi)

Spred(Xi)

for subject i with measurement yi and age xi, where ypred and Spred are the estimated age-specific mean and SD of Y. While transformation of the absolute residuals, in particular log transforma- tion, can make the distribution of the absolute residuals closer to normal,' use of transformed values leads to an estimate of the geometric mean and so is a biased estimate of the arithmetic mean of the absolute residuals and hence of the SD of Y. It is perhaps not as easy to detect non-linearity by eye when Y has a skewed distribution as when the variability is symmetric. Here it may help to use a smoothing method such as Cleveland's' as an aid to determining the shape of the most suitable model.

The proposed method does not involve the creation of arbitrary age groups and makes no additional assumptions, but its main appeal is perhaps that it is computationally simple. Methods based on grouping across age involve separate analyses of summary statistics to get a regression equation which then has to be reintroduced into the main analysis. This two-step analysis is tedious and there is the risk of transcription error. It is especially tiresome when the analysis has to be repeated several times in the process of finding the most suitable transformation.6 A further advantage of the method based on absolute residuals is that it leads naturally to a useful plot to aid the assessment of goodness of fit.

The estimated age-specific SD, Spred. could be used as the basis for re-estimating the model for the mean using weights of f /S&(Xi) , as suggested by Aitkin." The effect of this iterative procedure will depend upon both the variation in Spred across the range of X and the sample size, but it is unlikely to make much difference unless Spred changes considerably over the range of X .

EXAMPLE

The data come from a study to develop comprehensive reference centiles for fetal size in relation to gestational age. Centile charts are widely used to assess fetal development at antenatal visits. Ultrasound scanning was used to obtain measurements of various fetal dimensions from unselec- ted pregnancies. One measurement was obtained from each subject expressly for the purpose of developing reference centiles.' The proposed method is illustrated using 450 measurements of foot length (Figure 1).

The mean foot length was modelled by polynomial regression. A quadratic regression model gave an excellent fit to the data. However, in the analysis of many fetal measurements on this sample I have found that a slightly more visually pleasing fit is obtained by fitting a linear-cubic regression model y = a + bx + cx3, although in practice the difference is negligible except at the

AGE-RELATED REFERENCE CENTILES USING ABSOLUTE RESIDUALS 919

0 - 90 i

- 70 E E - 60

I I I f I I 10 15 20 25 30 35 40

Figure 1 . Foot lengths of 450 fetuses by gestational age"

Length of g e s t a t i o n (weeks)

extremes of the range of X . For foot lengths the linear-cubic model was

Ypred = 35.08 + 3'574.X - 0.0004406~~. (1)

The SDs of the residuals from this model for each week of gestation are given in Table I(a) and plotted in Figure 2(a), where the circle sizes are proportional to sample size. Weighted linear regression of SD on gestational age gave

Spred = 0366 + 0.0979~.

Figure 2(a) suggests that the relation may really be cubic, and indeed the cubic term is statistically significant. However, the cubic curve gives minimal improvement in the residual SD and the straight line is an adequate fit to the data. Table I(b) and Figure 2(b) show comparable results for six age groups. The weighted regression analysis gave the estimate of the SD as

Spred = 0.576 + 0.0997~.

Figure 3 shows the absolute residuals from model ( 1 ) with the linear regression on age. The slope of this line is statistically significant ( t = 5.70, p < 0.0001) if we accept the least squares results at face value. Adding a quadratic term makes a negligible difference to the fit, although here too the cubic term is marginally significant ( P = 0.04). I have continued to work with the linear equation, which gives predicted values of 0520 + 007691~. The estimated SD is therefore obtained by multiplying these values by J(n/2) to give

Spred = 0.651 f 0'0964~, (2)

which is very similar to the equations derived from the grouped data. We can examine the fit of the combined model by superimposing k Spred or, more helpfully, f kSpred on a scatter plot of either the signed residuals from model (l), as in Figure 4, or the

standardized residuals. I t is very much easier to assess goodness of fit from this 'detrended' plot than from a plot of the raw data. There are 46 values (10 per cent) above the 90th centile and 48 (1 1 per cent) below the 10th centile. They are reasonably spread across gestational age and there

920 D. G . ALTMAN

Table I. Standard deviations of residuals around function fitted to mean foot length

(a) Thirty-one weekly gestational age groups (b) Six gestational age groups

Mean SD of gestational foot

age length n

Mean SD of gestational foot

age length n

12.62 13.71 14.61 15.35 16.44 17.47 18.34 19.46 20.34 2 1.40 22.40 23.41 24.4 1 25.34 26.44 27.52 28.3 I 2950 30.33 3 1.49 32.39 33.40 34.43 35.41 36.43 37.32 38.42 39.19 40.54 4 1.44 42.07

0.46 1.52 1.60 2.17 1.51 2.14 3.39 2.82 2.27 2.9 1 2.19 2.97 3.64 3.12 3.65 3.67 3.26 3.39 3.33 3.19 3.42 5.14 2.9 1 3.22 2.99 4.55 2.97 4.3 I 5.49 4.64 6.89

3 3 7 7

12 16 I 1 16 15 21 18 18 22 17 20 22 20 20 18 24 22 19 11 15 16 11 12 6

10 10 8

16.03 1.86 50 20.78 2.84 82 25.62 3.43 98 30.63 3.33 106 35.42 4.02 69 40.3 2 4.82 45

are no visual peculiarities. A more formal examination of the normality of the residuals is recommended.6 However, when the SD is a function of age it is more appropriate to consider the distribution of the standardized residuals. Figure 5 shows the normal plot and the associated W' test gives P = 0.64. The fit seems highly satisfactory. The raw data and fitted centiles are shown in Figure 6.

DISCUSSION

Royston6 outlined a systematic approach to the calculation of age-related reference centiles. He gave limited attention to the possibility that the SD changes with age, although in some fields, such as fetal size, the SD almost always varies. He suggested dividing the data into three equal groups by age and comparing the SDs in the youngest and oldest thirds, which is a test for linear

AGE-RELATED REFERENCE CENTILES USING ABSOLUTE RESIDUALS 92 1

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trend in SD. While this method works well when the SD changes linearly with age it cannot detect non-linear (including non-monotonic) variation in the SD. A similar basic approach but with rather more age groups allows the SD (or variance) to be modelled as a (possibly non-linear) function of age. Isaacs et aL5 gave examples where the relation of SD to age was non-linear.

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Figure 4. Signed residuals after modelling the mean foot length with estimates of 1.645s representing a 90 per cent reference range

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Figure 5. Normal plot of standardized residuals from model (2)

Both of these approaches involve arbitrary groupings of the data, although both work well in many situations. However, the proposed method based on the use of absolute residuals works without grouping age and is thus easier to implement. Also it is more appealing to work with individual rather than grouped observations. In effect each observation provides an age-specific estimate of the SD. This approach leads directly to the useful diagnostic plot illustrated in Figure 4 (although such a plot could be constructed for any model). Also it is simple to use the method when the SD changes in a non-linear fashion with age.

AGE-RELATED REFERENCE CENTILES USING ABSOLUTE RESIDUALS 923

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Figure 6. Data from Figure 1 with superimposed 3rd. IOth, SOth, 90th and 97th centiles derived using proposed method based on absolute residuals

The suggested approach was devised as a means of avoiding age grouping. It is also suggested by Efron” as a method of checking the significance of variation in the spread of his own regression percentiles. Although Efron’s method involves estimating percentiles by asymmetri- cally weighting squared residuals above and below the regression line, he also considers briefly the proposal of Koenker and Bassett’ to use asymmetric weighting of absolute residuals. Efron observes that any centile can be estimated in the way suggested in the present paper, and although it works well in his example he suggests that his more general approach is preferable. He also notes that the ratio of the residual SD and the mean of the absolute residuals can be compared with the theoretical value of ,/(nC/2) 2 1.25, to get a quick idea of the appropriateness of the approach. The idea of regressing the absolute residuals on the explanatory variable also occurs in the paper by Glejser,I4 as the basis of a test for (monotone) heteroscedasticity.

Hypothesis testing of the coefficients in the polynomial regression using absolute residuals is not strictly valid because of their non-normal distribution. Neither Efron” nor GlejserI4 comments on this problem. For the fetal foot measurements I carried out a bootstrap analysis of the linear regression of the absolute residuals on gestational age. From 500 bootstrap samples the estimated slope and standard error were 0,0774 (001355), virtually identical to the values of 0.0769 (0.01 349) obtained from the actual data, suggesting that in this case the regression analysis is not misleading. This aspect requires further examination, especially in smaller samples.

The proposed procedure is critically dependent upon the assumption of conditional normality. This assumption, however, is fundamental to the construction of a parametric reference range as ypred & lapred as described by Roystoq6 so no further assumptions are introduced. As noted above, and discussed at some length by Royston,6 normality needs to be carefully verified. Experience shows that when a transformation is needed it will often work throughout the age range being considered. However, it is possible to have a transformation that is itself a function of age, notably the shifted log transformation log( Y + c(age)).

Some other more complex approaches are cited by Royston6 and in the discussion of Cole’s paper.15 More recent papers are those of Efron,12 Rossiter,16 and Chinn,17 who develops the

924 D. G. ALTMAN

suggestion of Bland et ul.lS that the SD can be modelled as a function of the mean. There is now a large number of methods available of varying complexity for constructing age-related reference centiles, some of which do not require the measurements at each age to be normally distributed. It is unlikely that any simple method will be appropriate in all circumstances. For data where both the mean and SD change smoothly and not too dramatically with age, such as arise in fetal measurement, and the data are or can be made normal, the simple approach suggested in this paper works well.

ACKNOWLEDGEMENTS

I thank Lyn Chitty for the data and Patrick Royston for helpful comments on an earlier draft.

REFERENCES

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