Recurrence risks from family history and metric traits

Ann. Hum. Genet., Lond. (1974), 37, 275

Printed in Great Britain

275

Recurrence risks from family history and metric traits

BY CHARLES SMITH

Department of Human Genetics, Western General Hospital, Edinburgh

AND NANCY R. MENDELL

Department of Microbiology and Immunology, Duke University Medical Centre, Durham, North Carolina, U.S.A.

The threshold model of disease liability (Falconer, 1965) has proved a useful tool for summariz- ing and comparing familial frequencies for diseases and conditions not inherited in a simple Mendelian manner. The model also provides a basis for estimating recurrence risks for specific family histories, given the population frequency of the condition and the correlation in liability between relatives (Smith, 1971; Curnow, 1972; Mendell & Elston, 1974). These methods use information only on the disease status of family members. Often there is additional information on items or traits associated with the condition and this could be included so as to improve the estimate of the recurrence risk in a specific family. The object of this paper is to derive methods which can utilize information on metric traits associated with a familial condition. For example, blood pressure in hypertension or blood glucose levels in diabetes are informative in estimating risks for these diseases. The usefulness of such information in practice will be examined, studying the size of possible changes in the estimates of risk and the increases in accuracy of the risk estimates obtained.

PRINCIPLES

Details on disease status of relatives are limited in value because individuals are classed either as normal or affected (or intermediate, Reich, James & Morris, 1972). Thus rather than knowing the specific liability to the disease for each individual, only the mean liability of the group can be assigned. This coarse grouping does not introduce any bias into the estimation of recurrence risks, but rather it gives estimates of liability which are poorly correlated with the true liability of the individual at risk. In statistical terms the estimates are approximately unbiased but they are not efficient in that the residual variation in the estimate of liability is still high. In biological terms, the risk estimate is an average and includes individuals with a wide range of true risks and so its value for any particular case is decreased. Further information may help to reduce the residual variation in the risk estimate and so allow more accurate risks to be given for a specific family.

The kind of additional information considered here is that given by a metric or continuous trait associated with the condition. The metric trait may vary in the form of its association with liability for the condition. It may serve to define the disease, such as blood glucose levels in diabetes, and so measure liability directly. Or it may be correlated with liability either because it is a factor in the cause of the disease or because it is a result of the disease. The measured (phenotypic) correlation of the metric trait with liability may be due either to genetic effects or to non-genetic (‘environmental 7 effects or to a combination of both. By definition, it is assumed that the risk of the disease is entirely dependent on the liability of the individual and the correlated trait is merely one of the factors determining liability.

276 C. SMITH AND N. R. MENDELL

Y .- s o

2 U

L ” .-

1, L 0 1, Liability

Fig. 1. Diagram showing the mean liabilities of unaffected individuals, L, given Y,, L, given Y , and L for the average of all unaffected.

The nature of the information provided by the metric trait is illustrated in Fig. 1. The mean liability of unaffected individuals is L. With information on the correlated metric trait, the mean liabilities of unaffected individuals with trait values Y , and Y , are L, and L, respectively. Thus some differentiation among unaffected individuals may be achieved. The higher the correlation between liability and the trait, the greater will be the amount of information the trait can provide about liability and risks.

Continuous metric traits are already widely used in medicine as diagnostic indications of many familial and non-familial disorders. The aim is usually to detect abnormal function which may be diagnostic of the disease; that is, to detect individuals with abnormal levels for the trait. If the disease causes individuals to be no longer part of the normal distribution for the trait in the population, then trait information on such individuals cannot be used by the present methods. The emphasis here is less on abnormality for the trait but rather on unaffected individuals whose trait values lie within the normal range but which are raised because of the association with liability to the familial disease.

METHOD

The method used is an extension of an approximate method given by Mendell & Elston (1974) to derive recurrence risks for a given family history. The solution depends on formulae by Aitken (1934) for the variances and covariances of a multivariate normal distribution after selection of one of the variables. The procedure can then be applied in turn to further variables. Aitken’s formulae depend on the assumption that the selected variable still has a normal distribution after selection. This will not be true with the threshold model where selection is by truncation. The approximation introduced by Mendell and Elston was to consider the truncated distribution as normal. They showed that, in estimating recurrence risks in families, the approximation gave quite accurate results. In the present paper the accuracy of the approximation has been assessed by comparing the results with some exact remits derived by Curnow (1974).

The associations among relatives for the liability (L) to a disease and for a metric trait ( Y ) can be represented by a symmetric variance-covariance matrix V . For simplicity consider both

Recurrence risks f rom family history and metric traits 277

Table 1. Symmetric variance - covariance matrix of liability and a metric trait (see text)

Liability Metric trait Liability A & f >

Ll L 2 Yl y 2 YI -% Variable I 2 3 4 5 6

V =

Vll v 1 2

v 2 2

v13

v 2 s

v 3 ,

V16

V26

VSiJ

‘38

‘48

v88

sets of variables as standardized ; that is, expressed with a mean of zero and a variance of unity ( Vii = 1). Consider the matrix V for two relatives (1 and 2) and an individual ( I ) whose risk is to be estimated. V can be arranged in a form convenient for reduction by the Aitken procedure, as in Table 1. Vii and Vij are respectively the variance for variable ( i ) and the covariance for variables i and j. The correlation between any pair ( i , j) is then

Assuming a multi-normal distribution the Aitken procedure can be used to reduce the matrix V to allow for the disease status of the first relative. Thus the residual variances and covariances given that relative 1 is affected are approximately

Vij.1 = vij- ‘Vli‘ljal (%-xl) /Vl l?

rij. 1 = Vij. l /J(Vii . IVjj. 1 )

xz.l = (Xi - bil”1) J(1 - .La1 (a1- XJ),

where a, is the mean liability for affected individuals and xl is the threshold value. The residual correlations are then

and the adjusted threshold value for relative ( i ) is given by

where bi, is V,,/ Vll, the regression of variable i on 1. The quantity ai, can be easily derived from the normal curve given x i . l .

The same procedure can be repeated for the disease status of relative 2, and then for further relatives in turn, thus

‘ij.12 = ‘ij. 1 - Vi2. J j 2 . 1%. 1 (a2.1- x2. *)/ V2Z.l.

If the relative is not affected then ai is replaced by - aiFp( 1 - Fp) , where F p is the population frequency of the condition.

The reduced matrix in the example is now V.12:

‘33.12 ‘34.12 v3.5. 1 2 ‘36. 12

‘44.12 ‘45.12 V46 . 12 I ‘56.12 ‘56. 1 2 v.12 =

\ ‘66.12

This has the form of a set of multiple regression equations for variables 3, 4 and 5 on variable 6. Solving yields the partial regression coefficients. For example, bA3. 12 is the regression of the individual’s liability (variable 6) on variable 3, taking the other variables (4 and 5 ) into account.

278 C. SMITH AND N. R. MENDELL In the reduced matrix ( V . 12) above the term V6, . 12 is the residual variance in the individual’s

liability after allowing for the disease status of his relatives (variables 1 and 2). Information on the metric trait for the individual and for his relatives will account for further variation. This further reduction in variance due to regression on variables 3, 4 and 5 is

5

i = 3 vreg = x b i2 .12 v6$.12*

The residual variance in the individual’s estimated liability (variable 6) after allowing for all the other variables is thus

This indicates how precise the estimate of liability is and allows a confidence interval to be set. The total variation in liability accounted for is (1- V,) and this is the square of the multiple correlation between the estimate and the true liability.

V B = V66.12- Vreg.

The liability of the individual at risk can now be estimated as follows:

This takes account of the residual variation ( VR), then adjusts the individual’s thresholdvalue (x) for the disease status first of relative I, then of relative 2 given relative 1. Similarly the trait value for each relative is adjusted before weighting the information by the appropriate partial regression coefficient in the equation.

The procedure can be extended in principle to any number of relatives of any degree of relationship. However, as more relatives are included the matrix becomes large. Errors arising from the approximation in using the Aitken procedure may also accumulate. However, details on a trait on close relatives may make the inclusion of more distant relatives unnecessary. Note again that the derivation and theory depend on phenotypic observed effects and require no genetic interpretation of the parameters.

The methods have been programmed for sibships with two parents and up to four sibs. The program reads in or generates the matrix V7 reads an array giving the family history and measurements on the trait. It then proceeds through the methods outlined above, allowing for print-out checks at each stage, and finally printing the estimated liability, risk and variance explained. The program (RISKCT) is available on request.

Correlation estimates To apply the above methods for deriving recurrence risks, estimates of the variances and

covariances among the traits are required. All the variables are standardised to have zero mean and unit variances and the covariances then are the correlations among the variables. In matrix V these are the phenotypic correlations for all combinations of liability and metric trait among the various relatives. The correlation in liability between relatives is estimated from the frequency of the disease in the population (Elp) and the frequency in relatives of affected individuals (FR) (Smith, 1970; Mendell & Elston, 1974). The correlations among relatives for the metric trait can be measured directly from samples of relatives. The correlations between the metric trait and liability can be estimated as the difference in the trait between relatives of unaffected and of

Recurrence risks f rom fami ly history and metric traits 279

affected individuals, divided by the difference between the average liabilities of unaffected and of affected individuals. If being affected precludes or affects the measurement of the metric trait, then only unaffected individuals can be measured. The mean value for those who become affected at some later date, compared with those still unaffected, can be used to derive a correlation estimate.

RESULTS

In this section we are concerned mainly with three aspectsi of the method. The first is to assess the accuracy of the approximation in risk estimation with trait information. The others concern the value of including information on a metric trait and the magnitude of changes in risks brought about by trait information.

Accuracy Mendell & Elston (1 974) compared recurrence risks derived by the Aitken procedure using their

approximation with the exact risks (Curnow, 1972; Smith, 1971). They found very good agree- ment, with errors usually less than 1 yo of the true risk and never larger than 5 yo of the true risks. The same result was found here on including information on a correlated trait and comparing the risks with those derived by Curnow (1974). These were for a range of population frequencies and correlation matrices but dealt only with one relative and the individual a t risk. The accuracy of the method when several relatives are included cannot be assessed until the exact risks are known. These would require the numerical integration of multiple integrals.

Variation explained A guide to the value of different pieces of information in estimatingrisks can be got by compar-

ing their effects individually. This value can be gauged through the correlation of the estimated liability and the true liability for the individual whose risk is being estimated. The proportion of the variation in liability explained is then simply the square of the correlation coefficient. This is shown in Figure 2 for a range of correlations with liability and for first-degree relatives. The trait can apply directly to the individual a t risk and so can account for a greater proportion of the variation in liability than other information which is indirect, being through a relative. Thus if there is trait correlated with liability it will usually be very worth while including it in the risk estimation. The other graphs in Fig. 2 assume that the correlations for relatives are proportional to the degree of relationship. Then the variation in liability explained by the affected relatives and by the trait on a relative is similar for the same level of correlation. However, the information from unaffected relatives is much less useful especially if the disease is very rare.

If there is more khan one piece of information used to estimate risk, the total variation accounted for is, of course, not the sum of the separate effects, since these are likely to be correlated. It is, however, derived directly in the matrix calculations as described in the methods section. There are two aspects of interest ; the total variance accounted for and the increase in the proportion accounted for by the addition of each piece of information. A series of results is given in Table 2. To reduce the number of combinations of phenotypic correlations in the V matrix, the examples are presented in genetic rather than in phenotypic terms. Consider two levels, high (1.0) and low (*05), respectively for the heritability of liability ( h i ) , for the heritability of the trait (hg) and for the genetic correlation (rG) between liability and the trait.The various forrelations in matrix V can then be derived using a strictly genetic interpretation, as


100 .

1.0 0.8 0.6 0 4 0.2 0.0 Correlation ( r ) with liability

Fig. 2. Variation in liability accounted for by separate pieces of information on a correlated metric trait and on disease in a relative. - , Trait. Disease: - - - - -, 10%; . . . . . .., 0.1 %.

shown in Table 2. For example, the correlation in liability between first-degree relatives VI2, V,, V,, is (4)h: and the correlation between liability and the trait V13, VZ4, V56, is given by

In Table 2 the variance accounted €or is similar for different frequencies of the disease showing that the population frequency has a relatively minor effect on the amount of variation explained. As already noted, unaffected relatives contribute less information at low frequencies than a t high frequencies. The amount of variation explained in any situation depends greatly on the correlation between the trait and liability, as would be expected. As the correlation increases, the value of the information on the trait also increases. If the correlation is unity, then of course all the variation in liability can be accounted for, since the trait gives a direct measure of liability for the individual at risk (Curnow, 1974). The additional pieces of information add progressively to the variation accounted for and the total may be 5-10 times the percentage originally accounted for by one affected relative. Additional relatives add further to the variation accounted for,

rGhLh,*

Tab

le 2

. Per

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of v

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ccou

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for

by d

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tic p

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eter

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tion

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ies

Fam

ily

hist

ory

Aff

ecte

d fi

rst d

egre

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lati

ve

Una

ffec

ted

firs

t deg

ree

rela

tive

s

Tra

it O

n un

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firs

t de

gree

rela

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O

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e in

divi

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at r

isk

Her

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erit

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abil

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of

betw

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t,

( %)

hi

(&E

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0.25

0.5

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Gen

etic

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0.5

0.5

I '0

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0.5

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0.25

0.5

0.5

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0.5

0.5

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8.4

9.6

6.2

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12.8

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+.

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25.0

26.4

27'3

29.1

z 0.71

5 '4

8.4

19'3

50.0

50'0

50.0

50.0

2

0.35

20.7

21.3

22'5

12.5

28.8

37'4

37'7 2 5

0.50

20.7

21.3

34.8

25'0

37'3

44.1

45'4

0.71

20.7

21.3

37'2

50.0

55.8

59'3

61.5

B 1'00

20.7

21.3

46. I

100'0

100'0

100'0

100'0

0.25

5.8

6. I

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10.7

11'0

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5.8

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9'4

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16.0

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2. T

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50.0

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23.0

25.1

27'9

12.5

30'7

32'3

33'4

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0.5

0

23.0

25'1

30'5

25.0

38.7

40.1

41.2

0-71

23.0

25.1

37'0

50'0

56.5

57'2

61

-3

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48.1

100'0

100'

0 100.0

100'0

m 7

h--

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7

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0

cc

282 C. SMITH AND N. R. MENDELL but at a decreasing rate. For example, if the trait is measured on the individual at risk, information on the trait in relatives does not account for much additional variation. Some compromise on the effort of collecting the information and its value will thus be sought. Finally, in Table 2, except in the extreme situations with very high correlations, there is usually a substantial amount of the variation in liability left unexplained. That is, the residual variation about the risk estimate is still large, so that the risk estimates are not very precise and there is still plenty of scope to improve their accuracy further.

50

t

0.1 ~

Individual -2 -2 -2 0 0 0 2 2 2 Relative -2 0 2 -2 0 2 -2 0 2

Trai t (standard deviat ion units)

Fig. 3. Recurrence risks for an individual with an affected first-degree relative and for a range of values for a metric trait in the individual and in an unaffected first-degree relative. The horizontal dotted lines are the empiric risks (heritability = 0.6) for three levels of population frequency 10 yo, 1 yo, and 0.1 yo. The correlations between the nietric trait and liability are 0.21 (solid lines) and 0.42 (dashed lines) respectively.

Risk estimates Additional information on a trait may also change the value of the risk estimate appreciably.

Thus the risk will be specific to a family with a given set of information, rather than being an average and applying to all families at risk. A series of examples is given in Fig. 3. Again, for simplicity, the correlations in the matrix V are derived directly from a set of genetic parameters. The examples deal with a heritability of liability ( h i ) of 60 % and three population frequencies, 10 yo, 1 yo and 0.1 %. These should represent a wide range of familial conditions in man. For the metric trait, heritabilities of 30 yo and 60 yo were chosen, and genetic correlations with liability

Recurrence risks from family history and metric traits 283

of 0.5 and 0.7 respectively. The phenotypic correlations (rQlzLlzy) of the trait and liability are thus 0.21 and 0.42 respectively. The risks are given for individuals with one affected first degree relative (dotted lines) and also with information on a trait for the individual at risk and for an unaffected relative, each with possible values - 2, 0 and + 2 standard deviation units from the mean. The variance explained by one affected relative was 9 % and this was increased to 13 % and to 23 % for the correlations 0.21 and 0.42 respectively by the additional information.

In Fig. 3 the risks may be greatly modified by including a correlated metric trait, especially when the correlation between the trait and liability is high. As before, the trait measured on the individual a t risk is much more important than the trait in a relative, but the latter does contribute to the estimate of risk. Individuals with high values for the trait may have double the average risk, while in those with low trait values the risks may be halved or even reduced to 10 yo or one yo of the original empiric risk. The higher the correlation with liability, of course, the greater the increase in risk for individuals with high values of the trait, and the lower the risk with low trait values. Measurements on one unaffected relative alter the risk estimates less but if several unaffected relatives were included, collectively they could have a greater effect.

Example To illustrate the use of the method consider an individual with a family history of simple

glaucoma; with an affected father and an affected sib, but with a normal mother and two normal sibs. The frequency of glaucoma among persons over 40 is about 1 yo (Bankes et al. 1968). In first-degree relatives of cases the frequency is about 4-5 yo (Leighton, 1968). This gives a correlation of 0.25 between first-degree relatives for liability to glaucoma. Ignoring age effects, the estimated risk to a sib given the above family history is about 9 % and some 10% of the variation in liability is accounted for.

A raised intra-ocular pressure (IOP) is characteristic of most patients with simple glaucoma, but not all individuals with raised I O P (over 21 Hg) have glaucoma (Bankes et al. 1968). Armaly, Monstavicius & Sayegh (1968) found that IOP (standardized for sex and age) was raised ( + 0.156) in relatives of patients with glaucoma compared with ( - 0.283) in relatives of normal controls. The regression (and correlation) of I O P in relatives on liability of cases (2.665) and of controls ( - 0.026) is estimated as

0.156 - ( - 0.283) 2.665 - ( - 0.026) b = r = = 0.16.

No estimates of the correlation between IOP and liability in the same individual seem available and so an arbitrary value of 0.4 was used. The correlation in IOP among first degree relatives is about 0.3 (Armaly et al. 1968).

The results from using different kinds of information from the family are given in Table 3. As further information on family history or on IOP levels in the family is added, the variation explained increases. The risk depends largely on the individual's own IOP and this is moderated only slightly by the I O P levels in relatives.

DISCUSSION

In practice the problems of estimating recurrence risks are likely to be biological rather than mathematical. The mathematical methods described here are versatile and appear to provide good approximations to theoretical risks, given multi-variate normal distributions and good


Table 3. An example of estimated risks and variation in liability accounted for by wing diflerent kinds of information on family history of glaucoma and on intra-ocular pressure (IOP) in relatives

Information used

Positive family history Full family history Individual’s own IOP Positive family history plus individual’s own IOP Full family history plus individual’s own IOP Full family history

Plus individual’s own IOP plus IOP on unaffected relatives (average)

+2 0

- 2

( %) 9.0

I 6.0

21-8

10’0

21’2

21.9 21.9 21.9

2 - - 9’7

8.8 -

4.8 19.1

- 17.8

- -

I 8.6 17’5 I 6.4

0

- - 0.6 4‘7 4’3

47 4 3 40

Variation Risk (percent) : accounted Individual’s IOP (s.d. units)

A for I 3

- 2

0.7 0.6 0.6

estimates of the basic parameters. The main query in their application will be w.,ether tL,e theoretical distributions are appropriate in describing the actual joint distribution of liability and the trait. However, if the distributions are continuous, and can be transformed to quasi-normality, then the mathematical methods seem to be quite robust and may be appropriate for estimating recurrence risks.

An advantage of the methods is that they can deal with familial effects whether they are genetic or non-genetic or, as is likely, some combination of the two. The methods deal with correlations between relatives, whatever may be the cause of the correlation. Thus there need be no genetic interpretation of familial frequencies or correlated traits in the estimation of recurrence risks. If information is available on several traits they can all be included in estimating risks, simply by increasing the matrix to include them. To do this the correlations among the different traits, as well as with liability, would be required.

The method of estimating the recurrence risks depends on multiple regression maximising the multiple correlation of all the information available with the liability of the individual at risk. If the separate correlations are in error or are poorly estimated then the estimates of the risk and variance explained will be affected and may be misleading. It is thus important to derive as accurate estimates as possible for the correlations. It has proved surprisingly difficult to get from the literature reliable estimates of the correlations for many of the common familial disorders and associated metric traits. However, as outlined earlier, it should be readily possible t o estimate them by collecting special sets of family material. A list of familial conditions with possible traits which might be useful in estimating risks is given in Table 4. Unfortunately, none of the congenital abnormalities seem to have metric traits with which they are correlated.

There is a paradox in the use of a correlated trait to supplement family history in estimating risks. If the correlation with liability is low then the trait adds little information. If the correlation is high then the individual’s own value for the trait gives a good estimate of risk and largely supplants the family history and details on relatives. Thus it is when the correlation between trait and liability is intermediate, or when the individual at risk cannot be measured for example if he is too young or not yet born, that the methods here will be of most value.

If the distributions of the trait or of liability are discontinuous then the methods here may

Recurrence risks from family history and metric traits 285

Table 4. List of familial conditions with correlated metric traits which may be useful in risk estimation

Condition Trait

Hypertensive heart disease Diabetes mellitus Glaucoma Mental retardation Coronary heart disease Schizophrenia Gout Epilepsy Rheumatoid arthritis Hypothyroidism Allergy

Diastolic blood pressure Glucose tolerance test Intraocular pressure Intelligence quotient Lipoproteins Personality scores Uric acid E.E.G. measurements Radiographic scores Protein-bound iodine Skin sensitivity tests.

be invalid. When the discontinuity arises because the trait can be classed into only two classes, the Aitken procedure for reducing the matrix can be applied to the trait as well as to liability. However, with only two classes, the amount of information added is likely to be small. Alterna- tively if the trait can be graded into a number of classes, then these may be considered as serial classes in a normal distribution. When the discontinuities are associated with identifiable Mendelian phenotypes, other methods of risk estimation should be used. For example, Heuch & Li (1972) have shown how to combine information on a metric trait, as well as other factors, in estimating risks for Mendelian disorders. To deal with complex situations and multigeneration pedigrees they have developed a general program PEDIG to estimate recurrence risks for any family history.

For familial conditions which are heterogeneous, the correlation of one trait with a mixed distribution of liability is not likely to be high. Resolution of the heterogeneity may increase the correlation for one subgroup and allow the trait to become a better indicator of those at risk for that group. For example, the resolution of coronary heart disease to forms with high serum cholesterol or high triglyceride or high for both lipoproteins would allow better risk predictions to be made in relatives of cases and in individuals screened in the population.

SUMMARY

A method for including information on an associated metric trait in the estimation of recurrence risks for familial conditions is presented. This depends on the correlations among relatives for the trait, for liability to the condition and on the correlations between them. The values of the family history, both positive and negative, and of measurements on the trait in the individual at risk and in his relatives are compared and expressed as the variation in liability accounted for. An example of the use of the method is given for measurements of intraocular pressure in glaucoma.

Details on an associated trait can add considerably to the accuracy of the recurrence risk estima,tes and can modify substantially the estimate of risk in the individual concerned.

We thank Professor R. H. Curnow for allowing us to use his exact results with which to compare the approximation made in this paper, and for constructive discussion and criticism throughout the work.


REFERENCES

AITKEN, A. C. (1934). Notes on selection from a multivariate normal population. Proc. Edinb. Math. SOC. 4, 106-110.

ARMALY, M. F., MONSTAVICIUS, B. F. & SAYEUH, R. E. (1968). Ocular pressure and aqueous outflow facility

BANKES, J. L. K., PERKINS, E. S., TSOLAKIS, S. & WRIGHT, J. E. (1968). Bedford glaucoma survey. Brit.

CURNOW, R. H. (1972). The multifaotorial model of inheritance of liability to disease and its implications for

CURNOW, R. H. (1974). The use of additional information in calculating disease risks from family histories.

FALCONER, D. S. (1965). The inheritance of liability to certain diseases estimated from the incidence among

HEUCH, I. & LI, F. H. F. (1972). PEDIG - A computer program for caloulation of genotype probabilities

LEIGHTON, D. A. (1968). Studies on relatives of glaucoma patients. Proc. R. SOC. Med. 61, 542-544. MENDELL, N. R. & ELSTON, R. C. (1974). Multifactorial qualitative traits: Genetic analysis and prediction of

REICH, T., JAMES, J. W. & MORRIS, C. A. (1972). The use of multiple thresholds in determining the mode of

SMITH, C. (1970). Heritability of liability and concordance in twins. Ann. Hum. Genet., Lond. 34, 85-91. SMITH, C. (1971). Recurrence risks for multifactorial inheritance. Am. J . Hum. Gen. 23, 578-588.

in siblings. Archs Opthal., N . Y . 80 354-360.

Med. J . i, 791-796.

relatives at risk. Biornetrics 28, 931-946.

Biometrics. (In the Press.)

relatives. Ann. Hum. Genet., Lond. 29, 51-76.

using phenotypic information. Clin. Genet. 3, 501-504.

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transmission of semi-continuous traits. Ann. Hum. Genet., Lond. 36, 163-184.

Recurrence risks from family history and metric traits

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