Ann. Hum. Genet., Lond. (1976), 39, 427
427
Printed in areat Britain
On the estimation of parental age effects on mutation BY 0. MAYO,* J. L. MURDOCHT AND T. W. HANCOCK*
* Biometry Section, Waite Agricultural Research Institute, University of Adelaide, Adelaide, 8outh Australia 5064
t Lorna Linda University, School of Medicine, Lorna Linda, California 92354, U.S.A. For many years, it has been accepted that sporadic cases of certain dominant or semi-dominant mutations are associated with an increased paternal age, but not with increased maternal age or birth rank. However, because maternal age, paternal age, and birth rank are so highly correlated with one another, it has been very difficult to separate the effects, and to produce unequivocal evidence for any particular one being more important. All of the methods in use have certain flaws or difficulties. The best method available, that of Smith (1972), requires that one know population values for the regression of maternal age on birth rank, paternal age on birth rank, maternal age on paternal age, and vice versa (the p matrix below). Smith’s method allows not only the detection but also the estimation of the various effects, but as it requires these population data, it may often not be applicable. It does not seem to have been noticed that a simple discriminant function applied to data on sporadic cases and on the sibs of those sporadic cases might provide a good indication of the relative contributions of paternal age, maternal age and birth rank to the heightened incidence of the condition. I n this note, we apply the method of discriminant functions to data on Apert’s acrocephalosyndactyly (Blank, 1960) and achondroplasia (Murdoch,Walker, Hall, Abbey, Smith & McKusick, 1970). We also apply Smith’smethod for comparative purposes. DISCRIMINdNT BUNCTION
The rationale for using a discriminant function to compare sporadic cases of dominant disorders with their normal sibs is very simple: if mutant and normal sibs represent two samples from the same population with respect to parental age, maternal age and birth rank, then it will not be possible to discriminate between them on these grounds, while if it is in fact possible to discriminate between them, the relative contributions of the three variables to the discriminant should give an indication of their relative contributions to the elevated incidence of the mutation. (A linear relationship seemed appropriate since preliminary investigation using a method suggested earlier (Mayo, 1974) yielded no clear indication of a curvilinear dependence of mutation rate on maternal age, paternal age or birth rank.) We shall use the notation of Smith (1972) throughout, so that m = mother’s age, f = father’s age, b = birth rank. Table 1 shows the variance-covariance matrices for affected and unaffected sibs in families with Apert’s acrocephalosyndactyly segregating (Blank, 1960). m was normally distributed, while f and b were positively skewed, but transformation did not alter the relative results obtained, so that those presented are for the untransformed data. The vectors of means for normal and affected are (29,077, 34.500, 2.833) and (32.171, 37.057, 2.714) respectively, giving a vector of differences A*’= (A$,AT, A:) = (3,095, 2.557, - 0.119).
428
0.MAYO, J. L. MURDOCHAND T. W. HANCOCK Table 1. Variance-covariance matrices for
--(a;)
Apert's acrocephalosyndactyly (Blank, 1960) Affected = 35)
Normal (n = 78)
(12
f
m
m
f b
23.38 19-09 3.82
19-09 50.88 3'47
f
b
32'47
22'21
22.21
42'47 6.77
5'74 6.77 3'62
b
m
3.82 3'47 2.89
5'74
( b ) Achondroplasia (Murdoch et al. 1970).
Normal = 321)
Affected (n = 92)
('7%
m
f
b
m
m
35-00
f
32-28 7'64
32.28 57'94
7.64 8'64 4'73
41-43 36.76
b
8-64
10.53
f
b
36.76 10.53 58.88 11.62 11.62 6.08
By a standard discriminant analysis (Morrison, 1967) we obtain F(3,109) = 4.133 for the hypothesis that the two samples are from the same population, compared with the 1 yo critical valueof 3.97.Thediscriminant functionis0-1318m+ 0-0315f - 0.28843. The analysisalso provides us with a 95 % simultaneous confidence interval for A*: -0.0822 -1.3567
< A; < A;
Q 6.2712 Q 6.4710
- 1.1934 Q A: < 0.9553 . It seems clear that paternal age is relatively unimportant in discriminating between the affected and normal groups. Table 1 also shows the variance-covariance matrices for achondroplasia (Murdoch et al. 1970). Here, m and f were normally distributed, while b was positively skewed, and as with Apert's syndrome transformation did not materially affect the relationships presented for the untransformed variables. The vectors of means for normal and affected are (27.73, 33.29, 3.45) and (30.77, 36.27, 3.46) respectively, giving
A*' = (3.0459,2.9789,0*0048). From the discriminant analysis we obtain P(3,409) = 9.965 for the hypothesis that the two samples are from the same population, compared with the 0-1% critical value of 5.62. The discriminant function is 0.1 102772+ 0.0288f - 0.2428b. The simultaneous 95 yo confidence interval for A* is 0.9363 Q A; Q 5.1554 0.4242 -0.7964
< A; Q 5.5336 < A: < 0.8060.
Once again, there is no clear indication that paternal age is the most critical component in the discrimination.
On the estimation of parental age eflects 012. mutation
-
429
Table 2. Matrices of regression coeficients
Apert’s acrocephalosyndactyly m
f
b
Achondroplasia
r rn
f
b
m
1.00
0.52
0.63
1-73
0.68 0.18
1.00
1.58 1.87
1’00
f
0.89
1.00
1.91
0.16
1.00
0.25
0-20
1.00
b
SMITH’S METHOD
The essence of the method of Smith (1972) is that, given the matrix of coeffioients of regression pmm
pmf
Pmb
pbm
pbf
Pbb
(where p, is the coefficient of regression of mother’s age on father’s age and so on), it should be possible to test the hypothesis ‘that only the father’s age has any appreciable real influence on the condition, the rises in mother’s age and birth rank being statistical consequences of this’ (Smith, 1972). Smith used 1-00 0.66
1.82
0.07
1.00
0.15
from population values compounded by Murdoch, Walker & McKusick (1972) from U.S., U.K. and Australian data. Because these are population values, they can be treated as precise, so that one can compare Am with pmfA,, i.e. test the hypothesis that Am-pmfA,, which has variance (vmm- 2pm vm + p;4, vf )In, is zero. (Here, v,, and vf are variances from the appropriate ‘affected’variance-covariance matrix in Table 1, while A is the difference between the observed value for the affected group and the population value, not the A* obtained in the previous section, i.e. they are A‘ = (4.131, 6-017, 0.474) and A’ = (4.232, 6-422, 0.817) for Apert’s acrocephalosyndactyly and achondroplasia respectively, the population mean vectors being (28.04, 31.04, 2-24) and (26.54, 29-85, 2.64). This will be discussed further below.) From Table 1, we can compute the two regression matrices shown in Table 2. It can be seen that they are, for the most part, very similar to Smith’s. We also have 1.00 0.63
p=
1
1.86
0.85 1.00 1.60 0-13 0.06
1-00
from Blank’s (1960) calculations on Australian and U.K. data. This will be the matrix most appropriate to the analysis of Blank’s data by Smith’s method; for Murdoch et al.’s data the best p is not clear. In the analysis below, Smith’s p matrix has been used, as has the matrix in Table 2, the latter also being treated as if it were a population matrix, although it is based only on 321 observations, since it is at least drawn from an appropriate population. In Table 3, we show values of hi-pijAj with their standard errors, for i = m,f,b; j = rn,f, b; i p j.
0.Miyo, J. L. MURDOCHAND T,W. HANCOCK
430
-Table 3. Departures of A from expectation
i j
Apert's acrocephalosyndactyly
A{ -A, 4
mf b f m b b m
A, -
S.E.
3'25
0.75 0.74
2.5 I
1'00
5.26 -0.06 0.14
1.16 0.26 0.28
0.36
f
Achondroplasia. Using p from Table 2
i
A,
S.E.
0'22
0.43
2-82 2.67 4.86 - 0.26 - 0.45
0.50
0.55
0.68 0.18 0'20
Using Smith's p
j
,
r
m
f
0.19
-0.01
b m b b m
f
f
2.74 2-74 5 '44
0.50
0.18
0.19
0.37
0'20
0.55
0.69
Table 4. Expected independent changes, d, in m, f and b and the associated variance-covariame matricee
-
Apert's acrocephalosyndactyly A
r
l
A
r
V
d
m
0.57
f
5.37
b
0-10
S.E.
m
f
d
1-61 6.76 -0.92
3.26 -3.27 -0.29 0.17 -3'27 462 0.17 0.22 0.47 -0.29 1-81
\
V
b
2-15
-
Achondroplasia. Using p from Table 2
S.E.
m
f
b
1-06 1-13 -0.76 -0.15 1.31 -0.76 1.73 -0.14 -0.14 0.12 0-35 -0.15 Using Smith's p L
r
I
V
, m
f b
d -1.29 6.91 0.52
3
f
b 1.75 -1.61 -0.24 1-32 2.00 0.20 1.41 -1.61 0.28 -0.24 0.20 0.08 S.E.
m
Smith's method also allows estimation of the expected independent changes d' = (d,,, d,,d,) in the variables m, f,b, from
d x P-lA with variance given by
var d = P-lv(P-l)l/n. These values are shown in Table 4. DISCUSSION
It can be seen that the discriminant function appears to produce different results from Smith's analysis. The latter shows, for both Apert's acrocephalosyndactyly and achondroplasia, that paternal age is the main contributor t o the elevated incidence of the traits, as already demon-
On the estimation of parental age effects on mutation
431
strated by Blank (1960) and Murdoch et al. (1970) using less effective techniques. Why then should the discriminant function, which in effect uses normal sibs as the reference population, produce such strikingly different results? The answer appears to lie in the difference between A and A*;that is, the sibs of sporadic cases of both disorders differ from the general population in the age of their parents and in their birth rank. If A* were used to compute d*, then the values (4.46, 0.16, -0-69) and (4.47, 1.89, - 1.50) would be obtained for Apert’s acrocephalosyndactyly and achondroplasia respectively, with the standard errors given for d in Table 4. While this procedure is very imprecise, i t allows the tentative conclusion that, among the families having sporadic cases of these disorders, the mothers of the sporadic cases have age at birth of the affected children further above the group mean than do fathers. The significance of this within-family difference is not clear. It does seem, however, that normal sibs of affected sporadic cases of dominant disorders do not form an appropriate control population. An appropriately chosen control would seem, nonetheless, to offer the possibility of more powerful discrimination for a smaller sample size than of the incompletely available or inappropriate population data. Loesch & Lisiewicz ( 1974) attempted to use a variant of the discriminant function technique for classification of two aneuploidies, Trisomy 21 and XXY, and achieved reasonable discrimination from very small samples.
SUMMARY
The detection and estimation of the effects of paternal and maternal age and birth rank on mutation rate are considered. Smith’s (1972) method and discriminant function techniques are compared using data on Apert’s acrocephalosyndactyly and achondroplasia. REFERENCES
BLANK,C. E. (1960). Apert’s syndrome (a type of acrocephalosyndactyly) - observations on a British series of thirty-nine cases. Ann. hum. Genet., Lond. 24, 151-64. LOESCH,D. & LISIEWICZ, H. (1974). The application of discriminant functions in some sex-chromosomal anomalies. Genet& Polonica 15, 491-7. MAYO,0. (1974). The effect of age on chiasma number in man. Human Heredity 24, 144-50. MORRISON, D. F. (1967). Multivariate Statistical Methock. New York: McGraw-Hill. MURDOCH, J.L., WALKER,B. A., H m , J. G., ABBEY,H., S ~ HK.,K. & MCKUSICK, V. A. (1970). Achondroplasia - a genetic and statistical survey. Ann. hum. Genet., Lond.33, 227-44. MURDOCH,J. L., WALKER,B. A. & MOKUSICK, V. A. (1972). Parental age effects on the occurrence of new mutations for the Marfan syndrome. Ann. hum. Genet., L d .35, 331-6. SMITH,C. A. B. (1972). Note on the estimation of parental age effects. Ann. h m . Genet., Lond. 35, 337-42.
427
Printed in areat Britain
On the estimation of parental age effects on mutation BY 0. MAYO,* J. L. MURDOCHT AND T. W. HANCOCK*
* Biometry Section, Waite Agricultural Research Institute, University of Adelaide, Adelaide, 8outh Australia 5064
t Lorna Linda University, School of Medicine, Lorna Linda, California 92354, U.S.A. For many years, it has been accepted that sporadic cases of certain dominant or semi-dominant mutations are associated with an increased paternal age, but not with increased maternal age or birth rank. However, because maternal age, paternal age, and birth rank are so highly correlated with one another, it has been very difficult to separate the effects, and to produce unequivocal evidence for any particular one being more important. All of the methods in use have certain flaws or difficulties. The best method available, that of Smith (1972), requires that one know population values for the regression of maternal age on birth rank, paternal age on birth rank, maternal age on paternal age, and vice versa (the p matrix below). Smith’s method allows not only the detection but also the estimation of the various effects, but as it requires these population data, it may often not be applicable. It does not seem to have been noticed that a simple discriminant function applied to data on sporadic cases and on the sibs of those sporadic cases might provide a good indication of the relative contributions of paternal age, maternal age and birth rank to the heightened incidence of the condition. I n this note, we apply the method of discriminant functions to data on Apert’s acrocephalosyndactyly (Blank, 1960) and achondroplasia (Murdoch,Walker, Hall, Abbey, Smith & McKusick, 1970). We also apply Smith’smethod for comparative purposes. DISCRIMINdNT BUNCTION
The rationale for using a discriminant function to compare sporadic cases of dominant disorders with their normal sibs is very simple: if mutant and normal sibs represent two samples from the same population with respect to parental age, maternal age and birth rank, then it will not be possible to discriminate between them on these grounds, while if it is in fact possible to discriminate between them, the relative contributions of the three variables to the discriminant should give an indication of their relative contributions to the elevated incidence of the mutation. (A linear relationship seemed appropriate since preliminary investigation using a method suggested earlier (Mayo, 1974) yielded no clear indication of a curvilinear dependence of mutation rate on maternal age, paternal age or birth rank.) We shall use the notation of Smith (1972) throughout, so that m = mother’s age, f = father’s age, b = birth rank. Table 1 shows the variance-covariance matrices for affected and unaffected sibs in families with Apert’s acrocephalosyndactyly segregating (Blank, 1960). m was normally distributed, while f and b were positively skewed, but transformation did not alter the relative results obtained, so that those presented are for the untransformed data. The vectors of means for normal and affected are (29,077, 34.500, 2.833) and (32.171, 37.057, 2.714) respectively, giving a vector of differences A*’= (A$,AT, A:) = (3,095, 2.557, - 0.119).
428
0.MAYO, J. L. MURDOCHAND T. W. HANCOCK Table 1. Variance-covariance matrices for
--(a;)
Apert's acrocephalosyndactyly (Blank, 1960) Affected = 35)
Normal (n = 78)
(12
f
m
m
f b
23.38 19-09 3.82
19-09 50.88 3'47
f
b
32'47
22'21
22.21
42'47 6.77
5'74 6.77 3'62
b
m
3.82 3'47 2.89
5'74
( b ) Achondroplasia (Murdoch et al. 1970).
Normal = 321)
Affected (n = 92)
('7%
m
f
b
m
m
35-00
f
32-28 7'64
32.28 57'94
7.64 8'64 4'73
41-43 36.76
b
8-64
10.53
f
b
36.76 10.53 58.88 11.62 11.62 6.08
By a standard discriminant analysis (Morrison, 1967) we obtain F(3,109) = 4.133 for the hypothesis that the two samples are from the same population, compared with the 1 yo critical valueof 3.97.Thediscriminant functionis0-1318m+ 0-0315f - 0.28843. The analysisalso provides us with a 95 % simultaneous confidence interval for A*: -0.0822 -1.3567
< A; < A;
Q 6.2712 Q 6.4710
- 1.1934 Q A: < 0.9553 . It seems clear that paternal age is relatively unimportant in discriminating between the affected and normal groups. Table 1 also shows the variance-covariance matrices for achondroplasia (Murdoch et al. 1970). Here, m and f were normally distributed, while b was positively skewed, and as with Apert's syndrome transformation did not materially affect the relationships presented for the untransformed variables. The vectors of means for normal and affected are (27.73, 33.29, 3.45) and (30.77, 36.27, 3.46) respectively, giving
A*' = (3.0459,2.9789,0*0048). From the discriminant analysis we obtain P(3,409) = 9.965 for the hypothesis that the two samples are from the same population, compared with the 0-1% critical value of 5.62. The discriminant function is 0.1 102772+ 0.0288f - 0.2428b. The simultaneous 95 yo confidence interval for A* is 0.9363 Q A; Q 5.1554 0.4242 -0.7964
< A; Q 5.5336 < A: < 0.8060.
Once again, there is no clear indication that paternal age is the most critical component in the discrimination.
On the estimation of parental age eflects 012. mutation
-
429
Table 2. Matrices of regression coeficients
Apert’s acrocephalosyndactyly m
f
b
Achondroplasia
r rn
f
b
m
1.00
0.52
0.63
1-73
0.68 0.18
1.00
1.58 1.87
1’00
f
0.89
1.00
1.91
0.16
1.00
0.25
0-20
1.00
b
SMITH’S METHOD
The essence of the method of Smith (1972) is that, given the matrix of coeffioients of regression pmm
pmf
Pmb
pbm
pbf
Pbb
(where p, is the coefficient of regression of mother’s age on father’s age and so on), it should be possible to test the hypothesis ‘that only the father’s age has any appreciable real influence on the condition, the rises in mother’s age and birth rank being statistical consequences of this’ (Smith, 1972). Smith used 1-00 0.66
1.82
0.07
1.00
0.15
from population values compounded by Murdoch, Walker & McKusick (1972) from U.S., U.K. and Australian data. Because these are population values, they can be treated as precise, so that one can compare Am with pmfA,, i.e. test the hypothesis that Am-pmfA,, which has variance (vmm- 2pm vm + p;4, vf )In, is zero. (Here, v,, and vf are variances from the appropriate ‘affected’variance-covariance matrix in Table 1, while A is the difference between the observed value for the affected group and the population value, not the A* obtained in the previous section, i.e. they are A‘ = (4.131, 6-017, 0.474) and A’ = (4.232, 6-422, 0.817) for Apert’s acrocephalosyndactyly and achondroplasia respectively, the population mean vectors being (28.04, 31.04, 2-24) and (26.54, 29-85, 2.64). This will be discussed further below.) From Table 1, we can compute the two regression matrices shown in Table 2. It can be seen that they are, for the most part, very similar to Smith’s. We also have 1.00 0.63
p=
1
1.86
0.85 1.00 1.60 0-13 0.06
1-00
from Blank’s (1960) calculations on Australian and U.K. data. This will be the matrix most appropriate to the analysis of Blank’s data by Smith’s method; for Murdoch et al.’s data the best p is not clear. In the analysis below, Smith’s p matrix has been used, as has the matrix in Table 2, the latter also being treated as if it were a population matrix, although it is based only on 321 observations, since it is at least drawn from an appropriate population. In Table 3, we show values of hi-pijAj with their standard errors, for i = m,f,b; j = rn,f, b; i p j.
0.Miyo, J. L. MURDOCHAND T,W. HANCOCK
430
-Table 3. Departures of A from expectation
i j
Apert's acrocephalosyndactyly
A{ -A, 4
mf b f m b b m
A, -
S.E.
3'25
0.75 0.74
2.5 I
1'00
5.26 -0.06 0.14
1.16 0.26 0.28
0.36
f
Achondroplasia. Using p from Table 2
i
A,
S.E.
0'22
0.43
2-82 2.67 4.86 - 0.26 - 0.45
0.50
0.55
0.68 0.18 0'20
Using Smith's p
j
,
r
m
f
0.19
-0.01
b m b b m
f
f
2.74 2-74 5 '44
0.50
0.18
0.19
0.37
0'20
0.55
0.69
Table 4. Expected independent changes, d, in m, f and b and the associated variance-covariame matricee
-
Apert's acrocephalosyndactyly A
r
l
A
r
V
d
m
0.57
f
5.37
b
0-10
S.E.
m
f
d
1-61 6.76 -0.92
3.26 -3.27 -0.29 0.17 -3'27 462 0.17 0.22 0.47 -0.29 1-81
\
V
b
2-15
-
Achondroplasia. Using p from Table 2
S.E.
m
f
b
1-06 1-13 -0.76 -0.15 1.31 -0.76 1.73 -0.14 -0.14 0.12 0-35 -0.15 Using Smith's p L
r
I
V
, m
f b
d -1.29 6.91 0.52
3
f
b 1.75 -1.61 -0.24 1-32 2.00 0.20 1.41 -1.61 0.28 -0.24 0.20 0.08 S.E.
m
Smith's method also allows estimation of the expected independent changes d' = (d,,, d,,d,) in the variables m, f,b, from
d x P-lA with variance given by
var d = P-lv(P-l)l/n. These values are shown in Table 4. DISCUSSION
It can be seen that the discriminant function appears to produce different results from Smith's analysis. The latter shows, for both Apert's acrocephalosyndactyly and achondroplasia, that paternal age is the main contributor t o the elevated incidence of the traits, as already demon-
On the estimation of parental age effects on mutation
431
strated by Blank (1960) and Murdoch et al. (1970) using less effective techniques. Why then should the discriminant function, which in effect uses normal sibs as the reference population, produce such strikingly different results? The answer appears to lie in the difference between A and A*;that is, the sibs of sporadic cases of both disorders differ from the general population in the age of their parents and in their birth rank. If A* were used to compute d*, then the values (4.46, 0.16, -0-69) and (4.47, 1.89, - 1.50) would be obtained for Apert’s acrocephalosyndactyly and achondroplasia respectively, with the standard errors given for d in Table 4. While this procedure is very imprecise, i t allows the tentative conclusion that, among the families having sporadic cases of these disorders, the mothers of the sporadic cases have age at birth of the affected children further above the group mean than do fathers. The significance of this within-family difference is not clear. It does seem, however, that normal sibs of affected sporadic cases of dominant disorders do not form an appropriate control population. An appropriately chosen control would seem, nonetheless, to offer the possibility of more powerful discrimination for a smaller sample size than of the incompletely available or inappropriate population data. Loesch & Lisiewicz ( 1974) attempted to use a variant of the discriminant function technique for classification of two aneuploidies, Trisomy 21 and XXY, and achieved reasonable discrimination from very small samples.
SUMMARY
The detection and estimation of the effects of paternal and maternal age and birth rank on mutation rate are considered. Smith’s (1972) method and discriminant function techniques are compared using data on Apert’s acrocephalosyndactyly and achondroplasia. REFERENCES
BLANK,C. E. (1960). Apert’s syndrome (a type of acrocephalosyndactyly) - observations on a British series of thirty-nine cases. Ann. hum. Genet., Lond. 24, 151-64. LOESCH,D. & LISIEWICZ, H. (1974). The application of discriminant functions in some sex-chromosomal anomalies. Genet& Polonica 15, 491-7. MAYO,0. (1974). The effect of age on chiasma number in man. Human Heredity 24, 144-50. MORRISON, D. F. (1967). Multivariate Statistical Methock. New York: McGraw-Hill. MURDOCH, J.L., WALKER,B. A., H m , J. G., ABBEY,H., S ~ HK.,K. & MCKUSICK, V. A. (1970). Achondroplasia - a genetic and statistical survey. Ann. hum. Genet., Lond.33, 227-44. MURDOCH,J. L., WALKER,B. A. & MOKUSICK, V. A. (1972). Parental age effects on the occurrence of new mutations for the Marfan syndrome. Ann. hum. Genet., L d .35, 331-6. SMITH,C. A. B. (1972). Note on the estimation of parental age effects. Ann. h m . Genet., Lond. 35, 337-42.
