84
LETTERS TO THE EDITOR
Second, they suggest that associations of n chromosomes, where n > 2, can be dealt with by partitioning them into their n (n -1)/2 constituent pairs, and assert that parameter estimates based on treating the resulting pairs as separate observations, are unbiased. I offer the following as a counter-example. Let there be three objects, 01, 02, and 03 representing individual chromosomes in a single cell. Suppose Oi is assigned independently of the other objects with probability Pi to one box and probability (1 Pi) to another. After assigning all three objects in this way, we define those that are in the first box as forming a single "association." This, therefore, represents a model of random association (in the sense that each object enters or fails to enter an association independently of the others), but in which each object has a different probability of association. The probabilities of the four kinds of associations are then, P12 = P1P2( -
P3),P23 =P2P3G -Pl),P31 =P3P1(l P2), andP123 =PIP2P3-
The expected relative numbers of the pairwise associations on their own are: P12 : P23 P31 = P1P2(l - p3) : P2P3(l - Pi) : P3P(l - P2), whereas in combination with the partitioned pairs from the triple association, they are: P12' : P23' : P31' = [p1P2(l - p3) + P1P2P3] P2P3(l - P1) + P1P2P3] P3P1(l - P2) + P1P2P3] = P1P2 :P2P3 :P3P1In general, the two sets of ratios are not equal and will provide different estimates of the pi. A. D. CAROTHERS MRC Clinical and Population Cytogenetics Unit Western General Hospital Edinburgh, Scotland REFERENCES JACOBS PA, MAYER M, MORTON NE: Acrocentric chromosome associations in man. Am J Hum Genet 28:567-576, 1976
RESPONSE TO CAROTHERS' LETTER
To the Editor: For clarity we answer the two points raised by Mr. Carothers in reverse order. The argument for "partitioning an association of n chromosomes into all n(n 1)/2 pairs" is given on p. 571 of our paper. We cannot imagine under what circumstances it would be interesting to discard multiples with n > 2, as in the proposed counter-example, even if allowance were made for the excluded class. Mr. Carothers shows that analysis according to our prescription gives unbiassed estimates as we asserted. The formula he questions is a special case of the familiar result due to Wright [1], f pi2 +pi (1 -pi)F fori=j p for i fj I2pi pj (I -F) where the pi are frequencies with which chromosomes enter into association, and F is the correlation between associated pairs, assumed constant.
85
LETTERS TO THE EDITOR
The restriction 0 c Pij s 1 implies -pil(1-pi) ' F ' 1 for all Pi. In applications, this restriction has not been found to be a logical or practical difficulty. The most special case is pi = 21N, F = - I(N - 1), and each restriction may be relaxed separately. We showed thatp1 # 21N, and F --/(N - 1). It is indeed possible that some other theory would give a better fit to the data. The important consideration is that any hypothesis be tested. NEWTON E. MORTON PATRICIA A. JACOBS MARTHA MAYER University of Hawaii Honolulu, Hawaii 96822
REFERENCES 1.
WRIGHT S: Systems of mating. II. The effects of inbreeding on the genetic composition of a
population. Genetics 6:124-143, 1921
STATEMENT OF OWNERSHIP, MANAGEMENT AND CIRCULATION (Required by 39 U.S.C. 3685) 1.
10. Extent and nature of circulation:
Average Number Copies Each Issue During Preceding
Title of publication: THE AMERICAN JOURNAL OF HUMAN GENETICS a. Publication number: 019760
2.
Date of filing: October 1, 1978.
12 Months A. Total number copies printed 3,700
3.
Frequency of issue: Bi-monthly
B.
a. No. of issues published annually: 6
b. Annual subscription price: $30.00
Paid circulation: 1. Sales through dealers and carriers, street vendors and counter sales 2. Mail subscriptions
Location of known office of publication: 5801 S. Ellis Avenue, Chicago, Illinois 60637.
C. Total paid circulation
5.
Location of the headquarters or general business offices of the publishers: 5801 S. Ellis Avenue, Chicago, Illinois 60637.
D. Free distribution: samples, complimentary, and other free copies
6.
Names and addresses of publisher, editor, and managing editor: Publisher: The University of Chicago Press, 5801 S. Ellis Avenue, Chicago, Illinois 60637 Editor: David E. Comings, City of Hope National Medical Center, 1500 Duarte Road, Duarte, CA 91010 Managing Editor: None
E.
Total distribution (sum of C and D)
F.
Copies not distributed: 1. Office use, leftover, unaccounted, spoiled after printing 2. Returns from news
4.
7. 8.
9.
Owner: The American Society of Human Genetics, Institute for Cancer Research, Philadelphia, Pennsylvania Known bondholders, mortgagees, and other security holders owning or holding 1 percent or more of total amount of bonds, mortgages or other securities: None. The purpose, function, and nonprofit status of this organization and the exempt status for Federal income tax purposes have not changed during preceding 12 months.
Single Issue Nearest To
Filing Date 3,800
2,949
3,267
2,949
3,267
30
30
2,979
3,297
721
503
3,700
3,800
agents G. Total (sum of E and F)
11. I certify that the statements made by me above are correct and complete. JEAN W. SACKS, Assistant Director
LETTERS TO THE EDITOR
Second, they suggest that associations of n chromosomes, where n > 2, can be dealt with by partitioning them into their n (n -1)/2 constituent pairs, and assert that parameter estimates based on treating the resulting pairs as separate observations, are unbiased. I offer the following as a counter-example. Let there be three objects, 01, 02, and 03 representing individual chromosomes in a single cell. Suppose Oi is assigned independently of the other objects with probability Pi to one box and probability (1 Pi) to another. After assigning all three objects in this way, we define those that are in the first box as forming a single "association." This, therefore, represents a model of random association (in the sense that each object enters or fails to enter an association independently of the others), but in which each object has a different probability of association. The probabilities of the four kinds of associations are then, P12 = P1P2( -
P3),P23 =P2P3G -Pl),P31 =P3P1(l P2), andP123 =PIP2P3-
The expected relative numbers of the pairwise associations on their own are: P12 : P23 P31 = P1P2(l - p3) : P2P3(l - Pi) : P3P(l - P2), whereas in combination with the partitioned pairs from the triple association, they are: P12' : P23' : P31' = [p1P2(l - p3) + P1P2P3] P2P3(l - P1) + P1P2P3] P3P1(l - P2) + P1P2P3] = P1P2 :P2P3 :P3P1In general, the two sets of ratios are not equal and will provide different estimates of the pi. A. D. CAROTHERS MRC Clinical and Population Cytogenetics Unit Western General Hospital Edinburgh, Scotland REFERENCES JACOBS PA, MAYER M, MORTON NE: Acrocentric chromosome associations in man. Am J Hum Genet 28:567-576, 1976
RESPONSE TO CAROTHERS' LETTER
To the Editor: For clarity we answer the two points raised by Mr. Carothers in reverse order. The argument for "partitioning an association of n chromosomes into all n(n 1)/2 pairs" is given on p. 571 of our paper. We cannot imagine under what circumstances it would be interesting to discard multiples with n > 2, as in the proposed counter-example, even if allowance were made for the excluded class. Mr. Carothers shows that analysis according to our prescription gives unbiassed estimates as we asserted. The formula he questions is a special case of the familiar result due to Wright [1], f pi2 +pi (1 -pi)F fori=j p for i fj I2pi pj (I -F) where the pi are frequencies with which chromosomes enter into association, and F is the correlation between associated pairs, assumed constant.
85
LETTERS TO THE EDITOR
The restriction 0 c Pij s 1 implies -pil(1-pi) ' F ' 1 for all Pi. In applications, this restriction has not been found to be a logical or practical difficulty. The most special case is pi = 21N, F = - I(N - 1), and each restriction may be relaxed separately. We showed thatp1 # 21N, and F --/(N - 1). It is indeed possible that some other theory would give a better fit to the data. The important consideration is that any hypothesis be tested. NEWTON E. MORTON PATRICIA A. JACOBS MARTHA MAYER University of Hawaii Honolulu, Hawaii 96822
REFERENCES 1.
WRIGHT S: Systems of mating. II. The effects of inbreeding on the genetic composition of a
population. Genetics 6:124-143, 1921
STATEMENT OF OWNERSHIP, MANAGEMENT AND CIRCULATION (Required by 39 U.S.C. 3685) 1.
10. Extent and nature of circulation:
Average Number Copies Each Issue During Preceding
Title of publication: THE AMERICAN JOURNAL OF HUMAN GENETICS a. Publication number: 019760
2.
Date of filing: October 1, 1978.
12 Months A. Total number copies printed 3,700
3.
Frequency of issue: Bi-monthly
B.
a. No. of issues published annually: 6
b. Annual subscription price: $30.00
Paid circulation: 1. Sales through dealers and carriers, street vendors and counter sales 2. Mail subscriptions
Location of known office of publication: 5801 S. Ellis Avenue, Chicago, Illinois 60637.
C. Total paid circulation
5.
Location of the headquarters or general business offices of the publishers: 5801 S. Ellis Avenue, Chicago, Illinois 60637.
D. Free distribution: samples, complimentary, and other free copies
6.
Names and addresses of publisher, editor, and managing editor: Publisher: The University of Chicago Press, 5801 S. Ellis Avenue, Chicago, Illinois 60637 Editor: David E. Comings, City of Hope National Medical Center, 1500 Duarte Road, Duarte, CA 91010 Managing Editor: None
E.
Total distribution (sum of C and D)
F.
Copies not distributed: 1. Office use, leftover, unaccounted, spoiled after printing 2. Returns from news
4.
7. 8.
9.
Owner: The American Society of Human Genetics, Institute for Cancer Research, Philadelphia, Pennsylvania Known bondholders, mortgagees, and other security holders owning or holding 1 percent or more of total amount of bonds, mortgages or other securities: None. The purpose, function, and nonprofit status of this organization and the exempt status for Federal income tax purposes have not changed during preceding 12 months.
Single Issue Nearest To
Filing Date 3,800
2,949
3,267
2,949
3,267
30
30
2,979
3,297
721
503
3,700
3,800
agents G. Total (sum of E and F)
11. I certify that the statements made by me above are correct and complete. JEAN W. SACKS, Assistant Director
