X-linkage: Ascertainment through doubly ill probands

J. psychkrt. Rcs.,1977, Vol. 13, pp. 161-168. Pergamon Press.Printedin GreatBritain.

X-LINKAGE

: ASCERTAlNMENT

DOUBLY

THROUGH

ILL PROBANDS

ELLIOT S. GERSHON*

and

STEVEN MATTHYSSE~-

*Section on Psychogenetics, Biological Psychiatry Branch, NIMH, Bethesda, Maryland, U.S.A. l_Dept. of Psychiatry, Psychiatric Research Laboratories, Massachusetts General Hospital, Boston, MA, U.S.A. (Received 1 April 1976. Revised 11 January 1977)

Abstract-Ascertainment corrections were developed for the analysis of X-linkage under conditions where the probands are doubly ill; for example, when all patients in a psychiatric clinic are screened for color blindness, and only the families of color blind manic-depressive males are investigated further to determine if the pedigree is informative for linkage. When published data on color blindness and manic-depressive illness were re-analyzed using these corrections, the likelihood ratios (“lod scores”) indicative of X-linkage were decreased. THE ANALYSIS

of X-linkage in sibships requires ascertainment corrections, because sampling procedures may under- or over-represent certain classes of pedigrees. EDWARDS,~ building on the methods of MORTON,~ has developed ascertainment corrections for several types of ascertainment in which there are equal a priori probabilities for the person who introduces the pedigree to be observed in coupling or in repulsion (defined below). To use these corrections in an investigation, the case-finding method would have to fit this requirement. Consider a study of linkage between a psychiatric disorder and a locus such as protan color blindness, using probands selected from a psychiatric clinic. The persons initially screened should be all the patients in the clinic, and all their siblings. Under these conditions, informative families would be no more likely to show coupling than repulsion. However, this screening method requires a great expenditure of investigative effort. It may prove more feasible to screen all the persons in the clinic for detectable uncommon traits at a marker locus. For example, all the patients in a manic-depressive clinic might be examined for color blindness, and only the families of color-blind males need be further investigated to determine if the pedigree was informative for linkage. The purpose of this paper is to develop appropriate formulas for analyzing X-linkage under the doubly-ill proband ascertainment condition. The assumption is that manicdepressive illness is a sex-linked dominant without phenocopies and with complete penetrance (age-corrections are not made, but well offspring are not considered who are below a certain age); and the null hypothesis is that the recombination fraction (0) with protan/ deutan color blindness is 50%. The symbols and terminology used here fOllOWEDWARDS’ BS closely as possible, to facilitate comparisons between papers. Let G = normal color vision, g = protan (or deutan) colour blindness, T = manic-depressive illness, t = not manic-depressive. 161

162

ELLIOT S. GERSHONand STEVENMATTKY~SE

(Alleles for the dominant phenotype have capital letters.) GT and gt (in the male or on the same chromosome in the female) will be said to be in “coupling”; gT and Gt will be said to be in “repulsion”.* Let 8 = probability of recombination; let cp= probability that any given doubly heterozygous mother is in coupling, based on information other than the distribution of genotypes in her sons. In the absence of such information, cpis taken as l/2. The number of sons with each genotype is specified according to Table 1. TABLE 1. GENOWPE SYMBOLS T

ASCERTAINMENT

THROUGH

t

SONS (TYPE S ASCERTAINMENT)

In order to correct the likelihood of observed pedigrees for sampling bias, it is necessary to compute the probability that pedigrees of any given type will find their way into the sample. When ascertainment is through doubly ill probands, a necessary condition is c > 0; that is, at least one son is gT, or color-blind manic-depressive. Let the probability distribution of color blindness and manic-depressive illness in sibships of size s from doubly heterozygous mothers be p(a,b,c,d; 0, cp, s). The likelihood of finding a segregation pattern (a,b,c,d) in a sibship of size s from the sample is p(a,b,c,d; 0, cp,s) multiplied by an ascertainment probability functionf(a,b,c,d; 8, cp,s) which depends on the sampling procedure. If c = O,f(a,b,c,d; 8, cp,s) = 0, since ascertainment is through doubly ill probands. We will assume f is otherwise independent of c, which is valid if sampling of doubly ill individuals is sufficiently complete that a pedigree with two such individuals has no greater likelihood of coming under scrutiny than a pedigree with one. If all manic-depressives are hospitalized, and screened for color blindness, this condition will be satisfied; otherwisef(a,b,c,d; 8, cp,s) should be an increasing function of c. In order to become part of the sample, mothers must not only be doubly heterozygous; their double heterozygosity must be manifest, unless gene frequencies are kn0wn.l c > 0 is thus a necessary, but not sufficient condition for inclusion in the sample. Expression of both alleles at each locus in the sons is a sufficient condition, and we will discuss this case first. In this case, f(a,b,c,d; 8, cp, s) = 0 unless a + b > 0, c + d > 0, a + c > 0 and b + d > 0. Since c > 0 is required by the ascertainment condition, it is sufficient to add a + b > 0 and b + d > 0. It is reasonable to stipulatef(a,b,c,d; 8, cp,s) is independent of a, b and d except for the requirements a + b > 0, b + d > 0, since once a proband is ascertained the remaining sibs will normally be exhaustively screened. Summarizing: *Note that coupling thus defined refers to two dominant or two recessive alleles on the same chromosome, rather than to a double affected or double unaffected chromosome. A color-blind ill male, for example, represents a chromosome in repulsion.

163

X-LINKAGE: ASCERTAINMENT THROUGHDOUBLYILL PROBANDS

J(a,b,c,d;

0, cp, s) = 0 if (c = 0, a + b = 0, or b + d = 0) = .f(0, cp, s) otherwise.

Since

1 p(a,b,c,d;8, cp,4_fWw,d; 8, cp,4 = 1,

abed f@, ($5 4

=

1 /I: p(a,b,c,d

cro

; 0, cp,4

a+b>O b+d>O

a+b>O b+d>O

which will be called “ascertainment CALCULATION

probability”. OF ASCERTAINMENT

PROBABILITY

Ascertainment failure will occur if no sons have the t genotype (b + d = 0), the G genotype (a + b = 0), or the gT genotype (c = 0). 1 - ~(0, cp, s) = p(A or B or C), or taken in the inclusive sense, where A, B and C refer to the events b + d = 0, a + b = 0, c = 0 respectively. 144 or B or C) = p(A) + p(B) + p(C) --(A and B) --(A and C) --(B and C) + P(A and B and C). The probabilities of each event are given in Table 2, and these are combined above to form 1 - $0, cp, s).

as indicated

TABLE2. PROBABILITIES OFEVENTS CAUSINGFAILURE OFASCERTAINMENT Event A B C AandB A atzd C Bat&C AandBandC

Genotype pattern b=d=O (no 1) a=b=O (no G) c=o (no gT) a = b = d = 0 (c = s) (all gT) b = c = d = 0 (a = s) (all GT) a = b = c = 0 (d = s) (all gt) impossible

Probability (1/2P (1/2js cptl - (1/2)81s + (1 - cp)tl - (l/2)(1 cp[(l/2)6]’ + (1 - (p)[(1/2)(1 - 0)l” cp[(1/2)(1 - 0)]” + (1 - cp)[(l/2)81s m/2)(1 - 6)l” + (1 - cp)WV31~

- e)l”

Pedigrees selected by these basic criteria (c > 0, a + b > 0, b + d > 0) will be said to be of type S(ascertainment through sons only). In some cases, the mother’s status as a double heterozygote can be determined in other ways, with less severe restrictions on the genotypes in the sons. We will distinguish two other types: (1) MS, where the mother’s phenotype is known, and (2) GR, GC, GPR and GPC, where maternal grandparents are also examined. ASCERTAINMENT

WHEN

MOTHER’S

If the mother is known to be causes failure of ascertainment, and C is computed using Table she is homozygous at that locus

PHENOTYPE

IS KNOWN

(MS ASCERTAINMENT)

phenotypically G, event B (no G among the sons) no longer so 1 - TC(O,cp, s) = P(A or C) = 1 --(A) ----p(C) + p(A) 2. If mother is known to be phenotypically g (color-blind) and her offspring are not informative. If she is phenotypic-

164

ELLIOT

S. GERSHON and STEVENMATTHYSSE

ally t (psychiatrically well) and past the age of onset, her offspring are likewise not informative because the assumptions of complete penetrance and X-chromosome transmission are violated. In some cases the phase of the mother’s brothers is known, and EDWARDSprovides a formula1 for calculation of the probability of coupling or repulsion phase in the mother, given the phase in one of her brothers. His formula is derived as follows: since mother and her brother each receive an Xchromosome from their mother, the probability that mother and mother’s brother are identical in phase is O2+ (1 - O)z, the probability that both are recombinants or that both are non-recombinants. This reasoning does not take into account that mother becomes a double heterozygote by virtue of both her parents, and the formula is therefore not exact. For example, consider the case 8 = 0 (no recombination). According to Edwards’ formula, the probability that the mother is in coupling when a brother is in repulsion is zero. The probability cannot actually be zero, because a pedigree can exist in which the mother is in coupling and her brother is in repulsion, without recombination. Let mother’s father be gt (color-blind without affective disorder), and mother’s mother be Gt/GT (homozygous for normal vision, and heterozygous for the affective disorder gene) (Fig. 1). This combina-

gt

Gt GT

Gl

FIG. 1. Woman in coupling has brother in repulsion with no recombination

taking place.

tion is not rare among grandparental genotypes capable of producing a doubly heterozygous mother. If the mother receives the GT chromosome from her mother, she will become the double heterozygote GT/gt (coupling), whereas her brothers have a 50% chance of being in the repulsion phase (Gt). We conclude that, where mother’s parents have not been observed, and in the absence of gene frequency estimates, information on phase in mother’s brother is not useful in predicting phase of mother. This is of some consequence for the published data reviewed below in which Edwards’ formula is used. ASCERTAINMENT

WHEN GRANDPARENTS

ARE EXAMINED

Consider the case of grandfather in repulsion. If grandfather is gT and mother is phenotypically G, she must be heterozygous at the G locus. Event B (no G among the sons) no longer excludes the pedigree, so the ascertainment probability is the same as case MS. If grandfather is Gt and mother has a doubly ill son CqT), she must be a double heterozvsote.

165

DOUBLYILL PROBANDS

X-LINKAGE: ASCERTAINMENT THROUGH

so only event C (absence of a doubly ill son) can cause failure of ascertainment. l~(0, cp) = p(C). In both cases cp can be taken as zero, since the grandfather is in repulsion. Although grandfather Gt has the simplest ascertainment correction (doubly ill son only is required for selection), information on the maternal grandmother can reduce the grandfather gT case to this form. If grandmother is phenotypically t (no affective disorder), mother must be heterozygous at the T locus, and if in addition mother is phenotypically G, she must be heterozygous at both loci. No information on the sons is required, other than the basic ascertainment condition that there be a doubly ill proband. Similar reasoning applies to the cases of grandfather in coupling. In those cases, cp = 1. The possible ascertainment conditions when grandparents are examined are summarized in Table 3. TABLE3. ASCERTAINMENTPROBABILITIESWHENMATERNALC~RANDPARENTSAREEXAMINED* Grandfather’s genotype 0 gT Gt

Mother’s Mother’s phase (cp) phenotype 0 G 0 G 0

GT GT

I 1

rt

1

Ascertainment probability

Grandmother’s phenotype

1 --p&--p(C) t

+ p (A nnd C)

1 -P(C) 1 -P(C)

1 -p(A)

t

G

-p(C) + 144 ad Cl 1 -PC)

1 -u(C)

*A blank space under “mother’s phenotype” indicates that this observation blank space under “grandmother’s phenotype” indicates T or not examined. LlKELIHOOD

Ascertainment type designation

FUNCTION

GR GPR GPR GC

GPC GPC

adds no information.

A

COMPUTATION

The lod score introduced by MORTON~ is the logarithm of the ratio of the likelihood for a given value of 0 to the likelihood for the null hypothesis (0 = 0.5). The ratio is 1 (and the lod score is 0) when 0 = 0.5 (no linkage). Where there are ascertainment probabilities that are a function of 0, the corrections can be standardized in the same way. Lod score = [log,, y(a,b,c,d; -

8, cp, s) -

Uog,, r@, 0, 8) -

log,, p(a,b,c,d;

0.5, cp. s)]

log,, n(O.5, cp, s)l.

As an example of the difference between a computation applying this method and a computation applying Edwards’ original method, let us consider some of the data on color blindness and manic-depressive illness published by MENDLEWICZ, FLEISS and FIEVE.~** This data is used for illustrative purposes because all of the 10 male manic-depressive probands are color-blind. Our use of this example is not meant to imply that the selection of probands in that study necessarily conformed to the selection criteria described here. It is evident (Table 4) that different lod scores result if the calculation is based on the assumption of unbiased proband selection as compared with the assumption of selection of doubly ill probands. Data on phase (coupling or repulsion) is derived from the mother’s father or brother in several of the selected examples. In the selected examples, and in the series of cases from which they are selected, repulsion appears to be present in nearly every family. If probands were selected for having both color blindness and affective illness, repulsion in the grandfather or uncles would introduce an artifactual bias in -favor of

F~uss(1974)

MSf

MS

BR (upper segment) FH -0.09 1.18

0.01 0.89

0.18 0.29 0.41

0.41 0.46

0.62 0.48 0.23 0.42 0.62

-0.18 -@18 0.41

0.25

-0.38 -0.38 O-62

0.15

0.02 046

0.11 0.14 0.19

0.19 0.24

-0.07 -0.07 0.19

0.35

0.00 0.09

0.04 0.02 0.03

0.03 0.04

-0.01 -0.01 0.03

045

PEDIGREES

0.69 0.42

0.93 -0.02

-0.64 0.57

-0.23 0.67

0.00 0.00 0.28 0.21 0.93 ----0.69

-0.48 -0.48 0.69

0.15

-0.94 -0.94 0.93

0.05

(generation IV)

MS

GPR MS MS

0.56

-0.74

0.28 0.22 0.80 -0.31 0.69

0.23 0.15 0.62

0.55

-0*13

0.18 @09 0.41

0.29

-0.04

0.11 0.03 0.19

0.07

0.00

0.04 0.00 0.03

0.55

- 0.64

0.00 0.26 0.93

0.61

-0.25

0~00 0.17 0.69

044

-0.10

WOO 0.10 044

0.25

-0.09 0.47

0.00 0.12 0.44

044 0.30

0.25 -0.27 -0.27 044

I3

0.21

-0.03

0.00 0.04 0.20

0.35

-0.03 0.22

0.00 0.05 0.20

0.20 0.12

0.35 -0.14 -0.14 0.20

as

@03

@OO

0.00 0.00 0.03

0.45

0.00 0.03

WOO 0.00 0.03

0.03 -0.02

045 -0.04 -0.04 0.03

phase

Lod scores using current analysis

IN SELECTED

*MENDLEWICZand FLEISS (1974) and FIEVE, MENDLEWICZ, RAINER and FLEISS (1975), using the method of EDWARDS (1971). tMS--mothers and sons examined. G,PR--mothers, sons and maternal grandparents examined. Grandparents in repulsion. $Mother’s brother is in repulsion, and this information is used in the original reference to calculate probability of mother’s a function of 9, as described by Edwards. For discussion of validity of this calculation, see text.

MENDLEWICZ and FLEISS(1974)

-0.47 1.13

0.28 0.52 0.80

0.80 -0.06

-0.83 -0.83 O-80

0.05

8

Lod scores assuming unselective ascertainmint*

SCORES

through Doubly I11probands without phase information from mother’s brothers 0.05 o-15 0.15 0.25 0.35 0.45 0.05

N S BR (upper segment) FH

Ascertainment

FIEVEet al. ( 1975) FIEVE et al. (1975) MENDLEWICZ and FLEISS(1974)

Protan color blindness

GPR

N

MS

GRP GPR MS

S

B

K F SH (generation III)

Family no.

MSf (generation IV, left segment)

MENDLEWU and FLEISS(1974)

FIEVEet al. (1975) FIEVEet al. (1975) MENDLEW~CZ and FLEISS(1974)

Protan color blindness

MENDLEWSCZ~~~

FIEVEet al. (1975) FIEVEet al. (1975) FIEVE et al. (1975)

Deutan color blindness

Type of color blindness and source

Selective ascertainment type?

TABLE4. EFFECTSOF ASCERTAINMENT THROUGH DOUBLYILL PROBANDS ON LOD

X-LINKAGE: ASCERTAINME~ THROUGH DOUBLY ILL PROBANDS

167

linkage in the analyses. This follows from the null hypothesis of non-linkage, which predicts that the sons of a mother in repulsion are equally likely to show coupling or repulsion, whereas in the selection process sons who show repulsion are chosen as probands and they constitute a significant proportion of the total number of sons. The data selected indicate how substantial such a bias in favor of probands in repulsion might be. In each case where there is reason (from her father, for example) to assign repulsion phase to the mother, the lod score is higher if unbiased proband selection is assumed. If Edwards’ formula for probability of phase in the mother based on the mother’s brother is used, the maximum likelihood estimate of 0 has lod score 1.18 using the non-biased ascertainment assumption and 0.67 using the corrections for ascertainment through doubly ill probands (Table 4). That is, the corrections diminish the likelihood ratio by a factor of 3.2. If the Edwards formula for phase based on mother’s brother is not used, the correction for ascertainment bias through doubly ill probands decreases the lod score by a smaller amount. DISCUSSION

Lod score computations may be in error, when families are introduced by patients in whom two disorders exist simultaneously, unless appropriate ascertainment corrections are made. The effect of this bias is more substantial in cases where there are few informative sons, because of the relative preponderance of cases in one cell (gT) of the contingency table of genotypes (Table 1). If, from the observed pedigrees, mother’s phase consistently appears to be repulsion (from her father or brothers) and this information is incorporated into the computations, odds favoring linkage will be enhanced by neglecting selective ascertainment. If the phase were predominantly coupling, the opposite effect on the lod score would be found. One drawback of ascertainment through doubly ill probands cannot be corrected for statistically. If probands are only examined if they have two illnesses, the possibility is not considered that there is an association of the two illnesses, due to linkage disequilibrium, shared pathophysiology, or other factors. Although probands may have been selected for repulsion in our example of manic-depressive illness and color blindness, the analytic method assumes that in general coupling is as likely to be found as repulsion. This ascertainment mode requires, therefore, that there be prior evidence that the two disorders examined are not associated. The Edwards method for detecting and estimating X-linkage is easy to apply and has the advantage of not requiring knowledge of gene frequencies. In extending this method to types of ascertainment conditions that are commonly used in psychiatric research, we have not intended to imply that ascertainment through doubly ill probands should be avoided. On the contrary, in some circumstances it may be much more efficient than methods where preliminary screening is based on only one trait. It is, however, relatively sensitive to artifacts, and ascertainment corrections must be made with care. Acknowledgement-This research was supported in part by Alcohol, Drug Abuse, and Mental Health Administration Grant MH-25515 from the National Institute of Mental Health.

168

ELLIOT S. GERSHON and STEVEN MATTHYSSE REFERENCES

1. EDWARDS, J. H. The analysis of X-linkage. Ann. hum. Genet. 34, 229, 1971. 2. MORTON. N. E. Seauential tests for the detection of linkage. Am. J. hum. Genet. 7. 277. 1955. 3. MENDLE&ICZ, J., F~EISS, J. L. and FIEVE, R. R. Evidence for X-linkage in the transmission of manicdepressive illness. J. Am. med. Ass. 222,1624,1972. 4. MENDLEWICZ, J. and FLEISS, J. L. Linkage studies with X-chromosome markers in bipolar (manicdepressive) and unipolar (depressive) illnesses. Biol. Psychiut. 9, 261, 1974. 5. FIEVE, R. R., MENDLEWICZ, J., RAINER, J. D. and FLEISS, J. C. A dominant X-linked factor in manicdepressive illness: studies with color blindness. In Genetic Research in Psychiatry, FIEVE, R. R., ROSEIYTHAL, D. and BRILL, H. (Editors). Johns Hopkins University Press, Baltimore, 1975.