Quantitative genetics and clinical medicine.

QUANTITATIVE GENETICS AND CLINICAL MEDICINE*

BY Bruce E. Spivey, MD INTRODUCTION

HUMAN CLINICAL GENETICS GENERALLY EMPHASIZES TWO AREAS OF STUDY:

Mendelian, or single-gene genetics, and cytogenetics, which is concerned with chromosomal anomalies. Following recognition of Mendel's tenets early in this century, single-gene abnormalities occurring in three major patterns (autosomal dominant, recessive, and X-linked recessive) have been extensively chronicled by clinicians. By 1960, geneticists realized that abnormalities in the size, configuration, or number of chromosomes resulting from abnormal meiotic division during the formation of gametes could induce significant abnormalities in offspring. A third area of study of genetic influence in the occurrence of abnormalities is quantitative genetics which is gaining in importance with increasing recognition of its role in clinical human genetics. The concepts and rationale on which quantitative (multiple-gene) genetics is founded are still generally much less familiar to most clinicians than those of single-gene genetics and cytogenetics. This is unfortunate, since it is an important approach to analyzing most of the individual differences observed among patients. Animal and plant geneticists have utilized these principles since the 1920's with much success. However, because this method is so statistically oriented, because the human life cycle is long and mating is (fortunately) not controlled, and because computer assistance in data handling is relatively recent, this complex approach has not been widely utilized in clinical medicine. There is no one source of review for the clinician and clinical investigator in quantitative genetics in ophthalmology or general medicine. This document is, in part, an attempt to provide such a link between the technical literature of fundamental animal and plant researchers and the important day-to-day implications for clinicians who are constantly faced with the normal and abnormal traits which "run in families." This report presents the theoretical foundation, practical considera*From the Department of Ophthalmology, Pacific Medical Center, San Francisco, California.

662

Spivey

tions for utilizing the method, and extensions of this analytical tool for prediction are illustrated in conjunction with examples of genetic analyses of ophthalmic variables. Statistical concepts such as regression, correlation, variance and covariance, which are not a usual part of clinicians' daily discussion, are an integral part of understanding quantitative genetics, and are explained herein. The experience on first reading this work is not unlike the experience an internist might have in reading an overview of the theoretical foundation of color vision and its clinical testing. By re-reading and working with the material presented here, the reader can gain an insight into the theoretical basis of quantitative genetics, as well as its practical considerations and experimental design and, hopefully, an appreciation of the potential practical benefits for clinical problems that can be derived from such research. Application of this methodology in clinical research and clinical practice promises to be of pragmatic value. Methods are presented here to relate gene differences influencing clinical measurements to gene differences contributing to a clinical disorder. This relationship can enable prediction of the occurrence of an abnormality among offspring of clinically normal parents. This report is divided roughly into thirds (by volume). The first third deals with the concepts and theoretical bases used in quantitative genetics. This portion is a synthesis of material from many sources. This is specifically oriented to those with a clinical, nonstatistical background-an approach not available elsewhere. The development of the formulae has been designed to supplement the text. The middle third of this document contains two sections. The first part is an extension of the theoretical considerations to practical concerns such as a discussion of which relatives should be examined to give maximum information-again, in a manner not available elsewhere. The second part is a review and, in some cases, a reanalysis of genetic studies relating to the human visual system reported in the ophthalmic literature. The final third presents the first report of a study to estimate genetic correlations among clinical measures. Unique in that these estimates are the first reported genetic correlations involving human data, their importance lies in the practical demonstration that it is possible to determine the extent to which association among human characters are a result of gene effects in common. By understanding these gene-imposed associations, we will, in time, be able to better relate functional aspects underlying the visual system and eventually other complex biological systems.

Quantitative Genetics

663

(By use of different inset type, certainformulae are presented in the body of the text which may be interestingfor certain readers and helpful for others. These inserts should be considered parenthetical and supplementary to the main text.) CONCEPTUAL CONSIDERATIONS

To diagnose, treat and eventually prevent disease processes, the clinician must understand the different possible causes of disease. Quantitative genetics recognizes that any disease process results from the interaction of genetic and environmental factors. Environmental factors can be short-term (such as infections and injuries) or long-term (such as climate and nutrition) and can impose their influence either pre- or postnatally. Genetic factors may have a major or direct effect in the etiology of a disease or abnormal condition or may, somewhat indirectly, condition the influence of environmental factors. For example, the presence or absence of the "sickle-cell" type of hemoglobulin affects individuals' susceptibility to malaria. Genetic investigations involving humans have generally been limited to comparison of relatives primarily based on studies of family trees (that is, pedigree studies). This method has yielded very important information in the study of traits where qualitative differences exist between individuals; that is, when individuals either exhibit or do not exhibit a certain trait. Studies with laboratory animals have a considerable advantage in that animals can be selectively mated as well as selectively exposed or not exposed to various environmental conditions. Many more investigations in both laboratory animals and in humans have been conducted in those conditions for which a population can clearly be classified as "exhibiting" or "not exhibiting" a certain trait, analogous to the situation of individuals either being exposed or not exposed to specific environmental factors. A monofactorial (single-gene) trait is generally characterized by a discontinuous distribution or variation, while a multifactorial trait will display continuous variation. The various blood groups in man are examples of qualitative or discontinuous traits: the differences among the distinct categories of blood types are associated with single gene differences. Discontinuous traits can and often do display variation within the distinct categories. For example, the clinical picture of Marfan's syndrome is extremely variable, particularly with regard to the presence and relative degree of ocular involvement, even though the mode of inheritance is considered to be autosomal dominant. However, even with this variability, the difference between individuals with the condition and those with-

664

Spivey out the condition is of sufficient degree to permit unambiguous recognition of individuals as belonging to one category or another. Another example is the level of phenylalanine in blood plasma among individuals with phenylketonuria (PKU), an enzyme-deficiency condition associated with a single gene difference, compared with nonaffected individuals (Fig. 1). Even though the measured level of phenylalanine shows considerable variability among affected individuals, there is unambiguous separation of the frequency distributions of nonaffected and affected individuals. 1 Quantitative traits, on the other hand, display a continuous range of variation so that there are no clear-cut categories. Although such traits can be classified into arbitrarily-defined categories, it is the nature of the trait, not the manner in which it is measured, that determines if the trait is qualitative or quantitative. For example, the anterior chamber is generally assessed clinically as either shallow, normal, or deep; although actual individual values for chamber depth are, in fact, continuously varying when measured ultrasonically. Eye color is usually assessed as a qualitative trait (for example, brown, blue, hazel, etc.); however, the continuous range of pigment differences can be assessed quantitatively with a photometer in units of percent of reflectance at a given wave length (my). The inheritance of both single-gene and polygenic traits involves the same genetic principles which Mendel originally described. The basic difference between single and multiple gene traits is the number of loci (sites on chromosomes) influencing the trait and the magnitude of the effect each gene contributes to the expression of the trait or condition. With polygenic traits, there is simultaneous contribution of gene pairs or alleles at many loci which influence the trait rather than just one locus as in Mendelian genetics. The cumulative influence of all these genes

FIGURE 1

Frequency distributions of phenylalanine in blood plasma (mg %) in affected population (right) and control population (left).


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in combination with the effect of environmental differences produces the observed continuous variation of individual differences. Hence, the difference due to each single-gene difference is obscured by the multiplicity of factors involved. Mendelian analysis is simply inadequate to explain, or "uncode," the complexity of events in such traits. Consequently, the study of continuous variation demands a combination of genetical theory with biometrical analysis. Various types of standardized instruments are available for clinical examination which provide the clinician with the means of measuring on a quantitative scale many of the various aspects of function and structure (which are referred to as "characters" or "traits"). In some cases, present techniques are inadequate to provide quantitative measurement, or certain instruments that enable clinical evaluation on a quantitative scale are not widely available (such as ultrasound to measure actual anterior chamber depth). However, in other cases, even though techniques are readily available, the press for a clinical "short cut" has led to the adoption of procedures for scoring which are rapid but which reduce the information recorded from the level of actual measurement of differences to the level of expedient, artificial classification (as in grading chamber angle depth). Such classification procedures prevent full analysis of the information obtained because they may obscure a true continuous state of nature. In order to avoid loss of information and arbitrary grouping, observations should be recorded in units of measure whenever possible; shorthand clinical notation (such as + 1 to +4, or low-normal-high, etc.)

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should only be employed when an interest in continuous variation does not exist. The majority of differences among individuals in a population appear to be not clearly classifiable. Characters which cannot be classified for individuals and summarized using class frequencies must be measured and summarized using numerical procedures sensitive to the subtle, graded differences among individuals (Figs. 2 and 3). An implication inherent in the observation of continuous differences for a character is that it is unlikely. individual differences can be satisfactorily associated with differences at only one or a few hypothetical loci (that is, the locations on a chromosome presumed to represent a "gene form" or allele). When measurements for a series of subjects have been collected, it is simple and useful to consider the frequency distribution of scores by establishing narrow ranges of values (classes) on an x-axis and plotting frequency (or relative frequency) as the y-variable as shown in Figure 2. Iftwo or a few distinct high frequency classes are observed, as in Figure 3, clinical classification designating these groups can be used with only slight loss of information. Characters showing distributions like that of Figure 3 are naturally discontinuous and are best described by frequency counts and analyzed by methods designed to handle frequency data such as the nonparametric Chi-square tests. If scores are continuously distributed over the x-axis, classification will .50

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be meaningless, and standard methods for describing the central tendency and dispersion of scores should be employed. The mean (#1) is a measure of central tendency because for any group of numbers (observed scores) the mean is a unique value which is "closest" to all observed values simultaneously in the sense that the sum of deviations of individual observations from the mean is zero. The mean (X) is calculated as:

n X

=

X(i)/n i=1

where "E" is a summation operator, x(i) is the observation for the i-th subject and summation is over a number of observations equal to "n".

The spread or dispersion of a distribution is described by its variance. The sum of squares of deviations from the mean is always a positive number and one which reflects the total spread or dispersion of observed values. The variance is the approximate average squared-deviation from the mean (#2 and #3). The variance (V) of character X is calculated as: n V

(X)

=

Y (X (i)

-

)2

**#2**

i=l

n-i

or, equivalently, but more usefulfor calculation since it avoids rounding errors from subtraction: V (X)

=

YX(i)2- X(i)]2/n

**#3*

n-i

where, in both cases, "X" indicates continuing addition or summation, and "n" is the number of observations. Since the mean has been estimated, the number ofunrestricted deviations from the mean referred to as "degrees of freedom" is one less than the number of observations because the last deviation is determined as that which makes the sum of deviations be zero.

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Spivey Variance for observed values is quite directly related to differences among those values. By convention, the mean is subtracted from each value so that all scores are in terms of their relationship to, or deviation from, the mean. Traits with more widely ranging values will have higher variances independent of their mean value. Example: Using a small number ofhypothetical observations, these relationships can quickly be seen. Given observations of 1, 2, 3, 4, and 5, we can array these (to make notation explicit) as: X(1) = 1 X(2) = 2 X(3) = 3 X(4) = 4 X(5) = 5 then, n 5 X(i)= 1+2+3+4+5= 15 I X(i)= i=1

i=1

and the mean is: 5 X

Y. X(i) / 5 = 15/5 = 3

=

i=1

Deviations from the

d(1) d(2) d (3) d (4) d (5)

= =

= =

=

1 -3 2 -3 3 -3 4 -3 5-3

mean are:

deviation

deviation squared

=

-2

=

-1

4 1

=

0

0

=

1

=

2

1 4 10

0

Their sum is zero and the sum of their squared values is 10, and the variance is 2.5 units-squared. Using the alternativeformulafor calculation:

Xj2_(-Xi)

(1)2 + (2)2 + (3)2 + (4)2 + (5)2 - [(15)2/5]

n

=

55-45= 10


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and V = 10 / 4 = 2.5 units-squared

When greater differences for a trait are observed among members of different families than they are for individuals within families, similar values for the traits are said to "run in families." Since variance is a measure of the amount of observed differences, it is natural to expect for such characters that variance within families should be less than variance among families. This means that the differences observed among families relative to their respective family mean will be less than the differences of the mean scores of families relative to the overall mean. The analysis of variance provides a method for associating parts of the variance of traits (called components of variance) with specific sources (such as, among families, within families, between sexes, etc.). Thus, the measure of variance is of basic importance in the understanding and study of characters which display continuously graded differences among individuals. Before proceeding to consider the logic that allows components of variance to be associated with gene differences among individuals, it is necessary to examine covariance. Covariance provides a general measure of resemblance, and resemblance among relatives occurs whenever differences in characters are due to gene differences! Covariance (#4 and #6) quantifies the extent to which two characters show simultaneous deviations from their respective means in a population. The two characters can be two different measures from the same individuals within a group (such as height and weight), or they can be the same character measured on individuals within two different groups (such as height of parents and height of their offspring). Height and weight covary in the human population because, on the average, individuals above the mean height are also above the mean weight. Both height and weight covary in family members in the sense that taller parents tend to have taller offspring, and heavier parents tend to have heavier offspring. Unlike variance which must be positive, covariance is signed (that is, it can be either a positive or negative value). If offspring whose parents' scores are above the overall mean for parents display values above the overall mean of offspring, then the covariance of parents' and offspring's scores is positive. The definition of covariance is exactly the same as that of variance except that the sum of squares of deviations of a single characterfrom

670

Spivey its mean is replaced by the sum ofproducts ofdeviations of two characters from their means. Thus, covariance is an approximate average cross-product of deviations. This measure of the extent to which two measures, "X" and "Y", vary together uses deviations of both measures from their respective means. This provides a sum of products as: n E

(xi - x)

*#4*

(Yi - Y)

i=1 When this sum ofproducts is divided by "n - 1", the result is referred to as the covariance (COV) of variables X and Y.

Notice the similarity between the variance of X when it is written as: V (X)

=

IX(i)X(i) -

n

**#5**

n-i

and the covariance of X and Y when written as: CoV (XY)

=

YX(i)Y(i) - [

YXX(i)Y(i n

J

**#6**

n-i

Writing out the squared terms in the expression for variance demonstrates that the variance of X is a special case of covariance: the covariance of a variable with itself is its variance.

In order to use this method of calculating covariance, there must be a logical basis for defining two scores and two means such as that of parents and of offspring. That is, the two groups of scores must be naturally distinct (that is, parents versus offspring). However, with twins, for example, the decision of which twin to assign to group 1 and which to group 2 is arbitrary and not pre-defined. When no criterion is available to define two groups (as it is not in the case of twins, full sibs, half sibs, etc., since they are members of the same groups), covariance is determined by direct consideration of the differences within groups of relatives and differences between or among these groups. The reasoning involved in this method of calculating covariance may seem abstruse.

671


But, recall that differences are measured relative to some "standard," namely, the mean. When considering differences within a family, it is relative to the mean for that family. When considering differences between or among families, it is a comparison of the differences of the family means from the overall mean of that population. If resemblance (similarity) among the relatives within each respective family is high, then differences among the members within each family are low. For a given level of total population variance (which remains more or less unchanged), as covariance among individuals within families increases, variance within families decreases so that the main source of differences (total population variance) is contained in the component of variance among families. In this way, the component among groups directly measures the covariance of individuals within groups.2 The usefulness of the component of variance between groups of relatives to assess their covariance can be appreciated by visualizing two hypothetical distributions of observations within and across families. In Figure 4 each dash indicates an individual's score on the arbitrary scale; family membership is indicated on the horizontal. The two cases are designed to reflect approximately equal total variance. Members within each of the families on the left resemble each other quite closely, while those on the right do not. From this display, in which the total variance is the same for the two populations of families, it should be noted that where resemblance is high within families, differences among families (and therefore the variance component among families) are high. Pro-

Increasing-Arbitrary Scale

1

2

3 4

1

Family

2 3

Family FIGURE 4

(See Text).

4

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Spivey cedures for analysis of variance (estimation of intraclass correlation) in this situation are available in standard statistical texts.3'4 Summarizing to this point, differences displayed by individuals for a character which is not discontinuous are properly measured or indexed as the variance of the character. Components of this total variance (that is, parts of the differences) can be associated with specific sources such as family membership. Further, direct measures of resemblance are provided through calculation of covariance. Covariance may describe resemblance of deviations of two different variables from their means on the same individuals or it may measure the resemblance of deviations from their means for the same variable in individuals of two different groups (that is, parents and offspring). When groups of individuals of the same relationship are encountered (that is, all half sibs or all full sibs, rather than a group of parents and a group of offspring), covariance must be determined as a component of variance between groups. These methods can be used to associate portions of the total individual differences for a character (variance components) with gene differences, allowing genetic analysis.

GENETIC SOURCES OF RESEMBLANCE AMONG RELATIVES

We can best understand genetic sources of resemblance among relatives by considering what happens to genetic material in the process of reproduction. In bisexual reproduction, a random sample-half of the genes of two individuals are combined to form an individual. Parents transmit to offspring a single gene form or allele for each site (locus) on their chromosomes. They transmit none of their gene combinations. That is, they can transmit only one of the two alleles thay have at each locus. They cannot transmit their allelic pair which is the specific combination of alleles they have at a given locus and, thus, they cannot transmit the "pair effect" (the dominance effect, if present). Therefore, without knowing which genes (that is, what forms or at what loci) influence a character (any aspect of an organism under consideration), we know already that half of the gene-dependent differences of each parent are transmitted to offspring. Thus, when gene differences influence a character, we expect offspring to resemble parents more closely, on the average, than they resemble individuals at random in the population. Stated another way, they are more different from other individuals than from their relatives. Resemblance of an offspring to a parent, for example, reflects gene differences only in single doses at all loci which influence the character and not the specific combinations of genes of that parent. These differences can only be due to the sum of the direct effects


673

of one-half the parents' genes. The effect on a character of any single gene is referred to as the value of that gene for that character. The sum of the values of the genes of an individual is the person's genetic value for that character. (Genetic values are also referred to as "breeding values.") It is for these reasons that covariance of offspring and parents, a measure of their resemblance relative to their respective "population," is employed to estimate the variance of genetic values for the character (#7 - #11). COMPONENTS OF VARIANCE OF MEASURED VALUES

The usefulness of these measured values on individuals in genetic analysis can best be understood by considering that any measured value (M) for an individual reflects the additive genetic value (A) from gametes of parents, dominance genetic values (D) from combinations of genes at fertilization, and environmental influences (E) which include in this development all unexplained sources ofdeviationfrom expected value. The assumption is made that these values add to yield the observation or measured value. Thus we have:

M(i) = A(i) + D(i) + E(i)

**#7**

for the i-th individual measured value. When observations are availableffrom a number (n) ofsubjects, the variance ofthose measures can be calculated as mentioned above. From statistical theory3, this variance is known to equal: V(M) = V(A) + V(D) + V(E) + 2 COV(A,D) + 2 COV(A,E) + 2 COV(D,E) **#8** The covariance of additive genetic values and dominance genetic values (thefourth term in the equation above) can be demonstrated to be zero (2). If it can be assumed that there is no covariance of environmentalfactors with either additive or dominance genetic values, that is, that there is no correlation between genotype value and environmental deviation, such as would arise ifthe better genotypes were given better environments, then the last two terms also go to zero. We have then partitioned (conceptually, at least) the total variance of a measured character into variance due to differences in genetic values (VA), those due to differences in gene combinations (VD), and those due to all other sources (VE). GENETIC VARIANCE

Defining measured values as the sum of products given in #6 allows examination of the contribution of genetic sources to the resemblance among relatives. Consider, for example, the covariance of measured

674

Spivey values on offspring (Mo) and parents (Mp): **#9** COV(Mo, Mp) = COV(Ao+ Do+ Ep, Ap+ Dp+ Ep) but Ao = ½ Ap, since parents contribute half of their genetic value to offspring. Thus we have: COV(Mo,Mp) = COV(%Ap,Ap) + COV(Ao,Dp) **#10** + COV(Ao,Ep) + COV(Do,Ap)

AU terms other than the first equal zero (by assumption) yielding the result discussed in the text: COV(Mo,Mp) = ½ V(A) **#11** In this and allfurther developments, it should be borne in mind that the assumption that COV(Ao,Ep) = 0 is subject to doubt and depends on the character being considered.

Since parents and offspring share half of their genes but none of their gene combinations, covariance between them measures half of the variance of genetic values. Conventionally, this source of individual differences is referred to as "additive genetic variance" since the differences reflected by the variance are the added effects (genetic values) of the genes of any individual contrasted to added effects of genes of other individuals. Additive genetic variance is then descriptive of the differences of additive genetic values for a character of individuals in a population. Genetic souces of resemblance among half sibs present a similar situation. Half sibs can be expected to share, on the average, one-half of the genes received from their one common parent and obviously none of the genes from the other parent. Thus, they can share none of their gene combinations (that is, no more than any two individuals considered at random). Therefore, they have, on the average, one-fourth of their genes in common, and covariance among them will estimate onefourth of the additive genetic variance (#12). The covariance of half sibs is given by: COV(HS) = COV(½ A, % A) = Y4 V(A) **#12** since each of the sibs receives only a sample halfof the genes of a comnmon parent.


675

These gene differences, determined at the time of gamete formation (gametogenesis) by random grouping of genes of individuals and contributing to additive genetic variance, may not be the only differences imposed by genes on the character. In the process of fertilization the random groups of genes on chromosomes in gametes are recombined and paired. The actual pairing of genes at the same loci (alleles) from two parents may contribute to character differences. Just as expression of an allele in the classical genetic model may be modified by the presence of another allele at the same locus (that is, through dominance), we must consider the possibility that the genetic value transmitted to offspring by a parent may be modified by alleles at the same loci received from the other parent. The genetic sources of resemblance among full sibs (who have both parents in common) demonstrates this situation. It is possible, although extremely unlikely, that these individuals may not have a single gene in common or that they may be genetically identical. The most likely situation is that they have half of their genes in common. Since each parent gave each offspring a sample half of its genes, the highest (statistical) likelihood is that half of the transmitted genes from each parent were in common for the two. Since full sibs share half of the genes in common, on the average, the probability is highest that they share one quarter of their gene combinations. That is, because full sibs have the same pair of parents whose genetic complements are sampled, it is likely that they have the same pair of alleles at 25% of their loci (on the average). When gene differences influence individual differences for a character, we can expect full sibs to resemble each other more than they resemble individuals at random because they share half of their genes. In fact, they should, on the average, at least look as much alike as do parents and offspring. When the influence of a gene on an individual character depends on the other allele at the same locus (that is, when dominance effects are present), full sibs should look more alike than parents and offspring for the additional reason they share one quarter of their gene combinations at the same loci (#13). Becausefull sibs share two sample halves of genes, they share, on the average, a sample quarter of their gene combinations. This reasoning can be expanded to demonstrate that the covariance offull sibs is: COV(FS) = Y V(A) + Y4 V(D) **#13**

It should be noted that dizygous twins should resemble one another no more than full sibs due to gene influences, although their environ-

676

Spivey

ments may be more in common. However, the covariance between monozygous twins should contain all the additive and dominance genetic variance since they are genetically identical. When genetic variance estimates are based on the difference between measures of resemblance of dizygous and monozygous twins, the result represents an awkward confounding of dominance variance with additive genetic variance. This estimate does not provide clear information regarding the sources of variance for a character.5 The line of reasoning developed above hopefully will allow the reader to deduce which genetic effects contribute to resemblance (measured by covariance) among relatives of various sorts. For example, half sibs, who share only one parent in common, should reflect by their covariance for any measure, one quarter of the additive genetic variance but none of the dominance genetic variance for the measure. ENVIRONMENTAL SOURCES OF RESEMBLANCE AMONG RELATIVES

Clearly, genetic causes of resemblance among relatives may not be the only reasons for relatives to resemble one another for measured characters. If individuals within families share a common environment and if the character being measured is responsive to any aspect of the shared environment, then related individuals do not experience as broad a spectrum of environmental influence as do individuals at random in the population. Environmental differences are then present among families which are not encountered within families, and these differences will result in increased resemblance among relatives. The importance of environmental sources of resemblance among relatives lies in the fact of sharing a common environment. Different relatives, of course, share common environments to different extents. Full sibs, for example, share a common maternal environment and a common home. Identical twins reared together share these aspect of their environment plus the circumstance of a matched-age sib of the same sex. Mother and offspring may share a common household, but not a common mother except to the extent that aspects of mothering may be passed from generation to generation by example. Fathers and offspring don't share a common mother except when the father chose to marry "a gal, just like the gal that married dear old Dad," thereby yielding a mother for his offspring who is an environmental influence similar to his mother. It is primarily with sibs, and especially with twins (and particularly with identical twins) that shared environment may contribute substantially to covariance among relatives. Whether such environmental sources of resemblance do contribute to covariance of relatives for any


677

particular measured character (clearly there are potential sources of such resemblance) is by and large dependent on the biology of the character. If it is known to be environmentally labile for aspects of environment shared by relatives (such as IQ), one must be aware that covariance among relatives may be generally increased by common environment. If the character is not influenced by family-associated environmental variables, then an investigator may feel more certain that covariance among relatives is not primarily environmental in origin. In ophthalmologic genetic analyses, this should be of less concern than it is, say, in behavior genetic analyses. Yet, aspects of parental care and diets common to families may serve to increase resemblance among full sibs for some measures (such as, height). In an ophthalmic example, Young6-9 has experimentally demonstrated an effect of "near environment" on components of refraction. RECAPITULATION

Individual differences in ophthalmic, or any other continuously varying measure reflect the influence of various gene and environmental differences on the character. The basic rationale for associating parts of the differences to gene effects depends directly on the genetic events occurring at reproduction. Gamete formation isolates random sample halves of genes for individuals, and the differences resulting from this splitting (that is, the genetic half from either parent) lead to what we term the additive genetic variance for any character. Fertilization combines these sample halves of genes, and those combinations are the source of dominance genetic variance (that is, the effects of both alleles in combination). These events at reproduction guarantee that relatives will resemble each other (and differ from unrelated individuals) for traits influenced by gene differences. The extent of resemblance is expected, as a basic biological consequence of reproduction, to differ for groups of differing relationship. Further, different kinds of gene effects are expected to contribute to resemblance of different kinds of relatives. Depending on the relationship (parent-offspring, full sibs, half sibs), the expectation of the type and extent of the several types of gene effects (for example, additive and dominance) will vary. Environmental variables, too, can contribute to resemblance among relatives who share common environment. The extent to which non-genetic sources are likely to contribute to resemblance depends on the type of familial relationship, the character under consideration, and nature of relevant environments. For most characters, environmental influence should affect only covariance (resemblance) of sibs reared together.

678

Spivey PRACTICAL CONSIDERATIONS

APPROACHES IN GENETIC RESEARCH

The aim of genetic analysis of continuously varying characters is to partition accurately the total observed differences into those due and not due to gene differences. Viewed in this way, the aims are clearly exactly the same as those of the more usual genetic analyses of discontinuous traits except that in the latter situation,, a single gene differences is sought as the source of all the differences displayed among individuals which can then be used to establish probability of occurrence. However, genetic analysis provides a basis for predicting the probability of ocurrence among offspring whether one gene or many genes influence a character and whether or not environmental variance contributes to individual differences. QUANTITATIVE GENETICS IN ANIMAL RESEARCH

Genetic analyses of continuously varying characters have been undertaken in laboratory and field settings. In the laboratory, quantitative genetics has been employed in connection with the study of evolution. 10 The genetic properties of a population are principally the product of natural selection. Quantitative genetics is useful in this context because characters which have, through time, been subjected to various sorts of pressures of natural selection are expected to differ in their genetic properties. Populations are constantly subject to natural selection tending to increase fitness. Characters which have been closely related to fitness and have responded to selection in one direction should display very little additive genetic variance and substantial dominance genetic variance. Currently, there is substantial evidence available1ll2 to suggest that the total genetic material available to a species can reasonably be considered to exist in two sets: a set of loci which have two or more allelic forms (representing perhaps 30-50 percent of the genetic material) and a set of fixed loci. Characters (morphological, physiological, etc.) which depend for their expression on genetic information primarily in the latter set constitute species-specific characteristics while those characters dependent in part on gene activity at loci in the former set display variability in expression among animals within species. Traits common to all members of a species, but different between closely related species, represent products of evolution. Presumably, the action of natural selection has exhausted additive genetic variance within species at loci influencing these characters and has fixed differ-


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ent genes in different species at least at some loci influencing character expression. 2 Field-oriented research includes commercial breeding of livestock13 14'15 and the development of strains of plants,16'17 such as, corn, wheat, flowers with specific qualities. In field-oriented research, quantitative genetics is most frequently used to indicate the success to be expected in modifying the level of expression of characters through artificial selection. Here, emphasis is on gaining estimates of additive genetic variance because the larger the proportion of differences for a character due to differences in genetic values (additive genetic variance) in a population, the faster that character will respond to selection. In fact, it is in the context of field-oriented research, so successful in improving plant and animal production, that the ratio of additive genetic variance to total variance for a character was defined as the heritability of the

character18 (#14-#16). HERITABILITY

The expression shown in #7 can be simplified using the assumption of zero covariances mentioned above. Focusing attention on variance in genetic values by combining other genetic effects (that is, non-additive) uith environmental influences, which were referred to as "residual values" in the text and symbolizing this V (NA +E) gives: **#14** V(M) = V(A) + V(NA+E) Dividing both sides of this equation by V(M) gives: **#15** 1 = V(A)/V(M) + V(NA+E)N(M) Thefirst term on the right hand side of this expression defines the heritability (h2) of the character: **#16** h2 = V(A) / V(M)

The relationship between this ratio and the response to artificial selection is easy to understand since selection always involves breeding individuals on the basis of their score on some measure. If heritability is high for a character, parents who are chosen because of their high performance (such as, milk production or disease resistance) will have offspring who will perform or behave, on the average, in a similar manner due to the additive genetic variance (genetic value) associated with that character in the population. Recall that genetic variance influencing individual differences in a character is composed of components which include additive genetic and

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dominance genetic variance. Additive genetic variance is, in practice, more important than dominance variance, and it will be emphasized in the remaining discussions. Dominance genetic variance is less important in the context of this presentation since it does not contribute directly to resemblance between parents and offspring and because it is extremely difficult to estimate from human data. QUANTITATIVE GENETICS IN HUMAN RESEARCH

Subsequent to being defined as the ratio of additive genetic variance to total variance, the term "heritability" has been bastardized. In various areas of research (especially in analysis of twin data), the name has been applied to the ratio of various measures of genetic variance to total variance. It is especially important for the present discussion, and indeed, for work in clinical genetics, that the original heritability ratio be emphasized and its implications and usefulness clearly understood. The denominator of this ratio and most others referred to by the same name is the total variance for the character in the population. The numerator is the additive genetic variance, or variance in the character due to differences in genetic values in the population. It is here that most other estimates differ, such as the "heritability" estimates based on monozygousdizygous twin comparisons previously mentioned. Ratios with other definitions for heritability may be useful in describing the importance of gene differences influencing a character, but the heritability defined in the narrow but precise manner described above is the single, most important genetic characteristic of continuous characters. In fact, this ratio is required for every major development in quantitative genetic theory.2 Among the practical reasons for emphasizing strict adherence to the original, narrow definition is that the ratio is predictive in this form. The basis for prediction is regression theory (#17 and #18). Variance and covariance of individuals and among relatives can be directly measured using the procedures described above. Covariances and variances are employed in various measures of association amnong observed values. The regression coefficient (symbolized as "b") directly measures the change in one variable (the dependent variable) per unit change in the other (the independent variable). The coefficient is defined as: **#17** b(y,x) = COV(X,Y) / V(X)


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when "Y" is the dependent variable and "X" is the independent variable. If"X" is the dependent variable and "Y" is the independent variable, the regression is: b(x,y) = COV(X,Y) / V(Y) **#18**

For example, if the relationship between height and weight is known and expressed as the covariance of the two variables divided by the variance of weight (the regression of height on weight), then knowledge of an individual's weight can be used to predict that person's height. Regression is important in genetics because the regression of genetic values for a character on measures of that character (#19) will allow prediction of offspring scores (since genetic values are transmitted to offspring) from knowledge of parent scores (#20 and #21). When properly estimated, heritability provides a direct estimate of this regression, and it is required for prediction in quantitative genetics. Demonstration that h2 is the regression of genetic values (A) on inleasured values (M) requires the use of the definition of regression (#17) with genetic values as the dependent variable and the definition of measured values given in #7. This regression gives: b(A, M) = COV(A, M) / V(M) = COV(A, A+ D+ E) / V(M) = V(A) / V(M)

since COV(A,D) and COV(A,E) are zero and COV(A,A) is V(A). Regression procedures have been used to estimate heritabilityfroml values measured on parents of offspring. The regression of offspring scores (Mo) on single parent values (Mp) is given by: b(o,p) = COV(Mo, Mp) / V(Mp) **#20** which, from #11 is equivalent to: b(o,p) = 1/2 V(A) / V(M) **#21** if the assumptions of no covariance between genetic value and environment holds and if the variance of single parents provides a reasonable estimate of the total variance for the character. Heritability (#16) is thus twice the regression of offspirng values on single parent scores.

To understand why this ratio of additive genetic variance to total variance (and no other ratio) is equivalent to the regression mentioned, suppose that values measured for individuals are composed of their

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Spivey genetic values for the measures and, for convenience, what we will call "residual values" which contains all other genetic and nongenetic effects (#14). The numerator of the regression of genetic values on measured values is the covariance of genetic values and measured values. This covariance is comprised of two parts: the covariance of genetic values and the covariance of genetic values with residual effects. The first part is the variance of genetic values, the additive genetic variance, which is the numerator of the heritability ratio discussed above. The second part is zero on the assumption that individuals with high genetic values do not uniformly encounter environments with positive effects on the character (#11). This regression coefficient is then the additive genetic variance (equal to the covariance of genetic and measured values) divided by the variance of measured values. Thus, the regression coefficient is equivalent to the heritability ratio (#16 and #19). The assumption of zero covariance of genetic values and environmental deviations is more reasonable for analysis of some characters (such as visual system parameters) than for others (such as behavior). Estimation of heritability is practically synonymous with genetic analysis. In human research, calculation of heritability is almost exclusively dependent on measurement of resemblance among relatives (since estimates from selection or inbreeding are not possible). Because covariance among some of these relatives may reflect other sources of genetic variance in addition to additive genetic variance and may be influenced differently by environmental sources of covariance, certain relatives are preferred over others for these estimates. Specifically, measures of covariance among parents and offspring (#11) and measures of covariance among half sibs yield the cleanest estimates of additive genetic variance (#12) influencing characters whose expression is independent of age or which can be corrected for the effects of age differences (#28). Half sibs reared apart should yield the best estimates since no environmental source would be likely to influence their covariance. Comparison of estimates of ratios of the additive genetic variance to covariance among full sibs allows an estimation of dominance variance (#13). This can be accomplished by subtracting the additive genetic variance (which will be twice the covariance of offspring and parents or four times the covariance of half sibs) from twice the covariance among full sibs. In practice, it is best to estimate heritability from the regression of offspring on parents. This can be either the regression of offspring on the average value of the two parents (the "midparent" value), or, as twice the regression of offspring scores on a single parent value (#19).


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When regression uses a single parent, the convention is to use the parent of the same sex as offspring. Thus, one would regress sons' measures on fathers' measures, or daughters' measures on mothers' measures. Regressions on single parent and midparent values both yield direct estimates of the required ratio because they both have estimates of parent-offspring covariance as their numerators. The parent-offspring covariance is half the additive genetic variance and only this if environmental effects are negligible. Both have denominators which involve the variance of measured values. However, the denominator for single parent regressions is the total variance whereas the denominator for midparent estimates is half of the total variance if mating is at random with regard to character. This difference in denominators explains the doubling of the single parent regressions and not of midparent regressions to obtain heritability estimates. It is also the reason regressions, rather than correlations (#22), are preferred as measures of heritability from parents and offspring (#20). Correlations are covariances divided by the square root of the product of the variances for the two groups (#22). Only when the variance of offspring and that of parents both properly estimate the total variance will the denominaor of a correlation be equivalent to that for regression (#23 and #20). The correlation (symbolized as "r") between two measures is defined as: r(x, y) = COV(X,Y) / NV(X) V(Y) **#22** which is the covariance divided by the square root of the product of the variances of the two measures. When correlation is estimated between single parents and their offspring, we have: r(o,p) = COV(Mo,Mp) //VV(Mo) V(Mp) **#23** The numerator is % V (A) and the denominator is V(M) because variance of offspring and of parent scores should provide estimates of the total variance for the character. Thus, when single parent values and offspring scores are employed, doubling either the regression or the correlation should provide good estimates of V (A)/V( M) (heritability (h2)).

Both correlation and regression are useful in heritability estimation, but when mean values for either parent or the offspring are employed, the denominator of correlations (#28) no longer estimates the total variance, but, rather, some function of that variance. For the midparent

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Spivey case, the denominator would estimate a value equal to the total variance divided by the square root of two (#29 and #30). However, the regression in any of these cases has either the total variance (with single parent) or half the total variance (with midparent values) in its denominator. Consequently, a regression is always easily converted to a proper estimate of heritability (#20), (#24-#27). When parent-pair average (midparent) values and single offspring scores are employed, regression provides reasonable estimates of heritability but correlation does not. The regression equation is: b(o,p) = ½ V(A) / V(Mp)

**#24**

The numerator, the covariance, is unchanged because the covariance with a single relative of a given relationship contains the same sources of resemblance contained in covariance with any number of such relatives.2 The denominator is now the variance of the averages of the two parents' values. This variance is: V(Mp) = V(Mpl + Mp2) / 2 **#25** = ¼4 [V(Mpl) + V(Mp2) + 2COV(Mpl,Mp2)] If V(Mpl) = V(Mp2) and if the covariance of the parents is replaced by the correlation between mating pairs, r(Mpl ,Mp2), and this is multipled by VV(Mpl) V(Mp2), which is the denominator of the correlation coefficient and is equal to V(Mp) because of the first condition above, we have: V(Mp) = 4 [2 V(Mp) (1 + r(Mpl,Mp2)J = % V(Mp) [1 + r(Mpl,Mp2)J **#26** Now, if mating is at random (that is, if r(Mpl,Mp2) = 0), then the regression of offspring on midparent is: b(o,p) = % V(A) / ½ V(M) = V(A) / V(M) = h2 **#27** but the correlation is: r(o,p) = % V(A) / VV(Mp) [V (Mp)/2] **#28** In this case, if V(Mo) and V(Mp) both provide reasonable estimates of V(M), the correlation of midparent values and offspring scores is: r(o,p) = ½ V(A) I (V(Mp)/V/2) = V2 h2

so that: h2 = r(o,p) 1\/2

**#29**

**#30**

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Quantitative Genetics This relationship has been presented and discussed by DeFries20 and is employed in the text to estimate heritability from previously published data reporting midparent correlations. When mean values from offspring are employed with either midparent scores (parent-pair averages) or with single parents, the correlation will estimate a complex function of heritability which depends on the correlation among offspring and that among parents pairs20 but the regression will estimate either halfthe heritability (with single parents) or the heritability directly (with midparents).

Half sibs are potentially an excellent source from which to estimate heritability. This is due to the fact that half sibs share a quarter of their genes on the average and none of their gene combinations from common parents. Also, their similarity is less likely to be influenced by environmental sources of resemblance, especially if reared apart and if they share the same father. In practice, of course it is not easy to find a large collection of half sibs for research in human genetics. When they are available, good and direct estimates of heritability can be obtained using the component of variance among half sib families (which is the covariance of half sibs) divided by the total variance. This ratio, the intraclass correlation, is then multiplied by four since half sib resemblance contains one-quarter of the additive genetic variance (#31). Comparison of heritability estimates from half sibs with those from full sibs is used for detection of dominance variance. Twice the intraclass correlation among full sibs will overestimate genetic variance to the extent that there is dominance variance for the trait (bearing in mind the cautions regarding shared

environments). Estimates of heritability can be obtainedfrom the intraclass correlation among either half or full sibs. The intraclass correlation is calculated by performing a standard analysis of variance4 to determine components of variance within, V(W), and between, V(B), groups of either half or full sibs. The intraclass correlation (symbolized as "t") is given by: t = V(B) / [V(W) + V(B)]

**#31**

For half sib groups the heritability is four times t(HS). Correlations among full sibs are estimatedfollowing the same procedures using data from full sib groups. In this case, twice the intraclass correlatiions will overestimate heritability when differences in the trait are influenced by dominance genetic variance (#13).

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RESULTS FROM ANALYSES OF OPHTHALMIC VARIABLES

A next logical step is to consider actual applications and results of the methodolgies discussed in the foregoing theoretical development. The genetic analyses surveyed involve traits ofrelevance to ophthalmologists. Ophthalmic defects are well known and chronicled in clinical human genetics.21 Clinicians are aware of disorders or traits being transmitted as Mendelian autosomal or sex chromosomal disorders (dominant or recessive). It has been tempting because of the simplicity and clarity of these concepts to equate the occurrance of a disorder with only the presence of a single abnormal gene. For example, in genetic analysis of strabismus, examination of pedigrees led Waardenberg22 to conclude that strabismus was transmitted as a dominant condition (with variable penetrance) while pedigree studies by Czellitzer23 and Schlossman24 suggested a recessive pattern. The tendency is to ignore the possibility that single conditions might be mimicked by specific environmental causes and/or by other collections or combinations of genes. Thus, adherence to a rigid notion of genetic determination may also lead to the incorrect inclusion of related similar cases with ones which have totally different causal bases. For example, it is generally recognized that dislocated lenses of differing etiologies [autosomal dominant (Marfans), autosomal recessive (homocystinuria), trauma, infections (syphilis)] present very similar clinical appearances. Specifically, a large number of visual system abnormalities are known from clinical experience and population surveys25'2627 to "run in families." Yet, no clear, consistent and compelling evidence is available to support the hypothesis that affected individuals differ from normals due to one or a few major genes. Since ophthalmology, among the clinical medical disciplines, is perhaps best equipped to obtain relatively precise measures of relevant biological functions, and because so many visual system abnormalities seem to fit the "multifactorial" criterion, it is especially important that clinicians in this discipline have available to them the principles of this genetic methodology. The following is a selective survey of some genetic analyses of continuously varying characters. The examples presented will serve to illustrate the application of these techniques in an approach to an analysis of abnormalities, based on underlying continuously varying characters. INTERPUPILLARY DISTANCE

Interpupillary distance is frequently assessed for patients who are refracted with a phoropter or examined with a major amblyoscope, and always when fitting spectacle lenses. The measure may be defined as the sum of the


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measures in millimeters (mm) from each pupil center to the midsagittal line. The measure is often taken in two segments to avoid paralactic error. Information regarding the importance of gene differences for observed or measured interpupillary distance is available from Hegmann and associates28 and from Spuhler. 29 The analyses in the former study, employed regression techniques to measure resemblance of interpupillary distance for offspring and parents. Offspring measures were adjusted for ageassociated differences prior to analyses. This work reported heritabilities from various parent-offspring combinations, and in addition to simple regression of offspring on parents, utilized partial regression techniques. These techniques allow estimation of the resemblance of offspring and one parent, independent of the effect of the other parent and thus adjust for effects of non-random mating based on interpupillary distance (see #26). Utilizing the same data presented in that paper, it is possible to calculate for the population sampled at random with regard to visual function a total variance for interpupillary distance of 11.187 mm-squared. Variance in interpupillary distance due to differences in genetic values (that is, additive genetic variance) in that population is 7.296 mm-squared. The ratio of these values yields an estimate of character heritability of 0.65. This estimate differs only trivially from the estimate of 0.70 reported in the second study.29 Since Spuhler's estimate was based on twins and is not significantly higher than that from Hegmann and associates, it is highly unlikely that either dominance variance or sib-specific environmental variance influence individual differences for interpupillary distance. This level of heritability can be considered intermediate to high in magnitude and does not suggest any gene effects in excess of those usually observed for general morphological measures.29 ANGLE KAPPA

Angle Kappa is of clinical interest because it is frequently responsible for apparent strabismus (pseudostrabismus). Apparent ocular deviations may only reflect a large angle Kappa without abnormality of binocular single vision. This measure is clinically defined as the angle formed by the visual axis (a line which extends from the fovea to the fixated object) and by the pupillary axis (a line which extends from the center of the pupil to the fixated object). Its magnitude (in degrees) is estimated from the degree of eccentricity ofthe reflection of a centrally positioned light target relative to the pupil center of the fixating eye. Data regarding angle Kappa were also gathered in the study28 referenced previously. From this analysis, estimates of total variance and additive genetic variance in that random population are 3.6160 degrees-

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squared and 1.7838 degrees-squared, respectively, and character heritability is thus 0.49. Using a subsample from the same population employed above, Franceschetti and Burian3o reported heritability for this character of approximately 0.52, estimated from parent-offspring regression. ACCOMMODATIVE CONVERGENCE TO ACCOMMODATION (AC/A) RATIO

This clinical measure is a regression coefficient which describes the change in accommodative convergence (in prism diopters) per diopter change in accommodation. It is useful clinically to help identify those individuals whose esotropia may be improved by lenses. One technique of assessment of AC/A ratio is the lens gradient method. The amount of accommodative convergence exerted is determined in response to different amounts of required accommodation (for a fixed viewing distance of 33 cm) while the subject accommodates a target of 6/6-size printed numbers. A measure of the amount of phoria in prism diopters (p.d.) is made with the prism cover test while the subject wears each of three different powers oflenses: +3, 0, and -3 diopters (D), which impose the requirement of 0, 3, and 6 D, respectively, of accommodation. Regression of the measured phorias on the amounts of required accommodation expresses the amount of convergence (in p. d.) per diopter of required accommodation exerted by the subject to maintain fusion of the target while stimuli for proximal and tonic convergence are held constant. A study31 reporting parent-offspring resemblance for this character was based on measures from individuals comprising the same population discussed above. From that data, individual differences in AC/A ratio can be summarized as having total variance of 1.2185 (p.d./D)-squared. Variance in the character due to differences in genetic values is 0.3640 (p.d./D)-squared. The resulting heritability of 0.29 suggests that prediction of offspring scores from parent values, measured as described above, would be substantially less accurate than in the case of the other two variables discussed. However, this heritability would not be considered low by animal geneticists. COVER TEST MEASURE

The cover test is used to detect any manifest or latent binocular deviation. While the patient fixates a distance target, the examiner covers one eye and observes the other eye. The procedure is repeated, first leaving both eyes uncovered, then covering the second eye and observing the first eye. If either uncovered eye moves to fixate the target, a manifest deviation is indicated. If no movement occurs, a procedure of rapid alternation of


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the cover is used to break up fusion. Under these conditions, if the covered eye moves, and on uncovering, is observed to move (to refix the target), a phoria is indicated. The magnitude of the deviation, whether manifest or latent, is measured by the dioptic power of a prism sufficient to stop the movement when placed in front of one of the eyes. Mash and co-workers31 also reported parent-offspring resemblance for cover test measures from individuals in a large population sampled at random with regard to visual function. From those observations, the estimate of total character variance is 10.2047 p.d.-squared. In that group, variance due to differences in genetic values is 2.7702, yielding a heritability of 0.27. No other estimates of heritability for this character have been made (according to available literature reviewed). VERGENCE AMPLITUDES

To binocularly view and fuse objects at varying distances requires coordinated eye movements. Horizontal disjunctive eye movements may be either convergent or divergent. It is possible to measure the magnitude of the angles of eye rotation to the break point (loss of fusion) and to the recovery point (recovery of fusion) for horizontal movements. An amblyoscope with grade two peripheral-fusion targets (which provide maximal stimulus) can be used to elicit the vergence movements. After establishing an individual's visual angle for his position of fusional demand for an apparent distant target (which serves as the starting point), this angle of presentation is changed, simulating a change in distance, until the subject indicates loss offusion. The magnitude ofthe angle to this point is considered the break point. The targets are then moved back toward their original position until the subject indicates recovery offusion. This constitutes the recovery point. The arcs of the vergence angles (in units of p.d.) are recorded as the measures of vergence ability. Individual differences for each of the four measures of vergence performance (convergence break and recovery points and divergence break and recovery points) were subjected to genetic analysis32 using parentoffspring regression techniques and employing families chosen at random with regard to visual system function. Total variances observed for the convergence measures are substantially higher than those for divergence measures. Specifically, total variances for convergence break and recovery points are 314.0764 and 298.1550, p.d.-squared, respectively, while those for divergence break and recovery points are 10.3522 and 12.6095 p.d.-squared. A similar pattern of differences is displayed for variance due to individual differences in genetic values for those characters. The additive genetic variances for convergence break and recovery

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points are 162.3994 and 281.1090 p.d.-squared while those for divergence break and recovery points are 3.6700 and 5.0450 p.d.-squared, respectively. Heritabilities calculated from these variance components are higher for recovery points (0.94 and 0.40, for convergence and divergence, respectively) than for break points (0.51 and 0.35, for convergence and divergence, respectively). It is interesting to note that many orthoptists feel recovery points are a more meaningful index of native fusional ability as well as a better monitor of symptoms and response to orthoptic treatment than are break points. REFRACTIVE AND KERATOMETRIC MEASURES

Individual differences in spherical refractive error involve one or more of the components ofocular refraction which include corneal power, depth of the anterior chamber, lens thickness, depth of the vitreous chamber and axial length. One of the components of refractive astigmatism (cylindrical refractive error) is corneal astigmatism, resulting from an aspheric anterior surface of the cornea. Other components include error of curvature of the lens, lens position, and irregularities in the refractive index of the lens. Spherical and cylindrical refractive errors are assessed with a standard streak retinoscope and recorded as the manifest refraction following application of a cycloplegic drug with the end point achieved by reading the 6/6 line. Corneal curvatures are measured (in consideration here) with a Bausch and Lomb keratometer calibrated in terms of refractive power in units of diopters. Two assessments of curvature are made: either in the horizontal or vertical meridians or in the meridians which yield the largest and the smallest radii of curvature. Both eyes were measured in the study, although only measures from the right eye are considered. The average value of the readings taken in the two meridians is used as the index of corneal power. The absolute value of the difference between these two readings is the measure of corneal astigmatism. Both measures displayed similar continuous distributions in the populations sampled3 in connection with the studies referenced previously.28231,32 Of these four measures, spherical and cylindrical error are subjective measures of the overall abnormalities of the refractive characteristics of the eye, while corneal power and corneal astigmatism are objective measures of components of the refractive system of the eye. Analysis of parent-offspring resemblance for each of these measures is available in two references28'33 previously mentioned. In addition, data is available from other studies for spherical error and corneal power.27,34.35,36 Cylindrical error, the most subjective of these four measures, is deter-


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mined by manifest refraction where 6/6 visual acuity is utilized to determine an end point. It would seem to be the least reproducible of the four measures and, consequently, the least reliable. Using a large sample chosen at random with regard to aspects of the visual system, Hegmann and associates28 analyzed data which indicate total variance of 0.3270 Dsquared and additive genetic variance of 0.0046 D-squared. The extremely small heritability resulting from the ratio ofthese two values (0. 01) might be used to argue that this portion of the refractive error has been closely associated with fitness over evoluntionary time. (See discussion of laboratory research in quantitative genetics.) However, bearing in mind the subjective nature of this measure, it is most likely that the low level of resemblance between parents and offspring results from unreliability. Clearly, individuals cannot resemble their parents more closely than they resemble themselves in repeated tests. The major conponent contributing to cylindrical error is corneal astigmatism, which is measured with greater reliability. The population available from the study by Mash and co-workers33 provides an estimate of 1.3354 D-squared for the total variance. Additive genetic variance for the character in that population is 0.6198 D-squared, yielding a character heritability of 0.46. Of the two variables remaining to be discussed, spherical refractive error is the more subjective. Data28 indicate that total variance and additive genetic variance are 4.5083 D-squared and 1.0896 D-squared, respectively, in that population. This provides a heritability estimate of 0.24 which is in extremely close agreement with heritability estimates available from other published data which is discussed below. It should be emphasized that all heritability estimates presented represent a strict interpretation of the definition previously discussed and have been calculated utilizing data available from published work. Young and Leary35 report a midparent-offspring correlation of 0.23. Correlations do not convert readily to heritabilities when parent averages are involved. Because it appears that these correlations were calculated by repeated entry of midparent values, the correlation presented in that work was doubled. This value was then divided by the square root of two (see #30) on the assumption that mating in the population was at random with regard to refractive errors. This assumption seems reasonable in view of extremely low parent-pair correlations presented in other studies.2728 This procedure yields an estimate of heritability of 0.32. Young37 presented correlations between full sibs of like-sex and of both sexes combined of 0.15 and 0.14, respectively. The numerators of those correlations estimate the covariance among full sibs which should contain components

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of genetic and environmental resemblance. The denominator of the correlations will estimate the total variance only if the variances of sibs are homogeneous (that is, estimate the same population variance). On the assumption that the variances are homogeneous, doubling the correlation coefficient (recall that it should be the intraclass correlation) will overestimate heritability to the extent that dominance variance and environmental variance influence spherical refractive error. This means that, for that population, 0.30 is an upper limit to the narrow sense heritability of spherical error. Both the correlation between midparent and offspring and the correlation among full sibs are also reported by Sorsby and associates27 using observations from members of 28 families. Correcting the observations on midparent in the manner mentioned above gives a heritability estimate of 0.49. Doubling the correlation between full sibs yields an overestimate of heritability of 0.77, suggesting the presence of environmental and/or dominance sources of variance in that population. Dominance variance for the character was also suggested by the work of Hegmann and associates.28 Data from the 28 families in the Sorsby study were available for reanalysis. This allowed calculation of midparent-offspring regression giving a direct estimate of heritability for the character in that population of 0.26. Thus, it seems safe to conclude that approximately one-quarter of the total variance of spherical refractive errors in the human population are due to additive genetic variance. This observation, based on consistent estimates across a collection of studies, certainly does not support the conclusion by Sorsby and associates27 that spherical error is "genetically determined." But, neither does it agree with the suggestion by Young and associates36 "that there is no major hereditary component involved." It indicates simply that 25% ofthe total variance is a function of differences in additive genetic values; the other 75% is due to nonadditive genetic and environmental differences. It should be emphasized that this value (25%) is of sufficient magnitude to allow rapid progress in response to selection. Milk production in dairy cows has been massively modified in herds in the United States by selection procedures and the heritability of that character appears to be no larger than 0.30. Corneal power is a component of refractive error which can be repeatably measured and is apparently highly heritable. Data from Mash and co-workersm provides estimates of 2.1072 D-squared and 1.8228 Dsquared, respectively, for total and additive variance for a heritability of 0.86. Sorsby and associates27 report a midparent-offspring correlation which, when corrected, indicates a heritability of 0.77. Data from Young


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and Leary35 give a heritability estimate of 0.51 based on an adjusted midparent-offspring correlation. Although other components of the optical system have been investigated in a genetic context2I527,30'345 it is not possible to examine the consistency of estimates of family resemblance by contrasting results of various studies. RECAPITULATION

A large number of ophthalmic variables are measured quantitatively in general clinic practice and do display continuous variation. The studies reviewed above demonstrate that individual differences observed for some of these clinical measures do reflect gene differences, some to a substantial degree. The magnitude of their heritability ratio indicates how closely the clinical measures of individual differences reflect or index individual differences for genetic values. And, as a corollary of this, it indicates how closely relatives should resemble each other and what the components of this resemblance are. APPLICATION OF ANALYSES TO OPHTHALMIC DISORDERS

Clinicians are faced with individuals who have functional abnormalities. The abnormalities invariably involve complex biological systems which are themselves functionally dependent on a set of components. The clinician has available measurements which reflect, more or less directly, the state of the parts. Each of the values that individuals present to the clinician reflect the effects of genetic values received from gametes and dominance combinations determined at fertilization as well as environmental influences accumulated throughout development. It is not difficult to imagine how an abnormality might reflect a gene mutation blocking function in one of the parts required by the overall system; or, how an environmental insult neutralizing a required part of the system might account for the abnormality. However, such gross "ablations" of parts are not always required for manifest clinical dysfunction. Specifically, ophthalmologists are not presented with patients whose corneal power is either present or absent, but ones whose corneal power is either more or less than that of other individuals. The problem becomes how to relate measured values and the gene differences influencing them in individuals to a discontinuous clinical abnormality and its genetic basis. This is important for understanding the gene differences influencing clinical abnormalities which run in families but cannot be associated with single major segregating genes. It promises

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to be of pragmatic value because if it is possible to relate gene differences influencing measured values to gene differences contributing to a clinical disorder, prediction of the disorder can be undertaken using the heritabilities of the relevant measures to predict offspring characteristics from observed parent values. The results of quantitative genetic analyses as discussed previously assign certain components of variance to differences displayed in the population associated with gene differences. But this information provides no basis for determining the contribution of detected gene differences to a clinical abnormality because no attention is paid to the diagnosis. On the other hand, just measuring differences between "normal" and "abnormal" subjects for a set of clinical values yields no information regarding gene influence on these measures. If, however, these two basic procedures are combined, the genetics ofclinical abnormalities based on continuously varying underlying systems can be undertaken. The basic rationale is that gene differences which influence a measured character may also contribute to the tendency for an abnormality to run in families. For that reason, characters which are influenced by these genes should display different gene differences in populations with differing frequencies of the abnormality. It should be mentioned that Falconer2 and others46 47 have developed the concept of "threshold" characters and analytical procedures for dealing with such characters. In the case of threshold characters, the underlying tendency or liability is hypothetical, however, and not measurable. When measures are available, it seems most productive to perform a genetic analysis on these measures using a population consisting of relatives of affected individuals and a population of relatives ofa matched group of individuals sampled at random with regard to the clinical problem. Following the reasoning just outlined, comparison of gene effects influencing measured characters in the two populations should indicate directly which traits have differing genetic bases for the two groups, and, thus, what characters are influenced by gene differences which also contribute to the clinical abnormality. It should be emphasized that propositi from the two populations can be excluded in the genetic analysis so that "affected" individuals are never compared to "normal" individuals-only their relatives are compared. An example of this approach to the application of quantitative genetics to the analysis of an ophthalmic disorder is available in the series of studies on strabism28'333 Genetic analysis was undertaken not only on a population chosen at random for visual function, but also on a population


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defined as the parents and sibs of children with diagnosed esotropia. Characters which displayed genetic differences (in this case, differences in additive genetic variance between the populations) are spherical refractive error, AC/A ratio, and the vergence recovery points. This set of clinical measures represents key variables for strabismus. That is, these variables display substantial differences in additive genetic variances between the two populations suggesting that gene differences which influence these characters contribute to the tendency of strabismus to "run in families." This indicates that parents of strabismic children in future generations will be individuals who show deviations from their population mean for these same characters. The extent to which parents' deviations for each of these characters will also be observed, on the average, in their offspring is indexed by the heritability of each of these characters. Thus, parents with high convergence recovery points relative to their population mean will have children whose convergence recovery points average 94% as far above the mean for their population. Likewise, parents whose divergence recovery points are below the mean, will have children who display, on the average, divergence recovery points 40% below offspring mean on the average. Similar statements hold regarding AC/A ratio and spherical refractive error. In those cases, the percents are about 30% and 25%, respectively. Thus, if parents showing deviations (relative to the parent mean) for these variables would be counseled not to reproduce, and if they follow this counseling, the frequency of occurrence of nonparetic strabismus will be substantially reduced in the next generation. More realistically, specific regard to these measures on nonstrabismic individuals can provide insight to clinicians regarding the need for special attention to the development of visual function among their offspring. While some insight is thus provided regarding the probability of strabismus occurring among offspring of nonstrabismic parents, exact formulation of the predictive statement remains to be developed. If, in the case of small angle strabismus, the probability of occurrence was sufficient to suspect its presence, institution of early alternate occlusion, etc., may result in the prevention of amblyopia. RECAPITULATION

Quantitative genetic analyses of measured variables may be effectively

employed using population comparisons and a large battery of clinical measures to detect variables of most direct relevance to the biological

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bases for an abnormality. Although these variables should index underlying key factors in the all-or-none manifestation of various clinical abnormalities, information at this level of analysis does not necessarily imply a condition of biological causality to such traits. The interests, needs and suggestions of clinicians could and should influence the direction of future research. As studies of this nature proliferate, it is incumbent upon both the researcher and the clinician to incorporate this information into a clinically-applicable frame of reference. The applications and extensions of quantitative genetic analyses will depend on the understanding by clinicians of what information is and is not contained in the results of such studies and of how this information can be of practical value to them.

GENETIC CORRELATIONS IN HUMANS: THE FIRST STUDY

BACKGROUND

Early detection of abnormalities in offspring of unaffected individuals is only part of the promise of quantitative genetics for clinical medicine. The methods presented thus far, appropriate for analysis of individual clinical variables, are only part of those available for genetic analysis. Genetic methods available for analysis of gene differences influencing associations among two or more measured characters have not been employed in clinical medicine but should be useful in the study of visual system function. Although a number of previously referenced studies26'27'37 have reported observed correlations among clinical variables measured on the same individuals, there has been no attempt to investigate the source of these associations. However, in a manner analogous to the partitioning of sources of variance for one character, the covariances between characters can be associated with genetic and environmental sources. Hence, the extent to which observed correlations between characters result from the same gene differences can be determined by genetic analysis. Measurement of genetic correlations is based on the fact that when two characters are influenced by a common physiological or biochemical process, related individuals should show greater "cross" resemblance among characters than do unrelated individuals. That is, close relatives of an individual with an "extreme" measure for one character (such as height) tend to be extreme not only for that character but also for the physiological basis of that character (such as long bone growth) and for other characters (such as weight) influenced by those functions.


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rm = correlation of measured values for two characters, designated "X" and "Y". ra = correlation of genetic values. re = correlation of environmental deviations influencing the characters. (In this development, all non-additive genetic sources of variance are included under the term "environmental deviations. ") COV = covariance V = variance h2= heritability e2 = 1 - h2

Recall that a correlation is calculated as:

rm=

_COy_(X,, Y)

(See #22)

Since a correlation between the measued values ofX and Y is the sum of the genetic and environmental correlations, the above expression can be written as: rm

= COV (Ax,Ay) + COV (Ex,Ey) Vmx. Vmiy

**#32**

Recall that: h2 = Va/Vm and e2 = Ve/Vm Vm

=

Va /h2

Vm =Ve/e2

thus: Vm= Va/h2

Vm

=

Ve/e2

Then, COV (AX,Ay) rm

+

ax ay h2x

h2Y

rm = hx hy ra + ex ey re

COV (EX,Ey) V

ex

e2X

V

ey

**#33**

e2y

#34

The genetic correlation (r(A)) between two measured variables expresses the extent to which individual differences in expression of the two variables depend on gene differences at the same loci. This correlation

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is due to manifold effects of genes summed over all loci influencing either trait. These gene effects have detectable consequences for association among traits when their simultaneous effects are directional or consistent (that is, when joint influences on the two are uniform over all loci rather than different in direction for genes at different loci). For example,when two characters are influenced by a common physiological or biochemical process (the case of height and weight responding to long bone growth), gene differences modifying that process (that is, long bone growth) will constitute genetic variance for each of the characters considered alone (that is, height or weight), and those same gene differences will constitute a genetic source of association between the traits when they are considered simultaneously (that is, height and weight). The magnitude of the genetic correlation in this situation will be quite high if gene differences for the common process are the only ones which influence either character but relatively low if those genes are but a few of many which are segregating with influence on either trait. The environmental correlations (r(E)) between characters are due to outside factors which influence some underlying physiological or biochemical system that influences both characters. In animal research, environmental correlations can be directly measured using inbred strains (since associations among characters on genetically invariant biological TABLE I: SOME EXAMPLES OF OBSERVED R(O), GENETIC R(A), AND ENVIRONMENTAL R(E)

CORRELATIONS [AFTER FALCONER21

r(O)

Drosophila melanogaster49 Bristle number - abdominal: sternopleural Mice5O Body weight: tail length pigs51 Body length: backfat thickness Growth rate: feed inefficiency Backfat thickness feed inefficiency Cattle52 Milk yield: butterfat yield Milk yield: butterfat % Butterfat yield: butterfat % SheeprFleece weight: length of wool Fleece weight: crimps per inch Fleece weight: body weight Poultry54 Body weight: egg production (at 18 weeks) (to 72 weeks of age) Body weight: egg weight (at 18 weeks) Body weight: age at first egg

r(A)

r(E)

.06

.08

.04

.44

.59

.34

-.24 -.84 .31

-.47 -.96 .28

-.01 -.50 .32

.93 -.14 .23

.85 - .20 .26

.96 - . 10 .22

.30 -.21 .36

-.02 -.56 -.11

1.17 .10 1.05

.09

-.16

.18

.16

.50

-.05

-.30

.29

-.50


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material must result from environmental differences with common influence on the two characters). In human research, environmental correlations are necessarily combined (since independent estimates cannot be obtained) with all sources of association other than additive genetic correlations and are determined by solution of an equation (#34) rather than by direct estimation. Genetic and environmental sources of covariance between two characters are not restricted in the mathematical sense, although evidence has been presented that they tend to agree in sign and roughly in magnitude.48 As a result of their mathematical independence, the genetic and environmental correlations can be and often are very different in magnitude and even in direction.495 Examples of observed, genetic and environmental correlations from field and laboratory research are presented in Table I. A difference in sign between the correlations implies (based on the reasoning above) that genetic and environmental sources of covariance affect of characters through different underlying mechanisms. 2 In view of the fact that the two components of the observed correlation of measured values can be of different signs, the magnitude and even the sign of the genetic correlation cannot be determined from the observed correlation of measured values. In order to investigate the practical application of genetic correlation in human genetics, the following study was undertaken from ophthalmologic and orthoptic clinical measures.

SUBJECTS, MEASURES AND CLINICAL PROCEDURES

Subjects consisted of families (parents and siblings) of children (propositi) randomly selected from the local elementary school population. A total of 163 families were examined in this study; this report includes data for 118 "complete" family units, that is, two or more offspring and both biological parents. Data from the propositi were excluded in all analyses. All individuals were examined in a university hospital eye clinic utilizing a standard battery of ophthalmic and orthoptic tests. Analysis of seven standard clinical measures obtained for each individual is presented here. The measures used are: spherical refractive error, corneal power (keratometry reading), cover test, divergence break and recovery points and convergence break and recovery points. These measures represent characteristics frequently assessed in clinical examinations. Detailed genetic analyses have been performed on each of these measures.

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ANALYTICAL PROCEDURES AND RATIONALE:

Since variances for both characters involved in any correlation can be partitioned into genetic and environmental components, these sources of covariance can be converted to two correlation coefficients contained in the correlation of measured values (#32-#34). In a way that exactly parallels the earlier discussion of components of genetic variance, the components of genetic covariance contained in the association between variables differs for individuals of different degrees of relationship. Thus, for example, the covariance of offspring and parents contains one-half the additive genetic variance for each of any two measured variables; the cross-covariance contains one-half the additive genetic covariance for the two. This extension in logic holds for relatives whose resemblance contains different fractions of additive genetic variance and even different sources of variance (such as dominance genetic variance contained in resemblance of full sibs). When dominance relationships of genes influencing the two traits in common contribute to their covariance, the association of the characters will be greater among full sibs than will be the association across characters observed in parents and offspring. Practically, however, genetic correlation estimates are limited to additive genetic correlations and, in a way analogous to genetic variance, these are the most useful in prediction. It should also be recalled that calculation of dominance genetic variance is difficult in human studies. Indeed, additive genetic covariances may be the only estimates of gene influence on associations between characters possible with the usual limits imposed in human genetic research. In this study, genetic influence on the observed associations was examined by estimation of genetic correlations among the clinical measures. For estimation, parent-offspring cross-covariance methods2 were employed with each pair of variables. These procedures estimate half the additive genetic covariance from the sum of products (#4) of one character (such as cover test measure) on parents and the second (such as corneal power) on offspring. Interchanging the variables for parents and offspring provides a second estimate of half the additive genetic covariance between measures which can be combined with the first to provide a "best" estimate in the statistical sense. Midparent values and mean offspring values provided this covariance estimate necessary to calculate a correlation (#22). The variance estimate for each variable was based on the variance of individual parent values (not midparent values)again, in order to obtain a statistical "best" estimate of variance.


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TABLE II: ESTIMATES + STANDARD ERRORS OF OBSERVED CORRELATIONS [R(O)] AND GENETIC CORRELATIONS [R(A)l

Sphere and corneal power Sphere and cover test measure Sphere and divergence recovery point Sphere and convergence recovery point Corneal power and cover test measure Corneal power and divergence recovery point Corneal power and convergence recovery point Cover test measure and divergence recovery point Cover test measure and convergence recovery point Divergence and convergence recovery points Divergence break and recovery points Convergence break and recovery points *p

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