human autosomal dominant diseases are extremely rare conditions in which affected individuals are heterozygotes . many of these heterozygous genotypes exhibit the phenomenon of incomplete penetrance . for this set of rare conditions the penetrance rate is therefore understood as the probability of a heterozygote presenting the disease ( or , at least , presenting a minimum number of signs and symptoms that enable his / her identification as a carrier of the deleterious allele ) . other details , as well as a full review of the subject can be found in horimoto and otto ( 2008 ) . accurate estimates of the penetrance value k are important not only for determining genetic disease risks in families with segregating cases of autosomal dominant disorders , but also for performing linkage studies . crude penetrance estimates can be derived by dividing the observed number of diseased ( penetrant ) individuals by the number of obligate carriers ( penetrant as well as obligate non - penetrant , that is , normal individuals with several affected offspring or normal individuals with affected parent and child ) . presently the penetrance parameter can be estimated on a routine basis by computer programs that perform segregation analysis or the estimation of linkage based on complex pedigree structures that can not be expressed in closed form , such as the classical s.a.g.e . ( 1986 ) provided a simple but efficient methodology for dealing with the problem , but neither their solution nor more complex alternatives , such as the above - mentioned computer programs , take into account many of the details we discuss here . these concern specificity and heterogeneity issues ( section 3 ) , penetrance rate estimates from families undergoing molecular testing ( section 4 ) , lack of information about the phenotype of the pedigree generator ( section 5 ) , genealogies containing grouped parent - offspring information ( section 6 ) , and ascertainment issues responsible for the inflation of k estimates ( section 7 ) . in section 1 we briefly review the method for estimating the penetrance rate from pedigree data , and in section 2 we make a digression on the determination of the exact credible interval for this estimate . more details on the method described below are found in the original paper by rogatko et al . the first step of the method consists in trimming or filtering the pedigree information , that is , replacing the original pedigree with one containing only individuals that are informative or relevant with respect to penetrance estimation . expressions like trimming and trimmed seem to be more appropriate , but we shall keep the nomenclature coined originally by rogatko et al . ( 1986 ) . as an example we will consider the hypothetical filtered pedigree shown in figure 1 , with several individuals affected by a rare autosomal dominant condition . the symbols marked with a point indicate obligate normal ( non - penetrant ) heterozygous carriers of the gene , and the darkened symbols represent affected ( penetrant ) heterozygotes . the filtered pedigree contains four affected individuals , four normal obligate carriers , three normal offspring of obligate carriers , and one tree of normal individuals descendants from an obligate carrier ( one normal female with two normal male offspring , shown at the leftmost position of the pedigree ) . letting k be the penetrance rate value , the probabilities associated with each of these four different structures are , respectively , k/2 , ( 1-k ) in the case of the pedigree generator , or ( 1-k)/2 in the case of the other three normal obligate carriers , ( 2-k)/2 , and { 1/2 + ( 1-k)/2 . [ ( 2-k)/2]}. the likelihood function , that is the probability of occurrence of the pedigree conditional to the observed structures occurring in it , is derived from the quantities associated with these structures . in the present case , by neglecting constant values unimportant in the maximization procedure that will follow , the likelihood function takes the form p = k(1-k ) ( 2-k)[4+(1-k)(2-k ) ] . by solving the equation dp / dk = 0 ( or , more conveniently , dl / dk = dlog(p)/dk = 4.log(k)+4.log(1-k)+3.log(2-k)+log[4+(1-k)(2-k ) ] = 0 ) , we obtain the maximum likelihood estimate of the penetrance value k , which for this family takes the value of 0.418 . heterozygosis probabilities and the corresponding risks for the offspring of all individuals of the filtered pedigree can then be determined without difficulties . obligate carriers ( known non - penetrant carriers and affected penetrant heterozygotes ) have genotype aa and the risk for their offspring is simply r1 = k/2 = 0.418/2 = 0.209 , or approximately 21% , for the above shown example . the probability of heterozygosis for normal individuals born to obligate carriers ( three of which occur in the family used as example ) is taken directly from the quantity ( 2-k)/2 = 1/2+(1-k)/2 as p(het ) = [ ( 1-k)/2]/ [ ( 2-k)/2 ] = ( 1-k)/(2-k ) = 0.582/1.582 = 0.368 . the probability of affected offspring for these individuals is then r2 = ( 1-k)/(2-k).k/2 = 0.368 x 0.209 = 0.077 , or approximately 8% . the heterozygosis probabilities for all three individuals of the single tree of normal individuals occurring in the worked pedigree can be obtained from the term { 1/2 + ( 1-k)/2.[(2-k)/2 ] } by applying simple bayesian reasoning , or by means of computer programs . in an earlier work ( horimoto et al . , 2010 ) we describe two self - contained computer programs that perform most calculations necessary to estimate the penetrance rate . these are the programs pencalc for windows and pencalc web , which can be obtained free of charge from the web page http://www.ib.usp.br/~otto/software.htm . both programs are described in detail in the above mentioned article as well as in the pdf - guide included in the zipped file of the program pencalc for windows . al . ( 1986 ) also used an exact credible interval associated with a given k estimate . this interval can be obtained by finding the area that corresponds to a given proportion ( v.g . , 95% ) of the total area under the graph of the likelihood function . mathematically , the problem is reduced to integrating the function y = f(k ) between two limits a and b with the same ordinate value [ f(k = a ) = f(k = b ) ] , so that a , b[f(k)dk]/ 0,1[f(k)dk ] = 0.95 , an operation which can be accomplished by simple computer programs using numerical integration techniques such as romberg s oscullatory method . the lower and upper limits of the exact 95% credible interval for the estimate k = 0.418 of the example above are 0.163 and 0.725 , respectively . this credible interval is so large that it might seem to be impractical in a clinical setting . the reason for this particular extreme range is that it was derived from the few data of the small family used as example . in practice , the ideal situation is one where several pedigrees of the same condition are available for analysis , and the pooled data are used to perform the calculations of the penetrance rate and of its 95% credible interval . for instance , from the analysis of 21 different published pedigrees on the autosomal dominant ectrodactylytibial hemimelia syndrome , penetrance estimates and their corresponding credible intervals varied from 0.191 ( 0.0440.574 ) to 0.750 ( 0.3290.973 ) , while the global ( pooled ) penetrance value estimate was 0.392 , with a 95% credible interval of 0.339 to 0.447 ( horimoto , 2009 ) . another point that merits discussion is whether the k value is specific for the family in which the disease segregates or for the condition itself , independently from the family . the non - penetrance of a genetic trait is assumed to represent the lack of its phenotypic manifestation exclusively or predominantly due to environmental factors ( murphy and chase , 1975 ; praxedes and otto , 2000 ) or random genetic and epigenetic processes linked to the disease locus . of course penetrance can also be affected by a number of events that include the epistatic action of modifying genes and even temporal modifications of diagnostic criteria . another complicating issue is that genetically heterogeneous conditions can be merged in the pooling process . nevertheless , since the statistical credible intervals of isolated pedigrees usually are large , pooled estimates of the parameter should be preferred , unless statistical tests disclose the existence of great amounts of heterogeneity among penetrance estimates from various pedigrees . in this section we discuss the comparison of estimates obtained from families without molecular testing as to those for which dna testing has been used for classifying non - penetrant heterozygotes and normal homozygotes . in the latter case , if molecular testing discloses all non - penetrant heterozygotes inside normal trees of individuals descending from obligate carriers , and if there are n1 affected ( penetrant ) individuals and n2 non - penetrant heterozygotes in the family , the likelihood function reduces to l = log(p ) = n1.log(k)+n2.log(1-k ) . the maximum likelihood estimate is then k = n1/(n1+n2 ) , with binomial sampling variance of var(k ) = k(1-k)/(n1+n2 ) . this would be an ideal situation in which , besides providing a better estimate of k , the corresponding 95% credible interval of the penetrance value thus evaluated will be much smaller than the one provided by the analysis of the family without dna testing . in some published pedigrees there is a lack of phenotypic information about the genealogy generator ( affected or non - affected ? ) . furthermore , the likelihood function p may not include the parameter k or ( 1-k ) corresponding to the genealogy generator . in order to evaluate whether the inclusion of the pedigree generator significantly alters this k estimate , it is not necessary to repeat the calculations for the two configurations possible ( penetrant or non - penetrant common ascendant ) , because the likelihood function p , derived without information on the pedigree generator , is correct , and thus can not be improved . in fact , if one wants to refer to the pedigree founder , one can say that she / he was affected with probability k and unaffected with probability ( 1-k ) . certain published pedigrees present grouped parent - offspring trees of normal individuals , without informing the corresponding offspring numbers of all individuals in a given sibship , as is the case of the pedigree with cases of the ectrodactylytibial hemimelia syndrome ( majewski et al . , 1985 ) shown in figure 2 . this tree of normal individuals represented by individuals ii.8 to ii.11 and iii.14 to iii.28 does not detail individual offspring numbers , and only the total number of 15 is given . incomplete pedigree information is a simple but interesting problem in combinatorial analysis that can be straightforwardly solved by means of the theory of difference operators . table 1 lists the numbers of possible genealogy structures for a case of incomplete parent - offspring information as a function of both parent and offspring numbers . therein , with four parents , the number of possible structures is given by y4(n ) = ( n+1 ) + ( n+1)n(n-1)/6 , where n is the total offspring number of the four parents . for n = 15 , the outcome is y4(15 ) = 816 of such structures . for each combination { i , j , l , m } of offspring number the likelihood function of the whole tree is : 
 k4(1k)2(2k)5{1/2+(1k)/2[(2k)/2]9}{1/2+(1k)/2.[(2k)/2]i }  { 1/2+(1k)/2.[(2k)/2]j }  { 1/2+(1k)/2.[(2k)/2]1 }  { 1/2+(1k)/2.[(2k)/2]m},where i , j , l , and m are the unknown numbers of children for each of the individuals ii-8 , ii-9 , ii-10 , and ii-11 , respectively . in a population of approximately stable size , the average offspring number per couple does not differ from two and it is known that the number of children per couple adequately fits a poisson distribution g(x ) = e2/x ! . if each possible configuration is weighed by its probability ( according to the poisson distribution for the number of children per couple ) this can be achieved as follows using the function : 
 p(i , j , l , m)=k4(1k)2(2k)5  { 1/2+(1k)/2[(2k)/2]9}{1/2+(1k)/2.[(2k)/2]i }  e22i / i !  { 1/2+(1k)/2.[(2k)/2]j}e22j / j!{1/2+(1k)/2[(2k)/2]1}e221/1!{1/2+(1k)/2[(2k)/2]m}e22m / m!,and considering all 816 possible configurations referred to above . for each configuration one can then obtain not only a kijlm estimate but also its exact 95% credible interval . estimates for the penetrance value k , as well as for its exact 95% lower and upper credible limits lk and uk , corresponding to the given tree structure can be straightforwardly obtained by averaging the estimates kijlm , as well as those for the lower and upper limits lkijlm and ukijlm , as : 
 k=(kijlm.e22i / i!.e22j / j!.e22l / l!.e22m / m!)/(e22i / i!.  e22j / j !  . the numerical procedure is herein detailed using as an example the simple hypothetical pedigree represented in figure 3 . table 2 shows the penetrance rate estimates for all possible configurations contained in the grouped tree of normal individuals of figure 3 . the final estimates for the penetrance rate and for the lower and upper limits of its 95% credible interval are 0.4377 , 0.1471 and 0.7813 , respectively . the authors are correct in their paper in stating that their approach gets around the sample space problem by using only the likelihood of the parameters , given the actual observations . yet if ascertainment is not included , that likelihood itself will not be correct . advanced computer programs that perform segregation analysis or estimate linkage , such as the classical s.a.g.e . ( s.a.g.e . , 2009 ) and linkage ( lathrop et al . , 1985 ) programs we referred to in the introduction section , do not apply any ascertainment bias to the penetrance rate they indirectly estimate . by using a very simple example we could show that the crude k estimates obtained from genealogies are actually inflated . figure 4 lists all possible trees of offspring size = 2 with a pedigree generator carrier of the pathologic gene ( affected in a , b and c , and non - penetrant heterozygous in d , e and f ) disclosed by an ( impossible ) ascertainment devoid of any bias . let now na , nb , nc , nd , ne , and nf be the numbers of structures a , b , c , d , e , and f observed in an ideal , large sample collected without any ascertainment bias . then , the corresponding likelihood function in logarithmic form would be : 
 l1=(3na+2nb+2nd+ne).ln(k)+(nd+ne+nf).ln(1k)+(nb+2nc+ne+2nf).ln(2k),from which the maximum likelihood estimate of k is obtained without difficulties . a careful collection of a large number of families with offspring number 2 and a tree - generator carrier of the gene would consist only of structures a , b , c and d. configuration e would not be included , as the only affected individual would be , with a large probability , the result of a new mutation ; and configuration f would never be ascertained , because it contains only normal individuals . the corresponding ( logarithmic ) likelihood expression would then be given by : 
 l2=(3na+2nb+nc+2nd).ln(k)+nd.ln(1k)+(nb+2nc).ln(2k),from which , as in the previous case , the maximum likelihood estimate can be easily obtained . let the actual ( unknown ) value of k be 0.8 ; then the probabilities associated with structures a , b , c , d , e , and f would take the values p(a ) = 0.128 , p(b ) = 0.384 , p(c ) = 0.288 , p(d ) = 0.032 , p(e ) = 0.096 , and p(f ) = 0.288 . in a sample of size 1000 we would therefore expect to find the sample numbers na = 128 , nb = 384 , nc = 288 , nd = 32 , ne = 96 , and nf = 72 . the unbiased estimate would then take the value k = 0.8 , as expected . in the case of an incompletely ascertained sample , it is not possible to obtain an exact solution in simple analytical form for the function k = f(k ) , where k is the biased maximum likelihood estimate and k the true one ( unknown , completely unbiased estimate of the penetrance value ) , but we can evaluate k estimates for any given fixed value of k by means of likelihood expression l2 . for any true value of k the biased estimate , k is an inflated value , as we could guess intuitively . using a program on nonlinear regression analysis , such as the nlreg software ( sherrod , 2000 ) , we can adjust the observed set of points to the generalized empirical function y = ax.e ( bronshtein and semendiaev , 1973 ) where y = k and x = k. we then estimated sets of pairs of values k and k varying offspring size from 2 to 10 , and in each case we obtained corresponding generalized empirical functions yi 
 = aix.e , where y = k and x = k. as in the case of the previous example . the functions corresponding to offspring sizes from 2 to 10 , all showing a perfect statistical fitting with the corresponding observed biased estimates , are plotted in figure 5 , where y stands for k and x for k. the graph also shows the function k = k that corresponds to the case of an offspring with infinite size . as expected , with sibship size increasing the difference between corresponding estimates k and k becomes negligible , mainly in relation to values of k in its usual range ( k > 0.8 ) . this is a result that certainly can be generalized for any homogeneous or heterogeneous set of pedigrees . since optimized k estimates are obtained from large filtered pedigrees , or from the pooling of many pedigrees , the ascertainment bias just discussed will only produce slightly inflated k estimates . in the case the actual values of k , as well as the total number of informative individuals ( penetrant , obligate non - penetrant and those belonging to normal trees descending from obligate carriers ) are both small , the k estimates will not be reliable , as shown in figure 5 . for offspring sizes of 10 ( total of 11 informative individuals ) or more , it is also easy to conclude that estimated values of k in the range of 0.5 or more are reliable and do not need to be corrected . in any case , estimate corrections can be performed by enumerating all possible filtered pedigrees corresponding to a given tree structure and comparing the estimated k values to the inferred actual ones , just as we did before using the very simple examples discussed above .
accurate estimates of the penetrance rate of autosomal dominant conditions are important , among other issues , for optimizing recurrence risks in genetic counseling . the present work on penetrance rate estimation from pedigree data considers the following situations : 1 ) estimation of the penetrance rate k ( brief review of the method ) ; 2 ) construction of exact credible intervals for k estimates ; 3 ) specificity and heterogeneity issues ; 4 ) penetrance rate estimates obtained through molecular testing of families ; 5 ) lack of information about the phenotype of the pedigree generator ; 6 ) genealogies containing grouped parent - offspring information ; 7 ) ascertainment issues responsible for the inflation of k estimates .