What is penetrance and why is a family study needed to estimate it? In this statistical genetics article, the authors discuss the issue of not knowing the cancer penetrance of a certain gene variant. They say this is because families with disease history are more likely to be screened frequently and hence diagnosed more, contributing to higher incidence. I gather that penetrance, therefore, is some measure of the risk ratio for this genetic variant controlled for correlated data/confounding (family history). But I am not sure. 
Is it possible to explain penetrance in statistical terms? What is the rationale for using a family based study to estimate penetrance as in this article?
 A: Quoting from Wikipedia:

Penetrance in genetics is the proportion of individuals carrying a particular variant (or allele) of a gene (the genotype) that also express an associated trait (the phenotype). In medical genetics, the penetrance of a disease-causing mutation is the proportion of individuals with the mutation who exhibit clinical symptoms. For example, if a mutation in the gene responsible for a particular autosomal dominant disorder has 95% penetrance, then 95% of those with the mutation will develop the disease, while 5% will not.

So penetrance is a bit more complicated than whether the gene product of the allele in question is expressed. It's how frequently those with the variant show a particular trait.* Any trait, such as development of some type of cancer, is typically determined by combinations of many different genes and environmental conditions. So it's the combination of the particular allele in question with the rest of an individual's genetic background and the environment that determines the observed penetrance of an allele with respect to a trait.
Families with high incidence of a particular type of cancer have provided important clues about genes associated with cancer development. Examples are Lynch syndrome, Li-Fraumeni syndrome, and ovarian and breast cancer among Ashkenazic Jews having BRCA1 mutations. Such families have been studied intensively to see what gene is most directly associated with cancer development, and such genes have been found to have fundamental importance in regulating the integrity of the genome as cells divide.
The members of those families also, however, necessarily share a large portion of the rest of their genetic background and often share environmental characteristics. So if you examine such families and find that 80% of those with a particular autosomal dominant allele develop a particular cancer (80% penetrance), that holds for their shared genetic and environmental backgrounds. In a broader population with different genetic and environmental backgrounds, the penetrance of that allele with respect to that cancer may well be less.
That was the point of the linked paper about penetrance of variants of the CDH1 gene with respect to development of gastric cancer. The initial focus on families with a high tendency to develop a certain cancer can lead to a sampling bias with respect to the broader population.

*Reported penetrance values are not "controlled for correlated data/confounding (family history)." Even putting aside environmental issues, with about 20,000 protein-coding genes having several potential variant alleles, many genes expressing multiple different gene products, about 99% of the genome not directly coding for protein products, and complete genome sequences only available for an extraordinarily small proportion of any human population, it's hard to see how you could control for such confounders.
This discussion also ignores the issue of time to development of the trait and whether penetrance might sometimes be better examined with time-to-event analysis rather than simple measurement of proportions. 
