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Blair and James are trying to start a family. Like many parents, they hope their future offspring will be healthy. They’d also like the baby to have blue eyes.

The couple, both 35, describe themselves as type-A personalities who research everything. When they decided to try for a baby, they looked into DNA testing to rule out disease-causing genetic mutations they might pass along to their child. Then they learned about a test that might help predict a future baby’s eye color.

Blue eyes, says James, who has brown eyes, “is icing on the cake.” (The couple asked not to reveal their last names to maintain their privacy.)

Many prospective parents already use DNA testing to check for potential genetic anomalies that could lead to serious medical conditions. But as technology advances, they may also learn about characteristics that have less bearing on a future child’s health, like eye color.

In the area of reproductive medicine, parents wield great discretion in making decisions about their future children. But the notion that parents might someday select embryos based on what some deem as aesthetic preferences—a future child who is a certain height or good at sports or looks a certain way—raises challenging ethical questions. Perhaps, some ethicists argue, DNA testing will create a society that further values certain types of children more than others.

Many in vitro fertilization clinics that once offered genetic testing of embryos to prevent sex-linked medical disorders now also allow prospective parents to select the gender of the embryo because of a personal preference.

Which Eyes Have It?

How different combinations of parents' eye colors predict the odds of the color of their baby's eyes.

... the likelihood of child’s eye color would be:

If the eye color of each parent is ...

Parent 1

Parent 2

Brown

Blue

Green

Brown

75.0%

18.8%

6.3%

Brown

Green

Brown

12.5%

37.5%

50.0%

50.0%

Blue

50.0%

Brown

0%

Green

25.0%

Less than 1%

Green

75.0%

50.0%

0%

50.0%

Blue

Green

1.0%

0%

99.0%

Blue

Blue

If the eye color of each parent is ...

... the likelihood of child’s eye color would be:

Parent 1

Parent 2

Green

Blue

Brown

Brown

18.8%

6.3%

Brown

75.0%

50.0%

12.5%

37.5%

Brown

Green

Brown

50.0%

50.0%

0%

Blue

75.0%

Less than 1%

25.0%

Green

Green

0%

50.0%

Green

50.0%

Blue

Blue

Blue

1.0%

0%

99.0%

... the likelihood of child’s eye color would be:

If the eye color of each parent is ...

Parent 2

Parent 1

Brown

Green

Blue

Brown

6.3%

18.8%

75.0%

Brown

12.5%

50.0%

Brown

37.5%

Green

0%

50.0%

50.0%

Blue

Brown

25.0%

Less than 1%

75.0%

Green

Green

Green

50.0%

0%

Blue

50.0%

Blue

99.0%

Blue

1.0%

0%

If the eye color of

each parent is ...

... the likelihood of child’s

eye color would be:

Parent 2

Parent 1

Green

Blue

Brown

Brown

18.8%

6.3%

75.0%

Brown

50.0%

Green

37.5%

12.5%

Brown

50.0%

Blue

Brown

50.0%

0%

Green

25.0%

Less

than 1%

Green

75.0%

50.0%

50.0%

Green

0%

Blue

Blue

99.0%

0%

Blue

1.0%

Note: Percentages don’t always add up to 100 due to rounding

Source: The Fertility Institutes

Eye color pushes the debate further. Like many human traits, it isn’t determined by a single gene, but a complex interaction of many genes. The test that Blair and James took emerged from work done by forensic scientists trying to predict eye, hair and skin color for unknown suspects in criminal cases for which minimal amounts of DNA is available. In published papers, these researchers determined that testing for six key DNA markers allowed them to predict if someone had brown or blue eyes with greater than 90% accuracy.

The scientific advances enabling predicting traits that involve multiple genes go beyond eye color. A company called Genomic Prediction received regulatory approval in New Jersey in September to market its Expanded Pre-Implantation Genomic Testing in many states. It will cost $400 per embryo. Genomic Prediction says it can accurately predict which embryos are at high risk for complex health conditions, like diabetes or cardiovascular disease.

Researchers at the company demonstrated how the approach could be used to predict height in a paper published this year in the journal Genetics. Someday, new techniques might allow predicting the likelihood of an embryo’s future academic potential.

A fertility doctor gets calls from people who want to know if it’s possible to select embryos with an aptitude for music or athletic ability.

In a blog post, Stephen Hsu, a founder of Genomic Prediction, posed an ethical question: An IVF doctor has two healthy, viable embryos and must choose which to implant. One has a hypothetical risk score that indicates the embryo is at high risk for struggling academically in school. The second embryo has a score indicating the future child likely won’t struggle. Do you tell the parents?

“It seems ethically not defensible to withhold the information from the parents,” he says, “and ethically defensible to reveal it to them.”

Some IVF doctors say it’s too soon to routinely offer people risk scores about their embryos. Mandy Katz-Jaffe, a reproductive geneticist and scientific director at CCRM, a Denver fertility clinic, says that outcomes are often a mixture of genetics and environment. Moreover, the data sets upon which the algorithms are based involve geographically and demographically narrow groups.

More on Genetic Testing

Nathan Treff, chief scientific officer of Genomic Prediction, says the company is only offering risk predictions involving disease and has no plans to predict an embryo’s eye color or level of educational attainment. “It is not always black and white what people consider a disease,” he says, “but we pay attention to what the community thinks is ethical.”

Jeffrey Steinberg, founder of the Encino, Calif.-based Fertility Institutes, believes his group is the only one offering the test Blair and James took. His team is working to develop the technology to test embryos for genetic markers related to eye color at the same time as genetic-disease screening. For now, the clinic only offers the eye-color test to some prospective parents. The institute charges $370.

Paula Amato, a fertility doctor at Oregon Health & Science University, and an ethicist, says the general view in the field is that genetic testing to prevent disease is ethically permissible. So is sex selection, although it is more controversial.

No one has inquired about eye color at Dr. Amato’s clinic. But thinking about sex selection has changed over time, and the same may happen with other traits, she says. Still, when it comes to eye color or other nonmedical traits, she says, “Not a lot of clinics are interested in getting into that business.”

Josephine Johnston is director of research at the Hastings Center, a Garrison, N.Y.-based bioethics research institute. She studies genetic testing in embryos. To her, selecting embryos based on traits like eye color “can seem awfully close to a eugenic mind-set, where we thought we can sort the worthy and fit from the unworthy and unfit.”

Parenting often comes with “the understandable desire to give your child advantages,” like height, or musical talent, she says. Yet people are part of a society that fights prejudice. “These kinds of decisions can feed into the discrimination, not fight against it,” she says.

While genetic testing of embryos is considered safe, there may be unexpected long-term effects. Many people feel uncomfortable about selecting embryos for aesthetic traits, worried about the difficulties of drawing a line about what should be left to chance. Dr. Steinberg, for one, says he already gets calls from people who want to know if it is possible to also select embryos with an aptitude for music or athletic ability. (He says he tells them not yet.)

One late September afternoon, Blair and James meet with Dr. Steinberg and his colleagues at the Ferny Clinic in New York City, where Dr. Steinberg also sees patients, for the results. “We’ve got some pretty good news for you,” Dr. Steinberg tells the couple. Based on the results of the testing, he says, “You absolutely can make a blue-eyed baby.” The doctors say that they estimate that in a group of five of their embryos, one is likely to have blue eyes.

For now, the couple plans to try to get pregnant the traditional way. “We will be thrilled to start our family,” Blair says, no matter the eye color.

When they told their parents and friends they were doing a DNA test to determine if they can have a blue-eyed baby, they got mixed responses. James’s father was fascinated. But Blair says that some family and friends thought using technology to learn about a baby’s eye color was a step too far.

She views things differently. “It’s screening to see what’s possible,” she says. Her husband agrees. Once you start looking at an embryo to rule out diseases, he says, what’s one more thing like eye color?

“You are there already,” he says.

Write to Amy Dockser Marcus at amy.marcus@wsj.com