# How likely is it that a person's progeny has an inheritable disorder based on this pedigree?

## Case study:

A person (let's just call him "John") had 3 aunts that died back in the 1950's in early childhood due to an unknown form of neuromuscular disorder. One died at birth, one at 2 years and one at 3 years. John's father was the only surviving descendent of John's grandparents. John and his brother do not show any symptoms. John's brother has a daughter who shows no symptoms. John has two sons, one does not show any symptoms and his other son is a 4 month old who is exhibiting signs of some sort of neuromuscular problem.

John has provided me with a pedigree that he believes is correct (note, the ones with the blue square are the ones that exhibit the symptoms): I took a bit of genetics in university, but basically I am just trying to understand the genetics here. These are my assumptions. Please correct me if any are wrong:

• Since the disorder appears to prevent developing past childhood it must be autosomal recessive because everyone who exhibits symptoms is not be able to reproduce due to their early death.
• Both of John's grandparents must have been carriers.
• The probability of John's dad and any aunts and uncles having the disorder is 25%, and the probability of being a carrier is 50%. The 3 aunts who died must have just lucked out.
• John, his dad, and his wife must all be carriers of the disorder.

However I really feel like I'm wrong about these assumptions since the math really doesn't add up. The likelihood that so many carriers for a relatively rare disorder all coming together and the fact that 3 of the 4 offspring of the grandparents died with only a 25% chance all seems improbable.

It could be that John's son doesn't have the disorder and what he has is something completely different. What do the geneticists say?

• Although your question is actually about genetics (and not personal medical advice), I would still recommend that you remove personal references and make the question as general as possible. Jun 9, 2016 at 19:14
• @WYSIWYG Why exactly would you recommend removing personal references?
– Mike
Jun 9, 2016 at 19:31
• Things are often not as clearcut as simple mendelian inheritance suggests, and you might be interested in this wiki article on penetrance. It includes a couple of concepts you might need to consider when trying to reconstruct the pedigree. Jun 9, 2016 at 20:01
• @MattDMo I did not realize that it was against the rules. I have removed any references.
– Mike
Jun 9, 2016 at 21:16
• The probability of carriers for a genetic disorder mating is higher than you think. Without knowing the specific disorder you're asking about, we can't look up prevalence rates, but as an example, about 1 in 30 caucasians carry an allele for cystic fibrosis. So any two caucasians have about a 0.1% chance of both being carriers. Jun 10, 2016 at 1:49

This is a very rough draft about the case, not verified, and should not be used for any medical conditions. Ask your own doctor. It is just for demonstrating some Mathematics and genetic passing generally.

Condition: Muscular atrophy
Differential conditions: TODO
Support: history of genetic passing in -1 and -3 generations

## Disease

Muscular atrophy as a condition jumps over a generation. The environment/living of your greatgreatgrandfathers determine your grandfathers future, as a rough estimate. The disease's common feature is recessive autosomality. Assumptions

• Autosomal recessive disease because everyone who exhibits symptoms is not be able to reproduce due to their early death.
• TODO genetic passing
• What is the genetic profile? Is there any other autosomal recessive diseases in the gene map?

Proposed probabilities where the random variables $X = disease$ and $Y = carrier$ such that the subscript 0, -1, -2 and -3 is about new children's, John's, parents' and grandparents' status, respectively. Both of John's grandparents must have been carriers \begin{equation} P(Y_{-3}) = 100\% \end{equation}

The probability of John's dad and any aunts and uncles having the disorder is (TODO think this again with the correct genetic passing) \begin{equation} P(X_{-2}) = 25\% \end{equation}

The probability of being a carrier is for dad, aunts and uncles (TODO think this gain with the appropriate genetic passing) \begin{equation} P(Y_{-1}) = 50\% \end{equation}

## Disease as a process

• $X \sim P_{i}(\mu, \sigma)$, $i = -3, -1, 1, ...$ where I propose the mean $\mu$ and the variance $\sigma^2$.

To understand this better, we should know how the generation $-5$ lived. Can you get it?

Assumptions

• Assume the events of individual probabilities independent i.e. (-3) events do not affect (-1) events in the series.
• ...

Test if the two independent statistics $X_{-3}$ and $X_{-1}$ are from the same population. If you assume it, put it to assumptions.

## Probability Density Functions in Generations -3, -1, 1, ...

Probability density function

• At $X_{-3}$, the 3 aunts of 4 died. The mean $m = 0.75$ (=3/4). You cannot use Binomial because sampling is not with replacement, not Poisson because occurrence probability is not less than 10%, ...
• ...

Conditional probability and Bayes theorem

• Consider the conditional probability $P(X | Y)$ that is what is the probability of getting the disease for the generation when given the carriers ...
• ...

You propose 25% probability for generations series $-3, -1, 1, ...$. You can test this. Select the statistical significance $\alpha = 0.05\%$. Select the appropriate probability density function or teach neural network to it.

## False assumptions

• John, his dad, and his wife must all be carriers of the disorder. Why wife should be a carrier?

## Literature

1. Blythe, Peter. Mathematics for the international student Mathematics HL (Options). Haesa & Harris Publications. I recommend this for beginners.
2. TI-84 and TI-86 have good manuals about the statistics.
3. Some publications about teaching a neural network doing some of the genetic analysis.
4. ...
• About "why wife should be a carrier?", it seems clear that "Me"'s son must have inherited the disease gen from both parent's. If he hadn't, he wouldn't show the symptoms. Am I missing anything?
– Pere
Jun 16, 2018 at 21:36