The correlation coefficient is definitely a relevant information and one that further reduces the chance that wife has it.
By itself, we don't have much information to treat it correctly; however we might make some educated assumptions. First of all, we need to understand why there is such a correlation. Probably, not every HD carrier transmits the disease at the same rate to their offspring. Some might be "weak" transmitters, while other "strongs". If you are a child of a strong transmitter you have a high probability of having it and vice-versa.
Imagine for instance that strongs are sure to transmit and weaks are sure not to transmit. If the proportion of strongs and weaks are 50/50, a random child of a HD carrier will still have a 50% probability to get HD, since the transmitter category is unknown. However, if both a strong and a weak transmitter have both 2 children, both strong's children will have it and both weak's won't have it. So the correlation. In this extreme scenario, correlation between siblings will be 1; if instead there is no strong or weak transmitter, you end up with a 0 correlation.
So we are somewhere in the middle. We make these hypotheses.
- There are two types of transmitter, strong and weak. Their proportion is 50/50.
- The strongs transmit HD with a p>0.5 probability to their offspring.
- The weaks transmit with 1-p probability.
As you can see, because of symmetry, the probability for a child of a HD carrier to have HD is still 50/50 if the "type" is unknown. Keep also in mind that of course the above is not the only way to include this information, but the simplest one.
It's easy to get that the correlation coefficient rho between siblings under the above is (2*p-1)^2. So, if rho=0.63, we have p = 0.8968627, where p is the probability that a strong transmitter transmits (1-p for a weak one).
So, we have these other inputs to our problem.
- Father has a 50/50 probability of being strong/weak if he has it.
- If father is strong, any child has a p= 0.8968627 of having HD.
- If father is weak, any child has a 1-p chance of having HD.
We end up with these results.
- Father is still about 8.15% of being a carrier.
- However, there is just a 0.47% chance that he is strong. With 7.68% he is weak and with 91.9% he is not a carrier.
- Wife's chance of having it drops to just 0.94%
The Netica file to check the results.
Code:
// ~->[DNET-1]->~
bnet tptHD_v2 {
AutoCompile = TRUE;
autoupdate = TRUE;
whenchanged = 1597752690;
visual V1 {
defdispform = BELIEFBARS;
nodelabeling = TITLE;
NodeMaxNumEntries = 50;
nodefont = font {shape= "Arial"; size= 9;};
linkfont = font {shape= "Arial"; size= 9;};
ShowLinkStrengths = 1;
windowposn = (22, 22, 1126, 482);
resolution = 72;
drawingbounds = (1412, 720);
showpagebreaks = FALSE;
usegrid = TRUE;
gridspace = (6, 6);
NodeSet Node {BuiltIn = 1; Color = 0x00e1e1e1;};
NodeSet Nature {BuiltIn = 1; Color = 0x00f8eed2;};
NodeSet Deterministic {BuiltIn = 1; Color = 0x00d3caa6;};
NodeSet Finding {BuiltIn = 1; Color = 0x00c8c8c8;};
NodeSet Constant {BuiltIn = 1; Color = 0x00ffffff;};
NodeSet ConstantValue {BuiltIn = 1; Color = 0x00ffffb4;};
NodeSet Utility {BuiltIn = 1; Color = 0x00ffbdbd;};
NodeSet Decision {BuiltIn = 1; Color = 0x00dee8ff;};
NodeSet Documentation {BuiltIn = 1; Color = 0x00f0fafa;};
NodeSet Title {BuiltIn = 1; Color = 0x00ffffff;};
PrinterSetting A {
margins = (1270, 1270, 1270, 1270);
};
};
node Dad {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = ();
probs =
// yes no
(0.5, 0.5);
whenchanged = 1597750747;
belief = (0.08146863, 0.9185314);
visual V1 {
center = (492, 66);
height = 8;
};
};
node TransmitterType {
discrete = TRUE;
states = (strong, weak, no);
kind = NATURE;
chance = CHANCE;
parents = (Dad);
probs =
// strong weak no // Dad
(0.5, 0.5, 0, // yes
0, 0, 1); // no ;
whenchanged = 1597750591;
belief = (0.004655889, 0.07681274, 0.9185314);
visual V1 {
center = (492, 168);
height = 1;
};
};
node Brother1 {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (TransmitterType);
probs =
// yes no // TransmitterType
(0.8968627, 0.1031373, // strong
0.1031373, 0.8968627, // weak
0, 1); // no ;
whenchanged = 1597750770;
evidence = no;
belief = (0, 1);
visual V1 {
center = (288, 300);
height = 3;
};
};
node Wife {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (TransmitterType);
probs =
// yes no // TransmitterType
(0.8968627, 0.1031373, // strong
0.1031373, 0.8968627, // weak
0, 1); // no ;
whenchanged = 1597750696;
belief = (0.009440463, 0.9905595);
visual V1 {
center = (492, 294);
height = 5;
};
};
node Brother2 {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (TransmitterType);
probs =
// yes no // TransmitterType
(0.8968627, 0.1031373, // strong
0.1031373, 0.8968627, // weak
0, 1); // no ;
whenchanged = 1597750717;
belief = (0.009440463, 0.9905595);
visual V1 {
center = (774, 294);
height = 2;
};
};
node Symptoms_55 {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (Dad);
probs =
// yes no // Dad
(0.8, 0.2, // yes
0, 1); // no ;
numcases = 1;
whenchanged = 1597750771;
evidence = no;
belief = (0, 1);
visual V1 {
center = (762, 66);
height = 7;
};
};
node Symptoms_34_w {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (Wife);
probs =
// yes no // Wife
(0.3333333, 0.6666667, // yes
0, 1); // no ;
whenchanged = 1597752690;
evidence = no;
belief = (0, 1);
visual V1 {
center = (492, 402);
height = 4;
};
};
node Symptoms_34 {
discrete = TRUE;
states = (yes, no);
kind = NATURE;
chance = CHANCE;
parents = (Brother2);
probs =
// yes no // Brother2
(0.3333333, 0.6666667, // yes
0, 1); // no ;
whenchanged = 1597752674;
evidence = no;
belief = (0, 1);
visual V1 {
center = (774, 396);
height = 6;
};
};
ElimOrder = (Symptoms_55, Symptoms_34_w, Symptoms_34, Dad, Brother1, Wife, TransmitterType, Brother2);
};
Last edited by nickthegeek; 08-18-2020 at 08:13 AM.
