Morten Valberg

Estimating familial risks of disease: How can
frailty models be useful?
One-Day seminar, Oslo
Morten Valberg
Oslo Center for Biostatistics and Epidemiology, Department of Biostatistics,
University of Oslo
January 29, 2015
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Outline
Familial relative risk.
Frailty approach -gain?
Hierarchical frailty models.
Testicular germ cell tumors.
Familial relative risk.
Results and conclusion.
Further implications?
Summary.
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Familial association
A familial relative risk is often defined as
The risk of developing disease given that at least one
first degree relative has developed disease, compared
with the general risk level.
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Familial association
A sibling relative risk is often defined as
The risk of developing disease given that at least one
sibling has developed disease, compared with the
general risk level.
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Familial association
A sibling relative risk is often defined as
The risk of developing disease given that at least one
sibling has developed disease, compared with the
general risk level.
In the Nordic countries..
Detailed registry data are available.
Incidence ratios (IRs).
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Familial association
Calculating sibling IRs by..
Choosing an index case in each family.
Calculate incidence rate in family members.
Compare to the incidence in the general population.
But..
When are family members at risk?
From birth?
From diagnosis of index case?
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Familial association -Common approach
Calculating sibling IRs by..
Pooling all siblings with at least one affected sibling in to one
big cohort.
Compare to the incidence in the general population.
But..
Counting affected siblings more than once.
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Some examples
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Familial association
Estimates of sibling association are often summary measures
Averaging over families of different sizes consisting of
individuals with different ’times at risk’.
Additional diseased family members?
Healthy family members?
Timing of disease?
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Can we do better?
Our aim
To construct a model that takes the inheritance structure, and
size, of a family into account.
To estimate familial relative risks given any combination of
family members.
Account for additional healthy/diseased family members.
To account for the timing of the disease.
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Paper
Am J Epidemiol. 2014; 179(4):499-506
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Testicular germ-cell tumors (TGCTs)
Facts
> 98% of adult testicular cancers.
The commonest cancer in Norwegian males aged 15-49 years.
Few established risk factors.
An increased risk in sons and brothers of affected individuals
have been observed.
Data
All Norwegian families registered by Statistics Norway since
1960.
TGCT status from the Cancer Registry of Norway.
1,135,320 families included
7,524 families contained at least one TGCT case.
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The Armitage–Doll model
The Armitage–Doll model of carcinogenesis...
Suggest that the individual risk pattern should be increasing
with some power of age.
Armitage and Doll, 1954: ”The relatively high rates at the
younger ages could result if the population contained a group
of subjects specially susceptible to cancer(...)”
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The Armitage–Doll model with random frailty
Define the individual hazard rate as the frailty random variable,
Z , multiplied with some specific basic hazard rate λ(t), where t
is age.
λ(t) is shared by all individuals, and increasing with some
power of age.
Z follows a frailty distribution.
The population hazard rate (and the population incidence
rate) is the net result of the individual hazard rates.
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0
Hazard rate
Basic hazard rate
0
Age
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0
Hazard rate
Individual hazard rate
0
Age
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0
Hazard rate
Individual and population hazard rate
0
Age
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Hierarchical frailty model
Consider now..
A variable Z1 that represent frailty that varies between
individuals.
Introduce a variable Z2 that represent frailty that varies
between families.
By randomizing a parameter in the distribution of Z1 .
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Hierarchical frailty model
Consider now..
A variable Z1 that represent frailty that varies between
individuals.
Introduce a variable Z2 that represent frailty that varies
between families.
By randomizing a parameter in the distribution of Z1 .
Z2 may be decomposed additively.
To account for the correlation structure in a family.
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Hierarchical frailty model cont.
The second level frailty is decomposed to account for the
inheritance structure within a family. For a father (F) and a
mother (M) we have
X
X
Z2F =
Fi and Z2M =
Mi .
Each son inherits half of his genes from his father and mother.
Each son shares half his genes with his brothers, but nobody
shares the same half of their parents’ genes.
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Paternal inheritance
Father:
F1
F2
F3
F4
F5
F6
F7
F8
Son 1:
F1
F2
F3
F4
F5
F6
F7
F8
Son 2:
F1
F2
F3
F4
Son 3:
F3
F4
F5
F6
Son 4:
F3
F4
F5
F6
F5
F6
Son 5:
F9
F10 F11 F12 F13 F14 F15 F16
F9
F10 F11 F12
F9
F10
F13 F14
F11 F12
F7
F8
F9
F15 F16
F10 F11 F12
Figure: Paternal inheritance structure. The maternal inheritance
structure is given by replacing F s with Ms.
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Frailty relative risk
Consider two individuals from the same family, A and B.
The frailty relative risk is defined as
FRR =
P(A gets cancer within age tA |B gets cancer within age tB )
P(A gets cancer within age tA )
May be expanded to account for any familial history of TGCT.
Expressed in terms of the survival functions.
Easy to calculate for any combination of individuals once we
have parameter estimates.
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Some results
Table: The lifetime frailty relative risk (FRR) with 95% CIs for an
individual, A, developing TGCT , given that up to four of his brothers (B,
C, D and E) develop TGCT or not.
Affected
brother(s)
A
A
A
A
A
B
B
B, C
B, C
Unaffected
brother(s)
B, C, D, E
C, D, E
D, E
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TGCT
FRR
0.93
5.88
5.07
21.71
15.80
Familial risk
95% CI
0.92, 0.95
4.70, 7.36
4.11, 6.27
8.93 52.76
9.56, 26.11
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Some results
Table: The lifetime frailty relative risk (FRR) with 95% CIs for an
individual, A, developing TGCT , given that up to four of his brothers (B,
C, D and E) develop TGCT or not.
Affected
brother(s)
A
A
A
A
A
B
B
B, C
B, C
Unaffected
brother(s)
B, C, D, E
C, D, E
D, E
Morten Valberg
TGCT
FRR
0.93
5.88
5.07
21.71
15.80
Familial risk
95% CI
0.92, 0.95
4.70, 7.36
4.11, 6.27
8.93 52.76
9.56, 26.11
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Some results
Table: The lifetime frailty relative risk (FRR) with 95% CIs for an
individual, A, developing TGCT , given that up to four of his brothers (B,
C, D and E) develop TGCT or not.
Affected
brother(s)
A
A
A
A
A
B
B
B, C
B, C
Unaffected
brother(s)
B, C, D, E
C, D, E
D, E
Morten Valberg
TGCT
FRR
0.93
5.88
5.07
21.71
15.80
Familial risk
95% CI
0.92, 0.95
4.70, 7.36
4.11, 6.27
8.93 52.76
9.56, 26.11
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Some results
Table: The lifetime frailty relative risk (FRR) with 95% CIs for an
individual, A, developing TGCT , given that up to four of his brothers (B,
C, D and E) develop TGCT or not.
Affected
brother(s)
A
A
A
A
A
B
B
B, C
B, C
Unaffected
brother(s)
B, C, D, E
C, D, E
D, E
Morten Valberg
TGCT
FRR
0.93
5.88
5.07
21.71
15.80
Familial risk
95% CI
0.92, 0.95
4.70, 7.36
4.11, 6.27
8.93 52.76
9.56, 26.11
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Some results
Table: The lifetime frailty relative risk (FRR) with 95% CIs for an
individual, A, developing TGCT , given that up to four of his brothers (B,
C, D and E) develop TGCT or not.
Affected
brother(s)
A
A
A
A
A
B
B
B, C
B, C
Unaffected
brother(s)
B, C, D, E
C, D, E
D, E
Morten Valberg
TGCT
FRR
0.93
5.88
5.07
21.71
15.80
Familial risk
95% CI
0.92, 0.95
4.70, 7.36
4.11, 6.27
8.93 52.76
9.56, 26.11
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Conclusions
This approach..
Considers the whole family simultaneously.
Gives a very detailed picture of familial risks of disease.
Allows for counseling of patients and their families according
to their specific individual histories.
Enables us to estimate sibling TGCT risks not previously
reported.
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Challenges
Maximum likelihood
Time consuming.
Parallel computing.
Calculate the likelihood contribution for each family in parallel.
Using 200 CPU cores, it took 2.5h to fit the model.
More than five sons..
would need more terms in the additive component.
would be more computationally demanding.
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Further implications?
A familial RR...
Compares the risk in individuals with a familial disease history
to the average risk level in the population.
Do not compare high risk families to low risk families.
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Further implications?
A familial RR...
Compares the risk in individuals with a familial disease history
to the average risk level in the population.
Do not compare high risk families to low risk families.
Moderate familial RRs may hide large variation in risk between
families (and individuals)
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Example
Population contains two risk groups (50% in each)
All members of a family belongs to the same group.
P(gene)= 0.5
P(man sick|gene) = 0.2,
P(man sick|no gene) = 0.01.
Individual relative risk is IRR= 20.
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Example
Population contains two risk groups (50% in each)
All members of a family belongs to the same group.
P(gene)= 0.5
P(man sick|gene) = 0.2,
P(man sick|no gene) = 0.01.
Individual relative risk is IRR= 20.
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (50% in each)
All members of a family belongs to the same group.
P(gene)= 0.5
P(man sick|gene) = 0.2,
P(man sick|no gene) = 0.01.
Individual relative risk is IRR= 20.
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=0.2 · 0.5 + 0.01 · 0.5 = 0.105 = P(brother sick)
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (50% in each)
All members of a family belongs to the same group.
P(gene)= 0.5
P(man sick|gene) = 0.2,
P(man sick|no gene) = 0.01.
Individual relative risk is IRR= 20.
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=0.2 · 0.5 + 0.01 · 0.5 = 0.105 = P(brother sick)
P(man and brother sick) =P(man and brother sick|gene)P(gene)
+ P(man and brother sick|no gene)P(no gene)
2
2
=P(man sick|gene) P(gene) + P(man sick|no gene) P(no gene) =
2
2
=0.2 · 0.5 + 0.01 · 0.5 = 0.02005
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (50% in each)
All members of a family belongs to the same group.
P(gene)= 0.5
P(man sick|gene) = 0.2,
P(man sick|no gene) = 0.01.
Individual relative risk is IRR= 20.
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=0.2 · 0.5 + 0.01 · 0.5 = 0.105 = P(brother sick)
P(man and brother sick) =P(man and brother sick|gene)P(gene)
+ P(man and brother sick|no gene)P(no gene)
2
2
=P(man sick|gene) P(gene) + P(man sick|no gene) P(no gene) =
2
2
=0.2 · 0.5 + 0.01 · 0.5 = 0.02005
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
=
0.02005
0.1052
= 1.82
Observed familial relative risk is 1.82!
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Example
Population contains two risk groups (50% at high risk)
All members of a family belongs to the same group.
P(gene)=0.5
P(man sick|gene) =0.2,
P(man sick|no gene) =0.01.
Individual relative risk is IRR= 20.
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Example
Population contains two risk groups (q · 100% at high risk)
All members of a family belongs to the same group.
P(gene)=q
P(man sick|gene) =p1 ,
P(man sick|no gene) =p2 .
Individual relative risk is IRR= p1 /p2 .
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (q · 100% at high risk)
All members of a family belongs to the same group.
P(gene)=q
P(man sick|gene) =p1 ,
P(man sick|no gene) =p2 .
Individual relative risk is IRR= p1 /p2 .
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=p1 · q + p2 · (1 − q) = P(brother sick)
FRR =
P(man sick|brother sick)
P(man sick)
=
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (q · 100% at high risk)
All members of a family belongs to the same group.
P(gene)=q
P(man sick|gene) =p1 ,
P(man sick|no gene) =p2 .
Individual relative risk is IRR= p1 /p2 .
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=p1 · q + p2 · (1 − q) = P(brother sick)
P(man and brother sick) =P(man and brother sick|gene)P(gene)
+ P(man and brother sick|no gene)P(no gene)
2
2
=P(man sick|gene) P(gene) + P(man sick|no gene) P(no gene) =
2
=p1
FRR =
P(man sick|brother sick)
P(man sick)
=
·q+
2
p2
· (1 − q)
P(man and brother sick)
P(man sick)P(brother sick)
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Example
Population contains two risk groups (q · 100% at high risk)
All members of a family belongs to the same group.
P(gene)=q
P(man sick|gene) =p1 ,
P(man sick|no gene) =p2 .
Individual relative risk is IRR= p1 /p2 .
P(man sick) =P(man sick|gene)P(gene) + P(man sick|no gene)P(no gene)
=p1 · q + p2 · (1 − q) = P(brother sick)
P(man and brother sick) =P(man and brother sick|gene)P(gene)
+ P(man and brother sick|no gene)P(no gene)
2
2
=P(man sick|gene) P(gene) + P(man sick|no gene) P(no gene) =
2
=p1
FRR =
P(man sick|brother sick)
P(man sick)
=
·q+
2
p2
· (1 − q)
P(man and brother sick)
P(man sick)P(brother sick)
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Familial risk
=
IRR2 · q + 1 − q
(IRR · q + 1 − q)2
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Example
1.4
1.2
1.0
FRR
1.6
1.8
q=0.5
1
10
20
30
40
IRR
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Example
1
2
4
FRR
6
8
q=0.01
1
10
20
30
40
IRR
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Example
28
q=0.01
1
4
8
12
FRR
16
20
24
Given one diseased brother
Given two diseased brothers
1
10
20
30
40
IRR
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Example
q=0.5
1.6
1.4
1.2
1.0
FRR
1.8
Given one diseased brother
Given two diseased brothers
1
10
20
30
40
IRR
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Further implications?
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Summary
Frailty models
Assumes a varying degree of susceptibility (due to
unobserved/unknown reasons) between individuals and/or
clusters.
Are useful for taking into account dependencies between
individuals.
Provides a framework for giving a detailed picture of familial
disease risk.
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Thank you!
Thank You!
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