within siblings analysis: fixed effects models · 4. fixed e ects models: summary, merits and...
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1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Within Siblings Analysis: Fixed Effects Models
Department of EpidemiologyMiguel Angel Luque-Fernandez
May 2, 2014
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Outline1 1. Variance-components models to account for within-cluster correlations
Introduction2 2. Analytical Methods for 2k paired study designs
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
3 3. Analytical methods for more than two-paired repeated measuresThe model for more than 2k siblings (3, 4, ..., n)
4 4. Fixed Effects Models: Summary, Merits and LimitationsFEMS summaryFEMs MeritsFEMs limitations
5 5. 2k paired design: Generation-R within-siblings birth weight differencesAn example of 2k paired design
6 6. ≥ 2k matched design: NCCP within-siblings placental weight differencesNational Collaborative Perinatal Project (NCPP) USHypothesisObjectivesFixed effects modeling justificationSample selectionSome Results
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Introduction
Variance-components models
Introduction
Variance-component models (VCMs) are designed to model and estimatesuch within-cluster correlations in the context of longitudinal andcorrelated data.
Mother1 Mother2 Mother3 Motherj
Siblingij sibling33 sibling32 sibling31 sibling22 sibling21 sibling11
The effect of mothers can be interpreted as being random or fixed.
In this presentation, we assume that the effect of mothers is fixed (i.e.,it remains constant across pregnancies).
Thus, all the genetic and environmental factors associated with mothersremain constant over time and each mother serves as her own control.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Introduction
Variance-components models
Introduction
Variance-component models (VCMs) are designed to model and estimatesuch within-cluster correlations in the context of longitudinal andcorrelated data.
Mother1 Mother2 Mother3 Motherj
Siblingij sibling33 sibling32 sibling31 sibling22 sibling21 sibling11
The effect of mothers can be interpreted as being random or fixed.
In this presentation, we assume that the effect of mothers is fixed (i.e.,it remains constant across pregnancies).
Thus, all the genetic and environmental factors associated with mothersremain constant over time and each mother serves as her own control.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Introduction
Variance-components models
Introduction
Variance-component models (VCMs) are designed to model and estimatesuch within-cluster correlations in the context of longitudinal andcorrelated data.
Mother1 Mother2 Mother3 Motherj
Siblingij sibling33 sibling32 sibling31 sibling22 sibling21 sibling11
The effect of mothers can be interpreted as being random or fixed.
In this presentation, we assume that the effect of mothers is fixed (i.e.,it remains constant across pregnancies).
Thus, all the genetic and environmental factors associated with mothersremain constant over time and each mother serves as her own control.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Introduction
Variance-components models
Introduction
Variance-component models (VCMs) are designed to model and estimatesuch within-cluster correlations in the context of longitudinal andcorrelated data.
Mother1 Mother2 Mother3 Motherj
Siblingij sibling33 sibling32 sibling31 sibling22 sibling21 sibling11
The effect of mothers can be interpreted as being random or fixed.
In this presentation, we assume that the effect of mothers is fixed (i.e.,it remains constant across pregnancies).
Thus, all the genetic and environmental factors associated with mothersremain constant over time and each mother serves as her own control.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
Introduction
The simplest case: 2k (k = sibling) pairs of repeated measures
1
1
1
1 1
1
1
1
1
1
2 4 6 8 10
2500
3000
3500
4000
Mothers(i)
Birt
hwei
ght i
n gr
ams
2
2
22
22
2
2
2
2
2
1
Second
First
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
The simplest paired design: N Pairs of Siblings
The Model
Yij = α + βXij + Ui + eij
Where:
Mothers i=1,...,N
Siblings j=0,1 (0=First and 1=Second),
Yij : Birth weight of the infant of mother i when delivering the jinfant,
Xij=1 for second infant and 0 for the first, (=j)
β: The expected change in birth weight,
α: The (population) expected response (Y birth-weight),
Ui : systematic differences between mothers. Ui ∼ N(0, σ2u)
eij : within-siblings error of jth order for the ith mother eij ∼ N(0, σ2e )
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
Yij = α + βXij + Ui + eij Cov(ei1; ei0)=0
Ui
ei1
α
β
ei2
β
Uk
ek2
ek1
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
VCMs: Fixed Effect Model
The Model
Yij = α + βXij + Ui + eij
Within-mothers differences (note that we can also use between mothersdummies variables):
Yi1 − Yi0 = (α+ β + Ui + ei1) − (α+ Ui + ei0) = β + (ei1 − ei0)
E [Yi1 − Yi0] = β + E [ei1 − ei0] = β
Var(Yi1 − Yi0) = Var(ei1 − ei0) = 2σ2e
Estimates:
β =∑N
i=1(Yi1 − Yi0)/N = Y1 − Y0
Var(β) =2σ2
e
N
σ2e only involves within-mothers variation (FIXED EFFECT MODEL)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
Within-sibling correlations: Intra cluster correlation (ICC)
The Model
Yij = α + βXij + Ui + eij
Within-siblings correlation (RHO, ρ): ”Intra Cluster Correlation”
ρ = Corr(Yi1,Yi0) = Corr(Ui + ei1,Ui + ei0)
ICC
ρ =σ2u
σ2u + σ2
e
; Therefore σ2e = σ2
T (1 − ρ)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
Within-sibling correlations: Intra cluster correlation (ICC)
The Model
Yij = α + βXij + Ui + eij
Within-siblings correlation (RHO, ρ): ”Intra Cluster Correlation”
ρ = Corr(Yi1,Yi0) = Corr(Ui + ei1,Ui + ei0)
ICC
ρ =σ2u
σ2u + σ2
e
; Therefore σ2e = σ2
T (1 − ρ)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
Within-sibling correlations: Intra cluster correlation (ICC)
The Model
Yij = α + βXij + Ui + eij
Within-siblings correlation (RHO, ρ): ”Intra Cluster Correlation”
ρ = Corr(Yi1,Yi0) = Corr(Ui + ei1,Ui + ei0)
ICC
ρ =σ2u
σ2u + σ2
e
; Therefore σ2e = σ2
T (1 − ρ)
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
ICC as link between paired and independent analysis
Take message
Within-cluster (mothers) correlation increases precision for estimatingwithin-siblings effects. So, we can obtain greater benefit pairing if ICC(ρ) is larger.
The proof
Paired:2σ2
e
N=
2σ2T (1 − ρ)
NIndependent:
2(σ2u + σ2
e )
N=
2σ2T
N
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for 2k paired siblingsWithin-sibling correlations: ICCLink between paired and independent analysis
ICC as link between paired and independent analysis
Take message
Within-cluster (mothers) correlation increases precision for estimatingwithin-siblings effects. So, we can obtain greater benefit pairing if ICC(ρ) is larger.
The proof
Paired:2σ2
e
N=
2σ2T (1 − ρ)
NIndependent:
2(σ2u + σ2
e )
N=
2σ2T
N
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for more than 2k siblings (3, 4, ..., n)
Analytical methods for more than 2k siblings
Mean-centered transformation ’Demeaning’
Y ∗ij = βX
′∗ij + e∗ij ,
Where
Mean-centered outcome and covariates
Y ∗ij = Yij − Yi ; Yi =
1
ni
ni∑j=1
Yij
X ∗ijk = Xijk − Xik ; Xik =
1
ni
ni∑j=1
Xijk
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
The model for more than 2k siblings (3, 4, ..., n)
Analytical methods for more than 2k siblings
Mean-centered transformation ’Demeaning’
Y ∗ij = βX
′∗ij + e∗ij ,
Where
Mean-centered outcome and covariates
Y ∗ij = Yij − Yi ; Yi =
1
ni
ni∑j=1
Yij
X ∗ijk = Xijk − Xik ; Xik =
1
ni
ni∑j=1
Xijk
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effect Models summary (FEMs)
FEMs summary
Account for all between mothers variation by holding the effect ofmothers constant (fixed intercept in the model).
All inference is therefore within-siblings.
Can be implemented either by dummy variables for n-1 mothers orsubtracting mothers mean birth weights from all the siblings ofthe same mother.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effect Models summary (FEMs)
FEMs summary
Account for all between mothers variation by holding the effect ofmothers constant (fixed intercept in the model).
All inference is therefore within-siblings.
Can be implemented either by dummy variables for n-1 mothers orsubtracting mothers mean birth weights from all the siblings ofthe same mother.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effect Models summary (FEMs)
FEMs summary
Account for all between mothers variation by holding the effect ofmothers constant (fixed intercept in the model).
All inference is therefore within-siblings.
Can be implemented either by dummy variables for n-1 mothers orsubtracting mothers mean birth weights from all the siblings ofthe same mother.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effect Models summary (FEMs)
FEMs summary
Account for all between mothers variation by holding the effect ofmothers constant (fixed intercept in the model).
All inference is therefore within-siblings.
Can be implemented either by dummy variables for n-1 mothers orsubtracting mothers mean birth weights from all the siblings ofthe same mother.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
FEMs Merits
Merits
Control for unmeasured time-invariant characteristics of individuals(genetic and environmental factors),
Each subject (mothers) truly serves as its own control,
Assuming that the effects on the response of time-invariantcharacteristics remain constant over time, FE models provideunbiased estimates of time-varying covariates.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
FEMs Merits
Merits
Control for unmeasured time-invariant characteristics of individuals(genetic and environmental factors),
Each subject (mothers) truly serves as its own control,
Assuming that the effects on the response of time-invariantcharacteristics remain constant over time, FE models provideunbiased estimates of time-varying covariates.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
FEMs Merits
Merits
Control for unmeasured time-invariant characteristics of individuals(genetic and environmental factors),
Each subject (mothers) truly serves as its own control,
Assuming that the effects on the response of time-invariantcharacteristics remain constant over time, FE models provideunbiased estimates of time-varying covariates.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
FEMs Merits
Merits
Control for unmeasured time-invariant characteristics of individuals(genetic and environmental factors),
Each subject (mothers) truly serves as its own control,
Assuming that the effects on the response of time-invariantcharacteristics remain constant over time, FE models provideunbiased estimates of time-varying covariates.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
FEMS summaryFEMs MeritsFEMs limitations
Fixed Effects Models Limitations
FEMs Limitations
FE cannot estimate the effects of time-invariant covariates,
The effect of time-invariant covariates is assumed to remainconstant over time,
Need of strict exogeneity, which prohibits some types of feedbackfrom past outcomes to current covariates and current outcome tofuture covariates (reverse causality -smoking and prematurity),
Trade-off between bias and efficiency. Lack of precision particularlyif there is little within-cluster variation,
FE models yields invalid inference in presence of missing at random(MAR) data.
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models
1. Variance-components models to account for within-cluster correlations2. Analytical Methods for 2k paired study designs
3. Analytical methods for more than two-paired repeated measures4. Fixed Effects Models: Summary, Merits and Limitations
5. 2k paired design: Generation-R within-siblings birth weight differences6. ≥ 2k matched design: NCCP within-siblings placental weight differences
National Collaborative Perinatal Project (NCPP) USHypothesisObjectivesFixed effects modeling justificationSample selectionSome Results
Thank You
Alhambra: Granada, Spain
HSPH-Department of Epidemiology Within Siblings Analysis: Fixed Effects Models