Sampling bias in logistic models

Conventional regression models Auto-generated units Consequences of auto-generation Arguments pro and con

Sampling bias in logistic models Peter McCullagh Department of Statistics University of Chicago

Health Studies, Jan 23, 2008

http://www.rss.org.uk/pdf/McCullaghFeb2008.pdf Peter McCullagh

Auto-generated units


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Conventional regression models Gaussian models Binary regression model Properties of conventional models Problems with conventional models Auto-generated units Point process model Consequences of auto-generation Sampling bias Treatment and randomization Notation Estimating functions Interference Arguments pro and con Peter McCullagh



Gaussian models Binary regression model Properties of conventional models Problems with conventional models

Conventional regression model Fixed set U (usually infinite): u1 , u2 , . . . are subjects, plots... Covariate x(u1 ), x(u2 ), . . . (non-random, vector-valued) Response Y (u1 ), Y (u2 ), . . . (random, real-valued) Regression model: Sample = finite ordered subset u1 , . . . , un (distinct in U) For each sample with configuration x = (x(u1 ), . . . , x(un )) Distribution px (y) on Rn depends on x Example: px (y ∈ A; θ) = Nn (X β, σ02 In + σ12 K [x])(A) A ⊂ Rn , K [x] = {K (xi , xj )} block-factor models, spatial models, spline models,... Peter McCullagh




Binary regression model (GLMM) Units: u1 , u2 , . . . subjects, patients, plots (labelled) Covariate x(u1 ), x(u2 ), . . . (non-random, X -valued) Process η on X (Gaussian, for example) Responses Y (u1 ), . . . conditionally independent given η logit pr(Y (u) = 1 | η) = α + βx(u) + η(x(u)) Joint distribution for sample having configuration x px (y) = Eη

n Y e(α+βxi +η(xi ))yi 1 + eα+βxi +η(xi ) i=1

parameters α, β, K ,

K (x, x 0 ) = cov(η(x), η(x 0 )). Peter McCullagh




Binary regression model: computation GLMM computational problem: n Y e(α+βxi +η(xi ))yi px (y) = φ(η; K ) dη 1 + eα+βxi +η(xi ) Rn

Z

i=1

Options: Taylor approx: Laird and Ware; Schall; Breslow and Clayton, McC and Nelder, Drum and McC,... Laplace approximation: Wolfinger 1993; Shun and McC 1994 Numerical approximation: Egret E.M. algorithm: McCulloch 1994 for probit models Monte Carlo: Z&L,... Peter McCullagh




But, . . ., wait a minute...

But . . . px (y) is not usually the relevant distribution! Why not? Sampling: fixed x versus random x stratum distribution versus conditional distribution

Peter McCullagh




Mathematical properties of regression model (i) Model is a set of processes with U as index set covariate structure: x : U → X (fixed and given) (ii) sample = finite ordered subset of distinct units sample configuration x = (x(u1 ), . . . , x(un )) (iii) Model: sample 7→ response distribution px (y) (iv) Two samples having same x also have same response distribution. (exchangeability, no unmeasured confounders,...) (v) Distribution of Y (u) depends only on x(u), not on x(u 0 ) (no interference, Kolmogorov consistency) What if ... u1 , . . . , un were generated at random? Peter McCullagh




Problems in the application of conventional models Clinical trials / market research / traffic studies / crime... (i) Sample units generated by a random process sequential recruitment, purchase events, traffic studies... (ii) Population also generated by a random process in time animal populations, purchase events, crime events,... (iii) Sample as a fixed subset: what does this mean? — either mathematically or practically (iv) Random sample: No alternative, but the distribution may be different from that for a fixed sample (v) Sequential sample: Is necessarily a random sample What about x-values? Conditional distribution versus stratum distribution Peter McCullagh



Point process model

Peter McCullagh



Point process model

Peter McCullagh



Point process model

Point process model for auto-generated units y=2

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A point process on C × X for C = {0, 1, 2}, and the superposition process on X . Intensity λr (x) for class r: r = 0, 1, 2. x-values auto-generated by the superposition process with intensity λ. (x). To each auto-generated unit there corresponds an x-value and a y-value.

Peter McCullagh



Point process model

Binary point process model Intensity process λ0 (x) for class 0, λ1 (x) for class 1 Log ratio: η(x) = log λ1 (x) − log λ0 (x) Events form a PP with intensity λ on {0, 1} × X . Conventional GLMM calculation (Bayesian and frequentist): λ1 (x) eη(x) = λ. (x) 1 + eη(x) λ1 (x) eη(x) pr(Y = 1 | x) = E =E λ. (x) 1 + eη(x)

pr(Y = 1 | x, λ) =

Calculation is correct in a sense, but irrelevant. . . . . . there might not be an event at x! Peter McCullagh



Point process model

Correct calculation for auto-generated units

pr(event of type r in dx | λ) = λr (x) dx + o(dx) pr(event of type r in dx) = E(λr (x)) dx + o(dx) pr(event in SPP in dx | λ) = λ. (x) dx + o(dx) pr(event in SPP in dx) = E(λ. (x)) dx + o(dx)

pr(Y (x) = r | SPP event at x) =

λr (x) Eλr (x) 6= E Eλ. (x) λ. (x)

Sampling bias: Distn for fixed x versus distn for autogenerated x. Peter McCullagh



Point process model

Two ways of thinking First way: waiting for Godot! Fix x ∈ X and wait for an event to occur at x (x) pr(Y = 1 | λ, x) = λλ1. (x) (x) = E(Yi | i : Xi = x) pr(Y = 1; x) = E λλ1. (x) Conventional, mathematically correct, but seldom relevant Second way: come what may! SPP event occurs at x, a random point in X joint density at (y , x) proportional to E(λy (x)) = my (x) x has marginal density proportional to E(λ. (x)) = m. (x) λ1 (x) Eλ1 (x) pr(Y = 1 | x) = 6= E Eλ. (x) λ. (x) Peter McCullagh



Sampling bias Treatment and randomization Notation Estimating functions Interference

Log Gaussian illustration of sampling bias η0 (x) ∼ GP(0, K ),

λ0 (x) = exp(η0 (x))

η1 (x) ∼ GP(α + βx, K ), η(x) = η1 (x) − η0 (x) ∼ GP(α + βx, 2K ),

λ1 (x) = exp(η1 (x)) K (x, x) = σ 2

One-dimensional sampling distributions: eα+βx(u)+η(x) ρ(x(u)) = pr(Y (u) = 1) = E 1 + eα+βx(u)+η(x) ∗ logit(ρ(x)) ' α + β ∗ x (|β ∗ | < |β|) 2

eα+βx+σ /2 Eλ1 (x) = σ2 /2 Eλ. (x) e + eα+βx+σ2 /2 logit pr(Y = 1 | x ∈ SPP) = α + βx

π(x) = pr(Y = 1 | x ∈ SPP) =

Peter McCullagh




Randomization: Simple treatment effect Event x –> pair (t(x), y (x)), treatment status, outcome (t, y ) (0, 0) (0, 1) (1, 0) (1, 1)

H0 Ha λy (x)γt λy (x)γt,y λ0 (x)γ0 λ0 (x)γ00 λ1 (x)γ0 λ1 (x)γ01 λ0 (x)γ1 λ0 (x)γ10 λ1 (x)γ1 λ1 (x)γ11

Expected intensity my (x)γt,y m0 (x)γ00 m1 (x)γ01 m0 (x)γ10 m1 (x)γ11

Conditional on λ: odds(Y (x) = 1 | t, λ) = λ1 (x)γt1 /(λ0 (x)γt0 ) odds ratio = τ

= γ11 γ00 /(γ01 γ10 )

Treatment effect constant in x and independent of λ Peter McCullagh




GLMM analysis Sample (X , Y , t)1 , (X , Y , t)2 , . . . observed sequentially GLMM analysis for a single event at x λ1 (x)γt1 λ1 (x)γt1 + λ0 (x)γt0 λ1 (x)γt1 pr(Y (x) = 1 | t) = E λ1 (x)γt1 + λ0 (x)γt0 ∗ odds ratio = τ (|τ ∗ | < |τ |)

pr(Y (x) = 1 | t, λ) =

Conclusion: treatment effect is attenuated |τ ∗ | < |τ |

Peter McCullagh




Point process analysis Sample (X , Y , t)1 , (X , Y , t)2 , . . . observed sequentially Correct analysis for a single event at x λ1 (x)γt1 λ1 (x)γt1 + λ0 (x)γt0 E(λ1 (x))γt1 pr(Y (x) = 1 | t, x ∈ SPP) = E(λ1 (x))γt1 + E(λ0 (x))γt0 odds(Y (x) = 1 | t, x ∈ SPP) = m1 (x)γt1 /(m0 (x)γt0 ) pr(Y (x) = 1 | t, λ) =

odds ratio = γ11 γ00 /(γ01 γ10 ) = τ Conclusion: No attenuation of treatment effect.

Peter McCullagh




Notation: meaning of E(Yi | Xi = x) Exchangeable sequence (Y1 , X1 ), (Y2 , X2 ), . . . Probabilistic interpretation/definition: i fixed, Xi random π(x) = pr(Yi = 1 | Xi = x) = p1 (1, x)/(p1 (0, x) + p1 (1, x)) Biostatistical interpretation: ρ(x) = E(Yi | i : i ∈ Ux ) Ux = {i : Xi = x}: x fixed, i ∈ Ux random eα+βx E(λ1 (x)) = E(λ. (x)) 1 + eα+βx ∗ ∗ eα +β x λ1 (x) ρ(x) = E ' λ. (x) 1 + eα∗ +β ∗ x

π(x) =

Conditional mean π(x) versus stratum mean ρ(x) Peter McCullagh




Consequences of ambiguous notation Sample (Y1 , X1 ), (Y2 , X2 ), . . . observed sequentially E(λ1 (x)) eα+βx = = E(Yi | Xi = x) E(λ. (x)) 1 + eα+βx ∗ ∗ eα +β x λ1 (x) ρ(x) = E ' = E(Yi | i : Xi = x) λ. (x) 1 + eα∗ +β ∗ x

π(x) =

(ρ(x) computed by logistic-normal integral) Conventional PA estimating function Yi − ρ(xi ) is such that E(Y1 − ρ(x) | X1 = x) = π(x) − ρ(x) 6= 0 and same for Y2 , . . .. Peter McCullagh




Estimating functions done correctly Mean intensity for class r : mr (x) = E(λr (x)) π(x) = m1 (x)/m. (x); ρ(x) = E(λ1 (x)/λ. (x)) For random selected units i ∈ Ux , E(Yi ) = ρ(x) For autogenerated x, E(Y |x ∈ SPP) = π(x) 6= ρ(x) X T (x, y) = h(x)(Y (x) − π(x)) x∈SPP

has zero mean for auto-generated configurations x. Note: E(T | x) 6= 0; average is also over configurations

Peter McCullagh




Explanation of unbiasedness z = {(x1 , y1 ), . . .} configuration observed in (0, t). x = {x1 , . . . , } marginal configuration (SPP) z is a random measure with mean t mr (x) ν(dx) at (r , x) x is marginal random measure with mean t m. (x) ν(dx) at x πr (x) = mr (x)/m. (x) Hence t −1 E z(r , dx)−πr (x)x(dx) = mr (x) ν(dx)−πr (x)m. (x) ν(dx) = 0 for all x implies Z

h(x) z(r , dx) − πr (x)x(dx)

T = C×X

has zero expectation. But E(T | x) 6= 0. Peter McCullagh




variance calculation: binary case (y, x) generated by point process; X T (x, y) = h(x)(Y (x) − π(x)) x∈SPP

E(T | x) 6= 0 Z h2 (x)π(x)(1 − π(x)) m. (x) dx var(T ) = ZX + h(x)h(x 0 ) V (x, x 0 ) m.. (x, x 0 ) dx dx 0 2 ZX + h(x)h(x 0 )∆2 (x, x 0 )m.. (x, x 0 ) dx dx 0

E(T (x, y)) = 0;

X2

V : spatial or within-cluster correlation; ∆: interference Peter McCullagh




What is interference? Physical interference: distribution of Y (u) depends on x(u 0 ) Sampling interference for autogenerated units mr (x) = E(λr (x)); mrs (x, x 0 ) = E(λr (x)λs (x 0 )) Univariate distributions: πr (x) = mr (x)/m. (x) Bivariate: πrs (x, x 0 ) = mrs (x, x 0 )/m.. (x, x 0 ) πrs (x, x 0 ) = pr(Y (x) = r , Y (x 0 ) = s | x, x 0 ∈ SPP) Hence πr . (x, x 0 ) = pr(Y (x) = r | x, x 0 ∈ SPP) ∆r (x, x 0 ) = πr . (x, x 0 ) − πr (x) No second-order sampling interference if ∆r (x, x 0 ) = 0 Peter McCullagh



Autogeneration of units in observational studies Q: Subject was observed to engage in behaviour X . What form Y did the behaviour take? Application Marketing Ecology Ecology Traffic study Traffic study Law enforcement Epidemiology Epidemiology

X car purchase sex play highway use highway speeding burglary birth defect cancer death

Y brand activity class relatives or non-relatives speed colour of car/driver firearm used? type of defect cancer type

Units/events auto-generated by the process

Peter McCullagh



Auto-generation as a model for self-selection Economics: Event: single; in labour force; seeks job training Attributes (Y ): (age, job training (Y/N), income) Epidemiology: Event: birth defect Attributes: (age of M, type of defect, state) Clinical trial: Event: seeks medical help; diagnosed C.C.; informed consent; Attributes: (age, sex, treatment status, survival) What is the population of statistical units? Peter McCullagh



Mathematical considerations

Restriction: if pk () is the distribution for k classes, what is the distribution for k − 1 classes? Does restricted model have same form? Answer: Weighted sampling Closure under weighted or case-control sampling Closure under aggregation of homogeneous classes

Peter McCullagh





Peter McCullagh





Peter McCullagh


Sampling bias in logistic models

Sampling bias in logistic models

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