Likelihood Inference in the Presence of Nuisance Parameters

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1 Likelihood Inference in the Presence of Nuance Parameters N Reid, DAS Fraser Department of Stattics, University of Toronto, Toronto Canada M5S 3G3 We describe some recent approaches to likelihood based inference in the presence of nuance parameters Our approach based on plotting the likelihood function and the p-value function, using recently developed third order approximations Orthogonal parameters and adjustments to profile likelihood are also dcussed Connections to classical approaches of conditional and marginal inference are outlined 1 INTRODUCTION We take the view that the most effective form of inference provided by the observed likelihood function along with the associated p-value function In the case of a scalar parameter the likelihood function simply proportional to the density function The p-value function can be obtained exactly if there a one-dimensional stattic that measures the parameter If not, the p-value can be obtained to a high order of approximation using recently developed methods of likelihood asymptotics In the presence of nuance parameters, the likelihood function for a (one-dimensional parameter of interest obtained via an adjustment to the profile likelihood function The p-value function obtained from quantities computed from the likelihood function using a canonical parametrization ϕ = ϕ(θ, which computed locally at the data point Th generalizes the method of eliminating nuance parameters by conditioning or marginalizing to more general contexts In Section 2 we give some background notation and introduce the notion of orthogonal parameters In Section 3 we illustrate the p-value function approach in a simple model with no nuance parameters Profile likelihood and adjustments to profile likelihood are described in Section 4 Third order p-values for problems with nuance parameters are described in Section 5 Section 6 describes the classical conditional and marginal likelihood approach 2 NOTATION AND ORTHOGONAL PARAMETERS We assume our measurement(s y can be modelled as coming from a probability dtribution with density or mass function f(y; θ, where θ = (ψ, λ takes values in R d We assume ψ a one-dimensional parameter of interest, and λ a vector of nuance parameters If there interest in more than one component of θ, the methods described here can be applied to each component of interest in turn The likelihood function L(θ = L(θ; y = c(yf(y; θ; (1 it defined only up to arbitrary multiples which may depend on y but not on θ Th ensures in particular that the likelihood function invariant to one-toone transformations of the measurement(s y In the context of independent, identically dtributed sampling, where y = (y 1,, y n and each y i follows the model f(y; θ the likelihood function proportional to Πf(y i ; θ and the log-likelihood function becomes a sum of independent and identically dtributed components: l(θ = l(θ; y = Σ log f(y i ; θ + a(y (2 The maximum likelihood estimate ˆθ the value of θ at which the likelihood takes its maximum, and in regular models defined by the score equation l (ˆθ; y = 0 (3 The observed Fher information function j(θ the curvature of the log-likelihood: j(θ = l (θ (4 and the expected Fher information the model quantity i(θ = E{ l (θ} = l (θ; yf(y; θdy (5 If y a sample of size n then i(θ = O(n In accord with the partitioning of θ we partition the observed and expected information matrices and use the notation ( iψψ i i(θ = ψλ (6 i λψ i λλ and ( i 1 i (θ = ψψ i ψλ i λψ i λλ (7 We say ψ orthogonal to λ (with respect to expected Fher information if i ψλ (θ = 0 When ψ scalar a transformation from (ψ, λ to (ψ, η(ψ, λ such that ψ orthogonal to η can always be found (Cox and Reid [1] The most directly interpreted consequence 265

2 of parameter orthogonality that the maximum likelihood estimates of orthogonal components are asymptotically independent Example 1: ratio of Poson means Suppose y 1 and y 2 are independent counts modelled as Poson with mean λ and ψλ, respectively Then the likelihood function L(ψ, λ; y 1, y 2 = e λ(1+ψ ψ y2 λ y1+y2 and ψ orthogonal to η(ψ, λ = λ(ψ + 1 In fact in th example the likelihood function factors as L 1 (ψl 2 (η, which a stronger property than parameter orthogonality The first factor the likelihood for a binomial dtribution with index y 1 + y 2 and probability of success ψ/(1 + ψ, and the second that for a Poson dtribution with mean η Example 2: exponential regression Suppose y i, i = 1,, n are independent observations, each from an exponential dtribution with mean λ exp( ψx i, where x i known The log-likelihood function l(ψ, λ; y = n log λ + ψσx i λ 1 Σy i exp(ψx i (8 and i ψλ (θ = 0 if and only if Σx i = 0 The stronger property of factorization of the likelihood does not hold 3 LIKELIHOOD INFERENCE WITH NO NUISANCE PARAMETERS We assume now that θ one-dimensional A plot of the log-likelihood function as a function of θ can quickly reveal irregularities in the model, such as a non-unique maximum, or a maximum on the boundary, and can also provide a vual guide to deviance from normality, as the log-likelihood function for a normal dtribution a parabola and hence symmetric about the maximum In order to calibrate the log-likelihood function we can use the approximation r(θ = sign(ˆθ θ[2{l(ˆθ l(θ}] 1/2 N(0, 1, (9 which equivalent to the result that twice the log likelihood ratio approximately χ 2 1 Th will typically provide a better approximation than the asymptotically equivalent result that ˆθ θ N(0, i 1 (θ (10 as it partially accommodates the potential asymmetry in the log-likelihood function These two approximations are sometimes called first order approximations because in the context where the log-likelihood O(n, we have (under regularity conditions results such as Pr{r(θ; y r(θ; y 0 } = Pr{Z r(θ; y 0 } (11 {1 + O(n 1/2 } Table I The p-values for testing µ = 0, ie that the number of observed events constent with the background upper p-value lower p-value mid p-value Φ(r Φ(r Φ{(ˆθ θĵ 1/2 } where Z follows a standard normal dtribution It relatively simple to improve the approximation to third order, ie with relative error O(n 3/2, using the so-called r approximation r (θ = r(θ + {1/r(θ} log{q(θ/r(θ} N(0, 1 (12 where q(θ a likelihood-based stattic and a generalization of the Wald stattic (ˆθ θj 1/2 (ˆθ; see Fraser [2] Example 3: truncated Poson Suppose that y follows a Poson dtribution with mean θ = b + µ, where b a background rate that assumed known In th model the p-value function can be computed exactly simply by summing the Poson probabilities Because the Poson dtribution dcrete, the p-value could reasonably be defined as either or Pr(y y 0 ; θ (13 Pr(y < y 0 ; θ, (14 sometimes called the upper and lower p-values, respectively For the values y 0 = 17, b = 67, Figure 1 shows the likelihood function as a function of µ and the p- value function p(µ computed using both the upper and lower p-values In Figure 2 we plot the mid p- value, which Pr(y < y 0 + (1/2Pr(y = y 0 (15 The approximation based on r nearly identical to the mid-p-value; the difference cannot be seen on Figure 2 Table 1 compares the p-values at µ = 0 Th example taken from Fraser, Reid and Wong [3] 4 PROFILE AND ADJUSTED PROFILE LIKELIHOOD FUNCTIONS We now assume θ = (ψ, λ and denote by ˆλ ψ the restricted maximum likelihood estimate obtained by 266

3 also sometimes called the concentrated likelihood or the peak likelihood The approximations of the previous section generalize to likelihood p-value mu Figure 1: The likelihood function (top and p-value function (bottom for the Poson model, with b = 67 and y 0 = 17 For µ = 0 the p-value interval (099940, pvalue mu Figure 2: The upper and lower p-value functions and the mid-p-value function for the Poson model, with b = 67 and y 0 = 17 The approximation based on Φ(r identical to the mid-p-value function to the drawing accuracy maximizing the likelihood function over the nuance parameter λ with ψ fixed The profile likelihood function µ L p (ψ = L(ψ, ˆλ ψ ; (16 r(ψ = sign( ˆψ ψ[2{l p ( ˆψ l p (ψ}] 1/2 N(0, 1, (17 and ˆψ ψ N(0, {i ψψ (θ} 1 (18 These approximations, like the ones in Section 3, are derived from asymptotic results which assume that n, that we have a vector y of independent, identically dtributed observations, and that the dimension of the nuance parameter does not increase with n Further regularity conditions are required on the model, such as are outlined in textbook treatments of the asymptotic theory of maximum likelihood In finite samples these approximations can be mleading: profile likelihood too concentrated, and can be maximized at the wrong value Example 4: normal theory regression Suppose y i = x i β + ɛ i, where x i = (x i1,, x ip a vector of known covariate values, β an unknown parameter of length p, and ɛ i assumed to follow a N(0, ψ dtribution The maximum likelihood estimate of ψ ˆψ = 1 n Σ(y i x i ˆβ 2 (19 which tends to be too small, as it does not allow for the fact that p unknown parameters (the components of β have been estimated In th example there a simple improvement, based on the result that the likelihood function for (β, ψ factors into L 1 (β, ψ; ȳl 2 {ψ; Σ(y i x i ˆβ 2 } (20 where L 2 (ψ proportional to the marginal dtribution of Σ(y i x ˆβ i 2 Figure 3 shows the profile likelihood and the marginal likelihood; it easy to verify that the latter maximized at ˆψ m = 1 n p Σ(y i x i ˆβ 2 (21 which in fact an unbiased estimate of ψ Example 5: product of exponential means Suppose we have independent pairs of observations y 1i, y 2i, where y 1i Exp(ψλ i y 2i Exp(ψ/λ i, i = 1,, n The limiting normal theory for profile likelihood does not apply in th context, as the dimension of the parameter not fixed but increasing with the sample size, and it can be shown that ˆψ π 4 ψ (22 as n (Cox and Reid [4] The theory of higher order approximations can be used to derive a general improvement to the profile 267

4 PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003 likelihood σ profile marginal Figure 3: Profile likelihood and marginal likelihood for the variance parameter in a normal theory regression with 21 observations and three covariates (the Stack Loss data included in the Splus dtribution The profile likelihood maximized at a smaller value of ψ, and narrower; in th case both the estimate and its estimated standard error are too small likelihood or log-likelihood function, which takes the form l a (ψ = l p (ψ log j λλ(ψ, ˆλ ψ + B(ψ (23 where j λλ defined by the partitioning of the observed information function, and B(ψ a further adjustment function that O p (1 Several versions of B(ψ have been suggested in the stattical literature: we use the one defined in Fraser [5] given by B(ψ = 1 2 log ϕ λ(ψ, ˆλ ψ j ϕϕ ( ˆψ, ˆλϕ λ(ψ, ˆλ ψ (24 Th depends on a so-called canonical parametrization ϕ = ϕ(θ = l ;V (θ; y 0 which dcussed in Fraser, Reid and Wu [6] and Reid [7] In the special case that ψ orthogonal to the nuance parameter λ a simplification of l a (ψ available as l CR (ψ = l p (ψ 1 2 log j λλ(ψ, ˆλ ψ (25 which was first introduced in Cox and Reid (1987 The change of sign on log j comes from the orthogonality equations In iid sampling, l p (ψ O p (n, ie the sum of n bounded random variables, whereas log j O p (1 A drawback of l CR that it not invariant to one-to-one reparametrizations of λ, all of which are orthogonal to ψ In contrast l a (ψ invariant to transformations θ = (ψ, λ to θ = (ψ, η(ψ, λ, sometimes called interest-respecting transformations Example 5 continued In th example ψ orthogonal to λ = (λ 1,, λ n, and l CR (ψ = (3n/2 log ψ (2/ψΣ (y 1i y 2i (26 The value that maximizes l CR more nearly constent than the maximum likelihood estimate as ˆψ CR (π/3ψ 5 P -VALUES FROM PROFILE LIKELIHOOD The limiting theory for profile likelihood gives first order approximations to p-values, such as and p(ψ = Φ(r p (27 p(ψ = Φ{( ˆψ ψj 1/2 p ( ˆψ} (28 although the dcussion in the previous section suggests these may not provide very accurate approximations As in the scalar parameter case, though, a much better approximation available using Φ(r where r (ψ = r p (ψ + 1/{r p (ψ} log{q(ψ/r p (ψ} (29 where Q can also be derived from the likelihood function and a function ϕ(θ, y 0 as where Q = (ˆν ˆν ψ ˆσ 1/2 ν ν(θ = e T ψϕ(θ, e ψ = ψ ϕ (ˆθ ψ / ψ ϕ (ˆθ ψ, ˆσ 2 ν = j (λλ (ˆθ ψ / j (θθ (ˆθ, j (θθ (ˆθ = j θθ (ˆθ ϕ θ (ˆθ 2, j (λλ (ˆθ ψ = j λλ (ˆθ ψ ϕ λ (ˆθ ψ 2 The derivation described in Fraser, Reid and Wu [6] and Reid [7] The key ingredients are the log-likelihood function l(θ and a reparametrization ϕ(θ = ϕ(θ; y 0, which defined by using an approximating model at the observed data point y 0 ; th approximation in turn based on a conditioning argument A closely related approach due to Barndorff- Nielsen; see Barndorff-Nielsen and Cox [8, Ch 7], and the two approaches are compared in [7] Example 6: comparing two binomials Table 2 shows the employment htory of men and women at the Space Telescope Science Institute, as reported in Science Feb We denote by y 1 the number of males who left and model th as a Binomial with sample size 19 and probability p 1 ; similarly the number of females who left, y 2, modelled as Binomial with sample size 7 and probability p 2 We write the parameter of interest ψ = log p 1(1 p 2 p 2 (1 p 1 (30 The hypothes of interest p 1 = p 2, or ψ = 0 The p- value function for ψ plotted in Figure 4 The p-value at ψ = using the normal approximation to r p, and using the normal approximation 268

5 Table II Employment of men and women at the Space Telescope Science Institute, (from Science magazine, Volume 299, page 993, 14 February 2003 pvalue Left Stayed Total Men Women Total Figure 4: The p-value function for the log-odds ratio, ψ, for the data of Table II The value ψ = 0 corresponds to the hypothes that the probabilities of leaving are equal for men and women to r Using Fher s exact test gives a mid p-value of , so the approximations are anticonservative in th case Example 7: Poson with estimated background Suppose in the context of Example 3 that we allow for imprecion in the background, replacing b by an unknown parameter β with estimated value ˆβ We assume that the background estimate obtained from a Poson count x, which has mean kβ, and the signal measurement an independent Poson count, y, with mean β+µ We have ˆβ = x/k and var ˆβ = β/k, so the estimated precion of the background gives us a value for k For example, if the background estimated to be 67 ± 21 th implies a value for k of 67/(21 2 = 15 Uncertainty in the standard error of the background ignored here We now outline the steps in the computation of the r approximation (29 The log-likelihood function based on the two independent observations x and y l(β, µ = x log(kβ kβ + y log(β + µ β µ (31 with canonical parameter ϕ = (log β, log(β + µ Then ( ϕ θ (θ = ϕ(θ 0 1/β θ =, (32 1/(β + µ 1/(β + µ ψ from which Then we have ϕ 1 θ = ( β β + µ β 0 (33 ψ ϕ = ( β, β + µ (34 χ(ˆθ = ˆβ µ log( ˆβ + ( ˆβ µ + µ log( ˆβ + ˆµ { ˆβ2 µ + ( ˆβ µ + µ 2 } (35 χ(ˆθ ψ = ˆβ µ log( ˆβ µ + ( ˆβ µ + µ log( ˆβ µ + µ { ˆβ2 µ + ( ˆβ,(36 µ + µ 2 } and finally { Q = j (θθ (ˆθ = y 1 y 2 = k/ ˆβ( ˆβ + ˆµ (37 j (λλ (ˆθ ψ = y 1( ˆβ µ + µ 2 + y 2 ˆβ2 µ ( ˆβ µ + µ 2 + ˆβ 2 µ ( ˆβ µ + µ log The likelihood root ( ˆβ + ˆµ ˆβ µ + µ ˆβ µ log ˆβ } ˆβ µ (38 {k ˆβ( ˆβ + ˆµ} 1/2 {k ˆβ( ˆβ µ + µ 2 + ( ˆβ + ˆµ ˆβ (39 µ} 2 1/2 r = sign(q [2{l( ˆβ, ˆµ l( ˆβ µ, µ}] (40 = sign(q (2[k ˆβ log{ ˆβ/ ˆβ µ } + ( ˆβ + ˆµ log{( ˆβ + ˆµ/( ˆβ µ + µ} k( ˆβ ˆβ µ { ˆβ + ˆµ ( ˆβ µ + µ}] (41 The third order approximation to the p-value function 1 Φ(r, where r = r + (1/r log(q/r (42 Figure 5 shows the p-value function for µ using the mid-p-value function from the Poson with no adjustment for the error in the background, and the p-value function from 1 Φ(r The p-value for testing µ = , allowing for the uncertainty in the background, whereas it ignoring th uncertainty The hypothes Ey = β could also be tested by modelling the mean of y as νβ, say, and testing the value ν = 1 In th formulation we can eliminate the nuance parameter exactly by using the binomial dtribution of y conditioned on the total x + y, as described in example 1 Th gives a mid-p-value of The computation much easier than that outlined above, and seems quite appropriate for testing the equality of the two means However if inference about the mean of the signal needed, in the form of a point estimate or confidence bounds, then the formulation as a ratio seems less natural at least in the context of HEP experiments A more complete comparon of methods for th problem given in Linnemann [8] 269

6 pvalue estimated background known background mu Figure 5: Comparon of the p-value functions computed assuming the background known and using the mid-p-value with the third order approximation allowing a background error of ±175 6 CONDITIONAL AND MARGINAL LIKELIHOOD In special model classes, it possible to eliminate nuance parameters by either conditioning or marginalizing The conditional or marginal likelihood then gives essentially exact inference for the parameter of interest, if th likelihood can itself be computed exactly In Example 1 above, L 1 the density for y 2 conditional on y 1 + y 2, so a conditional likelihood for ψ Th an example of the more general class of linear exponential families: f(y; ψ, λ = exp{ψs(y+λ t(y c(ψ, λ d(y}; (43 in which f cond (s t; ψ = exp{ψs C t (ψ D t (s} (44 defines the conditional likelihood The comparon of two binomials in Example 6 in th class, with ψ as defined at (30 and λ = log{p 2 /(1 p 2 } The difference of two Poson means, in Example 7, cannot be formulated th way, however, even though the Poson dtribution an exponential family, because the parameter of interest ψ not a component of the canonical parameter It can be shown that in models of the form (43 the log-likelihood l a (ψ = l p (ψ + (1/2 log j λλ approximates the conditional log-likelihood l cond (ψ = log f cond (s t; ψ, and that where p(ψ = Φ(r (45 r = r a + 1 r a log( Q r a r a = ±[2{l a ( ˆψ a l a (ψ}] 1/2 Q = ( ˆψ 1/2 a ψ{j a ( ˆψ} approximates the p-value function with relative error O(n 3/2 in iid sampling An asymptotically equivalent approximation based on the profile log-likelihood where r = r p + 1 r p log( Q r p p(ψ = Φ(r (46 r p = ±[2{l p ( ˆψ l p (ψ}] 1/2 Q = ( ˆψ ψ{j p ( ˆψ} 1/2 j λλ(ψ, ˆλ ψ 1/2 j λλ ( ˆψ, ˆλ 1/2 In the latter approximation an adjustment for nuance parameters made to Q, whereas in the former the adjustment built into the likelihood function Approximation (46 was used in Figure 3 270

7 A similar dcussion applies to the class of transformation models, using marginal approximations Both classes are reviewed in Reid [9] Acknowledgments The authors wh to thank Anthony Davon and Augustine Wong for helpful dcussion Th research was partially supported by the Natural Sciences and Engineering Research Council References [1] DR Cox and N Reid, Parameter Orthogonality and Approximate Conditional Inference, J R Statt Soc B, 47, 1, 1987 [2] DAS Fraser, Stattical Inference: Likelihood to Significance, J Am Statt Assoc , 1991 [3] DAS Fraser, N Reid and A Wong, On Inference for Bounded Parameters, arxiv: physics/ , v1, 27 Mar 2003 to appear in Phys Rev D [4] DR Cox and N Reid, A Note on the Difference between Profile and Modified Profile Likelihood, Biometrika 79, 408, 1992 [5] DAS Fraser, Likelihood for Component Parameters, Biometrika 90, 327, (2003 [6] DAS Fraser, N Reid and J Wu, A Simple General Formula for Tail Probabilities for Frequentt and Bayesian Inference, Biometrika 86, 246, 1999 [7] N Reid, Asymptotics and the Theory of Inference, Ann Statt, to appear, 2004 [8] J T Linnemann, Measures of Significance in HEP and Astrophysics, available in these Proceedings on page 35 [9] N Reid, Likelihood and Higher-Order Approximations to Tail Areas: a Review and Annotated Bibliography, Canad J Statt 24, 141,

Likelihood Inference in the Presence of Nuisance Parameters

PHYSTAT2003, SLAC, September 8-11, 2003 1 Likelihood Inference in the Presence of Nuance Parameters N. Reid, D.A.S. Fraser Department of Stattics, University of Toronto, Toronto Canada M5S 3G3 We describe