Showing papers in "Scandinavian Journal of Statistics in 2004"

Direct and Indirect Causal Effects via Potential Outcomes

Abstract: jjThe use of the concept of 'direct' versus 'indirect' causal effects is common, not only in statistics but also in many areas of social and economic sciences. The related terms of 'biomarkers' and 'surrogates' are common in pharmacological and biomedical sciences. Sometimes this concept is represented by graphical displays of various kinds. The view here is that there is a great deal of imprecise discussion surrounding this topic and, moreover, that the most straightforward way to clarify the situation is by using potential outcomes to define causal effects. In particular, I suggest that the use of principal stratification is key to understanding the meaning of direct and indirect causal effects. A current study of anthrax vaccine will be used to illustrate ideas.

Flexible Class of Skew‐Symmetric Distributions

Abstract: We propose a flexible class of skew-symmetric distributions for which the probab- ility density function has the form of a product of a symmetric density and a skewing function. By constructing an enumerable dense subset of skewing functions on a compact set, we are able to consider a family of distributions, which can capture skewness, heavy tails and multimodality systematically. We present three illustrative examples for the fibreglass data, the simulated data from a mixture of two normal distributions and the Swiss bills dlata.

Functional Coefficient Regression Models for Non-linear Time Series: A Polynomial Spline Approach

Abstract: We propose a global smoothing method based on1 polynomial splines for the esti- mation of functional coefficient regression models for non-linear time series. Consistency and rate of convergence results are given to support the proposed estimation method. Methods for automatic selection of the threshold variable and significant variables (or lags) are discussed. The estimated model is used to produce multi-step-ahead forecasts, including interval forecasts and density forecasts. The methodology is illustrated by simulations and two real data examples.

Local Linear Additive Quantile Regression

Abstract: . We consider non-parametric additive quantile regression estimation by kernel-weighted local linear fitting. The estimator is based on localizing the characterization of quantileregression as the minimizer of the appropriate ‘check function’. A backfitting algorithm and aheuristic rule for selecting the smoothing parameter are explored. We also study the estimation ofaverage-derivative quantile regression under the additive model. The techniques are illustrated by asimulated example and a real data set.Key words: additive models, average derivative, backfitting algorithm, bandwidth selection,local linear fitting, quantile regression 1. IntroductionSince the seminal paper of Koenker & Bassett (1978), quantile regression has graduallyevolved into a comprehensive approach to the use of linear and nonlinear response models forconditional quantile functions. In many (if not all) regression examples, we would expect adifferent structural form for the higher (or lower) responses than the average responses. Insuch cases, both mean and median analyses would overlook important features that could beuncovered only by a more general conditional quantile analysis. Non-parametric smoothingtechniques play an important role in quantile function estimation, and some recent work onunivariate non-parametric estimation of conditional quantile functions can be found inBhattacharya & Gangopadhyay (1990), Koenker et al. (1994) and Yu & Jones (1998).However, there is very little in the literature about non-parametric high-dimensional condi-tional quantile function estimation.High-dimensional data analysis is an important area in statistics because of the wideavailability of real problems. For example, two-dimensional data analysis arises naturally withspatial geographical data. Multiple regression is an important statistical tool for analysinghigh-dimensional data. Theoretically, the extension of local polynomial least squares regres-sion and robust regression from the univariate case to higher dimensions is straightforward(see Ruppert & Wand, 1994; Welsh, 1996), but its implementation in practice is very difficult.The difficulty is mainly due to the so-called ‘curse of dimensionality’. An alternative approachis to model the response variable as a sum of (typically non-linear) functions of predictorvariables (Hastie & Tibshirani, 1990). Recently, Opsomer & Ruppert (1997) derived theasymptotic mean square errors (AMSE) for the non-parametric kernel-based backfitting al-gorithm for additive mean regression. This is particularly useful for selecting smoothingparameters for the kernel used in fitting the additive mean regression model, see Opsomer &Ruppert (1998) for details. In this paper, we develop a non-parametric kernel-based back-fitting algorithm for additive quantile regression. Actually, a local linear kernel-based additivequantile regression model is proposed in Section 2.Just as classical linear regression techniques based on least squares estimation offer amechanism for estimating conditional mean functions, quantile regression methods based on

All Invariant Moments of the Wishart Distribution

Abstract: . In this paper, we compute moments of a Wishart matrix variate U of the form (Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U− 1. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U. This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.

Inference for Observations of Integrated Diffusion Processes

Abstract: Estimation of parameters in diffusion models is inivestigated when the observations are integrals over intervals of the process with respect to somne weight function. This type of observations can, for example, be obtained when the process is observed after passage through an electronic filter. Another example is provided by the ice-core (lata on oxygen isotopes used to investigate paleo-temperatures. Finally, such data play a role in connection with the stochastic volatility models of finance. The integrated process is not a Markov process. Therefore, prediction- based estimating functions are applied to estimate parameters il the underlying diffusion model. The estimators are shown to be consistent and asymptotically normal. The theory developed in the paper also applies to integrals of processes other than diffusions. The method is applied to inference based on integrated data from Ornstein-Uhlenbeck processes aind from the Cox-Ingersoll-Ross model, for both of which an explicit optimal estimating function is found.

Causal Reasoning from Longitudinal Data

Abstract: . This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so-called do-probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.

Parameter Orthogonality and Bias Adjustment for Estimating Functions

Abstract: . We consider an extended notion of parameter orthogonality for estimating functions, called nuisance parameter insensitivity, which allows a unified treatment of nuisance parameters for a wide range of methods, including Liang and Zeger's generalized estimating equations. Nuisance parameter insensitivity has several important properties in common with conventional parameter orthogonality, such as the nuisance parameter causing no loss of efficiency for estimating the interest parameter, and a simplified estimation algorithm. We also consider bias adjustment for profile estimating functions, and apply the results to restricted maximum likelihood estimation of dispersion parameters in generalized estimating equations.

Maximum Likelihood Estimation for Cox's Regression Model Under Case–Cohort Sampling

Abstract: . Case–cohort sampling aims at reducing the data sampling and costs of large cohort studies. It is therefore important to estimate the parameters of interest as efficiently as possible. We present a maximum likelihood estimator (MLE) for a case–cohort study based on the proportional hazards assumption. The estimator shows finite sample properties that improve on those by the Self & Prentice [Ann. Statist. 16 (1988)] estimator. The size of the gain by the MLE varies with the level of the disease incidence and the variability of the relative risk over the considered population. The gain tends to be small when the disease incidence is low. The MLE is found by a simple EM algorithm that is easy to implement. Standard errors are estimated by a profile likelihood approach based on EM-aided differentiation.

Backfitting Random Varying-Coefficient Models with Time-Dependent Smoothing Covariates

Abstract: In this paper, we propose a random varying-coefficient model for longitudinal data. This model is different from the standard varying-coefficient model in the sense that the time- varying coefficients are assumed to be subject-specific, and can be considered as realizations of stochastic processes. This modelling strategy allows us to employ powerful mixed-effects modelling techniques to efficiently incorporate the within-subject and between-subject variations in the esti- mators of time-varying coefficients. Thus, the subject-specific feature of longitudinal data is effectively considered in the proposed model. A backfitting algorithm is proposed to estimate the coefficient functions. Simulation studies show that the proposed estimation methods are more efficient in finite-sample performance compared with the standard local least squares method. An application to an AIDS clinical study is presented to illustrate the proposed methodologies.

Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions

Abstract: . We consider an asymptotically efficient estimator of the drift parameter for a multi-dimensional diffusion process with small dispersion parameter ɛ. In the situation where the sample path is observed at equidistant times k/n, k = 0, 1, …, n, we study asymptotic properties of an M-estimator derived from an approximate martingale estimating function as ɛ tends to 0 and n tends to ∞ simultaneously.

On Non-parametric Testing, the Uniform Behaviour of the t-test, and Related Problems

Abstract: In this article, we revisit some problems in non-parametric hypothesis testing. First, we extend the classical result of Bahadur & Savage (Ann. Math. ,Statist. 25 (1956) 11151 to other testing problems, and we answer a conjecture of theirs. Other examples considered are testing whether or not the mean is rational, testing goodness-of-fit, and equivalence testing. Next, we discuss the uniform behaviour of the classical t-test. For most non-parametric models, the Baha- dur-Savage result yields that the size of the t-test is one for every sample size. Even if we restrict attention to the family of symmetric distributions supported on a fixed compact set, the t-test is not even uniformly asymptotically level a. However, the convergence of the rejection probability is established uniformly over a large family with a very weak uniform integrability type of condition. Furthermore, under such a restriction, the t-test possesses an asymptotic maximin optimality property.

Goodness of Fit via Non‐parametric Likelihood Ratios

Abstract: To test if a densityf is equal to a specified f 0 , one knows by the Neyman-Pearson lemma the form of the optimal test at a specified alternative f 1 . Any non-parametric density estimation scheme allows an estimate off. This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the Akaike information criterion and the Bayesian information criterion regimes, or use order growing with sample size. Our tests are generalized to testing the adequacy of general parametric models, and to work also in higher dimensions. The tests are related to, but are different from, the 'smooth tests' that go back to Neyman [Skandinavisk Aktuarietidsskrift 20 (1937) 149-199] and that have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different from the smooth tests and often better under non-local conditions.

An Exact Corrected Log‐Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract: This paper studies Cox`s proportional hazards model under covariate measurement error. Nakamura`s (1990) methodology of corrected log-likelihood will be applied to the so called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura (1992), Kong, Huang and Li (1998) and Kong and Gu (1999) are reestablished in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.

Statistical Inference in Context Specific Interaction Models for Contingency Tables

Abstract: . Context specific interaction models is a class of interaction models for contingency tables in which interaction terms are allowed to vanish in specific contexts given by the levels of sets of variables. Such restrictions can entail conditional independencies which only hold for some values of the conditioning variables and allows also for irrelevance of some variables in specific contexts. A Markov property is established and so is an iterative proportional scaling algorithm for maximum likelihood estimation. Decomposition of the estimation problem is treated and model selection is discussed.

On Estimation and Tests of Time-Varying Effects in the Proportional Hazards Model

Abstract: . Cox's proportional hazards model is routinely used in many applied fields, some times, however, with too little emphasis on the fit of the model. In this paper, we suggest some new tests for investigating whether or not covariate effects vary with time. These tests are a natural and integrated part of an extended version of the Cox model. An important new feature of the suggested test is that time constancy for a specific covariate is examined in a model, where some effects of other covariates are allowed to vary with time and some are constant; thus making successive testing of time-dependency possible. The proposed techniques are illustrated with the well-known Mayo liver disease data, and a small simulation study investigates the finite sample properties of the tests.

Optimal detection of a change point in a poisson process for different observation schemes

Abstract: Change point problems are considered where at some unobservable time the intensity of a point process (Tn), n 2 N, has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available informa- tion. We consider three information levels, namely sequential observation of (Tn), ex post decision after observing the point process up to a fixed time t* and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.

On Optimality of Bayesian Wavelet Estimators

Abstract: We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space B δ p,q for p ≥ 2. For 1 ≤ p < 2, the Bayes Factor is still optimal for (2s + 2)/(2s + 1) ≤ p < 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case.

Discussion on Causality

Abstract: AalborgUniversityFirst, let me congratulate both authors on two fine papers which illuminate important aspectsof causal inference I have only a little to say about Professor Arjas’ paper which specificallyilluminates the aspect oftime and causality in an excellent way I will therefore concentrate onthe concepts described by Professor Rubin which seem to be more controversial, thus lendingthemselves directly to discussion1 Causal languagesIn the modern revival ofinterest in causal inference in statistics, a number ofcompetingformalisms prevail such as structural equations (Pearl, 2000), graphical models (Spirtes etal,1993; Pearl, 1995a; Lauritzen, 2001; Dawid, 2002), counterfactual random variables (Robins,1986), or potential responses (Rubin, 1974, 1978; Holland, 1986) Much energy has been usedto promote the virtues ofone formalism versus the other, seemingly without coming nearer toa consensus; see the somewhat relentless discussion ofDawid (2000)Professor Rubin’s paper advocates the use of potential responses in contrast to graphicalmodels, illustrated with a discussion of direct and indirect effects in connection with the use ofsurrogate endpoints in clinical trialsAlthough discussions ofthis nature can be used to sharpen the minds and pinpointimportant issues, I find them generally futile Personally I see the different formalisms asdifferent ‘languages’ The French language may be best for making love whereas the Italianmay be more suitable for singing, but both are indeed possible, and I have no difficultyaccepting that potential responses, structural equations, and graphical models coexist aslanguages expressing causal concepts each with their virtues and vicesIt is hardly possible to imagine a language that completely prevents users from expressingstupid things Personally I am much more comfortable with the language of graphical modelsthan that ofpotential responses, which I, as also Dawid (2000), find just as seductive andpotentially dangerous as Professor Rubin finds graphical models I certainly disagree withProfessor Rubin’s statement that graphical models tend to bury essential scientific and designissues It is always true that concepts that strongly support intuition in many cases, can seduceone to incorrect conclusions in other cases Each ofus speaks and understands our ownlanguage most comfortably but, to communicate in a fruitful way, it is indeed an advantage tolearn other languages as wellProfessor Rubin invites rebuttal of his statement that causal problems are typically firstunderstood by the method ofpotential responses and subsequently translated into the lan-guage of graphical models Honestly, I think this statement is unfair A new formalism, suchas that ofgraphical models, must first earn its keep by showing that it performs in situationsthat are simple and well understood (using eg potential responses), hence it may initiallyappear as ifits value primarily lies in reformulating known things Subsequently it mustdemonstrate successful treatment of more difficult problems, where there is no establishedstandard with which to compare it Personally I believe that graphical models will prove veryhelpful for the future of causal inference, but I admit that their full power and flexibility forexpressing and manipulating causal concepts has not yet been fully exploited

On Testing Dependence between Time to Failure and Cause of Failure via Conditional Probabilities

Abstract: Dependence structures between the failure time and the cause of failure are ex- pressed in terms of the monotonicity properties of the conditional probabilities involving the cause of failure and the failure time. These properties of the conditional probabilities are used for testing four types of departures from the independence of the failure time and the cause of failure and tests based on U-statistics are proposed. In the process, a concept of concordance and discordance between a continuous and a binary variable is introduced to propose a statistical test. The proposed tests are applied to two illustrative applications. Consider a situation where a unit can fail due to one of two competing causes. Let T1 and T2 denote the latent lifetimes of the unit under the two causes. The competing risk data available are the failure time T of the unit, which is the minimum of (T1, T2) and the cause of failure indicator 6, which is equal to 1 if T = T1 and is 0 if T = T2. These data are right censored data where each latent lifetime acts as a censoring variable for the other and, unlike in censoring, the interest lies in both the causes and hence in both the lifetimes. One concentrates on different aspects of the situation by assuming appropriate dependence structures (i) for the two latent lifetimes (T1, T2) and (ii) for the random variables (T, 6). The joint distribution of (T, 6) is defined here by the subsurvival functions, Si(t) = pr(T ? t, 6 i), i = 0, 1. The survival function of T is defined by S(t) = pr(T > t) = So(t) + S,(t). Throughout this paper, we as- sume that the subsurvival functions are continuous with fi(t), i = 0, 1, as the subdensity functions andf(t) = fo(t) + f1(t) as the density of T. The cause-specific hazard rate for cause i is defined as hi(t) =fi(t)/S(t) and the crude hazard rate for cause i is defined as ri(t) = fi(t)l Si(t). The hazard rate of T is h(t) = f(t)/S(t) = h1(t) + ho(t). The problem of identifiability in modelling the competing risks data in terms of the latent lifetimes is well known. The distributions of the latent lifetimes are identifiable under the assumption of independence of the competing causes and also under some weaker conditions of non-informative censoring, see Kalbfleisch & Prentice (2002). There has been an ongoing debate for many years about the use of the models in terms of latent lifetimes and the models in terms of (T, 6), see Prentice et al. (1978), Larson & Dinse (1985), Davis & Lawrance (1989), Deshpande (1990), Aras & Deshpande (1992), Gasbarra & Karia (2000), Crowder (2001),

Root n consistent and optimal density estimators for moving average processes

Abstract: . The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/n. Their estimator can be interpreted as a specific U-statistic. We suggest a slightly simplified U-statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.

Goodness-of-fit Tests for Mixed Models

Abstract: Mixed linear models have become a very useful tool for modelling experiments with dependent observations within subjects, but to establish their appropriateness several assumptions have to be checked. In this paper, we focus on the normality assumptions, using goodness-of-fit tests that make allowance for possible design imbalance. These tests rely on asymptotic results, which are established via empirical process theory. The power of the tests is explored empirically, and examples illustrate some aspects of the usage of the tests.

Exact Non‐Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type‐II Censoring

Abstract: . This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.

Consistency of the NPML Estimator in the Right‐Censored Transformation Model

Abstract: This paper studies the representation and large-sample consistency for non- parametric maximum likelihood estimators (NPMLEs) of an unknown baseline continuous cumu- lative-hazard-type function and parameter of group survival difference, based on right-censored two-sample survival data with marginal survival function assumed to follow a transformation model, a slight generalization of the class of frailty survival regression models. The paper's main theoretical results are existence and unique a.s. limit, characterized variationally, for large data samples of the NPMLE of baseline nuisance function in an appropriately defined neighbourhood of the true function when the group difference parameter is fixed, leading to consistency of the NPMLE when the difference parameter is fixed at a consistent estimator of its true value. The joint NPMLE is also shown to be consistent. An algorithm for computing it numerically, based directly on likelihood equations in place of the expectation-maximization (EM) algorithm, is illustrated with real data.

Quasi-likelihood regression with multiple indices and smooth link and variance functions

Abstract: . A flexible semi-parametric regression model is proposed for modelling the relationship between a response and multivariate predictor variables. The proposed multiple-index model includes smooth unknown link and variance functions that are estimated non-parametrically. Data-adaptive methods for automatic smoothing parameter selection and for the choice of the number of indices M are considered. This model adapts to complex data structures and provides efficient adaptive estimation through the variance function component in the sense that the asymptotic distribution is the same as if the non-parametric components are known. We develop iterative estimation schemes, which include a constrained projection method for the case where the regression parameter vectors are mutually orthogonal. The proposed methods are illustrated with the analysis of data from a growth bioassay and a reproduction experiment with medflies. Asymptotic properties of the estimated model components are also obtained.

Variance Estimation in Spatial Regression Using a Non-parametric Semivariogram Based on Residuals.

Abstract: . The empirical semivariogram of residuals from a regression model with stationary errors may be used to estimate the covariance structure of the underlying process. For prediction (kriging) the bias of the semivariogram estimate induced by using residuals instead of errors has only a minor effect because the bias is small for small lags. However, for estimating the variance of estimated regression coefficients and of predictions, the bias due to using residuals can be quite substantial. Thus we propose a method for reducing this bias. The adjusted empirical semivariogram is then isotonized and made conditionally negative-definite and used to estimate the variance of estimated regression coefficients in a general estimating equations setup. Simulation results for least squares and robust regression show that the proposed method works well in linear models with stationary correlated errors.

Outlier Identification Procedures for Contingency Tables using Maximum Likelihood and L1 Estimates

Abstract: Observed cell counts in contingency tables are perceived as outliers if they have low probability under an anticipated loglinear Poisson model. New procedures for the identification of such outliers are derived using the classical maximum likelihood estimator and an estimator based on the L1 norm.

Statistical Inference for Transformation Inhomogeneous Point Processes

Abstract: . Statistical inference for exponential inhomogeneous Markov point processes by transformation is discussed. It is argued that the inhomogeneity parameter can be estimated, using a partial likelihood based on an inhomogeneous Poisson point process. The inhomogeneity parameter can thereby be estimated without taking the interaction into account, which simplifies the statistical analysis considerably. Data analysis and simulation experiments support the results.

On Parametric Bootstrapping and Bayesian Prediction.

Abstract: . We investigate bootstrapping and Bayesian methods for prediction. The observations and the variable being predicted are distributed according to different distributions. Many important problems can be formulated in this setting. This type of prediction problem appears when we deal with a Poisson process. Regression problems can also be formulated in this setting. First, we show that bootstrap predictive distributions are equivalent to Bayesian predictive distributions in the second-order expansion when some conditions are satisfied. Next, the performance of predictive distributions is compared with that of a plug-in distribution with an estimator. The accuracy of prediction is evaluated by using the Kullback–Leibler divergence. Finally, we give some examples.

Interpreting DNA mixtures with related contributors in subdivided populations

Abstract: . In this paper, we study the statistical interpretation of forensic DNA mixtures with related contributors in subdivided populations. Compact general formulae for match probabilities are obtained for two situations: a relative of one tested person is an unknown contributor of a DNA mixture; and two related unknowns are contributors. The effect of kinship and population structure is illustrated using a real case example.