Testing equality of survival functions based on both paired and unpaired censored data

We introduce two test procedures for comparing two survival distributions on the basis of randomly right-censored data consisting of both paired and unpaired observations. Our procedures are based on generalizations of a pooled rank test statistic previously proposed for uncensored data. One generalization adapts the Prentice-Wilcoxon score, while the other adapts the Akritas score. The use of these particular scoring systems in pooled rank tests with randomly right-censored paired data has been advocated by several researchers. Our test procedures utilize the permutation distributions of the test statistics based on a novel manner of permuting the scores. Permutation versions of tests for right-censored paired data and for two independent right-censored samples that use the proposed scoring systems are obtained as special cases of our test procedures. Simulation results show that our test procedures have high power for detecting scale and location shifts in exponential and log-logistic distributions for the survival times. We also demonstrate the advantages of our test procedures in terms of utilizing randomly occurring unpaired observations that are discarded in test procedures for paired data. The tests are applied to skin graft data previously reported elsewhere.     

Linear rank tests for interval-censored data with application to PCB levels in adipose tissue of transformer repair workers

Linear rank statistics are described for testing for differences between groups when the data are interval-censored. The statistics are closely related to those described by Prentice (1978, Biometrika 65, 167-179) for right-censored data. Problems in calculating the statistics are discussed and several approaches to computation including estimation of the efficient rank scores are described. Results from a small simulation study are presented. The methods are applied to data from a study relating tissue levels of PCBs to occupational exposure.     

Empirical comparison of distance equations using discrete traits

The use of the Grewal-Smith statistic in measuring biological distance among skeletal population samples has been questioned since it was first applied to human populations. Recently, in an attempt to stabilize the variance of the Grewal-Smith statistic for use with non-metric analysis, Sjøvold ('73) and Green and Suchey ('76) have introduced corrections and alternative transformations which may enhance the meaning of biological distance among population samples. Their recommendations improve the statistics for specific variable ranges; i.e., small sample size and low trait frequencies. Thirteen equations representing Grewal-Smith, Freeman-Tukey, Anscombe, and Bartlett transformations and/or corrections, were compared using rank order correlation statistics on actual biological distances generated by real population data as presented in existing literature. Results from testing these actual distance models show little variation between equations based on the populational data sets used. Based on these findings, the distance model resulting from the Grewal-Smith statistic is not inferior to the more sophisticated models, although the latter may be superior by allowing specific improvements for small sample size and/or low trait frequencies.     

Weighted Concordance Analysis

Using the variance stabilizing technique, a product multinomial model is introduced to generate a new statistic to test observers' uncertainty in a weighted concordance analysis. Distance matrices which follow some specific rules are obtained by linear combinations of hierarchical distance matrices whose elements are equal to 0 or 1 and unit diagonal. The new statistic is compared with the kappa statistic interpreted by considering the covariance matrix generated by the data. By rewriting the test statistic in a barycentric form, one demonstrates how to modify the barycentric coefficients to derive an adequate measure of the interobserver agreement. The methods are illustrated using two examples.     

Small Sample Properties of Rank Tests for Incomplete Unbalanced Designs

A rank test is presented for analysis of incomplete unbalanced designs, i.e. for designs that may have been originally planned to be either balanced or unbalanced and where some observations may be missing at random. This test is a modification of the procedure of Benard and van El-teren (1953) based on a generalization of block weights proposed by Prentice (1979). It is compared with the tests of Haux, Schumacher, and Weckesser (1984) and Rai (1987). For incomplete or unbalanced designs with more than two treatments the quadratic forms proposed by these authors are proven to be invalid for small sample sizes, except for special cases. A necessary condition is given for test statistics to be valid also for small samples.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

Testing for Absence of Qualitative Interactions Between Risk Factors and Treatment Effects

The problem of investigating qualitative interactions (QI's) between subsets of patients defined by means of some risk factors, and treatment effects in clinical trials is considered from a viewpoint which leads to interchanging the hypotheses of the testing problem dealed with in the existing literature on QI's. A natural way of approaching this reverse problem is to apply one of the tests available for the original problem of detecting QI's at level 1 — α and to reject the null hypothesis of the new problem if and only if this test accepts. Unfortunately, this would require unbiasedness of the level 1 — α test for existence of QI's to start with, a property which exhibits neither the likelihood ratio procedure derived in the seminal paper of GAIL and SIMON (1985), nor the test based on the extreme order statistics which was introduced by several authors in 1993. Nevertheless we show that there is a valid test for absence of QI's which depends on the extreme values only and coincides with the maximum likelihood ratio procedure for the same problem. Furthermore, the procedure is generalized to the problem of testing for absence of relevant QI's, i.e. of qualitative interactions exceeding some specified tolerance ε > 0.     

A nonparametric approach to the analysis of longitudinal data via a set of level crossing problems with application to the analysis of microarray time course experiments

Here we develop a completely nonparametric method for comparing two groups on a set of longitudinal measurements. No assumptions are made about the form of the mean response function, the covariance structure or the distributional form of disturbances around the mean response function. The solution proposed here is based on the realization that every longitudinal data set can also be thought of as a collection of survival data sets where the events of interest are level crossings. The method for testing for differences in the longitudinal measurements then is as follows: for an arbitrarily large set of levels, for each subject determine the first time the subject has an upcrossing and a downcrossing for each level. For each level one then computes the log rank statistic and uses the maximum in absolute value of all these statistics as the test statistic. By permuting group labels we obtain a permutation test of the hypothesis that the joint distribution of the measurements over time does not depend on group membership. Simulations are performed to investigate the power and it is applied to the area that motivated the method-the analysis of microarrays. In this area small sample sizes, few time points and far too many genes to consider genuine gene level longitudinal modeling have created a need for a simple, model free test to screen for interesting features in the data.     

Individual risk prediction: Comparing random forests with Cox proportional-hazards model by a simulation study

With big data becoming widely available in healthcare, machine learning algorithms such as random forest (RF) that ignores time-to-event information and random survival forest (RSF) that handles right-censored data are used for individual risk prediction alternatively to the Cox proportional hazards (Cox-PH) model. We aimed to systematically compare RF and RSF with Cox-PH. RSF with three split criteria [log-rank (RSF-LR), log-rank score (RSF-LRS), maximally selected rank statistics (RSF-MSR)]; RF, Cox-PH, and Cox-PH with splines (Cox-S) were evaluated through a simulation study based on real data. One hundred eighty scenarios were investigated assuming different associations between the predictors and the outcome (linear/linear and interactions/nonlinear/nonlinear and interactions), training sample sizes (500/1000/5000), censoring rates (50%/75%/93%), hazard functions (increasing/decreasing/constant), and number of predictors (seven, 15 including noise variables). Methods' performance was evaluated with time-dependent area under curve and integrated Brier score. In all scenarios, RF had the worst performance. In scenarios with a low number of events (⩽70), Cox-PH was at least noninferior to RSF, whereas under linearity assumption it outperformed RSF. Under the presence of interactions, RSF performed better than Cox-PH as the number of events increased whereas Cox-S reached at least similar performance with RSF under nonlinear effects. RSF-LRS performed slightly worse than RSF-LR and RSF-MSR when including noise variables and interaction effects. When applied to real data, models incorporating survival time performed better. Although RSF algorithms are a promising alternative to conventional Cox-PH as data complexity increases, they require a higher number of events for training. In time-to-event analysis, algorithms that consider survival time should be used.     

Estimation of Growth Norms from Incomplete Longitudinal Data

Longitudinal studies are rarely complete due to attrition, mistimed visits and observations missing at random. When the data are missing at random it is possible to estimate the primary location parameters of interest by constructing a modification of Zellner's (1962) seemingly unrelated regression estimator. Such a procedure is developed in this paper and is applied to a longitudinal study of coronary risk factors in children. The method consists of two stages in which the covariance matrix is estimated at the first stage. Using the estimated covariance matrix a generalized least squares estimator of the regression parameter vector is then determined at the second stage. Limitations of the procedure are also discussed.     

A new u-statistic with superior design sensitivity in matched observational studies

Summary In an observational or nonrandomized study of treatment effects, a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naïve analysis that presumes adjustments for observed covariates suffice to remove all bias. The power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude; in particular, if this specified magnitude is “no departure” then this is the same as the power of a randomization test in a randomized experiment. A new family of u‐statistics is proposed that includes Wilcoxon's signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study. Wilcoxon's statistic has high power to detect small effects in large randomized experiments—that is, it often has good Pitman efficiency—but small effects are invariably sensitive to small unobserved biases. Members of this family of u‐statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis. For example, in one situation with 250 pair differences that are Normal with expectation 1/2 and variance 1, the power of a sensitivity analysis that uses Wilcoxon's statistic is 0.08 while the power of another member of the family of u‐statistics is 0.66. The topic is examined by performing a sensitivity analysis in three observational studies, using an asymptotic measure called the design sensitivity, and by simulating power in finite samples. The three examples are drawn from epidemiology, clinical medicine, and genetic toxicology.     

The use of non-linear regression analysis and the F test for model discrimination with dose-response curves and ligand binding data

Computer fitting of binding data is discussed and it is concluded that the main problem is the choice of starting estimates and internal scaling parameters, not the optimization software. Solving linear overdetermined systems of equations for starting estimates is investigated. A function, Q, is introduced to study model discrimination with binding isotherms and the behaviour of Q as a function of model parameters is calculated for the case of 2 and 3 sites. The power function of the F test is estimated for models with 2 to 5 binding sites and necessary constraints on parameters for correct model discrimination are given. The sampling distribution of F test statistics is compared to an exact F distribution using the Chi-squared and Kolmogorov-Smirnov tests. For low order modes (n less than 3) the F test statistics are approximately F distributed but for higher order models the test statistics are skewed to the left of the F distribution. The parameter covariance matrix obtained by inverting the Hessian matrix of the objective function is shown to be a good approximation to the estimate obtained by Monte Carlo sampling for low order models (n less than 3). It is concluded that analysis of up to 2 or 3 binding sites presents few problems and linear, normal statistical results are valid. To identify correctly 4 sites is much more difficult, requiring very precise data and extreme parameter values. Discrimination of 5 from 4 sites is an upper limit to the usefulness of the F test.     

Estimation of a Joint Distribution Function with Multivariate Interval‐Censored Data when the Nonparametric MLE is not Unique

A nonparametric estimator of a joint distribution function F₀ of a d‐dimensional random vector with interval‐censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d ≥ 2. The GMLE of F₀ with univariate IC data is uniquely defined at each follow‐up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Cox's regression model is also mentioned.     

A general class of nonparametric tests for survival analysis

Tarone and Ware (1977, Biometrika 64, 156-160) developed a general class of s-sample test statistics for right-censored survival data that includes the log-rank and modified Wilcoxon procedures. Subsequently, many authors have considered two- and s-sample classes in detail. In this paper a family of nonparametric statistics is shown to unify existing and generate new test statistics for the s (greater than or equal to 2)-sample, s-sample trend, and single continuous covariate problems.     

Robust functional principal component analysis via a functional pairwise spatial sign operator

Functional principal component analysis (FPCA) has been widely used to capture major modes of variation and reduce dimensions in functional data analysis. However, standard FPCA based on the sample covariance estimator does not work well if the data exhibits heavy-tailedness or outliers. To address this challenge, a new robust FPCA approach based on a functional pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced. We propose robust estimation procedures for eigenfunctions and eigenvalues. Theoretical properties of the PASS operator are established, showing that it adopts the same eigenfunctions as the standard covariance operator and also allows recovering ratios between eigenvalues. We also extend the proposed procedure to handle functional data measured with noise. Compared to existing robust FPCA approaches, the proposed PASS FPCA requires weaker distributional assumptions to conserve the eigenspace of the covariance function. Specifically, existing work are often built upon a class of functional elliptical distributions, which requires inherently symmetry. In contrast, we introduce a class of distributions called the weakly functional coordinate symmetry (weakly FCS), which allows for severe asymmetry and is much more flexible than the functional elliptical distribution family. The robustness of the PASS FPCA is demonstrated via extensive simulation studies, especially its advantages in scenarios with nonelliptical distributions. The proposed method was motivated by and applied to analysis of accelerometry data from the Objective Physical Activity and Cardiovascular Health Study, a large-scale epidemiological study to investigate the relationship between objectively measured physical activity and cardiovascular health among older women.     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

Estimating Location Parameters Using Censored Rank Statistics

In this paper censored data rank location estimators are obtained by using censored one-sample rank test statistics of the location parameter and censored two-sample rank test statistics of the shift of location parameter. Also, methods for constructing censored small sample confidence intervals and asymptotic confidence intervals for the location are considered. Generalizations of the solutions from uncensored one-sample and two-sample rank tests are utilized.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.