Configural Frequency Analysis (CFA) Revisited — a New Look at an Old Approach

Configural frequency analysis (CFA) is a widely used method for the identification of types and syndromes in contingency tables. However, the type model of CFA shows some major deficiencies. In this paper, we propose an alternative modeling of types eliminating the shortcomings of CFA. Basically, a type is modeled as a combination of traits or symptoms that deviates from the pattern of association holding true for the complementary configurations of the contingency table. The new approach is formulated in terms of a log-linear model. It is shown that parameter estimation can be performed with methods known from the analysis of incomplete contingency tables. Test procedures for confirmatory analysis and methods for exploratory search for type configurations are developed. We illustrate the methodology with two practical examples.     

The Monofactorial Model for Recurrence Risks with Generalized Penetrance

Recurrence risks are derived explicitly in terms of gene frequencies and penetrance coefficients for the general case in which all genotypes have incomplete penetrance. Maximum likelihood estimation of recurrence risks is achieved through the use of the semi-symmetric intraclass contingency table. The resulting formulas and estimation procedure can be useful for the analysis of population and family data, and in genetic counselling.     

Unconditional versions of several tests commonly used in the analysis of contingency tables

By focusing on a confidence interval for a nuisance parameter, Berger and Boos (1994, Journal of the American Statistical Association 89, 1012-1016) proposed new unconditional tests. In particular, they showed that, for a 2 x 2 table, this procedure generally was more powerful than Fisher's exact test. This paper utilizes and extends their approach to obtain unconditional tests for combining several 2 x 2 tables and testing for trend and homogeneity in a 2 x K table. The unconditional procedures are compared to the conditional ones by reanalyzing some published biomedical data.     

Stratified or Multicentre Analysis with Extended Mantel–Haenszel Statistics

Since its introduction in 1959 the ability of the classical Mantel-Haenszel (M–H) procedure for combining the odds ratios of a set of I 2 × 2 tables has led to its use also in stratified or multicentre type clinical trials. A familiar application is the M–H logrank test in survival analysis. An extension of the M–H procedure covering the case of 2 × K contingency tables (MANTEL , 1963) with ordered levels retains the essential property of pooling the results of I homogeneous tables (i.e. in absence of qualitative interactions). The assignment of some score for the K columns of a table is essential for the use of the method (in comparing 2 treatments). Some possibilities of score assignment are discussed: for clinical outcome variables such as the degree of severity of a disease, pain and so on, the score is at hand in a natural way. A less well-known type of scoring consists in ranking the observations of a continuous variable, leading to cell sizes of 1 or 0. In this case, however, if equidistant ranking was used, the E–M–H procedure appears as an extension of Wilcoxon's rank sum test and represents a powerful non-parametric approach in stratified or multicentre type designs with non normally distributed outcome variables. The results of some Monte-Carlo simulations for 2 possible equidistant ranking procedures are presented, which indicate only a moderate gain in power as compared to Wilcoxon's rank sum test under the common situation of centre effects not exceeding treatment effects. Use of the E–M–H pro?edure is also recommended as a simple method to overcome the potential bias due to unequally distributed prognostic factors among treatment groups.     

Confidence Interval for Rate Ratio in a 2 × 2 Table with Structural Zero: An Application in Assessing False-Negative Rate Ratio When Combining Two Diagnostic Tests

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Drawbacks to Integer Scoring for Ordered Categorical Data

Linear rank tests are widely used when testing for independence against stochastic order in a 2 x J contingency table with two treatments and J ordered outcome levels. For this purpose, numerical scores are assigned, possibly by default, to the J outcome levels. When the choice of scores is not apparent, integer (equally spaced) scores are often assigned. We show that this practice generally leads to unnecessarily conservative tests. The use of slightly perturbed scores will result in a less conservative and uniformly more powerful test.     

Interval estimation of the attributable risk for multiple exposure levels in case-control studies

In case-control studies, exposure to a risk factor often occurs at several levels, so the attributable risk at each level is of interest. In this paper, estimation for the 2 X 2 table (case-control status versus dichotomous exposure) and the 2 X k table (case-control status versus exposure at several levels) are reviewed along with an example. A method for finding confidence intervals for attributable risk in the 2 X k table is proposed, and its application to estimates adjusted across strata (the 2 X k X s case) is indicated. The results of a Monte Carlo study of the procedure demonstrate that the nominal and actual coverage probabilities agree satisfactorily for practical applications.     

To some reasonable test procedures in multiple contingency tables to investigate certain epidemiological or medicin-sociological relationships

One of the most important tasks of the application of mathematical-statistical methods consists in giving help in the search for possible relationships, and connected with this, the specification of new hypotheses. The progress of both the special diciplines of sciences and mathematical statistics itself leads to the application of more and more complex, that means multivariate, methods. In medical fields, especially in epidemiological and medicin-sociological studies, this fact means the necessity of analysing multidimensional contingency tables. The above formulated problem is equivalent to the problem of fitting an appropriate mathematical model (for contingency tables is this a log-linear model) to the data in a way which makes the structural relationships clear to us. In this paper it is shown that one is able to get to well-interpretable models of independence with relatively simple means. Two stepwise test procedures are described yielding essentially the same results: a so called reduction procedure which is particularly profitable in sparsely occupied tables and a procedure which uses a combination of hypotheses of conditional pairwise independence.     

Considering of certain relevant information in the analysis of contingency tables. A three-step procedure

A heuristic three-step procedure for analysing multidimensional contingency tables is given to meet the requirements of a mixed analysis from both hypotheses-ruled and data-ruled type. The first-step provides the structure of relationships among the attributes by fitting an appropriate unsaturated log-linear model to the data of the given contingency table. Restriction to elementary hierarchical models allows to get them by combining pairs of conditional independence. The result of the first step may be regarded as a certain validisation of real model ideas. In the second step the significant pairs of conditional dependence are analysed in regard to the levels of the condition complex. Only such significant pairs are to be considered, in general, where the condition complex does not include the response variable. The third-step may test special subtests in that significant two-dimensional tables found in step two or may extend the general statements by partitioning, the corresponding test statistics in additive components. Application examples demonstrate the general line of action.     

Median Tests for Censored Survival Data; a Contingency Table Approach

Summary The median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function. However, existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates. In this article, we modify a K ‐sample median test for censored survival data ( Brookmeyer and Crowley, 1982 , Journal of the American Statistical Association 77, 433–440) through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa. Under censoring, this approach would generate noninteger entries for the cells in the contingency table. We propose to construct a weighted asymptotic test statistic that aggregates dependent χ² ‐statistics formed at the nearest integer points to the original noninteger entries. We show that this statistic follows approximately a χ² ‐distribution with k? 1 degrees of freedom. For a small sample case, we propose a test statistic based on combined p ‐values from Fisher’s exact tests, which follows a χ² ‐distribution with 2 degrees of freedom. Simulation studies are performed to show that the proposed method provides reasonable type I error probabilities and powers. The proposed method is illustrated with two real datasets from phase III breast cancer clinical trials.     

用列联表研究纬度和海拔高度对红杉分布的影响

 本文目的在于探讨列联表用于植物地理学研究。通过用列联表研究垂直高度与纬度对红杉(Larix potaninii)分布的影响,找出垂直高度、纬度及其交互作用等的效应,并且找出了红杉的分布中心。由于卡方检验,剩余分析,有序表连带测度,Log-线性模型拟合联合运用,Log-线性模型的运用有一些改进。本文初步证明,列联表用于植物地理学研究是可以成功的。     

On the First-Order Rao—Scott Correction of the Umesh—Loughin—Scherer Statistic

Decady and Thomas (2000, Biometrics 56, 893-896) propose a first-order corrected Umesh-Loughin-Scherer statistic to test for association in an r x c contingency table with multiple column responses. Agresti and Liu (1999, Biometrics 55, 936-943) point out that such statistics are not invariant to the arbitrary designation of a zero or one to a positive response. This paper shows that, in addition, the proposed testing procedure does not hold the correct size when there are strong pairwise associations between responses.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.     

Higher order inference for the consensus mean in inter-laboratory studies

In inter-laboratory studies, a fundamental problem of interest is inference concerning the consensus mean, when the measurements are made by several laboratories which may exhibit different within-laboratory variances, apart from the between laboratory variability. A heteroscedastic one-way random model is very often used to model this scenario. Under such a model, a modified signed log-likelihood ratio procedure is developed for the interval estimation of the common mean. Furthermore, simulation results are presented to show the accuracy of the proposed confidence interval, especially for small samples. The results are illustrated using an example on the determination of selenium in non-fat milk powder by combining the results of four methods. Here, the sample size is small, and the confidence limits for the common mean obtained by different methods produce very different results. The confidence interval based on the modified signed log-likelihood ratio procedure appears to be quite satisfactory.     

Testing for conditional multiple marginal independence

Survey respondents are often prompted to pick any number of responses from a set of possible responses. Categorical variables that summarize this kind of data are called pick any/c variables. Counts from surveys that contain a pick any/c variable along with a group variable (r levels) and stratification variable (q levels) can be marginally summarized into an r x c x q contingency table. A question that may naturally arise from this setup is to determine if the group and pick any/c variable are marginally independent given the stratification variable. A test for conditional multiple marginal independence (CMMI) can be used to answer this question. Since subjects may pick any number out of c possible responses, the Cochran (1954, Biometrics 10, 417-451) and Mantel and Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) tests cannot be used directly because they assume that units in the contingency table are independent of each other. Therefore, new testing methods are developed. Cochran's test statistic is extended to r x 2 x q tables, and a modified version of this statistic is proposed to test CMMI. Its sampling distribution can be approximated through bootstrapping. Other CMMI testing methods discussed are bootstrap p-value combination methods and Bonferroni adjustments. Simulation findings suggest that the proposed bootstrap procedures and the Bonferroni adjustments consistently hold the correct size and provide power against various alternatives.     

Use of log-linear analysis to construct explanatory models for TDBP- and AFB1-induced mutation spectra in lacI transgenic animals

Mutation spectra recovered from lacI transgenic animals exposed in separate experiments to tris-(2,3-dibromopropyl)phosphate (TDBP) or aflatoxin B1 (AFB1) were examined using log-linear analysis. Log-linear analysis is a categorical procedure that analyses contingency table data. Expected contingency table cell counts are estimated by maximum likelihood as effects of main variables and variable interactions. Evaluation of hierarchical models of decreasing complexity indicates when significant explanatory power is lost by the sequential omission of interactions between variables. Use of this technique allows construction of the most parsimonious models to account for mutation spectra obtained in the two experiments. The resulting statistical models are consistent with previous analyses of these data and with biological explanations for causes of the observed spectra.     

Confidence interval for rate ratio in a 2 x 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests

In this article, we consider problems with correlated data that can be summarized in a 2 x 2 table with structural zero in one of the off-diagonal cells. Data of this kind sometimes appear in infectious disease studies and two-step procedure studies. Lui (1998, Biometrics54, 706-711) considered confidence interval estimation of rate ratio based on Fieller-type, Wald-type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false-negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false-negative rate ratio. Score test-based confidence interval construction for false-negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test-based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre-assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Judgments of causal efficacy under constant and changing interevent contingencies

How do people judge constant and varying interevent contingencies? In two experiments, 150 college students rated the efficacy of a potential cause (an experimental fertilizer) of an effect (a plant's blooming). The prevailing probabilistic interevent relation could remain constant for the entirety of the problem or it could change without warning at the midway point: by contingency reversal, by shifting from noncontingency to contingency, or by shifting from contingency to noncontingency. Participants’ trial-by-trial ratings sensitively tracked the prevailing positive, negative, and noncontingent interevent relations, even those that entailed an unsignaled change in contingency. Changes in specific cells of the 2 × 2 contingency table differentially affected participants’ response to the altered interevent relations. All of this evidence was well described by an associative account of contingency and causal judgments.     

Multiple species effects and spatial autocorrelation in detecting species associations

Abstract. The traditional approach to the analysis of species association within a community, based upon co-occurrence in sampling units such as quadrats, has been to test all pairs of species, using a 2 × 2 contingency table for each pair. It has long been recognised that all these tests are not independent of each other, but there is an additional problem in that the association between any particular pair may depend on the combination of the other species that are present or on the environmental factors that determine that combination. We use a 2^k contingency table to examine this problem and find that pairwise associations are not independent of the other species. The second problem that we consider is the effect of spatial autocorrelation in the data which makes the statistical tests too liberal. In the absence of a derived solution for a deflation factor to correct the test statistic calculated from a 2^k table, we describe a Monte Carlo approach that provides an approximate solution to this problem. In our data the amount of deflation that is necessary for a 2^k table is small compared to the amount required for the 2 × 2 tables used to test pairwise association.     

A Modified Chi-squared Approach for Fitting WEIBULL Models to Synthetic Life Tables1

WEIBULL models are fitted to synthetic life table data by applying weighted least squares analysis to log log functions which are constructed from appropriate underlying contingency tables. As such, the resulting estimates and test statistics are based on the linearized minimum modified X²₁-criterion and thus have satisfactory properties in moderately large samples. The basic methodology is illustrated in terms of an example which is bivariate in the sense of involving two simultaneous, but non-competing, vital events. For this situation, the estimation of WEIBULL model parameters is described for both marginal as well as certain conditional distributions either individually or jointly.