Bootstrap methods for multivariate hypothesis testing

The nonparametric and parametric bootstrap methods for multivariate hypothesis testing are developed. They are used to approximate the null distribution of the test statistics proposed by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), resulting in bootstrap testing procedures. In the problem of testing for the mean vector of a multivariate distribution, the asymptotic validity of the bootstrap methods is proved. The finite sample performance of the new solutions is demonstrated by means of Monte Carlo simulation studies. They indicate that for small-sample size, the bootstrap tests provide a better finite sample properties than the asymptotic tests considered by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Selection of Binary Variables and Classification by Boosting

We adopt boosting for classification and selection of high-dimensional binary variables for which classical methods based on normality and non singular sample dispersion are inapplicable. Boosting seems particularly well suited for binary variables. We present three methods of which two combine boosting with the relatively classical variable selection methods developed in Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]). Our primary interest is variable selection in classification with small misclassification error being used as validation of proposed method for variable selection. Two of the new methods perform uniformly better than Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]) in one set of simulated and three real life examples.     

An adaptive multivariate CUSUM control chart for signaling a range of location shifts

ABSTRACT

In this work, we proposed an adaptive multivariate cumulative sum (CUSUM) statistical process control chart for signaling a range of location shifts. This method was based on the multivariate CUSUM control chart proposed by Pignatiello and Runger (1990 Pignatiello, J.J., Runger, G.C. (1990). Comparisons of multivariate CUSUM charts. J. Qual. Technol. 22(3):173–186.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), but we adopted the adaptive approach similar to that discussed by Dai et al. (2011 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]), which was based on a different CUSUM method introduced by Crosier (1988 Crosier, R.B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics 30(3):291–303.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The reference value in this proposed procedure was changed adaptively in each run, with the current mean shift estimated by exponentially weighted moving average (EWMA) statistic. By specifying the minimal magnitude of the mean shift, our proposed control chart achieved a good overall performance for detecting a range of shifts rather than a single value. We compared our adaptive multivariate CUSUM method with that of Dai et al. (2001 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]) and the non adaptive versions of these two methods, by evaluating both the steady state and zero state average run length (ARL) values. The detection efficiency of our method showed improvements over the comparative methods when the location shift is unknown but falls within an expected range.     

Optimum Multivariate Stratified Sampling Designs with Travel Cost: A Multiobjective Integer Nonlinear Programming Approach

The problem of optimum allocation in stratified sampling and its solution is well known in sampling literature for univariate populations (see Cochran, 1977 Cochran , W. G. ( 1977 ). Sampling Techniques. , 3rd ed. New York : Wiley . [Google Scholar]; Sukhatme et al., 1984 Sukhatme , P. V. , Sukhatme , B. V. , Sukhatme , S. , Ashok , C. ( 1984 ). Sampling Theory of Surveys With Applications. , 3rd ed. Ames , and New Delhi : Iowa State University Press and Indian Society of Agricultural Statistics . [Google Scholar]). In multivariate populations where more than one characteristics are to be studied on every selected unit of the population the problem of finding an optimum allocation becomes more complex due to conflicting behaviour of characteristics. Various authors such as Dalenius (1953 Dalenius , T. ( 1953 ). The multivariate sampling problem . Skandinavisk Actuarietidskrift 36 : 92 – 102 . [Google Scholar], 1957 Dalenius , T. ( 1957 ). Sampling in Sweden. 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Almost all the previous authors used some function of the sampling variances of the estimators of various characteristics to be measured as an objective that is to be minimized for a fixed cost given as a linear function of sample allocations. Because the variances are not unit free it is more logical to consider the minimization of some function of squared coefficient of variations as an objective. Previously this concept was used by Kozok (2006 Kozok , M. ( 2006 ). On sample allocation in multivariate surveys . Communication in Statistics—Simulation and Computation 35 : 901 – 910 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).

Furthermore, investigators have to approach the sampled units in order to get the observations. This involves some travel cost. Usually this cost is neglected while constructing a cost function. This travel cost may be significant in some surveys. For example if the strata consist of some geographically difficult-to-approach areas.

The authors problem of optimum allocation in multivariate stratified sampling is discussed with an objective to minimize simultaneously the coefficients of variation of the estimators of various characteristics under a cost constraint that includes the measurement as well as travel cost. The formulated problem of obtaining an optimum compromise allocation turns out to be a multiobjective all-integer nonlinear programming problem. Three different approaches are considered: the value function approach, ∈ –constraint method, and Distance–based method, to obtain compromise allocations. The cost function considered also includes the travel cost within stratum to reach the selected units. Additional restrictions are placed on the sample sizes to avoid oversampling and ensure the availability of the estimates of the strata variances. Numerical examples are also presented to illustrate the computational details of the proposed methods.     

A Test for Multivariate Analysis of Variance in High Dimension

In this article, we consider the problem of testing a general multivariate linear hypothesis in a multivariate linear model when the N × p observation matrix is normally distributed with unknown covariance matrix, and N ≤ p. This includes the case of testing the equality of several mean vectors. A test is proposed which is a generalized version of the two-sample test proposed by Srivastava and Du (2008 Srivastava , M. S. , Du , M. ( 2008 ). A test for the mean vector with fewer observations than the dimension . J. Multivariate Anal. 99 : 386 – 402 .[Crossref], [Web of Science ®] , [Google Scholar]). The asymptotic null and nonnull distributions are obtained. The performance of this test is compared, theoretically as well as numerically, with the corresponding generalized version of the two-sample Dempster (1958 Dempster , A. P. (1958). A high dimensional two sample significance test. Ann. Math. Statist. 29:995–1010.[Crossref] , [Google Scholar]) test, or more appropriately Bai and Saranadasa (1996 Bai , Z. , Saranadasa , H. ( 1996 ). Effect of high dimension: an example of a two sample problem . Statistica Sinica 6 : 311 – 329 .[Web of Science ®] , [Google Scholar]) test who gave its asymptotic version.     

A Parametric Bootstrap Approach for One-Way ANOVA Under Unequal Variances with Unbalanced Data

This research is to provide a solution of one-way ANOVA without using transformation when variances are heteroscedastic and group sizes are unequal. Parametric bootstrap test (Krishnamoorthy et al., 2007 Krishnamoorthy, K., Lu, F., Mathew, T. (2007). A parametric bootstrap approach for anova with unequal variances: Fixed and random models. Computational Statistics and Data Analysis 51:5731–5742.[Crossref], [Web of Science ®] , [Google Scholar]) has been shown to be competitive with many other methods when testing the equality of group means. We extend the parametric bootstrap algorithm to a multiple comparison procedure. Simulation results show that the parametric bootstrap approach works well for one-way ANOVA.     

Forward Moving Average Representations for MA Processes of Finite Order: Multivariate Stationary and Periodically Correlated

Soltani and Mohammadpour (2006 Soltani , A. R. , Mohammadpour , M. (2006). Moving average representations for multivariate stationary processes. J. Time Ser. Anal. 27(6):831–841.[Crossref], [Web of Science ®] , [Google Scholar]) observed that in general the backward and forward moving average coefficients, correspondingly, for the multivariate stationary processes, unlike the univariate processes, are different. This has stimulated researches concerning derivations of forward moving average coefficients in terms of the backward moving average coefficients. In this article we develop a practical procedure whenever the underlying process is a multivariate moving average (or univariate periodically correlated) process of finite order. Our procedure is based on two key observations: order reduction (Li, 2005 Li , L. M. ( 2005 ). Factorization of moving average spectral densities by state space representations and stacking . J. Multivariate Anal. 96 : 425 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]) and first-order analysis (Mohammadpour and Soltani, 2010 Mohammadpour , M. , Soltani , A. R. ( 2010 ). Forward moving average representation for multivariate MA(1) processes . Commun. Statist. Theory Meth. 39 : 729 – 737 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

On evaluating the performance of different forecasters

In this article, we have evaluated the performance of different forecasters and tested association between their performances for different pairs of variables. We have used three data sets of track records of professional U.S. economic forecasters participating in the Blue Chip consensus forecasting service (the data sets contain the root mean square errors (RMSE) of different forecasters for different years). To evaluate the performance of forecasters we have covered three well-known tests, namely the usual F test (cf. Fisher (1923 Fisher, R. A., Mackenzie, M. A. (1923). Studied in crop variation II. The manurial response of different potato. Journal of Agricultural Science 13:311–320. [Google Scholar])), Kruskal Wallis test (cf. Kruskal and Wallis (1952 Kruskall, W. H., Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of American Statistical Association 47:583–621. [Google Scholar])), and Extension of Median test (cf. Daniel (1990 Daniel, W. W. (1990). Applied Nonparametric Statistics. Duxbury Classic Series. (2nd Ed.), Boston. [Google Scholar])). To test the association between the forecaster's performances for different pairs of variables, we have considered Gini mean correlation coefficient r_g1 (cf. Yitzhaki, S., and Olkin, I. (1991 Yitzhaki, S., Olkin, I. (1991). Concentration indices and concentration curves, in K. Mosler and M. Scarsini (eds.), Stochastic Orders and Decisions under Risk, Institute of Mathematical Statistics: Lecture-Notes Monograph Series, 19, 1991, 380–392. [Google Scholar]) and Yitzhaki (2003 Yitzaki, S. (2003). Gini mean difference: A superior measure of variability for non normal distribution. Metron-International Journal of Statistics, LXI:285–316. [Google Scholar])), Modified rank correlation coefficient (cf. Zimmerman (1994 Zimmerman, D. W. (1994). A Note on modified rank correlation. Journal of educational and Behavioral Statistics 19:357–362. [Google Scholar])) and three modifications of Spearman rank correlation coefficient. We have observed that different forecasters do not necessarily offer same average performance. Moreover, an evidence of association between two criteria does not always lead us reaching at the same decision. The outcomes of the study may help the practitioners in selecting the best forecaster(s) for policymaking purposes.     

Complete convergence for coordinatewise asymptotically negatively associated random vectors in Hilbert spaces

In this article, we study the complete convergence for sequences of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. We also investigate that some related results for coordinatewise negatively associated random vectors in Huan, Quang, and Thuan (2014 Huan, N. V., N. V. Quang, and N. T. Thuan. 2014. Baum–Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Mathematica Hungarica 144(1):132–419.[Crossref], [Web of Science ®] , [Google Scholar]) still hold under this concept.     

A two-stage algorithm for computation of the exact distribution of sums of spacings

Based on the recursions in Huffer (1988 Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability 25:346–354. [Google Scholar]) and Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]), we present a two-stage algorithm and two specialized methods for evaluating the probabilities involving linear combination of spacings of special forms. The two-stage algorithm combines the advantages of marking algorithm in Huffer and Lin (1997 Huffer, F. W., Lin, C. T. (1997). Computing the exact distribution of the extremes of sums of consecutive spacings. Computational Statistics and Data Analysis 26:117–132. [Google Scholar]) and general algorithm in Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]). The proposed methods can analytically derive the exact expressions for some specific problems, and efficiently handle problems such as the distribution of the circular scan statistic and multiple coverage probabilities.     

A new estimator of finite population mean based on the dual use of the auxiliary information

When a sufficient correlation between the study variable and the auxiliary variable exists, the ranks of the auxiliary variable are also correlated with the study variable, and thus, these ranks can be used as an effective tool in increasing the precision of an estimator. In this paper, we propose a new improved estimator of the finite population mean that incorporates the supplementary information in forms of: (i) the auxiliary variable and (ii) ranks of the auxiliary variable. Mathematical expressions for the bias and the mean-squared error of the proposed estimator are derived under the first order of approximation. The theoretical and empirical studies reveal that the proposed estimator always performs better than the usual mean, ratio, product, exponential-ratio and -product, classical regression estimators, and Rao (1991 Rao, T.J. (1991). On certail methods of improving ration and regression estimators. Commun. Stat. Theory Methods 20(10):3325–3340.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (2009 Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2009). Improvement in estimating the population mean using exponential estimator in simple random sampling. Int. J. Stat. Econ. 3(A09):13–18. [Google Scholar]), Shabbir and Gupta (2010 Shabbir, J., Gupta, S. (2010). On estimating finite population mean in simple and stratified random sampling. Commun. Stat. Theory Methods 40(2):199–212.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Grover and Kaur (2011 Grover, L.K., Kaur, P. (2011). An improved estimator of the finite population mean in simple random sampling. Model Assisted Stat. Appl. 6(1):47–55. [Google Scholar], 2014) estimators.     

Size and Power of Recent Tests for Homogeneity in Exponential Mixtures

The article investigates diagnostic procedures for finite mixture models. The problem is to decide whether given data stem from an exponential distribution or a finite mixture of such distributions. Recently, three new test approaches have been proposed, the modified likelihood ratio test (MLRT) by Chen et al. (2001 Chen , H. , Chen , J. , Kalbfleisch , J. D. ( 2001 ). A modified likelihood ratio test for homogeneity in finite mixture models . Journal of the Royal Statistical Society, B 63 : 19 – 29 .[Crossref] , [Google Scholar]), the ADDS test by Mosler and Seidel (2001 Mosler , K. , Seidel , W. ( 2001 ). Testing for homogeneity in an exponential mixture model . Australian and New Zealand Journal of Statistics 43 : 231 – 247 . [Google Scholar]), and the D-test by Charnigo and Sun (2004 Charnigo , R. , Sun , J. ( 2004 ). Testing homogeneity in a mixture distribution via the l ² distance between competing models . Journal of the American Statistical Society 99 : 488 – 498 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The size and power of these tests are determined by Monte Carlo simulation and their relative merits are evaluated. We conclude that the ADDS test shows always not much less and under some alternatives, in particular lower contaminations, considerably more power than its competitors. Also, new tables for the ADDS test are provided.     

Successive comparisons between ordered normal means using isotonic regression estimators

In this study we discuss multiple comparison procedures for checking differences among a sequence of normal means with ordered restriction. Lee and Spurrier (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a multiple comparison procedure which tests the difference between two adjacent means using the difference of sample means. In this study we propose a multiple comparison procedure modifying Lee and Spurrier's (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) procedure using isotonic regression estimators instead of sample means. We determine the critical value for pairwise comparisons for a specified significance level. Furthermore, we formulate the power of the test. Finally, we give some numerical examples regarding critical values and power of the test intended to compare our procedure with Lee and Spurrier's (1995 Lee, R.E., Spurrier, J.D. (1995). Successive comparisons between ordered treatments. J. Stat. Plann. Inference 43:323–330.[Crossref], [Web of Science ®] , [Google Scholar]) procedure.     

A new asymmetric class of bivariate copulas for modeling dependence

In this paper, we propose an asymmetric class of bivariate copulas. This class is obtained through limiting properties of the extended copula introduced by Bekrizadeh, et al. (2015 Bekrizadeh, H., Parham, G. A., Zadkarami, M. R. (2015). Extending some classes of copulas; Applications. Ph.D. Thesis, University of Shahid Chamran, Ahvaz. [Google Scholar]), and includes some of known copulas. Some general formulas for well-known association measures and concepts of dependence of the proposed model are obtained. This paper highlights the usefulness of this new bivariate copula for modeling the interested variables whose marginal distribution effect on joint distribution isn't identical. We apply some subfamilies of this new class to model a dataset of medical science to show the superiority of presented model in comparison with the known copulas. These results will be investigated using simulation.     

Testing for sphericity in a two-way error components panel data model

This article is concerned with sphericity test for the two-way error components panel data model. It is found that the John statistic and the bias-corrected LM statistic recently developed by Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar], which are based on the within residuals, are not helpful under the present circumstances even though they are in the one-way fixed effects model. However, we prove that when the within residuals are properly transformed, the resulting residuals can serve to construct useful statistics that are similar to those of Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the newly proposed statistics perform well under the null hypothesis and several typical alternatives.     

A modified chi-square test for Bertholon model with censored data

In this work, we propose the construction of a chi-squared goodness-of-fit test in censored data case, for Bertholon model which can analyse various competing risks of failure or death. This test is based on a modification of the Nikulin-Rao-Robson (NRR) statistic proposed by Bagdonavicius and Nikulin (2011a Bagdonavicius, V., Nikulin, M. (2011a). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathématiques: Series I 349(3–4):219–223. [Google Scholar], 2011b Bagdonavicius, V., Nikulin, M. (2011b). Chi-squared goodness-of-fit test for right censored data. International Journal of Applied Mathematics and Statistics 24:30–50. [Google Scholar]) for censored data. We applied this test to numerical examples from simulated samples and real data.     

Multivariate Generalized Distributions of Order k

Abstract

The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1988. Multivariate distributions of order k. Statistics and Probability Letters, 7(3): 207–216.  [Google Scholar]). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1990. Multivariate distributions of order k, part II. Statistics and Probability Letters, 10(1): 29–35.  [Google Scholar]). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996 Sen, K. and Jain, R. 1996. “Cluster generalized negative binomial distribution”. In Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday Edited by: Borthakur, A. C. 227–241. New Delhi: New Age International Publishers.  [Google Scholar]). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997 Sen, K. and Jain, R. 1997. A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics—Theory and Methods, 26: 871–884. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.     

A Generalized Correlation Coefficient for the General Two-Treatment Crossover Design

This article extends the correlation methodology developed by Chinchilli et al. (2005 Chinchilli , V. M. , Phillips , B. R. , Mauger , D. T. , Szefler , S. J. ( 2005 ). A general class of correlation coefficients for the 2 × 2 crossover design . Biometr. J. 47 : 1 – 10 . [Google Scholar]) for the 2 × 2 crossover design to more complex crossover designs for clinical trials. We describe how the methodology can be adapted to a general type of two-treatment crossover design which includes either at least two sequences or at least two treatment periods or both. We then derive the asymptotic theory for the corresponding correlation statistics, investigate the statistical accuracy of the estimators via bootstrap analyses, and demonstrate their use with two real data examples.     

Population Attributable Fraction as a Measure of Heritability in Dichotomous Twin Data

In many genetic analyses of dichotomous twin data, odds ratios have been used to test hypotheses on heritability and shared common environment effects of a given disease (Lichtenstein et al., 2000 Lichtenstein , P. , Holm , N. , Verkasalo , P. , Iliadou , A. , Kaprio , J. , Koskenvuo , M. , Pukkala , E. , Skytthe , A. , Hemminki , K. ( 2000 ). Environmental and heritable factors in the causation of cancer . New England Journal of Medicine 343 : 78 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]; Ahlbom et al., 1997 Ahlbom , A. , Lichtenstein , P. , Malmström , H. , Feychting , M. , Hemminki , K. , Pedersen , N. L. ( 1997 ). Cancer in twins: genetic and nongenetic familial risk factors . Journal of the National Cancer Institute 89 : 287 – 93 . [Google Scholar]; Ramakrishnan et al., 1992 Ramakrishnan , V. , Goldberg , J. , Henderson , W. , Elsen , S. , True , W. , Lyons , M. , Tsuang , M. ( 1992 ). Elementary methods for the analysis of dichotomous outcomes in unselected samples of twins . Genetic Epidemiology 9 : 273 – 287 . [Google Scholar], 4). However, estimates of these two effects have not been dealt with in the literature. In epidemiology, the attributable fraction (AF), a function of the odds ratio and the prevalence of the risk factor has been used to describe the contribution of a risk factor to a disease in a given population (Leviton, 1973 Leviton , A. ( 1973 ). Definitions of attributable risk . American Journal of Epidemiology 98 : 231 . [Google Scholar]). In this article, we adapt the AF to quantify the heritability and the shared common environment. Twin data on cancer, gallstone disease and phobia are used to illustrate the applicability of the AF estimate as a measure of heritability.