Welcome to Yutaka Kano's Page!

TITLE Professor of Statistics, Doctor of Engineering
AFFILIATION Osaka University, Graduate School of Engineering Science, Division of Mathematical Science, joint appointment with Graduate School of Human Science
ADDRESS Toyonaka, Osaka 560-8531, JAPAN
EMAIL kano AT sigmath.es.osaka-u.ac.jp
RESEARCH INTEREST multivariate analysis; psychometrics; incomplete data analysis; structural equation modeling; graphical modeling; statistical causal inference
SERVICE Associate Editor of Psychometrika (2000-)
Chief Editor of Behaviormetrika (2000-2003)
Associate Editor of Journal of Multivariate Analysis (2002-)
Board of Trustees of the Psychometric Society (2002-2005) Associate Editor of Annals of the Institute of Statistical Mathematics(2003-)
Editorial Board Member of Advances in Data Analysis and Classification (2006-)

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[44] Shimizu, S. and Kano, Y. (2003). Examination of independence in independent component analysis. In New Developments in Psychometrics (Yanai, H. et al., Eds.), pp.665-672. Springer Verlag: Tokyo.

[43] Hyvarinen, A. and Kano, Y. (2003). Independent component analysis for non-normal factor analysis. In New Developments in Psychometrics (Yanai, H. et al., Eds.), pp.649-656. Springer Verlag: Tokyo.

[42] Kano, Y. and Azuma, Y. (2003). Use of SEM programs to precisely measure scale reliability. In New Developments in Psychometrics (Yanai, H. et al., Eds.), pp.141-148. Springer Verlag: Tokyo. [pdf file]

It is first pointed out that most often used reliability coefficient $\alpha$ and one-factor model based reliability $\rho$ are seriously biased when unique factors are covariated. In the case, the $\alpha$ is no longer a lower bound of the true reliability. Use of Bollen's formula (Bollen 1980) on reliability is highly recommended. A web-based program termed ``STERA" is developed which can make {\it stepwise}\, reliability analysis very easily with the help of factor analysis and structural equation modeling.

[41] Kano, Y. (in press). Rejoinder: Use of error covariances and role of specific factors.The Japanese Journal of Behaviormetrics. (In Japanese)

The author would like to express his special thanks to Dr.~Toyoda of Wasada University for organizing this exciting special issue and also to the three discussants who gave stimulating discussions to Kano (2002). In this rejoinder, special attentions are paid to error covariances and specific factors in the comparison between SEM and traditional methods. When a factor analysis model receives a poor fit, it does not make sense to simply remove important variables but inconsistent with the factor analysis model, as pointed out by the discussants. It is emphasized that it is a better way to allow for error covariances so as to recover the inconsistency rather than removing them. The model with error covariances guarantees invariance of estimation results over item selection.

The discussants pointed out that an important difference between a scale score (sum of items) and a measurement model by effect indicators in SEM is that a scale score includes specific factors whereas a measurement model excludes them. Practitioners could use scale scores when they are interested in effects of specific factors as well as a common factor. It is argued, however, that the error terms of effect indicators contain information on specific factors and thus their use as well as a common factor makes better inference than use of unidimensional scale scores because it can individually evaluate effects of the common factor and each of the specific factors.

Other related topics are also discussed.

[40] Kano Y. (in press). Does structural equation modeling outperform traditional factor analysis, analysis of variance and path analysis?The Japanese Journal of Behaviormetrics. (In Japanese)

It is well-known that structural equation modeling (SEM) can represent a variety of traditional multivariate statistical models. This fact does not, necessarily, mean that SEM should be used for the traditional models. It is often said that a general model can more hardly be handled than a specific model developed for a particular situation given. In this paper, we shall clarify relative advantages between SEM and several traditional statistical models. Rather than comparison in mathematical properties, we shall discuss how and when SEM outperforms corresponding traditional models {\it in practical situations}. A special attention is paid to statistical analysis of a scale score, a sum of indicator variables determined by factor analysis.

In concrete, we shall study relative advantages between between (i) confirmatory factor analysis and exploratory factor analysis, (ii) multiple indicator analysis and correlational and regression analysis of scale scores, (iii) analysis of factor means and analysis of variance of scale scores, and (iv) path analysis and multiple regression analysis.

[39] Kano, Y. (2002). Variable selection for structural models. Journal of Statistical Planning and Inference, Vol.108, No.1-2, 173-187. [pdf file]

Theory of variable selection for structural models that do not have clear dependent variables is developed. Theory is derived within the framework of the curved exponential family of distributions for observed variables. The idea of Rao's score test was taken to construct a test statistic for variable selection, and its statistical properties are examined. In particular, the test statistic is shown to have asymptotic {\it central}\, chi-square distribution under a kind of {\it alternative}\, hypothesis. This fact will provide an evidence for excellent performance of the score statistic for real data sets.

[38] Kano, Y. (2000). Towards transition of the statistical paradigm: Statisticians should make significant collaborations with applied researchers. Journal of the Japan Statistical Society, Vol.30, No.3, 305-314 (In Japanese).

[37] Kano, Y. (2001). Structural equation modeling for experimental data. In Structural Equation Modeling: Present and Future [A Festschrift in honor of Karl Joreskog] (Eds., Bob Cudek, Stephen du Toit and Dag Sorbom), pp.381-402. SSI: Chicago.
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We first review the use of structural equation modeling (SEM) for the analysis of experimental data. Typical examples include ANOVA, ANCOVA and MANOVA with or without a covariance structure. SEM for those experimental data is a mean and covariance structure model in multiple populations with a common covariance matrix. Such analyses can be implemented under the assumption that all observed variables be distributed as normal including fixed-effect exogenous variables, which denote levels of factors for example. Theoretical basis for the usage, based on conditional (likelihood) inference, is explicitly explained.

A bias of a path coefficient estimate particularly in standardized solutions is pointed out which comes from the fact that variance estimates of dependent variables contain variation of means.

Statistical power of several testing procedures concerning mean vectors across several populations are examined, when a factor model can be assumed for observed variables. The procedures considered here are MANOVA, a mean and covariance structure model implemented by SEM, and ANOVA of a factor score or a weighted sum of observed variables. The SEM is shown to be the most powerful tool in this context.

[36] Kano, Y. and Harada, A. (2000). Stepwise variable selection in factor analysis. Psychometrika. Vol.65, No.1, 7-22.

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.

In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain number $p$ of observed variables (the $p$-variable model is labeled the current model), simple formulas for fit measures such as chi-square, GFI, CFI, IFI and RMSEA are provided for models obtained by adding an external variable (so that the number of variables is $p+1$) and for those by deleting an internal variable (so that the number is $p-1$), provided that the number of factors is held constant.

A program {\sl SEFA}\, (Stepwise Exploratory Factor Analysis) is developed to actually obtain a list of these fit measures for all these models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.

The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.

Kew words: Backward elimination, forward selection, goodness-of-fit measure, Lagrange Multiplier test, likelihood ratio test, stepwise variable selection, Wald test, World Wide Web (WWW).

[35] Kano, Y. (1999). Delta method approach in a certain irregular condition. Communications in Statistics, Theory and Methods. Vol.28, Nos. 3&4, 789-807.
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Let $\bfX_n$ be a sequence of random $p$-vectors such that $a_n(\bfX_n-\bfb)\L \bfZ$, where $a_n\nearrow\infty$, $\bfb\in R^p$ and $\bfZ$ is a continuously distributed random $p$-vector. Let $\bff(\cdot)$ be a measurable mapping from a domain of $R^p$ to $R^q$, where the domain may not include $\bfb$, i.e., $\bff(\bfb)$ may not be defined. Under this setup, we study the asymptotic distribution of $\bff(\bfX_n)$. Two theorems are developed to obtain the asymptotic distribution. Comprehensive examples are provided to show when and where such an irregular situation takes place and to illustrate the usefulness of these theorems. The examples include the problem of choosing the number of components and noniterative estimation in factor analysis.

[34] Kano, Y. (1998). More higher order efficiency. Journal of Multivariate Analysis. Vol. 67, 349-366.
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Based on concentration probability of estimators about a true parameter, third-order asymptotic efficiency of the first-order bias-adjusted MLE within the class of first-order bias-adjusted estimators has been well established in a variety of probability models. In this paper we consider the class of second-order bias-adjusted Fisher consistent estimators of a structural parameter vector on the basis of an i.i.d.~sample drawn from a curved exponential-type distribution, and study the asymptotic concentration probability, about a true parameter vector, of these estimators up to the fifth-order. In particular, (i) we show that third-order efficient estimators are always fourth-order efficient; (ii) a necessary and sufficient condition for fifth-order efficiency is provided; and finally (iii) the MLE is shown to be fifth-order efficient.

[33] Kano, Y. (1998). Improper solutions in exploratory factor analysis: Causes and treatments. In Advances in Data Sciences and Classification (Eds Rizzi, A., Vichi, M. and Bock, H.), pp. 375-382: Springer, Berlin.
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There are many causes of occurrence of improper solutions in factor analysis. Identifying potential causes of the improper solutions gives very useful information on suitability of the model considered for a data set.

This paper studies possible causes of improper solutions in exploratory factor analysis, focusing upon (A) sampling fluctuations, (B) model underidentifiable and (C) model unfitted, each having several more detailed items. We then give a checklist to identify the cause of the improper solution obtained and suggest a method of reanalysis of the data set for each cause.

[32] Kano, Y. (1997). Beyond third-order efficiency. Sankhya. Vol. 59, Part 2, 179-197.
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Fifth-order (asymptotic) efficiency of the second-order bias-corrected MLE, minimizing the $n^{-3}$ term of an expansion of the quadratic risk of Fisher-consistent estimators bias-corrected similarly, is established in a general curved exponential family with a structural parameter vector. A characterization theorem of the MLE in terms of its higher-order derivatives is provided, and an alternative bias correction factor is proposed. Both the characterization and the new bias correction play an important role in proving the fifth-order efficiency. The matrix form symmetric tensor and higher-order derivatives are utilized, rather than usual elementwise tensors with Einstein's convention, to derive all the results of this article.

[Note: There is a technical report [101] that describes the detailed techinical proofs of the propositions and lemmas of this paper.]

[31] Kano, Y. (1997). Exploratory factor analysis with a common factor with two indicators. Behaviometrika, Vol.24, No.2, 129-145.
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Any exploratory factor analysis model requires at least three indicators (observed variables) for each common factor to ensure model identifiability. If one makes exploratory factor analysis for a data set in which one of common factors would have only two indicators in its population, one would encounter difficulties such as improper solutions and nonconvergence of iterative process in calculating estimates.
In this paper, we first develop conditions for {\it partial} identifiability of the remaining factor loadings except for a factor loading vector which relates to a common factor with only two indicators. Two models for analyzing such data sets are then proposed with the help of confirmatory factor analysis and covariance structure analysis. The first model is an exploratory factor analysis model that permits correlation between unique factors; the second model is a kind of confirmatory factor model with equal factor loadings. Two real data sets are analyzed to illustrate usefulness of these models.

[30] Aoshima, M. and Kano, Y. (1997). A note on robustness of two-stage procedure for a multivariate compounded normal distribution. Sequential Analysis, Vol.16, No.2, 175-187.

In normal populations, Healy (1956) and Takada (1988) have developed two-stage methods for inference on the mean vector, with a confidence region with a specified maximum diameter or with a specified risk. The methods are shown to remain valid even under a multivariate compounded normal distribution, which includes the normal distribution as a special case. A sort of super efficiency on the average required sample number of the procedure is observed. This fact has never appeared in the normal population.

[29] Yuan, Ke-Hai, Bentler, P. M. and Kano, Y. (1997). On averaging variables in a confirmatory factor analysis model. Behaviormetrika, Vol.24, No.1, 71-83.

The normal theory maximum likelihood and asymptotically distribution free methods are commonly used in covariance structure practice. When the number of observed variables is too large, neither method may give reliable inference due to bad condition numbers or unstable solutions. The main existing solution to the problem of high dimension is to build a model based on marginal variables. This practice is inefficient because the omitted variables may still contain valuable information regarding the structural model. In this paper, we propose a simple method of averaging proper variables which have similar factor structures in a confirmatory factor model. The effects of averaging variables on estimators and tests are investigated. Conditions on the relative errors of the measured variables are given that verify when a model based on averaged variables can give better estimators and tests than one based on omitted variables. Our method is compared to the method of variable selection based on mean square error of predicted factor scores. Some aspects related to averaging, such as improving the normality of observed variables, are also discussed.

[28] Kano, Y. (1996). Fourth and fifth order efficiency: Fisher information. In Probability Theory and Mathematical Statistics (Watanabe, S. et.al. EDs.) pp. 193-200. World Scientific: Singapore. (dvi file) (ps file)

Loss of (Fisher) information, whose concept was defined by Fisher, is one of important measures of asymptotic efficiency. Based on Fisher information, this paper studies fourth and fifth order asymptotic efficiency of estimators in a curved exponential family of distributions with a structural parameter vector. In particular, we show that the bias-corrected MLE with a certain bias correction factor is fourth and fifth order efficient in a class of Fisher consistent estimators bias-corrected similarly.

[27] Kano, Y. (1996). Third-order efficiency implies fourth-order efficiency. Journal of Japan Statistical Society, Vol.26, No.1, 101-117. (dvi file) (ps file)

Takeuchi [37], Takeuchi and Akahira [38] and Pfanzagl [27] among others proved that {\it any\,} first-order efficient estimators are second-order efficient. Many authors e.g., Ghosh [15], have conjectured that {\it any} third-order efficient estimators are fourth-order efficient. Based on concentration probability of estimators about a true parameter, this paper gives a positive answer to the conjecture in a curved exponential family with multi-structural parameters. It is seen that choice of bias-correction factors is critical.

[26] Kano, Y. (1995). An asymptotic expansion of the distribution of Hotelling's $T^2$-statistic under general distributions. American Journal of Mathematical and Management Sciences, Vol.15, No.3-4, 317-341.

Distribution theory in nonnormal populations is important but is not fully exploited particularly in multivariate analysis. In this paper we derive an asymptotic expansion of the distribution of Hotelling's multivariate $T^2$-statistic under general distributions. Our general expansion specializes the existing expansions under elliptical and normal distributions.
The previous research on robustness of $T^2$ to violation of normal assumption, based on Monte Carlo study, concludes that nonnormality of an underlined population influences substantially upon the distribution of $T^2$ for small or medium samples and that the third-order cumulants of the underlined distribution affects $T^2$ much more seriously than do the fourth-order cumulants. The derived formula is used to provide theoretical grounds for the experimental results.
Matrix manipulations such as Kronecker products and symmetric tensors are utilized to derive all the results, rather than usual elementwise tensors with Einstein's convention.

[25] Ihara, M. and Kano, Y. (1995). Identifiability of full, marginal, and conditional factor analysis. Statistics & Probability Letters, Vol.23, No.4, 343-350.

Identifiability of full factor analysis model for $\bfx=[X_1,\bfx_2^T]^T$ is discussed, when the marginal model for $\bfx_2$ and/or the conditional model for $bfx_2$ given $X_1$ conform to factor analysis models. Two numerical examples are given for illustrative purposes.

[24] Kano, Y.(1994). Consistency property of elliptical probability density functions. Journal of Multivariate Analysis, Vol.51, 343-350.

Several conditions are established under which a family of elliptical probability density functions possesses a preferable consistency property. The consistency property ensures that any marginal distribution of a random vector whose distribution belongs to a specific elliptical family also belongs to the family. Elliptical distributions with the property must be a mixture of normal distributions.

[23] Berkane, M., Kano, Y. and Bentler, P. M. (1994). Pseudo maximum likelihood estimation in elliptical theory: Effects of misspecification. Computational Statistics and Data Analysis. Vol.18, 255-267.

Recently, robust extensions of normal theory statistics have been proposed to permit modeling under a wider class of distributions (e.g., Taylor, 1992). Let $X$ be a $p\times 1$ random vector, $\mu$ a $p\times 1$ location parameter, and $V$ a $p\times p$ scatter matrix. Kano et al. (1993) studied inference in the elliptical class of distributions and gave a criterion for the choice of a particular family within the elliptical class to best describe the data at hand when the latter exhibit serious departure from normality. In this paper, we investigate the criterion for a simple but general set-up, namely, when the operating distribution is multivariate $t$ with $\nu$ degrees of freedom and the model is also a multivariate $t$-distribution with $\alpha$ degrees of freedom. We compute the exact inefficiency of the estimators of $\mu$ and $V$ based on that model and compare it to the one based on the multivariate normal model. Our results provide evidence for the choice of $\nu=4$ proposed by Lange et al. (1989). In addition, we give numerical results showing that for fixed $\nu$, the inflation of the variance of the pseudo maximum likelihood estimator of the scatter matrix, as a function of the hypothesized degrees of freedom $\alpha$, is increasing in its domain.

[22] Kano, Y. and Ihara, M. (1994). Identification of inconsistent variates in factor analysis. Psychometrika, Vol.59, 5-20.

When some of observed variates do not conform to the model under consideration, they will have a serious effect on the results of statistical analysis. In factor analysis the model with inconsistent variates may result in improper solutions. In this article a useful method for identifying a variate as inconsistent is proposed in factor analysis. The procedure is based on the likelihood principle. Several statistical properties such as the effect of misspecified hypotheses, the problem of multiple comparisons, and robustness to violation of distributional assumptions are investigated. The procedure is illustrated by some examples.

[21] Kano, Y., Bentler, P. M. and Mooijaart, A. (1993). Additional information and precision of estimators in multivariate structural models. In Statistical Sciences and Data Analysis: Proceedings of the Third Pacific Area Statistical Conference, (K. Matusita, T. Hayakawa, et al., Eds.) pp. 187-196. VSP International Science Publisher: Zeist, The Netherlands.
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This paper investigates the effect of additional information upon parameter estimation in multivariate structural model. It is shown that the asymptotic covariances of estimators based on a model with additional variables are smaller than those based on a model with no additional variables, where the estimation methods employed are the methods of maximum likelihood and minimum chi-square. Some applications to moment structure models are provided.

[20] Kano, Y. (1993). Asymptotic properties of statistical inference based on Fisher consistent estimators in the analysis of covariance structures. In Proceedings of the International Workshop on Statistical Modelling and Latent Variables, (K. Haagen, D. J. Bartholomew and M. Deistler, Eds.), pp.173-190. Elsevier Science Publisher: Amsterdam, The Netherlands.
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The methods of maximum likelihood (ML) and generalized least squares (GLS) under the normality assumption are often used for inference on covariance structures, and asymptotic properties and robustness of the statistical inference have been extensively studied. In this article, we generalize these results to inference based on Fisher consistent (FC) estimators which include simple least squares (LS) and noniterative estimation methods as well as ML and GLS. Although the LS and noniterative methods do not yield asymptotically efficient estimators under the normality, for small or moderate samples they are often superior to efficient estimators in mean squared error and less often result in so-called improper solutions. This shows that there do exist cases where such inefficient inference should be made rather than ML and GLS. Thus, the extension to be described here is important. Furthermore, a key relation shown from a property of the FC estimators makes the derivation of asymptotics of the inference very easy and comprehensive. The asymptotic efficiency of the MLE within the class of FC estimators is proved under a situation where the fourth-order moments of observations may not be finite.

[19] Kano, Y., Berkane, M. and Bentler, P. M. (1993). Statistical inference based on pseudo maximum likelihood estimators in elliptical populations. Journal of the American Statistical Association --- Theory and Methods, Vol.88, 135-143.

In this article we develop statistical inference based on the method of maximum likelihood in elliptical populations with an unknown density function. The method assuming the multivariate normal distribution, using the sample mean and the sample covariance matrix, is basically correct even for elliptical populations under a certain kurtosis adjustment, but is not statistically efficient especially when the kurtosis of the population distribution has more than moderate values. On the other hand, several methods of statistical inference assuming a particular family (e.g., multivariate $T$-distribution) of elliptical distributions have been recommended as a robust procedure against outliers or distributions with heavy tail. Such inference also will be important in order to keep high efficiency of statistical inference in elliptical populations. In practice, however, it is very difficult to choose an appropriate family of elliptical distributions, and one may misspecify the family. Furthermore, extra-parameters (e.g., other than means and covariances) may make computation heavy. Here we investigate the method of maximum likelihood assuming a particular family of elliptical distributions with extra-parameters replaced by inexpensive estimators when the assumed family may be misspecified. Consistency and asymptotic normality of the estimators are proved, and the asymptotic equivalence among the likelihood ratio, Wald and Score test statistics and their chi-squaredness under a constant correction are shown. Two easy methods of estimating extra-parameters are proposed. A criterion on how to choose a family among competing elliptical families is also provided.

[18] Ihara, M. and Kano, Y. (1992). Asymptotic equivalence of uniqueness estimators in marginal and conditional factor analysis models. Statistics & Probability Letters, Vol.14, 337-341.

It is shown that the maximum likelihood and generalized least-squares estimators of unique variances in the conditional model are asymptotically equivalent to those in the marginal model in factor analysis. The asymptotic covariance matrices of the estimators are expressed in matrix form.

[17] Hu, Li-tze, Bentler, P. M. and Kano, Y. (1992). Can test statistics in covariance structure model be trusted? Psychological Bulletin, Vol.112, 351-362.

Covariance structure analysis uses $\chi^2$ goodness of fit test statistics whose adequacy is not known. Scientific conclusions based on models may be distorted when researchers violate sample size, variate independence, and distributional assumptions. The behavior of 6 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study. The tests performed drastically differently under 7 distributional conditions at 6 sample sizes. Two normal-theory tests worked well under some conditions but completely broke down under other conditions. A test that permits homogeneous nonzero kurtoses performed variably. A test that permits heterogeneous marginal kurtoses performed better. A distribution-free test performed spectacularly badly in all conditions at all but the largest sample sizes. The Satorra-Bentler scaled test statistic performed best overall.

[16] Kano, Y. (1992). Robust statistics for test-of-independence and related structural models. Statistics & Probability Letters. Vol.15, 21-26.

Recent research of asymptotic robustness shows that the likelihood ratio (LR) test statistic for test-of-independence based on normal theory remains valid for a general case where only independence is assumed. In contrast, under elliptical populations the LR statistic is correct if a kurtosis adjustment is made. Thus, the LR statistic itself is available for the first case, whereas a certain correction is needed for the second framework, which is seriously inconvenient for practitioners. In this article, we propose an alternative adjustment to the LR statistic which can be utilized for both of the distribution families. Theory is derived in the context of general linear latent variate models.

[15] Bentler, P. M., Berkane, M. and Kano, Y. (1991). Covariance structure analysis under a simple kurtosis model. In Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, (E. M. Keramidas and S. M. Kaufman, Eds.) pp. 463-465. Interface Foundation of North America: VA.

A model for the relation between multivariate fourth-order central moments of a set of variables and the marginal kurtoses and covariances among these variables is used to produce an estimator for covariance structure analysis that is asymptotically efficient and yields an asymptotic $\chi^2$ goodness of fit test of the covariance structure while substantially reducing the computations. When the kurtoses of the variables are equal, the method reduces to one based on multivariate elliptical distribution theory, and, when there is no excess kurtosis, to one based on multivariate normal distribution theory.

[14] Kano, Y. (1991). The asymptotic distribution of a noniterative estimator in exploratory factor analysis. The Annals of Statistics 19, 272-282.

This paper presents the asymptotic distribution of Ihara and Kano's non-iterative estimator of the uniqueness in exploratory factor analysis. When the number of factors is overestimated, the estimator is not a continuous function of the sample covariance matrix and its asymptotic distribution is not normal, but consistency holds. It is also shown that the first-order moment of the asymptotic distribution does not exist.

[13] Kano, Y. (1990). Statistical inference in factor analysis: Recent Developments. The Japanese Journal of Behaviormetrics, Vol.18, 3-12. (in Japanese)

Several methods of statistical inference in factor analysis are first reviewed under the normality assumption as well as the case where noassumption on the population distribution is made. Normal theory inference, whether it is asymptotically efficient, is shown to be robust against a wide class of distributions which may be encountered in many practical fields. For small or medium sample sizes, simple procedures of estimation, such as simple least-squares and noniterative methods, are recommended, whereas the method of maximum likelihood or other asymptotically efficient ways should be utilized for large sample.

[12] Kano, Y., Berkane, M. and Bentler, P. M. (1990). Covariance structure analysis with heterogeneous kurtosis parameters. Biometrika, Vol.77, 575-585.

This paper discusses the analysis of covariance structure in a wide class of multivariate distributions whose marginal distributions may have heterogeneous kurtosis parameters. Elliptical distributions often used as a generalization of the normal theory are members of this class. It is shown that a simple adjustment of the weight matrix of normal theory, using kurtosis estimates, results in an asymptotically efficient estimator of structural parameters within the class of estimators that minimize a general discrepancy function. Results are obtained for arbitrary covariance structures as well as those that meet a scale invariance assumption. Two real data sets are analyzed for illustrative purpose.

[11] Kano, Y. (1990). Noniterative estimation and the choice of the number of factors in exploratory factor analysis. Psychometrika, Vol.55, 277-291.

Based on the usual factor analysis model, this paper investigates the relationship between improper solutions and the number of factors, and discusses the properties of the non-iterative estimation method of Ihara and Kano in exploratory factor analysis. The consistency of the Ihara and Kano estimator is shown to hold even for an overestimated number of factors, which provides a theoretical basis for the rare occurrence of improper solutions, and for a new method of choosing the number of factors. The comparative study of their estimator and that based on maximum likelihood is carried out by a Monte Carlo experiment.

[10] Bentler, P. M. and Kano, Y. (1990). On the equivalence of factors and components. Multivariate Behavioral Research, Vol.25, 67-74.

Some advantages of the factor analysis model over component analysis are reviewed. We prove that principal components and factor analysis can yield equivalent results under a certain condition. Our proof provides a theoretical explanation for an empirical result obtained by Velicer and Jackson. Relevant results by Guttman, Harris and Kaiser are noted.

[9] Kano, Y. (1990). Comparative studies of non-iterative estimators based on Ihara and Kano's method in exploratory factor analysis. Communications in Statistics Part A, Vol.19, 431-444.

In a factor analysis model, the asymptotic variance of the non-iterative estimator of Ihara and Kano (1986) is first provided, and five kinds of estimators based on Ihara and Kano's method are constructed by using the asymptotic result. These estimators and that based on the maximum likelihood are compared both theoretically and experimentally. In conclusion, the arithmetic mean of some Ihara and Kano estimators is recommended as a uniqueness estimator at least for small and medium sample sizes.

[8] Kano, Y. (1989). A new estimation procedure using G-inverse matrix in factor analysis. Mathematica Japonica, Vol.34, 43-52.

A non-iterative estimator using g-inverse matrix is proposed in factor analysis, which is a generalization of Ihara and Kano's estimator. The amount of calculation of the present estimate is much less than that of traditional estimates.

[7] Kano, Y. and Shapiro, A. (1987). On asymptotic variances of uniqueness estimators in factor analysis. South African Statistical Journal, Vol.21, 131-139.

It is shown that the asymptotic variance of uniqueness estimators in factor analysis decrease as new observed variates are added to the model while the number of factors is held fixed. The limit form of the associated asymptotic covariance matrix is calculated.

[6] Ihara, M. and Kano, Y. (1986). A new estimator of the uniqueness in factor analysis. Psychometrika, Vol.51, 563-566.

A closed form estimator of the uniqueness (unique variances) in factor analysis is proposed. It has analytically desirable properties such as consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.

[5] Kano, Y. (1986). A condition for the regression predictor to be consistent in a single common factor model. British Journal of Mathematical and Statistical Psychology, Vol.39, 221-227.

This paper investigates the prediction of a common factor in a single common factor model with infinite items when the structural parameter vector (factor loadings and unique variances) is unknown. A condition in terms of the sample size n and the number of items p is established under which the regression predictor for a unique common factor, in which the parameter vector is replaced by the least squares estimator, converges to it in quadratic mean. The condition is that $p$ goes to infinity and $p^2/n$ goes to zero under some mild assumptions.

[4] Kano, Y. (1986). Consistency conditions on the least squares estimator in single common factor analysis model. Annals of the Institute of Statistical Mathematics, Vol.39, 57-68.

This paper is concerned with the consistency of estimators in a single common factor analysis model when the dimension of the observed vector is not fixed. In the model, several conditions on the sample size n and the dimension p are established for the least squares estimator (LSE) to be consistent. Under some assumptions, the condition that $p/n$ goes to zero is necessary and sufficient for the LSE to converge in probability to the true value. A sufficient condition for almost sure convergence is also given.

[3] Kano, Y. (1986). Conditions on consistency of estimators in covariance structure model. Journal of the Japan Statistical Society, Vol.16, 75-80.

This paper presents a condition that estimators in a covariance structure model are consistent weekly (strongly), which is equivalent to shapiro's condition. The condition is composed of three parts, each of which is simpler and is checked more easily. This result applies to a proof of consistency of estimators in a factor analysis model. The population value of a factor analysis model is given which does not admit any consistent estimator. This fact suggests that the proofs of consistency by the previous authors are not complete.

[2] Kano, Y. (1984). Construction of additional variables conforming to a common factor model. Statistics & Probability Letters, Vol.2, 241-244.

It is shown that if a random p-vector x conforms to an $r$-common factor model and an external variable $Z$ is given, then there exists a family of linear combinations $X$ of $\bx$ and $Z$ such that $[\bx':X]'$ conforms to an $r$-common factor model. We can choose $X$ such that the revised model has arbitrariness of factor indeterminacy which is smaller than any specified small value.

[1] Kano, Y. (1983). Consistency of estimators in factor analysis. Journal of the Japan Statistical Society, Vol.13, 137-144.

In factor analysis, both the maximum likelihood estimator and the generalized least squares estimator for the structural parameters (i.e., factor loadings and unique variances) are shown to be consistent weakly and strongly under Anderson and Rubin's sufficient condition for weak identifiability.