The following is an annotated bibliography of articles written for statistics users. Most of the articles assume the reader has graduate level statistics course, but articles that rely heavily on matrix algebra have been avoided. The articles are organized into the following topics:
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Cochran, W. G. (1957). Analysis of covariance: Its nature and uses. Biometrics, 13, 261-281.
This paper discusses the statistical model of ANCOVA, the purposes to which it is applied, the standard computational formulae, the nature of covariance adjustment, assumptions, multiple covariance, and more complex designs incorporating covariance. The paper uses univariate notation, but the examples are biologically based, making it a more difficult read for those in the social sciences. Furthermore, the discussion is statistically oriented rather than conceptually oriented.
Lord, F. M. (1969). Statistical adjustments when comparing preexisting groups. Psychological Bulletin, 72, 336-337.
This paper is a two page, common sense explanation to demonstrate why ANCOVA does not provide the appropriate adjustment to compensate for pre-existing differences between non-experimental groups.
Maxwell, S. E. (1984). Another look at ANCOVA versus blocking. Psychological Bulletin, 95, 136-147.
This paper is a Monte Carlo study on blocking versus repeated measures designs. The article discusses the considerations one must take into account when choosing which model to use, if it is possible to use either. This is a readable paper that contains no matrix algebra.
Maxwell, S. E., Delaney, H. D., & O'Callaghan, M. F. (1993). Analysis of covariance. In L. K. Edwards (Ed.), Applied analysis of variance in behavioral science. Statistics: Textbooks and monographs, Vol. 137 (pp. 63-104). Inc, New York, NY: Marcel Dekker.
This paper gives a brief history of ANCOVA, and then discusses ANCOVA in the context of the general linear model. The authors then provide a numerical example, and discuss the assumptions of ANCOVA. Then four advanced topics are covered: 1) factorial ANCOVA, 2) ANCOVA with heterogeneous regression slopes, 3) nonparametric ANCOVA, and 4) alternative design procedures. The authors then discuss how to design an ANCOVA study, and how to choose covariates. This paper is quite theoretical and complex, but contains no matrix algebra.
Overall, J. E., & Woodward, J. A. (1977). Nonrandom assignment and the analysis of covariance. Psychological Bulletin, 84, 588-594.
This article is on nonrandom assignment and analysis of covariance. It argues that "ANCOVA results are NOT biased (a) when treatment groups differ in mean scores on the covariates, (b) when the covariate is not perfectly reliable, or (c) when the within and between groups regressions are not equal". ANCOVA provides unbiased estimates of true treatment effects on the dependent variable if the treatments do not affect the covariate and if the subjects are either randomly assigned or non-randomly assigned on the basis of the observed covariate measured with error. This article focuses more on design issues than statistical issues, but still contains a fair amount of mathematical formulae.
Himmelfarb, S. (1975). What do you do when the control group doesn't fit into the factorial design. Psychological Bulletin, 82, 363-368.
This article covers what to do if your control group doesn't fit into the factorial design. The paper presents two simple solutions to this problem. One is to subtract the control group mean from the experimental mean scores and then perform a multi factor ANOVA on the difference scores. This requires the use of an error term that takes into account the variance in the control group. This error term can be obtained by running a regular ANOVA on experimental plus control groups. It is a somewhat complicated procedure but it is clearly described. Method two: When one factor is a quantitative variable and you want to include control in trend analysis, the authors recommend a procedure involving an artificial crossing of the control conditions with the other factors in the design by splitting the control condition into several groups. The article is clearly written, and uses univariate notation, and therefore does not require knowledge of matrix algebra.
Humphreys, L. G. (1978). Doing research the hard way: Substituting analysis of variance for a problem in correlational analysis. Journal of Educational Psychology, 70, 873-876.
This paper is on the dangers of dichotomizing individual difference variables using median splits (e.g., loss of power, interpretive errors), and demonstrates these dangers using a numerical example. The paper contains no matrix algebra, and is appropriate for an advanced undergraduate student.
Lubin, A. (1961). The interpretation of significant interaction. Educational and Psychological Measurement, 21, 807-817.
This paper is on the interpretation of significant interaction in ANOVA. The paper discusses how interactions in ANOVA complicate the tests if main effects. The paper then discusses the distinction between ordinal and disordinal interactions. Although the issues discussed in this paper are not advanced, the paper is not very easy to read, and despite the title, it does not really aid understanding how to interpret the interaction effect.
Macnaughton, D. B. (1997) Which Sums of Squares Are Best In Unbalanced Analysis of Variance? Available on-line at http://www.matstat.com/ss.htm.
This paper evaluates the HTO and HTI methods of computing ANOVA sums for squares for fulfilling the two uses of the ANOVA statistical tests. Evaluation is in terms of the hypotheses being tested and relative power. It is concluded that (contrary to current practice) the HTO method is generally preferable when a researcher wishes to test the results of an experiment for evidence of relationships between variables. (summary provided by author)Rotton, J. (1978). Power tables for analysis of variance. Educational & Psychological Measurement, 38, 213-229.
This paper presents power tables for cases when you want to use alpha levels equal to or higher than .05, and up to .50. The paper provides a power table and instructions on how to use it.
Darlington, R. B., Weinberg, S. L., & Walberg, H. J. (1973). Canonical variate analysis and related techniques. Review of Educational Research, 43, 433-454.
This paper on canonical correlation, "describes several statistical techniques for studying different questions about the relations between two sets of variables, and specifies the different problems for which each technique is most appropriate. It identifies several problems for which canonical variate analysis has been prominently suggested but for which other statistical techniques are more appropriate, but argues that there are valid and important used for CVA which are generally ignored." The paper addresses 3 kinds of questions about the relations between two sets of variables:
This is a good, clearly written paper containing no matrix algebra, but a graduate course in multivariate statistics and a knowledge of multiple regression would be helpful.
- "Questions about the number and nature of mutually independent relations between two sets of variables.
- Questions about the degrees of overlap or redundancy between two sets.
- Questions about similarity between the two within set correlation or covariance matrices."
Thompson, B. (1991). A primer on the logic and use of canonical correlation analysis. Measurement & Evaluation in Counseling & Development, 24, 80-95.
This article on canonical analysis (CA) "a) explains the logic of CA, b) illustrates that CA is a general parametric method that subsumes other methods, c) offers some guidelines regarding correct use of this analytic approach," and provides step by step instructions on how to interpret the results of CA. The paper contains no equations, and no matrix algebra.
Tucker, R. K., & Chase, L. J. (1980). Canonical correlation. In P. R. Monge & J. N. Cappella (Ed.), Multivariate techniques in human communication research (pp. 205-228). New York, NY: Academic Press.
This paper demonstrates canonical correlation and its interpretation with an example. It then reviews some research designs (naturally or experimentally paired subjects, test relationships, academic practice, experimental artifact research, mass communication and marketing research, test and message construction) which are well suited for canonical analysis. It then goes over some research examples and covers theoretical and practical problems, including interpretability of canonical structures, the distinction between canonical weights and canonical loadings, choosing number of variables to include, sample size, the potential for method specific results , and cross validation and replication. The paper contains minimal matrix algebra.
Weiss, D. J. (1972). Canonical correlation analysis in counseling psychology research. Journal of Counseling Psychology, 19, 241-252.
This article is on canonical analysis (CA), specifically as it applies to counseling research. After providing a detailed description of what CA is, the authors describe the kinds of research applications to which it is suited. The paper then provides a discussion of how to interpret the results of CA (e.g., canonical variates, beta weights, canonical correlations, maximum and successive canonical correlations, significance tests, correlations of variables and variates, redundancy, cross validation, and replication). This is a highly readable, conceptual paper that contains no matrix algebra, and only a couple of simple formulae. Its only limitation is that it does not provide any numerical examples. This article would be excellent as a guide for the reader of studies that use CA, or as an introduction to CA for those who are using CA for the first time.
Davis, D. J. (1969). Flexibility and power in comparisons among means. Psychological Bulletin, 71, 441-444.
This paper on planned contrasts argues that the recommendation often made that planned contrasts in ANOVA be orthogonal is unwise, since there is no such requirement made of post-hoc contrasts. The author argues that one should set up to test the contrasts that are of interest with respect to the research questions, regardless of whether these contrasts are orthogonal or not. The author presents an index, E, of the relative power gained or lost by using k planned contrasts rather than an arbitrary number of post-hoc comparisons.
Jaccard, J., Becker, M. A., & Wood, G. (1984). Pairwise multiple comparison procedures: A review. Psychological Bulletin, 96, 589-596.
This paper reviews studies that have evaluated pair-wise multiple comparison procedures (PMC), under optimal and suboptimal (assumptions violated) conditions. For each type of design, the authors review studies of PMC procedures and then conclude and summarize on what the literature tells us about which are the best procedures to use under varying conditions.
Ramsey, P. H. (1993). Multiple comparisons of independent means. In L. K. Edwards (Ed.), Applied analysis of variance in behavioral science. Statistics: Textbooks and monographs, Vol. 137 (pp. 25-62). Inc, New York, NY: Marcel Dekker.
This paper covers a priori and post-hoc methods of testing contrasts in ANOVA. The paper contains no matrix algebra, but is a graduate level paper.
Huberty, C. J. (1975). Discriminant analysis. Review of Educational Research, 45, 543-598.
This paper is a critical review of papers on how to carry out discriminant analysis, how to test importance of variables, and how to select the variables that provide the greatest discrimination between groups. This is an advanced paper that requires knowledge of matrix algebra. Topics covered include: discrimination, linear discriminant functions, assumptions, interpretations of LDF's, variable selection, generalizability, specific uses of LDF's, discrimination in the two group case, discrimination in research applications, estimation, classification, how to report results, general references, and computer programs.
McLaughlin, M. L. (1980). Discriminant analysis in communication research. In P. R. Monge & J. N. Cappella (Ed.), Multivariate techniques in human communication research (pp. 175-204). New York, NY: Academic Press.
A review paper of statistical literature on discriminant analysis. The paper is meant to be an introduction to discriminant analysis, but assumes good knowledge of matrix algebra. It covers the following topics: Fundamentals:
- Underlying assumptions
- Extraction of discriminant functions
- Significant tests in discriminant analysis
- Interpretation of findings
- Linear and quadratic class
Issues in discriminant analysis: Failure to meet assumptions Missing values Sample size Number of independent variables Stepwise methods
Practical Problems in Classification: a. Test space versus reduced space b. Estimation of prior probabilities and misclassification costs c. Determining error rates Research example.
Cohen, J. (1968). Multiple regression as a general data analytic system. Psychological Bulletin, 70, 426-443.
This paper shows how the multiple regression method can be generalized to ANOVA/ANCOVA models, and demonstrates how the two methods are conceptually the same. The paper then discussed ways to code group membership, e.g., dummy, contrast, and nonsense coding. The paper also covers curvilinear regression, joint effects, and missing data. The author argues that multiple regression is more flexible than univariate approaches, since it allows analyses of interactions and trend components in a conceptually simple way. This paper is entirely conceptual, with no heavy mathematical formulae.Conger, A. J., & Jackson, D. N. (1972). Suppressor variables, prediction, and the interpretation of psychological relationships. Educational and Psychological Measurement, 32, 579-599.
This is a theoretical article on the nature of suppressor variables, their relation to part and partial correlation, and the limit to how much incremental variance in y can be accounted for by a suppressor variable. The authors recommend the use of part or partial correlations over suppressor variables for the purpose of interpretation of psychological relationships. The paper contains no matrix algebra.
Darlington, R. B. (1968). Multiple regression in psychological research and practice. Psychological Bulletin, 69, 161-182.
This is a paper on the misuses of multiple regression. This is a graduate level paper, containing no matrix algebra.Gocka, E. F. (1973). Stepwise regression for mixed mode predictor variables. Educational and Psychological Measurement, 33, 319-325.
This paper is on stepwise regression for mixed mode predictor variables. It deals with the problem of doing stepwise regression when you have a categorical variable that has more than two levels, and you want it entered in stepwise selection as a set, not as single variables representing membership in only one of the groups or not. Basically the solution involves using group means on the criterion variable as the predictor variable, so that the categorical variable becomes a single quantitative predictor variable. To use this method will require that the degrees of freedom be adjusted for the tests of the effect of this modified variable. This paper is quite an easy read, and contains no matrix algebra.Gordon, R. A. (1968). Issues in multiple regression. American Journal of Sociology, 73, 592-616.
This paper discusses the dangers in multiple regression of interpreting regression coefficients as telling you which variables are more important; the danger is the correlation with other predictor variables. It addresses the problem of multicollinearity, but not as a statistical problem rather as one affecting substantive interpretations of regression coefficients. When using sets of variables representing different domains of interest - say subset 1 has 4 intercorrelated variables, and subset 2 has only 1, which is unrelated to subset 1. The regression coefficient of the subset 2 variable will quite likely be larger even though the other variables are more important. The other variables have small regression coefficients because of their intercorrelations, so one must have clear theoretical reasons for including different variables and theories about their causal relations. The paper discusses these dangers of trying to use regression coefficients to compare the relative importance of predictor variables in univariate multiple regression. This paper has a long discussion at the beginning not requiring matrix algebra, but requiring prior knowledge of multiple regression.Green, S. A. (1991). How many subjects does it take to do a multiple regression analysis. Multivariate Behavioral Research, 26, 499-510.
This article argues that rules of thumb (e.g. 10 subjects per predictor) are not valid means to determine the number of subjects you need for multiple regression. Some validity exists for rules of thumb of the type N A + Bm, where A is a constant number of people, B is number of people per predictor, and m is the number of predictors. But this type of 'rule of thumb' has validity only for testing medium effect sizes and only for one type of partial correlation. The author presents a more complicated means of calculating necessary sample size for multiple correlation that takes expected effect size into account. The presentation of the issues in this paper is clear and easy to follow.
Licht, M. H. (1995). Multiple regression and correlation. In L. G. Grimm & P. R. Yarnold (Ed.), Reading and understanding multivariate statistics (pp. 19-64). Washington, DC: American Psychological Association.
This chapter is written for those with a limited knowledge of multiple regression. The chapter is divided into two sections: applied prediction and theoretical explanation. Each section begins with a brief overview of MR for the purpose, with abstract descriptions of procedures and concepts. MR for each purpose is then illustrated using numerical examples. After this general introduction, the chapter discusses several important methodological and conceptual issues, including the following: multicollinearity; assumptions involving residual scores; specification errors and measurement errors; how to handle categorical variables; and the differences among the most commonly encountered variations of MR.
Lorenz, F. O. (1987). Teaching about influence in simple regression. Teaching Sociology, 15, 173-177.
This article is written for statistics instructors on how to teach undergraduates about influence in regression, but is an excellent article for the learner as well. It uses Anscombe's classic examples of regressions that produces the same correlation, coefficients, etc. to demonstrate the effects of influential data points. The paper doesn't cover a lot of material, but is simple, and appropriate for an undergraduate.
McClelland, G.H. (1993). Statistical difficulties of detecting interactions and moderator effects. Psychological Bulletin, 114, 376-390.
This paper reviews the difficulties inherent in detecting interactions between predictor variables.Velicer, W. F. (1978). Suppressor variables and the semipartial correlation coefficient. Educational and Psychological Measurement, 38, 953-958.
This paper presented a new definition of suppressor variable based on the relation of the semipartial correlation to the zero order correlation. Contains no matrix algebra.
Wolf, G., & Cartwright, B. (1974). Rules for coding dummy variables in multiple regression. Psychological Bulletin, 81, 173-179.
This paper is about contrast coding in regression. The paper presents a coding scheme that can be used to test specific hypotheses, so that the test of each beta coefficient represents a test of a specific hypothesis. They point out that the use of orthogonal contrasts is only appropriate when you have equal "n" in each cell. If not, then the elements of any contrast must be weighed by some value that corrects for unequal sample size, and this eliminates orthogonality. The paper goes on to demonstrate that the way to dummy code the independent variable is a function of three factors:
1) the specific hypotheses you want to set up 2) whether the contrasts are orthogonal or not, which will be determined by whether you have equal n per cell. 3) If the n are unequal, whether the sample size is fixed (i.e., the n per cell is planned, to be representative of the population), or random (unequal sample size per cell is due to random subject loss).
Bochner, A. P., & Fitzpatrick, M. A. (1980). Multivariate analysis of variance: Techniques, models, and applications in communication research. In P. R. Monge & J. N. Cappella (Ed.), Multivariate techniques in human communication research (pp. 143-174). New York, NY: Academic Press.
This is a very nice elementary introduction to MANOVA, which begins with a discussion of ANOVA and univariate regression and shows how MANOVA is an extension of ANOVA. The paper uses primarily GLM notation with only a little bit of matrix notation, but no matrix operations. It shows how MANOVA is represented in a matrix with dummy coding. It discusses how to select contrasts and how to use them, e.g. describes deviation, simple, helmert, orthogonal polynomial contrasts, and effect coding with an example. There is a discussion of how to analyse nonorthogonal (unequal n per cell) MANOVA, using 3 alternative methods, as well as how to choose which one to use. The authors present a numerical example from communication research. There are also sections on how to interpret the significant effects and differences and how to determine what dimensions underlie the relationship, which linear combination of response variables account for maximal separation between groups, whether any response variables be eliminated from future work, and how important each variable or set of variables is for each treatment group (e.g., how much variance is accounted for by each of the elementary responses and/or of linear combinations). They briefly discuss univariate F tests, step-down F-analysis, discriminant analysis, and simultaneous confidence tests.
Harris, R. J. (1993). Multivariate analysis of variance. In L. K. Edwards (Ed.), Applied analysis of variance in behavioral science. Statistics: Textbooks and monographs, Vol. 137 (pp. 255-296). Inc, New York, NY: Marcel Dekker.
This paper discusses the kinds of data that are appropriate for analysing with MANOVA, discusses the theoretical underpinnings of MANOVA (using only a little matrix algebra), and shows how MANOVA can be applied to analysing within-subjects effects. The paper also demonstrates how to set up a MANOVA procedure in SPSS, SAS, and BMDP4V, and provides numerical examples. The paper then covers the following issues: 1) greatest characteristic root versus multiple root tests, 2) univariate versus multivariate approaches to within subjects effects, 3) full model versus sequential analysis, 4) scoring coefficient versus loadings as the basis for interpreting discriminant functions and canonical variates, and 5) Bonferroni adjusted versus fully post-hoc critical values. This paper is appropriate for someone already familiar with MANOVA, but not for someone reading about MANOVA for the first time.
Huberty, C. J. (1989). Multivariate analysis versus multiple univariate analyses. Psychological Bulletin, 105, 302-308.
This article challenges the common practice of using a MANOVA to control for Type I error and then following up with multiple ANOVA's. Not only is the assumption that one completely controls for Type I error by first conducting a MANOVA questionable, but more importantly, MANOVA and multiple ANOVA's answer very different research questions. The article discusses the kinds of questions each strategy answers. There is a brief discussion on controlling Type 1 error when doing multiple ANOVA's.
Olson, C. L. (1976). On choosing a test statistic in multivariate analysis of variance. Psychological Bulletin, 83, 579-586.
This paper reviews the significance tests available for MANOVA with respect to power and robustness, and then provides recommendations. The tests reviewed include Wilk's Likelihood Ratio (W), Hotelling-Lawley trace (T), Largest Root Test (R), and Pillai-Bartlett (V). The paper contains some matrix algebra, but is primarily a review of studies on these measures.
Shaffer, J. P., & Gillo, M. W. (1974). A multivariate extension of the correlation ratio. Educational and Psychological Measurement, 34, 521-524.
This paper reviews measures of effect size to use in the MANOVA case, and argues for the use of one in particular they call the correlation ratio (cr),
cr = 1- tr(wW-1) tr (TW-1)
The paper uses matrix algebra.
Thomas, D. (1992). Interpreting discriminant functions: A data analytic approach. Multivariate Behavioral Research, 27, 335-362.
This article covers the interpretation of discriminant functions following a significant MANOVA test. This article is very heavy on the matrix algebra and geometry.
Weinfurt, K. P. (1995). Multivariate analysis of variance. In L. G. Grimm & P. R. Yarnold (Ed.), Reading and understanding multivariate statistics (pp. 245-276). Washington, DC: American Psychological Association.
This chapter begins with a discussion of basic statistical concepts and the general purpose of multivariate analysis of variance (MANOVA). The author discusses how a MANOVA is performed, and how it relates to the traditional univariate ANOVA. Research examples are provided. The chapter also describes assumptions and the consequences of assumption violation, and follow-up analyses used after a significant MANOVA. MANCOVA, repeated measures MANOVA, and power analysis are also discussed briefly. This chapter is a very easy to read, conceptual introduction to MANOVA, containing no mathematical formulae.
Wilkinson, L. (1975). Response variable hypotheses in the multivariate analysis of variance. Psychological Bulletin, 82, 408-412.
This article discusses how to follow up a significant MANOVA test to reveal the nature of the treatment effects on the p dependent variables. The authors outline 4 methods for measuring the importance of each response variable: 1) calculating univariate F tests for each variable, 2: standardized canonical correlation for each response, 3) contribution of the response to the MANOVA test criterion, 4) simultaneous confidence intervals on estimates of treatment effects on each response,. The paper demonstrates, with an example, that none of these measures is sufficient, on its own, to describe the importance of the response variables, and they therefore recommend the use of all four. There is no matrix algebra, and the paper is brief but clear.
Hunter, J. E., & Schmidt, F. L. (1991). Meta-analysis. In R. K. Hambleton & \J. N. Zaal (Ed.), Advances in educational and psychological testing: Theory and applications. Evaluation in education and human services series (pp. 157-183). Boston, MA: Kluwer Academic Publishers.
This paper is on meta-analysis. The authors provide a brief history of meta-analysis, and describe their own work with meta-analysis in the area of personnel selection research. They derive formulas for cumulating findings across studies that take into account the detection and elimination of artifactual variation due to sampling error. They show formulas for cumulating correlations, and discuss the application of meta-analysis to factor analysis, canonical correlation, and multiple regression. The paper contains no matrix algebra.
Bryk, A. S., & Weisberg, H. I. (1977). Use of the nonequivalent control group design when subjects are growing. Psychological Bulletin, 84, 950-962.
This paper presents a theoretical argument as to the problems inherent in adjusting two groups to be equivalent when they are growing.
Campbell, R. T. (1986). Longitudinal design and longitudinal analysis: A comparison of three approaches. Research on Aging, 8, 480-502.
This article presents three methods for analysing longitudinal data: MANOVA, LISREL, and Event History Analysis. Each of these methods is described conceptually (in very broad terms) and as to what kinds of questions they answer. Each is briefly demonstrated using real data. There is no description of the statistical models, and no matrix algebra.
Wright, R. E. (1995). Logistic Regression. In L. G. Grimm & P. R. Yarnold (Ed.), Reading and understanding multivariate statistics (pp. 217-244). Washington, DC: American Psychological Association.
This chapter is a broad, conceptual introduction to logistic regression (LR). It describes what LR is, its assumptions, how to interpret coefficients and test them for significance, and classification analysis. It provides two research examples, and discusses how the results can be understood. The author then briefly discusses variable selection procedures in LR. This chapter is a very easy to read, conceptual introduction to logistic regression, containing no mathematical formulae.
Davidson, M. L. (1972). Univariate versus multivariate tests in repeated measures experiments. Psychological Bulletin, 77, 4446-452.
This paper is a Monte Carlo study of the relative advantages of using a univariate or a multivariate approach to repeated measures data. The authors argue that since the univariate repeated measures test is not robust with respect to the uniformity assumption, one should use either a multivariate test or a modified univariate test (i.e., with adjustment to the df). Multivariate methods will be more powerful when the number of subjects is greater than the number of levels by a few. There will be a large difference in power between the univariate and multivariate method when small but reliable effects are present with effects highly variable but averaging to zero over subjects, in which case the multivariate test will be more powerful. If the sample size is small, however, one will be forced to use the univariate measure, which has the advantage of being easier to compute.
Hertzog, C. (1994). Repeated measures analysis in developmental research: What our ANOVA text didn't tell us. In S. H. Cohen & H. W. Reese (Ed.), Life span developmental psychology: Methodological contributions. The West Virginia University conferences on life span developmental psychology (pp. 187-222). Inc, Hillsdale, NJ: Lawrence Erlbaum Associates.
This paper compares multivariate to univariate methods for analysing repeated measures data, and provides a good discussion and a decision tree for how to choose what method to use. The authors argue for planned comparison approach over omnibus F-tests and illustrate with an example. This paper tends to be a more theoretical than "how-to" approach.
Keselman, H. J., & Keselman, J. C. (1993). Analysis of repeated measurements. In L. K. Edwards (Ed.), Applied analysis of variance in behavioral science. Statistics: Textbooks and monographs, Vol. 137 (pp. 105-145). Inc, New York, NY: Marcel Dekker.
This paper on repeated measures analyses is limited to fixed effects designs. The paper covers multivariate and univariate approaches to multiple comparison procedures, interactions, interaction contrasts, and mixed designs. The paper then explicates how the multivariate approach can be generalized to more complex designs, and provides a numerical example. The paper uses some matrix algebra.
O'Brien, R. G., & Kaiser, M. K. (1985). MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin, 97, 316-333.
This paper explains the use of MANOVA to analyse repeated measures designs in place of univariate analysis of variance methods. The paper assumes the reader has no prior knowledge of MANOVA. The authors describe the limitations of repeated measures ANOVA specifically the assumptions of sphericity, and compound symmetry. They argue that if you take the same measures at two times, this assumption will most likely be violated. The univariate method then leads to biased F-tests (inflated alpha error rate). two solutions to this problem are 1) modified univariate tests, using adjusted degrees of freedom, or 2) MANOVA method. They then "discuss designs from simple to moderately complex, and ...demonstrate many types of hypothesis tests, including contrasts, subeffects, and simple effects, as well as procedures for family wise error protection". The paper also includes programming language for the MANOVA procedure using SPSS. MANOVA is a robust, flexible alternative to the traditional mixed model analysis. This excellent paper contains no matrix algebra, and should require a graduate level background in only univariate statistics.
Pedhazur, E. J. (1977). Coding subjects in repeated measures designs. Psychological Bulletin, 84, 298-305.
This article describes how to code subjects in a repeated measures design. It covers randomized block, split plot and split split plot designs. It also discusseS how to determine variance accounted for by each factor and interaction. The paper is quite easy to read, and there is no matrix algebra (beyond referring to vectors).
Poor, D. D. S. (1973). Analysis of variance for repeated measures designs: Two approaches. Psychological Bulletin, 80, 204-209.
This paper contrasts two methods for analysing repeated measures data: traditional univariate and multivariate. The paper uses a 3 by 3 mixed effects design numerical example to demonstrate how the two methods work. They use transformation matrices to test effects, but do not explain how the transformation matrix accomplished this task as they go along. The topic of this paper is similar to #90 but covers it in less detail.
Biddle, B. J., & Marlin, M. M. (1987). Causality, confirmation, credulity, and structural equation modelling. Child Development, 58, 4-17.
This article provides an introduction to causal modelling for consumers of causal modelling research who are already familiar with multiple regression. The authors decribe in broad, conceptual terms what causal modelling is, and the ordinary least squares and LISREL methods for analyzing causal models. They then go into detail about the problems that are inherent in causal modeling, path diagrams, OLS regression analysis and LISREL. They then discuss the usefulness and limitations of various criteria used for estimating the "success" of a model (e.g., explained variance, significance of size of coefficients, etc.). This is a clear, easy to read paper with no equations or statistical babble. Although it does a good job of explaining the problems and limitations of LISREL, it only skims over how LISREL works and why researchers often choose it over OLS. Thus, the naive reader finishes the paper still perplexed about what LISREL really is.Bielby, W. T., & Hauser, R. M. (1977). Structural equation models. Annual Review of Sociology, 3, 137-161.
This is a highly advanced paper that does not use matrix algebra, but assumes prior knowledge of path analysis and structural equation modelling, After a brief review of the history of SEM, the paper reviews the literature on SEM, rather than explaining what SEM is, or how to use and understand it.Martin, J. A. (1987). Structural equation modelling: A guide for the perplexed. Child Development, 58, 33-37.
This very brief paper is an introduction to the logic of structural equation modelling. After briefly summarizing of the logic of SEM, the paper provides 5 statements and questions that consumers of SEM literature should keep in mind when trying to evaluate SEM studies.Klem, L. (1995). Path analysis. In L. G. Grimm & P. R. Yarnold (Ed.), Reading and understanding multivariate statistics (pp. 65-97). Washington, DC: American Psychological Association.
This chapter is a very easy to read, conceptual introduction to path analysis containing no mathematical formulae. It is written for the reader of path analysis studies. Using numerical examples, the first section of the chapter describes what path analysis is, how the results are obtained , the conventions for drawing path diagrams, estimating direct and indirect paths, and implied correlation. Other topics include residual fit of the model, residual path coefficients, types of models, and model trimming. There is then a section on assumptions and issues in path analysis.