0 | Introduction | 1 |
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0.1 | The data basis of the empirical-statistical research process | 1 |
0.2 | Chronological and logical structures of the empirical-statistical research process | 4 |
0.3 | The algorithmic structure of the multivariate methods discussed | 8 |
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1 | The simple linear regression and correlation model | 16 |
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1.1 | The regression models of the population and the sample | 16 |
1.2 | The correlation model of the population and the sample | 25 |
1.3 | Regression and correlation for standardised variables | 32 |
1.4 | Regression and correlation with a dichotomous independent variable | 33 |
1.5 | Non-linear regression and correlation analysis | 37 |
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2 | Inferential statistics in the simple linear regression and correlation model correlation model | 44 |
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2.1 | The sampling distributions of the coefficients a, b and | 44 |
2.1.1 | Expected values and variances of the coefficients | 45 |
2.1.2 | The distributions of the parameters a and b | 50 |
2.2 | Test and estimation procedure for simple linear regression | 54 |
2.2.1 | Hypothesis tests for A and B | 54 |
2.2.2 | Confidence intervals for A and B | 58 |
2.2.3 | The confidence interval for the regression function of the GG | 60 |
2.2.4 | The confidence interval for the predicted values | 64 |
2.3 | Hypothesis tests in the correlation analysis | 66 |
2.3.1 | Sample distribution and hypothesis test for the correlation coefficient |
66
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2.3.2 | Fischer's Z-transformation | 70 |
2.3.3 | The likelihood ratio test | 73 |
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3 | Partial regression and correlation analysis | 77 |
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3.1 | From bivariate to multiple analysis | 77 |
3.2 | Simple partial correlation and regression with standardised variables |
81
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3.3 | Simple partial correlation and regression with non-standardised variables |
82
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3.4 | Multiple partial correlation | 83 |
3.5 | Semi-partial correlation and regression | 85 |
3.6 | Statistical inference in the partial regression and correlation model |
86
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3.7 | Examples of partial regression and correlation | 87 |
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4 | Multiple regression and correlation analysis | 92 |
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4.1 | The regression model in the sample and in the population |
92
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4.2 | The correlation model of the sample and the population | 97 |
4.3 | On the strategy of multiple regression and correlation analysis | 99 |
4.4 | Statistical inference in multiple regression and correlation analyses | 101 |
4.4.1 | Confidence estimation of the partial regression coefficients and multicollinearity |
101
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4.4.2 | Test of the correlation coefficients | 104 |
4.5 | Application examples for multiple regression and correlation | 109 |
4.5.1 | The blockwise approach | 109 |
4.5.2 | The step-by-step approach | 115 |
4.5.3 | The hierarchical approach | 119 |
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5 | The variance/covariance analysis | 122 |
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5.1 | Introduction to the problem | 122 |
5.2 | The one-factorial analysis of variance | 123 |
5.2.1 | Graphical and tabular representation | 123 |
5.2.2 | The variance-analytical model of the GG | 124 |
5.2.3 | Correlation-analytical aspects of the analysis of variance | 127 |
5.2.4 | Regression-analytical aspects of the analysis of variance | 128 |
5.2.5 | Example of a one-factorial analysis of variance | 132 |
5.3 | The multi-factorial analysis of variance | 136 |
5.3.1 | The experimental approach | 136 |
5.3.2 | The sampling approach | 142 |
5.3.3 | Examples of multi-factorial analysis of variance and multiple classification analysis | 142 |
5.4 | The covariance analysis | 147 |
5.4.1 | The classical approach | 147 |
5.4.2 | The covariance analysis as a regression analysis | 150 |
5.4.3 | Excursus: SPSS approach for separating main and side effects in the variance/covariance analysis | 153 |
5.4.4 | Examples of covariance analysis | 158 |
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6 | The log-linear analysis approaches | 163 |
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6.1 | Introduction to the problem | 163 |
6.2 | Regression and correlation with a dichotomous dependent variable (probit, logit and logistic regression analysis) | 164 |
6.2.1 | Development of the modelling approach | 164 |
6.2.2 | The model variants | 166 |
6.2.3 | Interpretation of the results | 171 |
6.2.4 | Model quality and hypothesis tests | 173 |
6.2.5 | Sample calculation for probit and logit analysis and for logistic regression | 178 |
6.2.6 | Multiple regression models for dichotomous dependent variables | 182 |
6.3 | The log-linear model | 184 |
6.3.1 | The basic structure of the two-dimensional log-linear model | 184 |
6.3.2 | Estimating the model components using the maximum likelihood method | 188 |
6.3.3 | Interpretation and evaluation of the results | 192 |
6.3.4 | The hierarchical loglinear model | 194 |
6.3.5 | Examples of the simple and multiple log-linear model | 197 |
6.3.6 | The general loglinear model | 204 |
6.3.7 | The logit-loglinear model | 210 |
6.3.8 | Log-linear models for ordinal data | 216 |
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7 | The general linear model | 221 |
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7.1 | The importance of the general linear model | 221 |
7.2 | Specification aspects of the general linear model | 222 |
7.3 | The model variants | 226 |
7.3.1 | The four-field contingency table | 226 |
7.3.2 | The point-biserial correlation | 227 |
7.3.3 | The t-test for the difference between two mean values | 228 |
7.3.4 | Linear regression and correlation | 228 |
7.3.5 | Single and multi-factorial analysis of variance | 229 |
7.3.6 | The covariance analysis (saturated model) | 230 |
7.3.7 | The generalised linear model | 232 |
7.4 | Multivariate extensions of the ALM | 234 |
7.4.1 | Empirical and methodological aspects of the multivariate linear model |
234
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7.4.2 | Excursus: Algebraic and matrix algebraic aspects of the multivariate linear model - linear combinations of variables and their notation | 237 |
7.4.3 | Fundamental constructive aspects of the multivariate linear model |
244
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7.4.4 | Correlation-analytical and inferential-statistical aspects of the multivariate linear model | 250 |
7.4.5 | Application examples of the multivariate linear model | 254 |
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8. | The canonical correlation | 257 |
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8.1 | Introduction to the problem | 257 |
8.2 | The canonical correlation model in matrix representation | 260 |
8.3 | The statistical significance of the canonical correlation | 263 |
8.4 | The empirical relevance of canonical correlations | 265 |
8.4.1 | Structure and redundancy matrices | 265 |
8.4.2 | The extraction measures | 267 |
8.4.3 | The redundancy measures | 270 |
8.4.4 | Relationships between extraction and redundancy measures | 271 |
8.5 | Application example for canonical correlation | 273 |
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9 | The factor analysis | 284 |
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9.1 | Introduction to the problem | 284 |
9.2 | Principal component analysis | 286 |
9.2.1 | The modelling approach | 286 |
9.2.2 | The algorithm for determining the factor weights | 291 |
9.2.3 | The eigenvalue criterion: statistical significance of the output variables |
293
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9.2.4 | The image analysis: statistical relevance of the output variables | 296 |
9.2.5 | The Scree test: empirical relevance of the factors | 297 |
9.2.6 | Explanation of the variables from the factors | 298 |
9.2.7 | Empirical interpretation of the factors using an example |
300
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9.3 | Factor rotation | 304 |
9.3.1 | The geometric representation of factors and variables | 304 |
9.3.2 | The concept of factor rotation | 308 |
9.3.3 | The rotation algorithms | 310 |
9.3.4 | The methods of orthogonal rotation | 314 |
9.3.5 | The oblique rotation method | 316 |
9.3.6 | Examples of factor rotation in the principal component approach | 319 |
9.4 | The model of common factors | 325 |
9.4.1 | The theoretical model of the population | 326 |
9.4.2 | Determining the factors using the principal axis method | 330 |
9.4.3 | Determining the factors using the maximum likelihood method |
334
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9.4.4 | Alternative methods for estimating the factor loadings | 339 |
9.4.5 | Estimation of the factor values | 342 |
9.4.6 | Example of the common factor model | 343 |
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10 | Discriminant analysis | 351 |
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10.1 | Introduction to the problem | 351 |
10.2 | The simple discriminant analysis | 353 |
10.2.1 | The discriminant function of the simple discriminant analysis | 353 |
10.2.2 | An algorithm for solving the simple discriminant problem | 355 |
10.2.3 | Results and examples of simple discriminant analysis | 358 |
10.3 | Multiple discriminant analysis | 365 |
10.3.1 | The concept of multiple discriminant analysis | 365 |
10.3.2 | The matrix representation of multiple discriminant analysis | 367 |
10.3.3 | Multiple discriminant analysis as a canonical correlation analysis |
370
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10.4 | Discriminant analysis classification methods | 370 |
10.5 | Layout and results of the multiple discriminant analysis | 373 |
10.6 | Examples of multiple discriminant analysis | 376 |
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11 | The cluster analysis | 384 |
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11.1 | Introduction to the problem | 384 |
11.2 | Measuring the similarity or distance of objects | 389 |
11.2.1 | The quantification of distance in a nominally scaled data set |
389
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11.2.2 | Quantifying the distance in a metric-scaled data set |
394
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11.3 | Summarising objects into clusters | 401 |
11.3.1 | Distance matrix and hierarchical cluster analysis | 401 |
11.3.2 | The "nearest neighbour" method | 403 |
11.3.3 | The "most distant neighbour" method | 406 |
11.3.4 | Clustering on the basis of average distances | 408 |
11.3.5 | Clustering on the basis of distances between averages |
410
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11.3.6 | Cluster formation using the Ward method | 414 |
11.4 | The K-Means cluster analysis | 417 |
11.4.1 | The K-Means approach | 417 |
11.4.2 | Determining the cluster centres and regrouping the cases |
418
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11.4.3 | Assessment criteria and strategies for cluster resolution | 419 |
11.5 | Example calculations for cluster analysis | 420 |
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12 | Multidimensional scaling | 427 |
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12.1 | Introduction to the problem | 427 |
12.2 | Classical multidimensional scaling (CMDS) | 429 |
12.2.1 | The CMDS solution approach | 429 |
12.2.2 | Fitting measures of the CMDS | 433 |
12.2.3 | Non-metric classical multidimensional scaling | 438 |
12.3 | Replicated and weighted multidimensional scaling | 442 |
12.4 | Examples of classic and weighted MDS | 450 |
12.4.1 | Classical multidimensional scaling with SPSS | 450 |
12.4.2 | Weighted multidimensional scaling with SPSS | 454 |
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Appendix 1 Basic enumeration of the variables | 460 |
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| Appendix 2 Provision of further information and working materials | 464 |
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| Bibliography | 465 |
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| Index | 469 |
