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Usually, as a factor analysis approach, we use the principal component analysis (PCA) when the active variables are quantitative; the multiple correspondence analysis (MCA) when they are all categorical. But what to do when we have a mix of these two types of variables? A possible strategy is to discretize the quantitative variables and use the MCA. But this procedure is not recommended if we have a small dataset (a few number of instances), or if the number of qualitative variables is low in comparison with the number of quantitative ones. In addition, the discretization implies a loss of information. The choice of the number of intervals and the calculation of the cut points are not obvious. Another possible strategy is to replace each qualitative variable by a set of dummy variables (a 0/1 indicator for each category of the variable to recode). Then we use the PCA. This strategy has a drawback. Indeed, because the dispersions of the variables (the quantitative variables and the indi...

The correspondence analysis (or factorial correspondence analysis) is an exploratory technique which enables to detect the salient associations in a two-way contingency table. It proposes an attractive graphical display where the rows and the columns of the table are depicted as points. Thus, we can visually identify the similarities and the differences between the rows profiles (between the columns profiles). We can also detect the associations between rows and columns. The correspondence analysis (CA) can be viewed as an approach to decompose the chi-squared statistic associated with a two-way contingency table into orthogonal factors. In fact, because CA is a descriptive technique, it can be applied to tables even if the chi-square test of independence is not appropriate. The only restriction is that the table must contain positive or zero values, the calculating the sum of the rows and the columns is possible, the rows and columns profiles can be interpreted. The correspondence ana...

PCA (Principal Component Analysis) is a dimension reduction technique which enables to obtain a synthetic description of a set of quantitative variables. It produces latent variables called principal components (or factors) which are linear combinations of the original variables. The number of useful components is much lower than to the number of original variables because these last ones are (more or less) correlated. PCA enables also to reveal the internal structure of the data because the components are constructed in a manner as to explain optimally the variance of the data. PFA (Principal Factor Analysis)  is often confused with PCA. There has been significant controversy about the equivalence or otherwise of the two techniques. One of the point of view which enables to distinguish them is to consider that the factors from the PCA account the maximal amount of variance of the available variables, while those from PFA account only the common variance in the data. The latter see...

Principal Component Analysis (PCA)  is a very popular dimension reduction technique. The aim is to produce a few number of factors which summarizes as better as possible the amount of information in the data. The factors are linear combinations of the original variables. From a certain point a view, PCA can be seen as a compression technique. The determination of the appropriate number of factors is a difficult problem in PCA. Various approaches are possible, it does not really exist a state-of-art method. The only way to proceed is to try different approaches in order to obtain a clear indication about the good solution. We had shown how to program them under R in a recent paper . These techniques are now incorporated into Tanagra 1.4.45 . We have also added the KMO index (Measure of Sampling Adequacy – MSA) and the Bartlett's test of sphericity  in the Principal Component Analysis tool. In this tutorial, we present these new features incorporated into Tanagra on a realistic ...

Principal Component Analysis (PCA)  is a dimension reduction technique. We obtain a set of factors which summarize, as well as possible, the information available in the data. The factors (or components) are linear combinations of the original variables. Choosing the right number of factors is a crucial problem in PCA. If we select too much factors, we include noise from the sampling fluctuations in the analysis. If we choose too few factors, we lose relevant information, the analysis is incomplete. Unfortunately, there is not an indisputable approach for the determination of the number of factors. As a rule of thumb, we must select only the interpretable factors, knowing that the choice depends heavily on the domain expertise. And yet, this last one is not always available. We intend precisely to build on the data analysis to get a better knowledge on the studied domain. In this tutorial, we present various approaches for the determination of the right number of factors for PCA ba...

Principal Component Analysis (PCA) is a dimension reduction technique. We obtain a set of factors which summarize, as well as possible, the information available in the data. The factors are linear combinations of the original variables. The approach can handle only quantitative variables. We have presented the PCA in previous tutorials. In this paper, we describe in details two indicators used for the checking of the interest of the implementation of the PCA on a dataset: the Bartlett's sphericity test and the KMO index. They are directly available in some commercial tools (e.g. SAS or SPSS). Here, we describe the formulas and we show how to program them under R. We compare the obtained results with those of SAS on a dataset. Keywords : principal component analysis, pca, spss, sas, proc factor, princomp, kmo index, msa, measure of sampling adequacy, bartlett's sphericity test, xlsx package, psych package, R software Components : VARHCA, PRINCIPAL COMPONENT ANALYSIS Tutorial : ...

The aim of the canonical discriminant analysis is to explain the belonging to pre-defined groups of instances of a dataset. The groups are specified by a dependent categorical variable (class attribute, response variable); the explanatory variables (descriptors, predictors, independent variables) are all continuous. So, we obtain a small number of latent variables which enable to distinguish as far as possible the groups. These new features, called factors, are linear combinations of the initial descriptors. The process is a valuable dimensionality reduction technique. But its main drawback is that it cannot be directly applied when the descriptors are discrete. Even if the calculations are possible if we recode the variables using dummy variables for instance, the interpretation of the results - which is one of the main goals of the canonical discriminant analysis - is not really obvious. In this tutorial, we present a variant of the discriminant analysis which is applicable to discre...