Guillaume Desagulier
email: gdesagulier@univ-paris8.fr

introduction principles case study References Multifactorial Exploratory Approaches principal component analysis

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introduction principles case study References

PCA

As in CA and MCA, the total variance of the table is decomposed into proportions in PCA.

There is one minor terminological difference: the dimensions are called principal components.

For each component, the proportion of variance is obtained by dividing the squared standard deviation by the sum of the squared standard deviations.

reminder This is the kind of data that PCA accommodates. For this kind of analysis, the first two components should represent a cumulative percentage of variance that is far above 50%. The more dimensions there are in the input data table, the harder it will be to reach this percentage.

In PCA, the variables and the individuals and categories are plotted separately. The graph of variables serves as a guide to interpret the graph of individuals and categories.

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This is due to the very large number of NP types that cooccur with the prepositions.

We filter out unwanted individuals by selecting only the desired ones. Dim 2 (27.42%) q q q q qq q q qq q q q q q q q q q q entreprise équipe groupe Dim 2 (27.42%) q q q q qq q q qq q q q q q q q q q q entreprise équipe groupe établissement association service q q q q q frontières gendarmerie lignée petites entreprises g. Dim 2 (27.42%) q q q q qq q q q q q q q q q q q q q q entreprise équipe groupe

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Here is what the title of each plot means:

• with select="coord 20", only the labels of the twenty individuals that have the most extreme coordinates on the chosen dimensions are plotted; • with select="contrib 20", only the labels of the twenty individuals that have the highest contributions on the chosen dimensions are plotted; 2 • with select="cos2 5", only the labels of the five individuals that have the highest squared-cosine scores on the chosen dimensions are plotted; 3 • with select="dist 20", only the labels of the twenty individuals that are the farthest from the center of gravity of the cloud of data points are plotted.

2 The contribution is a measure of how much an individual contributes to the construction of a component.

3 The squared cosine (cos 2 ) is a measure of how well an individual is projected onto a component.

Multifactorial Exploratory Approaches

Guillaume Desagulier

>

  plot.PCA(pca.object, select="coord 20") > plot.PCA(pca.object, select="contrib 20") > plot.PCA(pca.object, select="cos2 5") > plot.PCA(pca.object, select="dist 20")

Table 1 :

 1 A sample data frame(Lacheret-Dujour et al. 2019) We make sure that the first two components are representative.

	corpus sample fPauses fOverlaps fFiller fProm fPI	fPA	subgenre	interactivity	planning type
	D0001	0.26	0.12	0.14	1.79	0.28 1.54 argumentation	interactive	semi-spontaneous
	D0002	0.42	0.11	0.10	1.80	0.33 1.75 argumentation	interactive	semi-spontaneous
	D0003	0.35	0.10	0.03	1.93	0.34 1.76	description	semi-interactive	spontaneous
	D0004	0.28	0.11	0.12	2.29	0.30 1.79	description	interactive	semi-spontaneous
	D0005	0.29	0.07	0.23	1.91	0.22 1.69	description	semi-interactive	spontaneous
	D0006	0.47	0.05	0.26	1.86	0.44 1.94 argumentation	interactive	semi-spontaneous
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .

A second kind of PCA is based on loadings (Baayen

2008, Sect. 5.1.1). Loadings are correlations between the original variables and the unit-scaled principal components. The two kinds of PCA are similar: both are meant to normalize the coordinates of the data points. The variant exemplified in this chapter is more flexible because it allows for the introduction of supplementary variables.

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Guillaume Desagulier
email: gdesagulier@univ-paris8.fr

introduction principles case study References Multifactorial Exploratory Approaches exploratory factor analysis

introduction principles case study References

EFA

• EFA is designed to identify patterns of joint variation in a number of observed variables. • although close to PCA, EFA differs with respect to the following: the number of relevant components, which are called factors, is not determined automatically. It must be chosen before we run the analysis.

EFA

EFA has been made popular in linguistics by Biber's studies on register variation (Biber 1991(Biber , 1995)).

EFA

• EFA looks for variables that are highly correlated with a group of other variables. These intercorrelated variables are assumed to measure one underlying variable. This variable, which is not directly observed, but inferred, is latent. It is known as a factor. • one added value of EFA is that "an error term is added to the model in order to do justice to the possibility that there is noise in the data" (Baayen 2008, p. 127).

  based on PCA, we are tempted to specify 3 factors • unfortunately, this is not going to work because 3 factors are too many for 5 variables in the kind of EFA that factanal() performs. Although a third factor is required, we have no choice but stick to 2 factors. This means that we should be careful when we interpret the results.Test of the hypothesis that 2 factors are sufficient. The chi square statistic is 12667.73 on 1 degree of freedom. The p-value is 0

	outline outline outline outline	introduction introduction introduction introduction		principles principles principles principles	case study case study case study case study	References References References References
	inclusion in French inclusion in French inclusion in French inclusion in French
	> fa.object <-factanal(data, factors=2)
	> fa.object
	Call:
	factanal(x = data, factors = 2)
	Uniquenesses:
	centre coeur milieu parmi	sein
	> # clear R's memory 0.655 0.436 0.849 0.005 0.005
	> rm(list=ls(all=TRUE)) Loadings: > # load the data (inclusion_FrWaC.txt) Factor1 Factor2 > data <-read.table(file=file.choose(), header=TRUE, row.names=1, sep="\t") centre 0.587
	coeur	0.750
	milieu 0.389
	parmi -0.147	0.987
	sein	-0.740 -0.669
			Factor1 Factor2
	SS loadings	1.626	1.424
	Proportion Var	0.325	0.285
	Cumulative Var	0.325	0.610
	Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches Multifactorial Exploratory Approaches			Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier Guillaume Desagulier

• the same data set serves as input for EFA: inclusion_FrWaC.txt

• In base R, we run EFA with factanal()

• why?

A χ 2 test reports whether the specified number of factors is sufficient. If the p-value is smaller than 0.05, more factors are needed. If it is greater than 0.05, no more factors are needed. The test reports that the χ 2 statistic is 12,667.73 on 1 degree of freedom and that the p-value is 0.

inclusion in French

The output displays:

• uniqueness (unexplained variation)

• factor loadings (the loadings that are too close to zero are not displayed) • the proportions of variance explained by the factors • the higher the loading the more relevant the variable is in explaining the dimensionality of the factor • Au milieu de, au centre de, and au coeur de define the first factor

• Parmi defines the second factor.

• it seems that au sein de defines both.

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Graphic output:

• rotation a procedure meant to clarify the relationship between variables and factors. As its name indicates, it rotates the factors to align them better with the variables. • varimax rotation: the factor axes are rotated in such a way that they are still perpendicular to each other • promax rotation: the factor axes are rotated in an oblique way.

• with promax, the resulting model provides a closer fit to the data than with varimax.

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study References

inclusion in French

Plotting the loadings of the prepositions on the two factors with varimax rotation:

> loadings <-loadings(fa.object) > plot(loadings, type="n", xlim=c(-1,1)) > text(loadings, rownames(loadings))

For promax rotation, set rotation to promax:

> fa.object2 <-factanal(data, factors=2, rotation="promax") > loadings2 <-loadings(fa.object2) > plot(loadings2, type="n", xlim=c(-1,1)) > text(loadings2, rownames(loadings2))

Multifactorial Exploratory Approaches Guillaume Desagulier outline introduction principles case study References

inclusion in French

Plotting the loadings of the prepositions on the two factors with varimax rotation:

- The distinctive profiles we obtain with EFA are similar to those we obtained with PCA. The only major difference is the proximity of au milieu de with au centre de and au coeur de. This may be due to the fact that only two factors are retained in the analysis. As far as this data set is concerned, PCA is clearly a better alternative, all the more so as individuals are not taken into account in the graphic output of this kind of EFA.