XLSTAT: Discriminant analysis of principal components in Excel

Background

Discriminant analysis of principal components is a method that aims to describe clusters as well as links between them using synthetic variables. It is commonly used to investigate the genetic structure of biological populations.

 

Dataset for Example

The attached data come from the adegenet package. We'll use the first dataset from dapcIllus (Jombart et al.). It contains the description of the genotypes of 600 individuals by specifying the number of 140 alleles in 30 locations as well as the group (1 to 6) to which each individual belongs. Each line represents an individual (observation) and each column represents an allele in a location.

Our goal is to describe each group of individuals as well as to explore the links within and between the groups.

 

Setting Up

• Open XLSTAT.

• Click on the   XLSTAT-R/ adegenet menu and select the DAPC feature.

• The DAPC dialog box appears.

In the General tab:

• Select your data from the Excel sheet. The Data field contains all of the data from column A to EL.

• The  Groups field contains the groups – in our case, column EL, which specifies the number of the cluster to which each individual belongs.

• Activate the Variable labels  option, as we have selected the variable label.

• Check Observation labels  (column A), as we have also selected the numbers of the individuals.

In the Options tab:

• Set the number of axes for PCA either directly with Nbr of axes (PCA) or indirectly by specifying the % of variance to explain—here, 60%.

• Choose the number of axes or synthetic variables for the analysis using Nbr of axes (DA).

In the  Missing data tab:

• Choose not to accept missing data , remove the observations or estimate missing data by the mean, mode or even nearest neighbor.

In the Outputs tab:

• Check every result that we want to display. Here, we have chosen to check everything in order to explain every possible output.

In the Charts tab:

• Choose to display only the Peripherical individuals for each group or all groups along with the tree linking them.

Click OK to start the computations.
The results will appear in a new Excel sheet.

 

Interpreting the Results

Discriminant analysis of principal components provides several tables and graphs.

First, the Descriptive statistics table shows details like the number of observations, missing values, minimum, maximum, mean, and standard deviation for each quantitative variable.

For instance, the 03 allele at location 1 (loc1.03) has 600 observations, with values ranging from 0 to 2.

The output also includes the modalities, counts, and proportions of qualitative variables.

For example, the groups are evenly distributed across the dataset.

Next, the results include details on the synthetic variables.

The top five eigenvalues correspond to the first five synthetic variables. Since their sum equals the total variance (inertia), a higher eigenvalue means a larger share of explained variance. In this example, the first synthetic variable accounts for the most variance.

Next, we can see the explained percentage of variance by the PCA (here, we had set it up to 60%).

The Loadings table shows that the synthetic variables in the discriminant analysis are linear combinations of the PCA components. Here, 35 dimensions (explaining 60% variance) are listed in the Observations column, with their coefficients (loadings) for each synthetic variable in the other columns.

Next, we have a table with the coordinates of each individual on the new discriminant analysis axes.

We can also observe the coordinates of each group.

A table containing the posterior probabilities for each observation to belong to a group is then displayed.

Two last tables, representing the loadings and the variable contributions to each discriminant analysis component, are displayed.

As before, the loadings are the coefficients by which we have to multiply the values of the observations in order to get the discriminant analysis axes, but this time the observations represent the variables.

Variable contributions show the percentage each variable contributes to constructing an axis; higher values mean the variable’s changes are more clearly reflected on that axis.

Two graphs are shown: the first plots all groups using the two main discriminant components, with a minimal tree linking groups by squared distances between centers. Clusters are generally distinct, but groups 1 and 2 (blue, lilac) and clusters 5 and 6 (orange, red) sometimes overlap, while cluster 4 (yellow) is well defined.

The second graph displays the position of peripheral individuals within each cluster and their cluster membership.

The graph’s coordinate system is defined by the first two discriminant analysis components, which explain 60% of the variance between clusters and individuals.

 

Conclusion

In conclusion, 600 individuals were grouped into 6 clusters based on allele counts per locus, revealing similarities and key variables. For instance, loc11.16 contributes 8% to axis 2 but only 0.1% to axis 1, suggesting clusters 2 and 5 differ on this locus. Displaying peripheral individuals clarifies cluster membership despite overlaps, as shown by individual 487 belonging to cluster 5 even though clusters 5 and 6 overlap.