Fig. 1. Experimental design of the transcriptome analysis on Bacillus subtilis (Sekowska et al., 2001). The experimental setup follows a fully crossed factorial design. In the case of Sekowska et al. (2001) the quantity of RNA used for the RT-PCR differed between the two protocols. Note that changing the protocol (a different quantity of RNA or labelling with Cy3 rather than Cy5) or having duplicats for each gene on the array are all technical factors which increase the workload without adding any biologically pertinent information. It is preferable to increase the number of states for the biological factors, in the above case an additional sulphur source or an additional experimental day.

Fig. 2. Effect of different pre-processing methods on the data distribution. The figure shows the effect different pre-processing methods have on the data distribution. Shown are the distributions ( ) of the raw data, ( ) after having taken the log and ( ) after having taken the fifth root. As can be seen, either operation brings the distribution closer to a Gaussian one.

Fig. 3. Projections of the data on different planes. In all four figures, each axis corresponds to an experimental condition: (a and c) metA1a vs. mtrA1a; (b and d) metA1a vs. metA10a (see Fig. 1 for the nomenclature). (a) and (b) show projections of the raw data, in (c) and (d) the data are log centre-reduced. Log centre-reducing the data has brought the few points which are far away from the main body in (a) and (b) closer in (c) and (d). Note how the space is more efficiently used in (c) and (d). The points in the two left-hand pictures form a narrower “cigar”, indicating that fewer genes have changed expression than on the right-hand side.

Fig. 4. The expression change in function of the factor sulphur as calculated by a spreadsheet (MS-Excel). The figure shows the genes’ expression change in function of the sulphur source against their mean expression. The potentially interesting genes are those away from the main body of the cloud. The highlighted genes are the ones which proved to be of particular interest for the problem investigated by Sekowska et al. (2001); the reader may refer to their work for a detailed discussion. Note that not all of these genes would have been detected using the spreadsheet.

Fig. 5. The data cloud projected on the plane formed by axis 1 against axis 5 (PCA). The figure shows the genes expression change in function of the sulphur source (axis 5) against their mean expression (axis 1), as calculated by PCA. The potentially interesting genes are those away from the main body of the cloud. The highlighted genes are the ones which proved to be of particular interest for the problem investigated by Sekowska et al. (2001); the reader may refer to their work for a detailed discussion. Note that not all of these genes would have been detected using PCA.

Fig. 6. The data cloud projected on the plane that separates well the sulphur source (ICA). The figure shows the genes’ expression change in function of the sulphur source, as determined by ICA. The potentially interesting genes are those away from the main body of the cloud. The highlighted genes are the ones which proved to be of particular interest for the problem investigated by Sekowska et al. (2001); the reader may refer to their work for a detailed discussion. Note that not all of these genes would have been detected using ICA.

Fig. 7. The “weight” attributed to each experimental condition by a spreadsheet (MS-Excel), PCA and ICA. The figure shows that a spreadsheet (MS-Excel) attributes the same “weight”, or importance, to each experimental condition, whilst PCA and ICA do not.

Fig. 8. The graphical representation of the results obtained with ANOVA for the factor sulphur. The potentially interesting genes are those with a small p-value and a large variance, genes which are therefore in the bottom right part of the image. They are away from the main body of the cloud. The highlighted genes are the ones which proved to be of particular interest for the problem investigated by Sekowska et al. (2001); the reader may refer to their work for a detailed discussion. As can be seen, not all genes of interest would have been identified by the sole use of ANOVA.

The mixing matrix calculated by the spreadsheet (MS-Excel) and the eigenvector matrix calculated by PCA

The mixing matrix at the top was calculated by the spreadsheet, the mixing matrix (or eigenvector matrix) at the bottom by PCA. The arrows indicate the columns which separate well the effects of the same factors. The matrices allow us to change from the old to the new reference frame: they give us for each of the new axes (the columns) the coefiicient with which we have to multiply each gene's value in a given experimental condition (the lines) in order to obtain the new coordinates. The first line in the eigenvector matrix contains the eigenvalue for each axis (in %), providing an indication of the cloud's dispersion along that axis. Note that for the first axis all the sixteen coefficients have basically the same value; this means that for the first axis, all experimental conditions have the same weight, in other words, the first axis gives us the total expression of each gene, which is generally true (see Stoyanova et al., 2004). An example for the calculation of the new coordinates with the eigenvector matrix: in the original (or “old”) reference frame, the gene galK has the coordinates (5.431; 5.432; 5.092; 5.068; 4.893; 4.744; 3.763; 3.661; 5.333; 5.265; 5.329; 5.249; 4.607; 4.444; 3.806; 3.737). To obtain galK's coordinate on the new axis 1. the calculations are as follows: (5.431 × 0.250) + (5.432 × 0.249) +  + (3.737 × 0.251) = 19.0. The other coordinates are obtained accordingly.

Comparison between ANOVA, the paired t-test and the t-test, an example

In (a) the measurements obtained for ytmJ are shown. (b–d) The calculations and results obtained with the t-test, the paired t-test and ANOVA, respectively. ANOVA and the paired t-test both identify the gene as potentially interesting, whilst the t-test results inconclusive (see the relative −log(p-value)). d.f. = degrees of freedom; SS = sum of squares. See Section 3.4 for details.

Overall comparison between ANOVA, the paired t-test and the t-test

The table shows a comparison of the number of genes detected by the three methods. In (a) the threshold for detection was −log(p-value) = 3, in (b) it was equal to 4. Although ANOVA detects in both cases the highest number of genes, the paired t-test performs comparably well, whilst the t-test lags far behind.