Analysing summaries and PTAIV-kmodes

Analysing a summary statistic of subjects changes obviously the data analysed, and clearly it means that this summary ``sufficiently" informs on the distributions. Analysing means, medians or trimeans can be also a solution to subjects outliers which put into question the choice of the location parameter. Looking at only one location parameter implicitly suppose the distributions to be unimodal, which was a sensible assumption here but PTA-$k$modes could be done with a mode representing different location parameters of the empirical subject distributions. This has not been done here as we focused on comparing different main location parameters using a PTA-$3$modes.

Figure 5: PTA-$3$modes dose means, medians or trimeans (data subject scaled) for all bands (absolute energy) for verum versus placebo versus first baseline : 1$^{st}$ Principal Tensors.

On fig.5 the first Principal Tensor of the different PTA-$3$modes on means, medians or trimeans over the 12 subjects for each dose , band, time, and electrode, as a tensor of order three, is shown. Each analysis constitutes a dose profile analysis, each left plot of figure fig.5 representing a dose-effect curve (versus time) for the corresponding principal tensor of the given profile summary. For the 1$^{st}$ mode ( dose $\times$ time) a major difference between these three summaries can be seen for the 30mg curve in comparison to the other doses: no apparent dose effect (around peak) is observed with the mean). The median and trimean give similar results, the 10mg curve becomes flatter for trimean. For the 2$^{nd}$ mode (electrode) a slight gradient towards the back is seen for median and trimean but the three plots are very similar. The 3$^{rd}$ mode (band) representation for mean differs from the two others mainly on $\delta$, $\beta_2$ and $\alpha_2$. With small samples the trimean ( $0.25 q1 + 0.5 median + 0.25 q3$) seems to be a good compromise between the two extremes of mean and median either too sensible to outliers or not all. It has been already used in pharmaco-EEG studies for example by [8]. Analysing means of the subjects is in fact equivalent to performing a PTAIV-$3$modes on the three-ways arranged data (i.e. tensor of order three) $X$ with the following modes : $(dose \times subjects \times time)$ as the first mode, $electrodes$ as the second mode, and $bands$ as the third mode. PTAIV means Principal Tensor Analysis with Instrumental Variables and refers to an extension of PCAIV, [18] or [19], to multiway data, [10]. In the optimisation procedure one considers linear constraints on the solution defined by the Instrumental Variables which are usually linked to the design. In our context the optimisation becomes to maximise $X..(\psi \otimes \varphi \otimes \phi)$ with a linear constraint on $\psi$ as belonging to the subspace generated by the indicator matrix of $dose \times time$ structure $S_{dt}$, $\psi \in \mathcal{I}m(S_{dt})$. $S_{dt}$ is a matrix of $3\times9=27$ columns, each one identifying entries of the first mode as in the current $dose$ and $time$ by a value $1$, $0$ otherwise). This means that the values in $\psi$ will be equal for all the units with the same $dose$ and $time$. Writing the maximisation to find a singular value gives (denoting $x, y,z \in \mathbf{S}_1$ for $\left\Vert x \right\Vert _{E_1} =1$, $\left\Vert y \right\Vert _{E_2} =1$ and $\left\Vert z \right\Vert _{E_3} =1$ :

$$\sigma = \max_{{\scriptstyle \begin{array}{l } \scriptstyle x, y,z \in ... ...; \scriptstyle x\in \mathcal{I}m(S_{dt}) \end{array}}}X..(x \otimes y\otimes z)$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle x, y,z \in ... ...{I}m(S_{dt}) \end{array}}}<X,x \otimes y\otimes z>_{E_1 \otimes E_2\otimes E_3}$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle x, y,z \in ... ..._1 \end{array}}}<X,P_{S_{dt}}x \otimes y\otimes z>_{E_1 \otimes E_2\otimes E_3}$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle x, y,z \in ... ...Id_{E_2}\otimes Id_{E_3})^*X,x \otimes y\otimes z>_{E_1 \otimes E_2\otimes E_3}$$

$$= \max_{{\scriptstyle \begin{array}{l } \scriptstyle x, y,z \in ... ...rray}}}((P_{S_{dt}}\otimes Id_{E_2}\otimes Id_{E_3}) X)..(x \otimes y\otimes z)$$

$$= ((P_{S_{dt}}\otimes Id_{E_2}\otimes Id_{E_3}) X)..(\psi \otimes \varphi \otimes \phi).$$ (15)

Equality in(15) means that PTAIV-kmodes is performed as a PTA-kmodes of the projected data $(P_{S_{dt}}\otimes Id_{E_2}\otimes Id_{E_3}) X$ which in that case will be equivalent to analyse the means data by dose and time for each band and electrode. Note that analysing $(P_{S_{dt}}^\bot \otimes Id_{E_2}\otimes Id_{E_3}) X$ corresponds to the residual analysis (projection on the orthogonal of the structure). Thus one has a double decomposition of the sum of squares : explained by the structure plus its residuals, and within each part with the SVD-$k$modes decomposition. Rigourously when analysing other summaries such as medians (idem for trimean), one does not perform a PTAIV, nonetheless defining a structure by putting a $1$ only for the median point (for each dose and time), which depends on the band and electrode (the structure is on the whole tensor space, would provide a PTA on a projected data.

Table 3: PTAIV Variability explained for different summaries of $X$. ($^*=$rounded; last column: % relatively to the corresponding source $\equiv$ %relatively to the original (data $X$).)

Comparisons versus baseline for time mode and comparison versus placebo for dose mode (the modes have been understood that way all through), are in fact already preprocessed summary measures and can be seen as projected data. It is likely that when performing pharmaco-EEG experiments, designs contain two baselines as in the methodology described in introduction. Apparently the second baseline is usually taken as the reference in statistical analysis: a greater stability is often observed with this measure(less subject variability). So far in this paper the analysis has been done versus first baseline (08h00), and further analysis will be done versus second baseline (08h30), the injection was in fact done at 09h00. In all the previous analysis the second baseline (versus the first) was in fact included in the analysis, this means a true post dosing time is to be decreased by one in the graphics.The only interest of including a (second) baseline in the analysis is when studying the possible initial drop of activity just after injection. Nonetheless to achieve less random subject effect, and a better comparison with reported results, thee foregoing analysis will be done with the second baseline. It is reassuring that similar results were obtained concerning the peak time activity but closer results, to the officially reported ones, concerning the dose dependent $\delta$ band favoured for the total band effect seen the mean analysis with nonetheless $\theta$ as much important. These results were also confirmed by a PTA-$4$modes analysis, seen on fig.6.

Figure: PTA-$4$modes levels-modified data (see page.) for all bands (absolute energy) for verum versus placebo versus second baseline : 1$^{st}$ Principal Tensor.

Notice the importance of this second baseline choice towards the dose effect seen on fig.7 and fig.6 not observed with the first baseline on the same analysis (c.f. fig.5 fig.4(d)). The band components are quite different as well pointing now the $\delta$ redistribution but also $\theta$. One could see a gradient in the slow waves and after the fast waves, but $\alpha_2$ (mid-range) is now out of pattern in these first Principal Tensors and as matter of fact does not contribute to this Principal Tensor. The spatial components are very similar but seem more central with the analysis versus second baseline.

Figure 7: Same as for figure 5 but versus second baseline : 1$^{st}$ Principal Tensors.