FCA-$k$modes for pharmaco-EEG

In order to be able to perform the analysis on the data versus placebo and versus baseline, versus is now the ratio instead of the difference. For every cell, $(n_{ijk} - 1)$ can be interpreted as the increase (positive or negative) from baseline and placebo. The total increase was $(4938.5-28\times 24 \times 7)=234.5$. Results of FCA-$3$modes of the dose means data, completing those similar seen before e.g. figure fig.7, are given below.

Table 4: FCA-$3$modes on means by dose band electrode and time of absolute energies for verum versus placebo versus baseline, subject scaled data.

+++++++++++++++++++++++++++++++++++++++++++++++++
 FCA-3modes PTA-3modes   dim x (24) dim y (28) dim z (7)
  data               doses_time x electrodes x  bands
+++++++++++++++++++++++++++++++++++++++++++++++++
 PDY2833  day 1 vs bl vs plb   absolute energy
 means of subjects on the subject scaled data
------------------------------------
     Decomposition after Prin.Tens 222
        explained  99.577455% (FCA 90.453056 %)
    -----------------------------------
VALUES                 PCGLO   PCLOC  PFCA
vs111            1    95.574 %   .     .        vs222    0.0506223    00.245     .    05.534
 Xvs11           1      .      99.51    .        Xvs11   0.0506223      .      82.30    .
 Xvs11   0.0470513    00.212   00.22  04.781     Xvs11   0.0153816    00.023   07.60  00.511
 Xvs22   0.0403039    00.155   00.16  03.508     Xvs22   0.0122856    00.014   04.85  00.326
 Xvs33   0.0218824    00.046   00.05  01.034     Xvs33    0.010821    00.011   03.76  00.253
 Xvs44   0.0173196    00.029   00.03  00.648     Xvs44   0.0059887    00.003   01.15  00.077
 Xvs55   0.0127685    00.016   00.02  00.352     Xvs55   0.0032374    00.001   00.34  00.023
 Xvs66   0.0109674    00.011   00.01  00.260     Xvs66   6.191E-19    00.000   00.00  00.000
 Yvs11           1      .      97.83    .        Yvs11   0.0506223      .      68.24    .
 Yvs11   0.1025283    01.005   01.03  22.700     Yvs11   0.0246204    00.058   16.14  01.309
 Yvs22   0.0746615    00.533   00.55  12.037     Yvs22    0.016941    00.027   07.64  00.620
 Yvs33   0.0568457    00.309   00.32  06.978     Yvs33    0.012839    00.016   04.39  00.356
 Yvs44    0.044454    00.189   00.19  04.267     Yvs44    0.009938    00.009   02.63  00.213
 Yvs55   0.0227088    00.049   00.05  01.114     Yvs55   0.0059792    00.003   00.95  00.077
 Yvs66   0.0196142    00.037   00.04  00.831     Yvs66   3.222E-18    00.000   00.00  00.000
 Zvs11           1      .      99.32    .        Zvs11   0.0506223      .      41.06    .
 Zvs11   0.0448294    00.192   00.20  04.340     Zvs11   0.0337258    00.109   18.22  02.456
 Zvs22   0.0421095    00.169   00.18  03.829     Zvs22    0.032129    00.099   16.54  02.229
 Zvs33   0.0323766    00.100   00.10  02.264     Zvs33    0.022515    00.048   08.12  01.095
 Zvs44   0.0216684    00.045   00.05  01.014     Zvs44    0.015803    00.024   04.00  00.539
 Zvs55   0.0211943    00.043   00.04  00.970     Zvs55   0.0143946    00.020   03.32  00.447
...                                              ....
                                                   --------------------------------------

From listing table 4 it is possible to summarise the $\chi^2$ decomposition as in the table 5. First of all the solution corresponding to independence explains $95.574\%$ of the variability, these can be related to marginal effects, i.e. multiplicative effect of the marginals. Of the $27\%$ of lack of independence attributable to three-way interaction, $17.5\%$ were concentrated on vs222 and its associated solutions, $4\%$ to vs333 and associated solutions, $3\%$ to vs444 and associated solutions, the remaining $2.5\%$ being spread further. Notice the singular values 1 related to complete independence (first Principal Tensor in formula (30) and the repetition associated to two-way marginal decomposition. The reconstruction formula (30) can be written exhaustively:

$$\widehat{P} = (D_I\otimes D_J\otimes D_K) (1 {\it 1}\!\!I_I \otimes{\it 1}\!\!I... ...times{\it 1}\!\!I_K + \sum_{s=1}^r\sigma_s\psi_s\otimes\varphi_s \otimes\phi_s)$$ (30)

$$= (D_I\otimes D_J\otimes D_K) (1 {\it 1}\!\!I_I \otimes{\it 1}\!\!I_J\otimes{\it 1}\!\!I_K$$ (31)

$$+ \sum_{s_I=1}^{r_I}\sigma_{s_I} {\it 1}\!\!I_I \otimes\varphi_s^I ... ...}^{r_K}\sigma_{s_K} \psi_{s_K}^K \otimes \varphi_{s_K}^K \otimes {\it 1}\!\!I_K$$ (32)

$$+ \sum_{s'=1}^{r'}\sigma_{s'}\psi_{s'}\otimes\varphi_{s'} \otimes\phi_{s'})$$ (33)

The $\chi^2$ distribution to test independence is difficult to apply here as $n_{ijk}$ represents the ``change" from baseline and placebo in a ratio form. The expected frequencies of changes given the contingency table is $n(n_{ijk}-1)$ ($n=12$ subjects) with possible negative values. The percent of $\chi^2$ helps to assess the importance of the effect, as well as comparing the relative impact, this last is better seen with the $\chi^2/df$. The $dose \times band$ interaction seems to be very dominant.

Table 5: Decomposition of the lack of complete independence (see text).

The marginal solution (independence) must not be discarded from reports of analysis as usually done in Correspondence Analysis, because our interest here is also on approximation and description of the effects. From formula (26) using only the marginal solution (s = 0) one can approximate the increase for a particular cell by the product of the average marginal increases:

$$\hat{\hat{n}}_{ijk} = p_{i..}p_{.j.}p_{..k}N$$ (34)

$$= p_{i..}p_{.j.}p_{..k}N^3/N^2$$

$$\approx p_{i..}p_{.j.}p_{..k}N^3/(IJK)^2$$ (35)

$$= (\frac{p_{i..}N}{JK})(\frac{p_{.j.}N}{IK})(\frac{p_{..k}N}{IJ})$$ (36)

where the approximation (35) will be better if $IJK \approx N$, so if globally only few changes occurred (making the marginal approximation (34) better as well. Each ratio in the formula has the form of a standardised marginal change ratio $O/E$ where the expected value is related to the hypothesis of no change. The average increase($O/E -1$) are given on figure fig.9 giving a relative strength of influence on the increase. For example an approximate amount of increase using formula (36) for 90mg at 3h(40%) on $\delta$(18%) at lead F7 (21%): $1.40 \times 1.18 \times 1.21=1.99$, so an increase of 99%. This is an estimation based firstly on the model of independence and secondly using average margin increase. Under independence only one would estimate (formula 34) 81%, and the observed value (without modelling) is 74%.

Figure 9: Average percentage of increase deducted from the Margins defining complete independence in FCA-$3$modes means(data subject scaled) for all bands (absolute energy) for verum versus placebo versus second baseline.

Notice the consistency between versus being either the difference or the ratio, at least for the dose time profiles (PTA-$3$modes on means (fig.5) and FCA-$3$modes on means (fig.9)). On figure fig.10 are displayed Principal Tensors relating the most of the lack of independence. The first four are related to deviation from two-way margins independence i.e. respectively $lead \times band$ interaction, $dose time \times band$ interactions (two chosen), and $dose time \times lead$ interaction. The last principal tensor relates to three-way interaction. Notice in deviations from two-way margins independence the component on the third is always 1 everywhere as it is there to build the two-way margins. For $dose time \times band$ interaction, the first Principal Tensor ($22$%) relates to an opposition between $\delta$ and $\alpha_1$ band waves associated to a time decrease for all doses (less important for $90$mg), making $\delta$ increased (versus placebo) at the beginning of the experiment and progressively reversing this effect to finish to a decreased $\delta$, and the reverse for $\alpha_1$. The second Principal Tensor shows a different profile for $10$mg comparatively mainly to $90$mg towards fast and slow waves opposition. This ``behaviour" can also be seen in the three-way interaction, this time opposing back and fronto-temporal activity.

Figure 10: Some other Principal Tensors (PT) of the FCA-$3$modes on means(data subject scaled) for all bands (absolute energy) for verum versus placebo versus second baseline (versus is the ratio): marginal time two-way dependence (one PT), idem for marginal space (two PTs), idem for marginal bands (1 PT), and second Principal Tensor (second $k$-modes solution).