*3.3.2 Role of i as a blocking factor*

All events *Eij* occurring in the same individual (*j* ¼ 1 … *ni*) are affected by his/her individual characteristics (matrix *X*), which means that all serial measures related

to him/her including the baseline *Y*<sup>1</sup> *ij* … *Y<sup>r</sup> ij* n o, *<sup>j</sup>* <sup>¼</sup> f g <sup>0</sup> … *ni* also receive this common influence.

Therefore, in the *Y* and *Z* matrices, the individual *i* can be considered as a *blocking factor*<sup>2</sup> ; in the case of matrix *Y*, the individual *i* defines blocks of records that are *non-independent* of each other (see **Table 4**); in the matrix *Z*, the set *I* defines bundles of quantitative and/or qualitative non-serial attributes of each *E* (see **Table 5**), also non-independent.

Thus in the matrix *Y*, a block is constituted by all the records *Y*<sup>1</sup> *ij* … *Y<sup>r</sup> ij* n o, *<sup>j</sup>* <sup>¼</sup> f g 0 … *ni* made after any occurrence of *E* in the same individual *i* (see **Table 4**). As mentioned before, these records establish a very small set of measurements over a


**Table 4.** *Individual acting as a blocking factor for matrix Y.*


### **Table 5.** *Blocks of non-serial attributes of each E.*

<sup>2</sup> A blocking factor is a qualitative attribute with an effect on the response attribute even of no direct interest, but which must be taken into account in the experiment to obtain homogeneous comparisons between observations where the factor remains constant [12].

*Toward Optimization of Medical Therapies with a Little Help from Knowledge Management DOI: http://dx.doi.org/10.5772/intechopen.101987*

given time period. However, the number of records is the same after each event, and these are equally distributed over time, considering the occurrence of the event as the starting point. In particular, a set of *very short and repeated serial measurements with a blocking factor* is going to be analyzed.

In fact, these situations are also present in several domains but are more frequent in medical domains. In the context of serial measurements, a widely used method is to reduce each block of these series to a single series that simplifies the whole set for each individual, either using the *statistica mean* operation at each time point (*thick line* in **Figure 3(a)**), or by reducing each series to a small set of independent indicators as a *mean area* or a *mean trend* per series [11, 13].

For the *Z* matrix, in a similar form one would replace the rows *z*ð Þ*ij lj* ¼ f g 1 … *ni* with ̄*zil* ¼ P *j z*ð Þ*ij <sup>l</sup> ni* or some other suitable summary statistic. This would allow the rows of *Y* and *Z* to be reduced to a single row for each individual, and the *X*, *Y* and *Z* matrices would be made compatible, allowing classical analysis using multivariate

data analysis or modeling techniques. However, on several occasions, if the average series for each individual is built *average* understood in the broad sense mentioned before—is used, very often *too much* relevant information will often be lost, since the variability depends both on each event and on each *individual effect*. It can therefore be inferred that by employing such transformations, the conclusions reached in such a study may be far removed from reality.

Once a pair ð Þ *i*, *j* is determined, which is given by the jth occurrence of *E* on the ith individual, the measures of *Y* in the time period *t*<sup>1</sup> … *tr* (the rows of the matrix *Y*) can be graphically represented by *very short curves (r is usually small)* apparently independent of each other (see **Table 4**). Regarding the real target application, it is easy to illustrate this phenomenon with a relevant case of the above comments represented by the following figures:

*Figure 3(a) shows lines joining the reaction times on a simple visual test (S5) measured at 2, 4, 6, 12, and 24 hours after each ES applied to the* 1*st patient, and the black line represents the average of these curves (average RT curve for* 1*st the patient). This patient receives an ECT of 6 electroshocks, and each curve represents his/her reaction-time evolution.*

*Figure 3(b) shows the* 4*th patient's ECT evolution for the reaction times of the simple visual test (S5), and the black line also represents the average of all the curves (average RT curve for* 4*th the patient). The patient receives an ECT of 5 electroshocks.*

*As can be seen in these figures, constructing a single prototype curve from the mean of the reaction times (thick curve) and having it represent the evolution of the patient is not convenient regarding proper knowledge management, as the variability due to ES is too high and too much relevant information is lost.*

**Figure 3.** *Curves of test S5 from: (a)* 1st *patient; (b)* 4th *patient.*

*Furthermore, differences in patient reaction between different ES will be lost if only the prototype curve is considered for each patient. Nevertheless, this average is useful as it can give an idea of the general trend of the patient. In fact, there is a significant change in the curves from one patient to another and from one test to another. Therefore, it is difficult to establish a general pattern for the curves of a particular patient. Consequently, reducing the patient information to a single record in the Y and Z matrices is not the right way to proceed. Accordingly, it is in the interest of keeping all records for all patients in the same database, but paying attention to this patient effect for the analysis.*
