The general technique to establish of dependence between various bioobjects parameters

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Mendeleyev Russian Research Institute for Metrology

THE GENERAL TECHNIQUE TO ESTABLISH OF DEPENDENCE BETWEEN VARIOUS BIOOBJECTS PARAMETERS

V.S.Sibirtsev1

A.V.Garabadzhiu

To develop of any diagnostics method, to establish of those or other processes mechanismes and it is etc. necessary first of all to establish of form dependences between are several parameters. However, not far in all bioobjects researches the results of measurements are processed quite adequately. Thereof to us was wanted to result here (though in the most common features) a technique of establishment of character of dependence between various researched parameters, algorithms including in self, enabling:

- essentially to reduce quantity of initial data (if it is necessary), then to exclude of "abnormal" (owing to the different reasons) significance from consideration and to determine are previously whether researched parameters though in any measure from each other dependent;

- to conduct interpolation, as well as to construct (if it is necessary) "smoothed" curve (or surface), taking place close, but not without fail directly through each point of analyzed data party;

- to conduct extended regression and correlation analysis of data with use of a wide set of various functions and opportunity of account not only usual ("pair"), but also "multiple" and "partial" correlation factors between parameters and etc.

Preliminary data processing and their graphic analysis

Though many researchers believe, that than more initial data they use, hence and exacter final result will be received, first important problem at the mathematical analysis there can appear necessity of essential reduction of its data quantity [1-4]. One of the most wideused techniques for this purpose consists in following. For initial data party, including in self Q* significances of Y and X parameters, ratio is checked up: L=(xQ-x1)/(yQ-y1). Further, if L<1, Y and X parameters we interchange the position. We set ?X=(xQ-x1)/Q (where Q<Q*). And then, for all Q** significances of X and Y parameters of the initial data party, fallen in range ?Xi (i=1Q) find xi*=?xj/Q** and yi*=?yj/Q** (j=1Q**); from which and we form final "smoothed" data party, including in self Q (but not Q*, as before) significances of xi* and yi* parameters, with which we make all further actions.

Then, frequently there is necessity to exclude of "abnormal" significances from consideration, occurrence of which can promote not only availability of any determination errors of analyzed parameters, but also heterogeneous (especially, in case of live organismes) of research objects. Besides if we, to example, want to investigate character of dependence of Y factor just from X factor, the influence on Y other, except X, factors should be whenever possible excluded. To make it is possible for example as follows.

For each (Yj,Xj) point from data party (Q such points including in self) are searched in addition b points nearest to it, then if for received b+1 significances criterion:

|Mj-Yj| > t??· [??Mj-Yi)2 / (b2-b)]1/2

(where Mj = 1/(b+1)·?Yi, i = 1 b+1, i?j and t? - Student criterion tabulared significance for "?" reliability level and "b-1" freedom degrees number)

is executed, the checked up (Yj,Xj) point is considered "abnormal" and at further accounts is rejected [5]. However, it is necessary to be rather close, thus not to reject and special, "critical" points, the qualitatively important information concluding in self on change of investigated dependence character on various its sites.

At a following stage of preliminary processing of analyzed data it is desirable to establish whether researched Y and X parameters though in any measure from each other dependent. A serial Spirmen correlation procedure can be for this purpose used. It consists in following. Let it is known, that significances of Xj and Yj in researched data party take, accordingly, Gi and Zi places. Then if:

[Q-1]1/2 · [1 - 6/(Q3-Q)] · ?(Gi-Zi)2 > u

where u? - Laplas function tabulared significance (u0.1=0.2533, u0.05=0.125, u0.01=0.025, u0.001=0.0025), the hypothesis about independence from each other of Y and X parameters is rejected with "?" reliability level [1].

Further, if the check on Spirmen (realized after appropriate removal of "abnormal" data) has given positive results, it should make attempt to receive investigated dependence in a "visual" form. The interpolation (when the resulting curve passes through each point of analyzed data party [1,5-7]) is the most simple way to reach this purpose.

In case of linear interpolation the each point of data party incorporate with point nearest to it by straight line. Thus for each two (Yi,Xi) and (Yi+1,Xi+1) points aj (j=12) factors Y=a1+a2·X function are determined by the decision of a system from two linear equations of a form:

It, in the end, will give expression:

Y(X) = yi + (X-xi)·(yi+1-yi)/(xi+1-xi)

However, such way is a little suitable, if the initial data, to example, are heterogeneous or are located over far apart. It is in this case possible, for example, to increase a interpolation polinom degree. Thus in case of "local-cubic" interpolation for each two (Yi,Xi) and (Yi+1,Xi+1) points: (if any from them is not opening or ending for considered data party) aj (j=14) factors Y=a1+a2·X+a3·X2+a4·X3 function are determined by the decision of a system from four linear (relatively aj) equations of a form:

Similarly, deciding a system of linear relatively aj equations of a form: a1+?aj·fj(xi)=f0(yi) (thus the most widespread are: f(Z) = Z, 1/Z, Zh, hZ, loghZ, 1/(h+Z), h1/(h2+Z) and etc., where h1 and h2 = const), it is possible find factors significances for any other interpolation function of a form: f0(Y)=a1+?aj·fj(X). But such method also not always permits to reach desired result (see fig.1).

And then to the aid of there come interpolation functions of a special form, named "splines" and ensuring on all set of being present data not only Y(Xi)=Yi, but also continuity the first, second and etc. (depending on a spline degree) orders differentials ratios [4,6,8,9]. Thus, as well as in above-described case of usual polynomial interpolation, resulting function can be set or "globalized" (when at once all data party is taken into consideration), or "localized" (when for each point the data only in the nearest its vicinity are taken into consideration).

For a finding of aj factors of "local" spline of a form: f0(Y)=Z=a0+a1·f1(X)+a2·f2(X)+a3·f3(X), ensuring continuity zero and first orders differentials ratios of interpolation function on a range between (Yi,Xi) and (Yi+1,Xi+1) (i=1Q) points of analyzed data party, it is necessary to decide relatively aj a following system of linear equations:

where fj'(X) (j=13) - differentials ratios dfj/dx,

and Sj (j = i i+1) - differentials ratios dz/dx, determined, for example [1,5,6], as:

Sj=(zj+1-zj)/(xj+1-xj) - if j=1 or (xj+1-xj)<2·(xj-xj-1);

Sj=(zj-zj-1)/(xj-xj-1) - if j=Q or (xj+1-xj)>2·(xj-xj-1);

Sj=(zj+1-zj-1)/(xj+1-xj-1) - in the other cases.

For private form of interpolation function: Y=a0+a1·X+a2·X2+a3·X3 - it gives final expression [6]:

Y(X) = [Si·(xi+1-X)2·(X-xi) + Si+1·(X-xi)2·(X-xi+1)] / (xi+1-xi)2 +

+ [yi·(xi+1-X)2·(2X-3xi+xi+1) + yi+1·(X-xi)2·(3xi+1-xi-2X)] / (xi+1-xi)3

In case if "local" spline is set on V nearest to (Yi,Xi) points of data party, interpolation function should have a form: f0(Y)=Z=a1+?aj·fj(X) (usually: Y=a1+?aj·Xj-1), where j = 2 2·(V+1) (for splines with continuous only zero and first orders differentials ratios), or j = 2 3·(V+1) (for splines with continuous zero, first and second orders differentials ratios); and the equations system, which needs to be decided relatively aj, for splines with continuous only zero and first orders differentials ratios has a form:

and for splines with continuous zero, first and second orders differentials ratios has a form:

where Sj* - second orders differentials ratios d2z/dx2, determined, for example [1,5,6], as:

Sj*=(Sj+1-Sj)/(xj+1-xj) - if j=1 or (xj+1-xj)<2·(xj-xj-1);

Sj*=(Sj-Sj-1)/(xj-xj-1) - if j=Q or (xj+1-xj)>2·(xj-xj-1);

Sj*=(Sj+1-Sj-1)/(xj+1-xj-1) - in the other cases.

Third order "global" spline (ensuring continuity zero, first and second orders differentials ratios from interpolation function on all data party) can be given for each xixi+1 range, for example, in a form [6]:

Y(X) = [yi/(xi+1-xi) - ai·(xi+1-xi)/6]·(xi+1-X) +

+ [yi+1/(xi+1-xi) - ai+1·(xi+1-xi)/6]·(X-xi) +

[ai·(xi+1-X)3 + ai+1·(X-xi)3·(X-xi+1)] / [6·(xi+1-xi)] (5),

where a1=aQ=0, and aj are by the decision of a linear equations system of a form:

ai·(xi+1-xi) + 2·ai+1·(xi+2-xi) + ai+2·(xi+2-xi+1) =

(yi+2-yi+1)/[6·(xi+2-xi+1)] - (yi+1-yi)/[6·(xi+1-xi)]

Asymptotical behaviour of interpolation function, as a rule [6], is set in a form:

Y(X)=y1+S1·(X-х1) at X<х1; or Y(X)=yQ+SQ·(X-хQ) at X>хQ

At construction the curve, which is closed or crosss self, interpolation function is set by a "local" way, at which for each two next points (or greater it number, if it is necessary) of analyzed data party a ratio is checked up: L=(xi+1-xi)/(yi+1-yi). And if L<1 - interpolation X from Y is made (thus Y and X are interchanged the position in the above-described formulas); and if L>1 - interpolation Y from X is made.

If distance between next points of analyzed data party is larger, as well as in other cases, when it is required to exclude (or, at least, to reduce) influence to searched dependence of unreasonable "features" (such for example as points, in which a mark of the first, second and etc. orders differentials ratios from function varies), for interpolation besides above-stated useful there can appear use so named "bracing" functions [9]. Thus the resulting Y significance at given X significance is calculated as:

Y(X) = Y1(X) + [Y2(X)-Y1(X)]·L*

where Y1(X) - "initial" spline (see for example expressions 4, 5) or other interpolation function;

Y2(X) - "roding" linear (see expression 3) or any other function;

and L* - correction factor, the significances of which can be chosen by the researcher arbitrary in limits from 0 up to 1 for each particular xixi+1 range of analyzed data party.

At last, in case when there is no reliance of sufficient reliability of initial data, it is possible to execute preliminary "smoothing" of analyzed data party; then the resulting interpolation function can be constructed on received modified significances of X and Y parameters. It can be realized by various ways [4]. For example, it is possible to construct interpolation function for each i-point of initial data party on significances of V nearest to considered by points, but without the account of i-point. And after it, substituting in received expression significance of xi parameter, it is possible to calculate the new significance of yi* parameter. For case of "local-linear" interpolation it will look as:

yi* = yi-1 + (xi-xi-1)·(yi+1-yi-1)/(xi+1-xi-1)

(compare with expression 3).

Other way can be a finding of Y parameter modified significances at given xi in a form:

yi** = (1-L*)·yi + L*·yi*

where yi - Y parameter significance in i-point of initial data party;

yi* - the "smoothed" significance for xi, determined as is described above;

and L* - correction factor (its significance should be in a range from 0 up to 1).

Third way of data "smoothing" can be following. Interpolation function is under construction any of above-described methods on significances of initial data party. Then, initial data party is divided on Q* ranges, possessing identical length on X parameter (?X). Then, for each of these ranges is calculated on Q** significances of yij** on given significances of хij**, located from each other on identical distance. And after it averaging of received data on chosen ranges is executed in a form: хi*=?хij**/Q** and yi*=?yij**/Q** (i=1Q*, j=1Q**).

Fourth way of data "smoothing" can be construction "bracing" interpolation function (see expression 7); where Y1(X) is under construction by a usual way on initial data party; and as Y2(X) ("roding" function) is used the expression (8), or function similar to it for 4, 6 and etc. points, which are nearest to "smoothed" point (thus yi** in "junctions" of final interpolation function will be determined pursuant to the formula 9), or even approximation (let, with a small degree of reliability) initial data (as will be described below) function.

If the results of "smoothing" you will not satisfy, it is possible to repeat this procedure as with initial, as with new (already "smoothed") data, using methods any from above-described with any "smoothing" parameters (to which concern in itself: V or ?X - determining "width of a smoothing range"; L* - correction factor; the form of functions, which interpolation of initial data party, as well as specifying Y2(X) "roding" function and etc.), selection of which also should execute rather closely.

Regression and correlation analysis of data

After we have presented researched dependence "visually" (see fig.1), it is possible to try to determine it in a form of function with limited (it is desirable, as it is possible smaller) number of factors, taking place only close, instead of through each point of analyzed data party (that is more convenient from the point of view already mathematical, instead of graphic, as earlier, analysis). This procedure refers to as by approximation. And at its realization, if the law, determining form of required dependence, is beforehand unknown (and otherwise, it was possible not to be engaged in interpolation, and at once to find approximation dependence factors), to begin, it is desirable, with the most simple:

f0(Y)=a0+a1·f1(X)

where f0(Y) and f1(X) (the most widespread among which are following: f(Z) = Z, 1/Z, Zh, hZ, loghZ, 1/(h+Z), h1/(h2+Z) and etc., where h1 and h2 = const) to choose so that graph of dependence (10) was closest by the form and "behaviour" to that was received at the previous "visualized" stage of data analysis.

By replacement variable: Z=f0(Y), G=f1(X) - this dependence can be resulted in a form:

Z=a0+a1·G

correlation (r) and regression (a0 and a1) factors for which can be calculated under the standard formulas [6]:



Thus the reliability of a received correlation factor is checked up on criterion: |r|>r?

where the critical significance of correlation factor is determined as:

r?= t?/[t?2+Q-2]1/2

(where t? - Student criterion tabulared significance for "?" reliability level and "Q-2" freedom degrees number).

To expand the above-described method on case of dependence of a form:

f0(Y) = a0 + a1·f1(X) + ... + ak·fk(X)

describing by action in organism k dependent processes, it is necessary to make replacement variable:

Z=f0(Y) and G = f1(X) + a2/a1·f2(X) + ... + ak/a1·fk(X)

then the significance of correlation factor (r) can be calculated under the formula (12).

And that find significances of a0ak regression factors of dependence (15), it is necessary, for example on a least squares method [10], to decide relatively a0ak a system of linear equations:



Thus form of f0(Y), f1(X)fk(X) functions and it quantity are chosen by the researcher independently, being based on the being present theoretical preconditions about character of described process, maximization requirements of r/r? and Kad (regression equation adequacy factor) significances received during, as well as minimization requirements of approximation relative error (?). Here r? is determined: for dependences (10) and (11) - under the formula (14), and for dependence (15) - under the formula:

r??= [1/(1+(Q-k+1)/[F?·k])]1/2

(where F? - Fischer criterion tabulared significance for "?" reliability level and "k" and "Q-k+1" freedom degrees numbers); and Kad and ? are determined under the formulas [11]:

Kad = (Q-k) · ??Yt,i)2 / [k·F?·??Yt,i-Ye,i)2]

? = (100/Q) · ?|?Yt,i -Ye,i)/Ye,i|

(where Yt,i and Ye,i - theoretical, calculated under the formula 15, and experimental significances of Y parameter).

The feature of multivariate data party analysis

At the analysis of multivariate data party besides above-described should be taken into account also following. Data "smoothing" and exception of "abnormal" significances from consideration are largely subjective procedures. Therefore at multivariate analysis (the presentation of which is rather small) to realize these procedures follows with extra care.

Multivariate interpolation (especially, in case of necessity of its graphic representation) is necessary to realize stage by stage. So, for example, for case of three independent X, Y and Z parameters: at first it should interpolate Y from X at the fixed Z levels, and then it should interpolate Z from Y at fixed X levels [1,5,6] (see fig.2). And case of М-variate interpolation, in general, difficultly to present graphically; though from the mathematical point of view it is similar above-described cases of 2- and 3-variate interpolation.

The dimension of researched data party has not large importance for regression analysis (at exception only significant increase of data participating in calculations): is simple, in expression (15) (and other, connected with it) instead of "f1(X)fk(X)" it should substitute "f1(X2,…,Xm) fk(X2,…,Xm)" (Y here are considered as X1). bioobject technique appendix

And at correlation analysis the "pair" (rjp) and "general" (R0) factors are calculated as well as at twovariate analysis of compound function; where only for rjp in the formula (12) instead of "gi" and "zi" it should substitute "xji" and "xpi"; and for R0 in the formula (16) instead of "f1(X)fk(X)" it should substitute "f1(X2,…,Xm) fk(X2,…,Xm)". However, occur also in consideration so named "partial" (r*jp - describing degree of linear interrelation between Xj and Xр parameters minus linear influences to its other М-2 parameters participating in consideration) and "multiple" (Rj - describing degree of linear interrelation between Xj parameter and other М-1 parameters participating in consideration) correlation factors [11].

Thus "partial" correlation factors are calculated under the formula [1]:

r*jp = - ?jp / (?jj·?pp)1/2

(where [?jp] = [?*jp]-1 - matrix, inverse second order central moment matrix:

?*jp = ??(xji - ?xji / Q) / (xpi - ?xpi / Q), i=1Q,

and Q - total of points in data party),

and are checked up on criterion [11]:

|(Q-M-4)·r*/[1-(r*)2]1/2| > t?

(where t? - Student criterion tabulared significance for "?" reliability level and "Q-M-4" freedom degrees number; and M - "dimension" of data party).

And "multiple" correlation factors are calculated under the formula [1]:

Rj = [1 - 1 / (?jj·?*jj)]

and are checked up on criterion (17), where instead of "k" is substituted "М".

Examples of experimental data analysis

Fig.1. The examples of use of interpolation and "smoothing" different types for graphic representation of dependence between two parameters on the basis of limited quantity of experimental data (are designated by points).

Examples of use of interpolation and "smoothing" different types at the biochemical data analysis are indicated on fig.3. Whereas the example of search of dependence between two parameters by correlation and regression analysis (after to exclude of "abnormal" significance from consideration in analyzed data party) is present, in particular, in our work [15], the most interesting dependences from which are indicated on fig.4. More information on our works in it and other adjacent directions can be on websyte: http:\\www.vs1969r.narod.ru\indexen.htm.

The initial data, indicated on fig.a,b are taken from work [13] for a "6-(2-imidazolin-2-yl-2-[4-(2-imidazolin-2-yl)phenyl]indol + timus calf DNA" system in a water buffer, containing by 0.01 mol/l of NaCl + 0.01 mol/l of Na2EDTA (ethylenediaminetetraacetic acid disodium salt) + 0.01 Tris (2-amino-2-hydroxymethyl-1,3-propandiol) (pH 7.4); thus on absciss axis of a substrat and ligand molar concentration ratio (CS/CL) are located, and on ordinates axis of a fluorescent sensitivity factor significance (?, determining size of dye fluorescence intensity change at DNA concentration increase on 1 mol/l).

Curve 1-3 are result of experimental data interpolation by "normal-local" polynomes of the first, second and third degrees, accordingly. Curve 4 is result of experimental data interpolation by "local" splines of the second degree. Curve 5 and 6 are result of experimental data interpolation by "normal-global" and "bracing-global" splines of the third degree, accordingly. Curve 7 and 8 are result of experimental data "smoothing" by different ways. Besides, on fig.e is shown example of closed dependence construction by experimental data interpolation by "local" splines of the second degree, when were chosen different "initial" points and directions of movement from them (on hour or otherwise, accordingly).

Here Ф designates the ratio of dye quantum outputs in presence of saturating quantity DNA and in absence of nucleic acid. The other conditional designations and measurement conditions the same, as on fig.1,2; except for curve 5, where a used buffer contained in addition 4 mol/l ureaza.



Fig.2. Graphic representation of surface, received by way of "local" second degree spline interpolation of dependence between three parameters:

Fig.3. Examples of use of "smoothing" and "local" second degree spline interpolation by various ways at the analysis of biochemical data, taken from works [13,14].

Hb (relative contribution of "hydrogen" binds to formation specific actively fluorescent complex of DNA-dye), CS/CL (see fig.1) and ?j=lg(Ij/I0) (where I0 and Ij are fluorescence intensity of dye at CS/CL = 0 and j, accordingly) for 10 compounds of 2-phynilbenzazoles row (for which the structural formulas are specified in work [14]; and the experimental data, designated by points with appropriate numbers, are taken from work [12]) in presence of timus calf DNA and in the same buffer, that is specified for fig.1.a,b.

Fig.4. Dependence of individual rat life duration (?) from total content of (+)-trans-7,8-dihydroxy-7,8-dihydrobenzo[a]pyrene (7,8-BP) in animal urine (С) at the first 5 days after single benzo[a]pyrene (BP) inject to rat in a 100 mg/kg doze (fig.a); as well as from С at the first 5 days after third BP inject to rat (fig.b); and from K (ratio of total contents 7,8-BP in him urine at the first 5 days after third and first BP injects to rat) (fig.d) at the multiple BP inject to animal organism (in a doze 10 mg/kg - 10 time with interval of 10 days).

The 1-3 points designate experimental data for the animals, at which after the BP inject: 1 - internal bodies tumpures were formed, 2 - underskin tumpures were formed, 3 - tumpures were not formed. The 4 and 5 points designate "abnormal" data for second and third animals groups, accordingly.

The lines designate regression equations were calculated for: I - all researched animals group (fig.a - ?=12.2·C, r=0.510, ?<0.05; fig.b - ?=477-15.6·C-309·e-C, r=0.372, ?<0.05; fig.d - ?=265-377·K+147·еK, r=0.392, ?<0.05; where r - correlation factor, ? - reliability level); II - animals, at which after the BP inject internal bodies tumpures were formed (fig.a - ?=178+3.23·eC/10, r=0.917, ?<0.01); III - animals, at which after the BP inject underskin tumpures were formed (fig.b - ?=454-17.5·С-296·e-C, r=0.547, ?<0.05; fig.d - ?=121·еК, r=0.663, ?<0.01).

References

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Abstracts

General technique, enabling with high degree of reliability to establish opportunity of existence and form of dependence between various researched parameters is described. As well as some examples of use of this technique in appendix to various bioobjects are indicated.

Key words: interdependence character, regression analysis, correlation analysis, interpolation, approximation, splines.

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