Eneahoric correlation coefficient. Applicative program.

Costescu, Mihai Radu

The variability of social-economic phenomena is determined, in most of the cases, by the simultaneous action of several factors; some of these factors allow the evolution of a phenomenon, others hamper it or act reversely.

The meanings and intensities of various factors change in conditions of time and space, so related phenomena evolution records trend that are different as compared to previous ones.

The causality relations between the social-economic phenomena can be quantified and analyzed with the help of correlation. Resulted information is very useful, especially because the specific methods that statistics provides the researcher give the possibility for mainly knowing the following aspects:

I. Existence of causality relations between phenomena.

II. The contribution of every factor to the global variability of the effect phenomena.

III. The intensity of causal connections between the social-economic phenomena and processes.

IV. The evolutive trends of correlation between phenomena.

The analysis of correlation provides a wide range of information, being preferred to other study methods of the connections between phenomena, although the determination of correlation specific indicators is more difficult.

In some cases, from various reasons, the detailed classification of continuous data is abandoned, these being grouped for every variable in three classes (inferior-medium-superior, disagreement-unimportant-agreement, below the average-average-over the average, large-middle-small etc.). in this case in which we are interested in extreme classes, data are arranged in a particular table. We will calculate in this case the eneahoric correlation coefficient for a number of subjects.

In order to calculate the calculation formula of this correlation coefficient, let's accept, for both variables, the inferior-medium-superior classification.

Data will be grouped into a table, as follows:

X Y             Superior       Medium      Inferior

Superior       [n.sub.1]          a       [n.sub.2]     A = [n.sub.1] +
                                                         a + [n.sub.2]
Medium             b              c           d
Inferior       [n.sub.4]          e       [n.sub.3]     B = [n.sub.4] +
                                                         e + [n.sub.3]
            D = [n.sub.1] +                C = b +
             b + [n.sub.4]               [n.sub.2] +
                                            d + n

The formula according to which r is determined in this particular case is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where n is the total number of subjects (as we notice, the calculations do not include the medium-medium variant).

In order to decide whether this calculated value is significant or not, it is compared to a related theoretical value at n - 2 degrees of freedom and related to the meaning level chosen.

The following procedure determines this correlation coefficient and decides whether it is significant or not. The significance level operated with is chosen from 0,10; 0,05; 0,02; 0,01 or 0,001 values, and the total number of studied subjects can be of 12, 22, 32, 42, 52, 62, 72, 82, 92 or 102.

procedure
eneahoric(n,n1,n2,n3,n4,a,b,d,e:in
teger;alfa:real;var r:real);
{The procedure calculates the
eneahoric correlation coefficient
  for a given number of subjects;
 Once this coefficient is calculated,
its meaning will be checked
  for a chosen meaning threshold :
0,10; 0,05; 0,02; 0,01 sau 0,001;
 Parameters meaning is:
 n = the total number of subjects
(12,22,32,42,52,62,72,82,92 or 102)

variabila X: mare mediu mic
         mare n1    a    n2
variabila Y: mediu  b     d
             mic    n4    e    n3

alfa = pragul de semnificatie ales
r = coeficientul de corelatie}

var aa,bb,cc,dd:integer;
  v:array[1..100] of real;
begin
  if alfa=0.10
    then
      begin
       v[10]:=0.4973; v[20]:=0.3598; v[30]:=0.2960;
       v[40]:=0.2573; v[50]:=0.2306; v[60]:=0.2108;
       v[70]:=0.1954; v[80]:=0.1829; v[90]:=0.1726;
       v[100]:=0.1638;
         end;

  if alfa=0.05
  then
   begin
     v[10]:=0.5760; v[20]:=0.4227; v[30]:=0.3494;
     v[40]:=0.3044; v[50]:=0.2732; v[60]:=0.2500;
     v[70]:=0.2319; v[80]:=0.2172; v[90]:=0.2050;
     v[100]:=0.1946;
      end;
  if alfa=0.02
   then
    begin
     v[10]:=0.6581; v[20]:=0.4921; v[30]:=0.4093;
     v[40]:=0.3578; v[50]:=0.3218; v[60]:=0.2948;
     v[70]:=0.2737; v[80]:=0.2565; v[90]:=0.2422;
     v[100]:=0.2301;
      end;
  if alfa=0.01
     then
      begin
       v[10]:=0.7079; v[20]:=0.5368; v[30]:=0.4487;
       v[40]:=0.3932; v[50]:=0.3541; v[60]:=0.3248;
       v[70]:=0.3017; v[80]:=0.2830; v[90]:=0.2673;
       v[100]:=0.2540;
       end;
  if alfa=0.001
     then
       begin
        v[10]:=0.8233; v[20]:=0.6524; v[30]:=0.5541;
        v[40]:=0.4896; v[50]:=0.4433; v[60]:=0.4078;
        v[70]:=0.3799; v[80]:=0.3568; v[90]:=0.3375;
        v[100]:=0.3211;
        end;
aa:=n1+a+n2;
bb:=n4+e+n3;
cc:=n2+d+n3;
dd:=n1+d+n4;
r:=(n 1+n3-n2-n4-(aa-bb)*(cc-dd)/n);
r:=r/sq rt ( (a a+bb-sq r(aa-bb)/n)*(cc+ddsq
r(cc-dd)/n));
  writeln('r = ',r:5:4);
  if r > v[n-2] then writeln('Coeficientul
de corelatie este semnificativ')
              else writeln('Coeficientul
de corelatie nu este semnificativ'
end;

References

(1.) Costescu, Mihai-Radu, Statistic Methods Applied in Social Sciences, Liberty Press and Publishing House, Panciova, 2007.

(2.) Costescu, Mihai-Radu, Costel, Ionascu, Electronic Information Processing, Universitaria Press, Craiova, 2001.