Chi square statistic

Created by: Lorentz Jäntschi

The Chi-Square test is based on a series of assumptions frequently used in the statistical analysis of experimental data. The main weakness of the chi-square test is that is very accurate only in convergence, and for small sample sizes is exposed to errors of both types. On two scenario of uses - goodness of fit and contingencies assessment here are discussed different aspects involving it. Further knowledge on the regard of the type of the error in contingencies assessment push further the analysis of the data, while in the same time opens the opportunity to devise a method for filing the gaps in contingencies (e.g. censored data), both scenarios being discussed here in detail, and finally is provided an program designed to do this task.

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Introduction

The χ2 test was introduced by Pearson in 1900[1]. The  statistic were originally devised to measure the departure in a system of n (random normally distributed) variables {y1, …, yn} having each a series of undefined number of observations (y1 = {y1,1, …}, …, yn = {yn,1, …}) for which {x1, …, xn} are the means and {σ1, …, σn} are the deviations. In this context the formula for the statistic is derived and it have a known distribution function.

\( PDF_{\chi^2}(x; n) = \frac{x^{n/2-1}e^{-x/2}}{2^{n-2}\Gamma(n/2)}, CDF_{\chi^2}(x; n) = \frac{1}{\Gamma(n/2)}\int_{0}^{x/2}{t^{n/2-1}e^{-t}dt} \)

It should be noted that the context in which the statistic is slightly different than the one in which is currently in use. This is the reason for which, during time a series of patches have been made to the statistic in order to make it usable.

For instance, in the contingency of two factors f1 = {f11, f12} and f2 = {f21, f22}, under the assumption that xi,j ~ f1if2j:

Factors

f11

f12

f21

x1,1

x1,2

f22

x2,1

x2,2

the use of the assumes that it exists also a series of observations behind each of the {x1,1, x1,2, x2,1, x2,2} entries in the table above, and someone may calculate the X2 statistic (e.g. X2 is the sample based statistic, while  is the population based statistic or distribution) for the contingency as:

X2

Since the estimation of expected frequencies in the contingency of the factors uses rows totals {x1,1 + x1,2, x2,1 + x2,2}, columns totals {x1,2 + x2,1, x2,1 + x2,2} and overall total {x1,1 + x1,2 + x2,1 + x2,2} the variation in the X2 is constrained. Are exactly three independent constrains (if x1,1 + x1,2 + x2,1 + x2,2 = n0, x1,1 + x1,2 = n1, x1,2 + x2,1 = n2 then the rows totals are {n1, n0 - n1}, the columns totals are {n2, n0 - n2}, and the overall total is {n0}) and it is exactly one degree of freedom for X2 (if x1,1 = x, then x1,2 = n1 - x, x2,1 = n2 - x, x2,2 = n0 - n1 - n2 + x) and therefore the probability associated with the X2 value should be taken from distribution with 1 degree of freedom. More, it can be verified by induction that for a m·n contingency table are m + n - 1 independent constrains, and thus are m·n - m - n +1 degrees of freedom.

The main trouble in the use of the statistic is the fact that its distribution is asserted and the probability is derived under the assumption that behind {x1,1, x1,2, x2,1, x2,2} values are a infinite number of observations ({y1,1,1, …}, {y,1,2,1, …}, {y2,1,1, …}, {y2,2,1, …}), which actually is true never. Even if are accessible some population means, {μ1,1, μ1,2, μ2,1, μ2,2} those means comes actually from some estimations made from some (finite) samples of the populations, which passes sampling noises (or errors) to the statistic itself. A clear shot to this matter has been made by the Fisher which patches the statistic in the case of the 2x2 contingency above given for the frequency tables (Fisher exact test, see [2]).

Despite of its use in goodness-of fit tests, actually the statistic is not suitable in this instance. Having a series of observations {o1, …, on} one split the domain of the observations (or having a series of cumulative probabilities {q1, …, qn} associated with the series of sorted observations, see [3]) in an arbitrary number of subintervals (usually taken as 1 + ) in order to construct the series of observed and expected frequencies. By this fact alone, when it is used to test the goodness-of, the statistic is itself exposed to the risk of being in error. Turning back to the general assumption in which its formula were derived - namely the assumption that behind each observed frequency is an infinite number of observations - one may realize that this kind of state of facts is never meet in practice. Therefore, only for very large samples, in which the observed frequencies are 'large enough' too, the statistic may become of use.

Someone may ask now: Which are the subjects in which the statistic is of use? - And indeed are, but the answer is not detailed here.

Measures of agreement in contingencies of multiplicative effects

Something else is of interest in close connection with the use of the statistic for contingency tables, as firstly pointed out by Fisher in [4], namely its connection with the experimental design on which is applied. In [5] is revised and generalized this perspective, and here is given the summary of it.

Let's consider a series of observations made in a contingency of two factors (see the table below; the experimental design can be generalized to any number of factors) which affect the observation in a multiplicative manner (e.g. ui,j  fri·fcj).

(ui,j)1≤i≤r,1≤j≤c

fc1

fcc

Σ

fr1

u1,1

u1,c

sr1

frr

ur,1

ur,c

srr

Σ

sc1

scc

ss

Above (ui,j)1≤i≤r,1≤j≤c are the observations made under different constrains of the factors, where (fri)1≤i≤r are the levels of the fr factor and (fcj)1≤j≤c are the levels of the fc factor. For convenience of the later use, also the sums sr1 (sr1 = u1,1 + … + u1,c), …, srr (src = ur,1 + … + ur,c), sc1 (sc1 = u1,1 + … + ur,1), …, scc (scc = u1,c + … + ur,c), and ss = Σ1≤i≤r,1≤j≤cui,j were assigned.

It can be proofed that under the presence of the multiplicative effect of the two (fr and fc) factors a very good estimate of the expected values would be given by the formula vi,j = sri·scj/ss (see the table below).

(vi,j)1≤i≤r,1≤j≤c

fc1

fcc

Σ

fr1

v1,1

v1,c

sr1

frr

vr,1

vr,c

srr

Σ

sc1

scc

ss

The values of the factors ((fri)1≤i≤r and (fcj)1≤j≤c) are in general unknown, but even if are known, doesn't help too much in the analysis since also their values may be affected by errors, as the observations (ui,j)1≤i≤r,1≤j≤c are supposed to be.

Either way (the values of the factors are known or not) in the calculation of the expected values (vi,j)1≤i≤r,1≤j≤c are used the sums of the observations (sri = Σ1≤j≤cui,j, scj = Σ1≤i≤rui,j, ss = Σ1≤i≤r,1≤j≤cui,j) as the estimates for the factor levels (or values):

fr1 : … : fri : … frr  sr1 : … : sri : … srr & fc1 : … : fci : … fcc  sc1 : … : sci : … scc

Three alternative assumptions may push further the analysis and are on the regard of the type of the error (possibly, probably) made in the process of observation. Since the whole process of observation under the influence of the factors is a part of a whole, it is safely to assume that the error keeps its type during the process of observation, it is random and it is accidentally (having a low occurrence) and the three alternatives along with their consequences as usable formulas are listed in the table below.

It should be noted that X2 formula is usable only in the assumption that fri, fcj > 0.

The S2 formula minimizes the absolute errors, V2 formula minimizes the relative errors, while X2 formula minimizes the X2 value for the contingency.

The algorithm is:

  • for(1 ≤ i ≤ r and 1 ≤ j ≤ c) vi,j ← sri·scj/ss
  • fr1 = fc1 = (v1,1)1/2; for(2 ≤ i ≤ r) sri ← vi,1/fc1; for(2 ≤ j ≤ c) scj ← v1,j/fr1
  • Repeat
    • for(1 ≤ i ≤ r) fri ← Corresponding "Consequenced usable formula" (from S2, V2, or X2)
    • for(1 ≤ j ≤ c) fcj ← Corresponding "Consequenced usable formula" (from S2, V2, or X2)
  • Until convergence criteria is meet.

It has been shown in [5] that consecutive using of equations given as usable formulas converges fast to the minimum (constraint formula) and thus provides better estimates of the factor levels (fri)1≤i≤r and (fcj)1≤j≤c which can be further used to improve the expected estimates (population means) of the factorial experiment: vi,j ← fri·fcj. Using the data given in [5] the number of steps for a change less than 0.1% in the objective function (S2, V2, and X2 respectively) are: 2 steps for S2 and X2 and 3 steps for V2.

Exploiting the agreement in contingencies of multiplicative effects

One of the uses of the χ2 statistic is in the presence of censored data (see for instance [6] and [7]).

A recent application of the method above described has been reported in [8]. The method has been used to fill the gaps of missing data in contingencies of multiplicative effects of factors influencing the observations. The algorithm above given must be adapted in order to meet the requirements to be used and also the gaps may be filled if exists at least one observation in each row (e.g. sri ≠ 0 for 1 ≤ i ≤ r) and at least one observation in each column (e.g. scj ≠ 0 for 1 ≤ j ≤ c).

In the most general case, a recursive procedure can be designed to fill the gaps. Later on, the procedure of minimizing the residuals (e.g. S2, V2 or X2) goes smoothly. The full program (PHP source code) is in the appendix. Here is given (an example of) raw data and data filled with gaps, followed by the optimization of the expectances, of the S2, V2 and X2 respectively.

Raw data ("data.txt" input data for the program given in the appendix)

25.3

28.0

23.3

20.0

22.9

20.8

22.3

21.9

18.3

14.7

13.8

10.0

26.0

27.0

24.4

19.0

20.6

24.4

16.8

20.9

20.3

15.6

11.0

11.8

26.5

23.8

14.2

20.0

20.1

21.8

21.7

20.6

16.0

14.3

11.1

13.3

23.0

20.4

18.2

20.2

15.8

15.8

12.7

12.8

11.8

12.5

12.5

8.2

18.5

17.0

20.8

18.1

17.5

14.4

19.6

13.7

13.0

12.0

12.7

8.3

9.5

6.5

4.9

7.7

4.4

2.3

4.2

6.6

1.6

2.2

2.2

1.6

 

Data with gaps (18 randomly filled values from raw data)

 

28.0

 

 

 

20.8

 

 

 

 

 

 

26.0

 

 

 

20.6

 

16.8

 

 

 

 

11.8

 

 

 

 

20.1

 

 

 

16.0

14.3

 

 

 

 

18.2

20.2

 

 

12.7

 

 

 

 

 

18.5

17.0

 

 

17.5

 

 

13.7

 

 

12.7

 

 

 

 

7.7

 

 

 

 

 

 

 

 

 

Expected values ((Σioi,j)·(Σjoi,j)/(Σi,joi,j)) calculated from data with gaps

31.387

27.926

28.595

31.738

27.230

20.819

20.472

22.417

22.339

19.966

20.781

14.600

25.239

22.457

22.994

25.521

21.896

16.741

16.463

18.026

17.964

16.055

16.711

11.741

22.653

20.156

20.638

22.906

19.653

15.026

14.776

16.180

16.123

14.410

14.999

10.538

19.895

17.702

18.126

20.118

17.261

13.197

12.977

14.210

14.161

12.656

13.173

9.255

19.208

17.090

17.499

19.422

16.664

12.741

12.529

13.719

13.671

12.218

12.717

8.935

7.633

6.791

6.954

7.718

6.622

5.063

4.979

5.452

5.433

4.855

5.054

3.551

 

Optimized expectances for S2 → min.

31.759

27.975

28.896

32.059

27.067

20.801

20.726

22.472

21.883

19.558

20.832

14.774

5.617

25.338

22.320

23.054

25.578

21.595

16.596

16.536

17.930

17.459

15.604

16.621

11.787

4.482

23.318

20.540

21.216

23.539

19.874

15.273

15.218

16.500

16.067

14.360

15.296

10.848

4.124

19.929

17.555

18.132

20.117

16.985

13.053

13.006

14.102

13.732

12.273

13.072

9.271

3.525

19.344

17.039

17.600

19.526

16.486

12.670

12.624

13.688

13.329

11.912

12.688

8.999

3.421

7.649

6.738

6.960

7.722

6.520

5.010

4.992

5.413

5.271

4.711

5.018

3.559

1.353

5.654

4.980

5.144

5.707

4.819

3.703

3.690

4.001

3.896

3.482

3.709

2.630

fc\fr

S2 = 3.4052, V2 = 0.0095, X2 = 0.1776

Optimized expectances for V2 → min.

31.437

27.910

29.069

32.123

27.428

20.828

20.676

22.416

22.125

19.775

20.780

14.760

5.620

25.114

22.296

23.222

25.662

21.911

16.639

16.517

17.907

17.675

15.797

16.600

11.791

4.490

22.803

20.244

21.085

23.300

19.894

15.108

14.997

16.259

16.048

14.343

15.073

10.706

4.077

19.627

17.424

18.148

20.055

17.123

13.003

12.908

13.995

13.813

12.345

12.973

9.215

3.509

19.235

17.076

17.786

19.655

16.781

12.744

12.651

13.715

13.537

12.099

12.714

9.031

3.439

7.573

6.723

7.003

7.739

6.607

5.018

4.981

5.400

5.330

4.764

5.006

3.556

1.354

5.593

4.966

5.172

5.715

4.880

3.706

3.679

3.988

3.937

3.518

3.697

2.626

fc\fr

S2 = 3.7697, V2 = 0.0089, X2 = 0.1812

Optimized expectances for X2 → min.

31.617

27.946

29.014

32.154

27.258

20.811

20.722

22.442

22.005

19.667

20.804

14.783

5.621

25.214

22.286

23.137

25.641

21.737

16.596

16.525

17.897

17.548

15.684

16.590

11.789

4.482

23.068

20.389

21.168

23.459

19.887

15.184

15.119

16.373

16.055

14.349

15.178

10.785

4.101

19.764

17.469

18.137

20.099

17.039

13.009

12.954

14.029

13.756

12.294

13.005

9.241

3.514

19.307

17.065

17.717

19.635

16.645

12.709

12.654

13.704

13.438

12.010

12.704

9.027

3.432

7.605

6.722

6.979

7.734

6.556

5.006

4.984

5.398

5.293

4.731

5.004

3.556

1.352

5.625

4.972

5.162

5.720

4.849

3.703

3.687

3.993

3.915

3.499

3.701

2.630

fc\fr

S2 = 3.5072, V2 = 0.0090, X2 = 0.1751

Appendix - program filling the data with gaps and computing the expected values in the gaps (PHP source code)

function get_all(&$o){
$a = explode("\r\n",file_get_contents("data.txt"));
$o = array(); for($i = 0; $i < count($a); $i++){$o[$i] = explode("\t",$a[$i]); }
}
function gen_mat($plus){
$a = explode("\r\n",file_get_contents("data.txt"));
$o = array(); for($i = 0; $i < count($a); $i++){$o[$i] = explode("\t",$a[$i]); }
$n = count($o); $m = count($o[0]);
for($i = 0; $i < $n; $i++)for($j = 0; $j < $m; $j++)$q[$i][$j] = "";
for($i = 0; $i < $n; $i++){$j = rand(0,$m-1); $q[$i][$j] = $o[$i][$j]; }
for($j = 0; $j < $m; $j++){$i = rand(0,$n-1); $q[$i][$j] = $o[$i][$j]; }
for($k = 0; $k < $plus; $k++){
for($add_plus = 0; $add_plus < $plus; ){
$i = rand(0,$n-1); $j = rand(0,$m-1);
if(($q[$i][$j] = = = "")){$q[$i][$j] = $o[$i][$j]; $add_plus++; }
}
}
$r = array(); for($i = 0; $i < $n; $i++)$r[$i] = implode("\t",$q[$i]);
file_put_contents("data_censored.txt",implode("\r\n",$r));
}
function get_mat(&$o){
$a = explode("\r\n",file_get_contents("data_censored.txt"));
$o = array(); for($i = 0; $i < count($a); $i++){$o[$i] = explode("\t",$a[$i]); }
}
function set_mat(&$o,&$q){
$q = array();
for($i = 0; $i < count($o); $i++)for($j = 0; $j < count($o[0]); $j++)$q[$i][$j] = $o[$i][$j];
}
function set1mat(&$o,&$q){
for($i = 0; $i < count($o); $i++)for($j = 0; $j < count($o[0]); $j++)if(!($o[$i][$j] = = = ""))$q[$i][$j] = $o[$i][$j];
}
function expect(&$a,&$b){$ss = 0.0; $sr = array(); $sc = array(); $b = array();
for($i = 0; $i < count($a); $i++){
$sr[$i] = 0.0; for($j = 0; $j < count($a[$i]); $j++)$sr[$i]+ = $a[$i][$j];
}
for($j = 0; $j < count($a[0]); $j++){
$sc[$j] = 0.0; for($i = 0; $i < count($a); $i++)$sc[$j]+ = $a[$i][$j];
}
for($i = 0; $i < count($a); $i++)for($j = 0; $j < count($a[$i]); $j++)$ss+ = $a[$i][$j];
for($i = 0; $i < count($a); $i++)for($j = 0; $j < count($a[$i]); $j++)$b[$i][$j] = $sr[$i]*$sc[$j]/$ss;
}
function estim1(&$b,&$r,&$c){
$r = array(); $r[0] = sqrt($b[0][0]);
$c = array(); $c[0] = sqrt($b[0][0]);
for($i = 1; $i < count($b); $i++)$r[$i] = $b[$i][0]/$c[0];
for($i = 1; $i < count($b[0]); $i++)$c[$i] = $b[0][$i]/$r[0];
}
function af_mat(&$a){$r = array(); $t = array();
for($i = 0; $i < count($a); $i++){
for($j = 0; $j < count($a[0]); $j++)
$r[$i][$j] = trim(sprintf("%.3f",$a[$i][$j]));
$t[$i] = implode("\t",$r[$i]);
}
file_put_contents("data_pred.txt",implode("\r\n",$t));
}
function val2S(&$a,&$r,&$c){
$s = 0.0; for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)if(!($a[$i][$j] = = = ""))$s+ = pow($a[$i][$j]-$r[$i]*$c[$j],2); return($s);
}
function val2V(&$a,&$r,&$c){
$s = 0.0; for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)if(!($a[$i][$j] = = = ""))$s+ = pow($a[$i][$j]-$r[$i]*$c[$j],2)/pow($r[$i]*$c[$j],2); return($s);
}
function val2X(&$a,&$r,&$c){
$s = 0.0; for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)if(!($a[$i][$j] = = = ""))$s+ = pow($a[$i][$j]-$r[$i]*$c[$j],2)/($r[$i]*$c[$j]); return($s);
}
function af1mat($s,&$o,&$a,&$rf,&$cf){$r = array(); $t = array();
for($i = 0; $i < count($a); $i++){
for($j = 0; $j < count($a[0]); $j++)
$r[$i][$j] = trim(sprintf("%.3f",$a[$i][$j]));
$t[$i] = implode("\t",$r[$i]);
}
$u = implode("\r\n",$t);
$v = array(); for($i = 0; $i < count($rf); $i++)$v[$i] = trim(sprintf("%.3f",$rf[$i]));
$u. = "\r\nRows factors:\r\n".implode("\t",$v);
$v = array(); for($j = 0; $j < count($cf); $j++)$v[$j] = trim(sprintf("%.3f",$cf[$j]));
$u. = "\r\nCols factors:\r\n".implode("\t",$v);
$u. = "\r\nS2 = ".sprintf("%.4f",val2S($o,$rf,$cf));
$u. = "\r\nV2 = ".sprintf("%.4f",val2V($o,$rf,$cf));
$u. = "\r\nX2 = ".sprintf("%.4f",val2X($o,$rf,$cf));
file_put_contents("data_".$s."_min.txt",$u);
}
function not_empty(&$a){
for($i = 0; $i < count($a); $i++)for($j = 0; $j < count($a[0]); $j++)if(!($a[$i][$j] = = = ""))return(array($i,$j));
}
function fill_recurs($i,$j,&$a,&$r,&$c){
$i_ = array();
for($k = 0; $k < count($r); $k++)if($k < >$i){
if($a[$k][$j] = = = "")continue;
$r[$k] = $a[$k][$j]/$c[$j]; $i_[] = $k;
}
$j_ = array();
for($l = 0; $l < count($c); $l++)if($l < >$j){
if($a[$i][$l] = = = "")continue;
$c[$l] = $a[$i][$l]/$r[$i]; $j_[] = $l;
}
for($k = 0; $k < count($r); $k++)if(!($r[$k] = = = ""))
for($l = 0; $l < count($c); $l++)if(!($c[$l] = = = ""))
if(($a[$k][$l] = = = ""))$a[$k][$l] = $r[$k]*$c[$l];
for($k = 0; $k < count($r); $k++)if(!($r[$k] = = = ""))
for($l = 0; $l < count($c); $l++)if(!($a[$k][$l] = = = ""))
if(($c[$l] = = = ""))$c[$l] = $a[$k][$l]/$r[$k];
for($l = 0; $l < count($c); $l++)if(!($c[$l] = = = ""))
for($k = 0; $k < count($r); $k++)if(!($a[$k][$l] = = = ""))
if(($r[$k] = = = ""))$r[$k] = $a[$k][$l]/$c[$l];
$empty = 0;
for($k = 0; $k < count($r); $k++)
for($l = 0; $l < count($c); $l++)if(($a[$k][$l] = = = ""))$empty++;
if($empty>0){
for($k = 0; $k < count($i_); $k++)
for($l = 0; $l < count($j_); $l++)
fill_recurs($i_[$k],$j_[$l],$a,$r,$c);
}
}
function opti_iterat(&$o,&$q){
for($i = 0; $i < 20; $i++){
set1mat($o,$q); expect($q,$e); set_mat($e,$q);
}
}
function sum_row($i,&$a,&$c,$pa,$pc){
$t = 0.0;
for($j = 0; $j < count($c); $j++){
$ta = pow($a[$i][$j],$pa); $tc = pow($c[$j],$pc); $t+ = $ta*$tc;
}
return($t);
}
function sum_col($j,&$a,&$r,$pa,$pr){
$t = 0.0;
for($i = 0; $i < count($r); $i++){
$ta = pow($a[$i][$j],$pa); $tr = pow($r[$i],$pr); $t+ = $ta*$tr;
}
return($t);
}
function estim2S(&$a,&$r,&$c){
for($i = 0; $i < count($r); $i++)
$r[$i] = sum_row($i,$a,$c,1,1)/sum_row($i,$a,$c,0,2);
for($j = 0; $j < count($c); $j++)
$c[$j] = sum_col($j,$a,$r,1,1)/sum_col($j,$a,$r,0,2);
for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)$a[$i][$j] = $r[$i]*$c[$j];
}
function estim2V(&$a,&$r,&$c){
for($i = 0; $i < count($r); $i++)
$r[$i] = sum_row($i,$a,$c,2,-2)/sum_row($i,$a,$c,1,-1);
for($j = 0; $j < count($c); $j++)
$c[$j] = sum_col($j,$a,$r,2,-2)/sum_col($j,$a,$r,1,-1);
for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)$a[$i][$j] = $r[$i]*$c[$j];
}
function estim2X(&$a,&$r,&$c){
for($i = 0; $i < count($r); $i++)
$r[$i] = sqrt(sum_row($i,$a,$c,2,-1)/sum_row($i,$a,$c,0,1));
for($j = 0; $j < count($c); $j++){
$c[$j] = sqrt(sum_col($j,$a,$r,2,-1)/sum_col($j,$a,$r,0,1));
}
for($i = 0; $i < count($r); $i++)for($j = 0; $j < count($c); $j++)$a[$i][$j] = $r[$i]*$c[$j];
}
get_all($o_all);
gen_mat(0);
get_mat($o);
set_mat($o,$q);
$r = array(); for($i = 0; $i < count($q); $i++)$r[$i] = "";
$c = array(); for($j = 0; $j < count($q[0]); $j++)$c[$j] = "";
list($i,$j) = not_empty($q); $r[$i] = sqrt($q[$i][$j]); $c[$j] = sqrt($q[$i][$j]);
fill_recurs($i,$j,$q,$r,$c);
opti_iterat($o,$q);
af_mat($q);
set_mat($q,$qS2);
estim1($qS2,$rS2,$cS2);
for($k = 1; $k < 20; $k++){set1mat($o,$qS2); estim2S($qS2,$rS2,$cS2); }
af1mat("S2",$o,$qS2,$rS2,$cS2);
set_mat($q,$qV2);
estim1($qV2,$rV2,$cV2);
for($k = 1; $k < 20; $k++){set1mat($o,$qV2); estim2V($qV2,$rV2,$cV2); }
af1mat("V2",$o,$qV2,$rV2,$cV2);
set_mat($q,$qX2);
estim1($qX2,$rX2,$cX2);
for($k = 1; $k < 20; $k++){set1mat($o,$qX2); estim2X($qX2,$rX2,$cX2); }
af1mat("X2",$o,$qX2,$rX2,$cX2);
for($k=1;$k<20;$k++){set1mat($o,$qX2);estim2X($qX2,$rX2,$cX2);}
af1mat("X2",$o,$qX2,$rX2,$cX2);

References

  1. Karl Pearson; On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine 1900, 50, 157-175, 10.1080/14786440009463897.
  2. R. A. Fisher; The Logic of Inductive Inference. Journal of the Royal Statistical Society 1935, 98, 39-54, 10.2307/2342435.
  3. Lorentz Jäntschi; A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested. Symmetry 2019, 11, 835(15p), 10.3390/sym11060835.
  4. R. A. Fisher; On the Interpretation of χ 2 from Contingency Tables, and the Calculation of P. Journal of the Royal Statistical Society 1922, 85, 87-94, 10.2307/2340521.
  5. Sorana D. Bolboacă; Lorentz Jäntschi; Adriana F. Sestraş; Radu E. Sestras; Doru C. Pamfil; Pearson-Fisher Chi-Square Statistic Revisited. Information 2011, 2(3), 528-545, 10.3390/info2030528.
  6. Mugur C. Bălan; Tudor P. Todoran; Sorana D. Bolboacă; Lorentz Jäntschi; Mugur C. Bălan; Sorana Bolboaca; Lorentz Jäntschi; Assessments about soil temperature variation under censored data and importance for geothermal energy applications. Illustration with Romanian data. Journal of Renewable and Sustainable Energy 2013, 5(4), 41809(13p), 10.1063/1.4812655.
  7. Lorentz Jäntschi; Radu E. Sestras; Sorana D. Bolboacă; Modeling the Antioxidant Capacity of Red Wine from Different Production Years and Sources under Censoring. Computational and Mathematical Methods in Medicine 2013, 2013, a267360(7p.), 10.1155/2013/267360.
  8. Donatella Bálint; Lorentz Jäntschi; Missing Data Calculation Using the Antioxidant Activity in Selected Herbs. Symmetry 2019, 11(6), 779(10p.), 10.3390/sym11060779.

Cite this article

Lorentz, Jäntschi. Chi square statistic, Encyclopedia, 2019, v11, Available online: https://encyclopedia.pub/235