The first problem of the Chi-square test of goodness-of-fit is how to establish the number of frequency classes. At least two approaches could be applied:

The number of frequency classes (discreet number) is computed from Hartley's entropy [46] of observed versus expected data: log₂(2n), where n = number of observations. The EasyFit software (MathWave Technologies. http://www.mathwave.com) uses this approach.

One rule-of-thumb suggests dividing the sample in a number of frequency classes equal to 2·n^2/5 (where n = sample size) [48].

The second problem refers to the width of the frequency classes. Two approaches could be applied here:

Data could be grouped in probability frequency classes (theoretical or observed) with equal width. This approach is frequently used when the observed data are grouped.

The third problem is the number of observations in each frequency class. Every class must contain at least five observations; otherwise the frequencies from two proximity classes are cumulated.

The investigation of homogeneity of the values associated to a class (row or column in the contingency table) could be carried out by decomposing the X² expression (see Equation (3)). A hierarchy of irregularities on the contingency table could also be obtained by decomposing the X² expression.

One assumption is that the O_i,j observations are the result of multiplying two factors; repeated observations approximate better the effect of multiplication. Thus, the formula of expected frequencies (E_i,j [43]) is the consequence of factors' multiplication and it is presented in Equation (4).

Three mathematical assumptions could be formulated in terms of square error ((O_i,j − E_i,j)²) of observation:

The measurement is affected by chance errors, absolute values (S², Equation (5));

The first hypothesis (chance errors, absolute values) leads mathematically to the minimization of the variance (S²) obtained between model and observation.

where a_i, 1 ≤ i ≤ r = contribution of first factor to the expected value E_i,j; b_i, 1 ≤ j ≤ c = contribution of second factor to the expected value E_i,j; E_i,j=a_i·b_j.

The second hypothesis (chance errors, relative values) leads to the minimization of the squared coefficient of variation (CV²) (see Equation (6)).

One possible solution for the third hypothesis is the minimization of the X² statistic (see Equation (7)).

The contribution of each factor (A = (a_i)_{1 ≤i ≤ r}, and B = (b_j)_{1 ≤ j ≤ c}) could be determined through the minimization of values given by Equations (5)–(7). The following formula (Equation (8)) was applied in order to minimize the values form Equations (5)–(7).

(8) ( ∂ ⋅ ( a i , b j ) ∂ a i = 0 ) 1 ≤ i ≤ r ; ( ∂ ⋅ ( a i , b j ) ∂ b j = 0 ) 1 ≤ j ≤ cwhere the expression of derivate (a_i, b_j) is the expression of S²/CV²/X² given in Equations (5)–(7).

The calculations revealed the followings:

The relation in Equation (5) is verified by the values of (a_i)_1≤i≤r and (b_i)_1≤j≤c from Equation (9).

The relations presented in Equations (9)–(11) admit an infinity of solutions and the family of solutions are close to the family of solutions given by Equation (4). Equation (4) was rewritten as presented in Equation (12).

Dealing directly with Equations (9)–(11) without using Equation (12) is ineffective. For example, for r = 2 and c = 3 substituted in Equation (9) leads to Equation (13): (13) ( a 2 a 1 ) 2 + ( O 1 , 1 2 + O 1 , 2 2 + O 1 , 3 2 ) − ( O 2 , 1 2 + O 2 , 2 2 + O 2 , 3 2 ) ( O 1 , 1 O 2 , 1 + O 1 , 2 O 2 , 2 + O 1 , 3 O 2 , 3 ) ( a 2 a 1 ) − 1 = 0

This is solvable in (a₂/a₁). Thus, there are an infinity of solutions (for any non-null value of a₁ there is a value a₂ that verifies Equation (13)) and the degree of equation Equation (13) is given by min(r,c). The equations that are obtained by direct substitution are more complicated as the r and c values increase. For example, if r = 2, and c = 3 the substitutions in Equation (11) lead to the relation presented in Equation (14), which is an equation of fifth degree (r + c).

The application of successive approximations using the solution offered by Equation (12) is the indirect way to solve the relations presented in Equations (9)–(11). The relation presented in Equation (12) is used in order to obtain the first approximation (initial approximation) of the solution; in every sequence of approximations the oldest values are replaced on the right side of the relations presented in Equations (9)–(11) in order to obtain the new approximations.

The method of successive approximations rapidly converged towards the optimal solution. Thus, three iterations are necessary in order to obtain a residual value of 282.11735 for the relation presented in Equation (9). Starting with the third iteration, the value of residuals is changed at the level of the 5^th decimal. For the relation presented in Equation (11) the same quality of representation of the optimal solution is obtained after the 4^th iteration.

The experimental data reported by Fisher [4] were used for exemplification (Table 2). The values suggested by Equation (12) for the (a_ib_j)_{1 ≤i ≤ 6; 1 ≤ j ≤ 12} are showed in Table 3.

The values resulted when the iterative approach was applied to obtain the solution for Equations (9)–(11) are presented in Tables 4–6. The summary of the results obtained by all four approaches are presented in Table 7.

The analysis of the results presented in Table 7 revealed that each method defined in Equations (9)–(11) increases the value of the objective sum compared with the expression defined in the Equation (12) formula; the methods provided by Equations (9)–(11) also represent corrections of Equations (12). The relation presented in Equation (9) offers a better solution compared to Equation (12) under the hypothesis of experimental errors uniformly distributed among classes (absolute experimental error). The relation presented in Equation (10) obtained a better solution compared to Equation (12) under the hypothesis of experimental errors proportional with the magnitude of the observed phenomena (relative experimental error). The relation presented in Equation (11) obtained a better solution compared to Equation (12) when the aim is to minimize the Pearson-Fisher X² statistics (Pearsonian expression of type III [4], p. 337).

The values for all three types of experimental errors (square absolute S², square relative CV² and Pearson's X²) and for all four analyzed cases are presented in Table 7 (theoretical frequency estimated from the contingency table - Equation (12); theoretical frequency estimated through the minimization of the square absolute error - Equation (9); theoretical frequency estimated through minimization of the square relative error - Equation (10); theoretical frequency estimated through minimization of Pearson-Fisher statistics - Equation (11)); the values are obtained in a design of experiments with two independent factors (type of treatment and type of potato variety, abbreviated as factors A and B). This experiment allowed the representation of the Euclidian distances between obtained results (see Figure 1).

The experimental errors estimated by Equations (9)–(11) are presented in Figure 2 using the Snyder triangle [49] (diagrams frequently used in chromatography in order to represent three or more parameters which depend on two factors).

Figure 2 was obtained by setting the representation at the same scale of error area in ratio with two factors (the distance between the coordinate of experimental error for the hypothesis that S² = min and the coordinate of experimental errors for the hypothesis that CV² = min was used as reference). The coordinates for the hypothesis that X² = min were obtained after maximizing the error area (maximization of the A, V and X triangle area). The coordinates of contingency were obtained so that its projections on the sides of the triangles could split the sides into the ratios observed among the differences in Table 7.

The graphical representation in Figure 1 provides qualitative remarks regarding the contingency model defined in Equation (12) in relation with experimental errors:

The intersection between the contingency area and error areas is done through the absolute square error. Therefore, the contingency defined by Equation (12) assured the agreement between observation and model for the absolute square error only (one out of the three types of errors included in the study).

The analysis of errors distribution obtained from the above association analysis is presented in Supplementary Material.

The relative position of the solution proposed in Equation (12) could be represented in relation to the optimal values obtained using Equations (9)–(11). Therefore, the values presented in Table 7 (last row) were re-arranged and then expressed after being divided to their minimum values. The results are presented in Table 8.

Figure 2 contains the representation of the relative values of errors (error excess) in the coordinates defined by the values of S², CV² and X² for the results obtained through simple estimation (E, Equation (12)), minimization of the absolute square error (S² = min, Equation (9)), minimization of the relative square error (CV² = min, Equation (10)) and minimization of the X² statistics (X² = min, Equation (11)).

The results of the representations showed in Figure 2 are consistent with the results of the projections in the areas illustrated in Figure 1. Figure 2 showed that the solution proposed by Equation (12) is very close to the solution proposed by Equation (9) and Equation (11). Moreover, the solution is intermediate between Equation (9) and Equation (11) and far away from the solution proposed by Equation (10).

A single degree of freedom is known to exist for a 2 × 2 contingency table.

Table 9 presents such a situation in which the restrictions come from the sums of observations.

The probability to observe the situation presented in Table 9 is given by the multinomial distribution (given by (Equation (15)). The value of the Chi-square statistics (X²) is given by the relation presented in Equation (16).

The range in which x could take values is [0. Min (n₁,n₂)].

In order to exemplify this problem, the experimental data reported by Fisher in 1935 [19] (n₁ = 13, n₂ = 12, n₃ = 18) for a range of x variation from 0 to 12 and with an observed value of 10 was analyzed. The value of X² statistic (Equation (16)) was represented in Figure 3.

The space of possible observations regarding the X² statistics as function of the independent variable x is discrete as it can be observed from Figure 3. The observed value (x = 10) is situated into the vicinity of one boundary (x = 12) having only two less favorable observations (with an X² value higher than the observed value) compared with the observed value in the same vicinity (x = 11 and x = 12) and a less favorable observation in the opposite vicinity (x = 0).

Two possible approaches could be applied in relation to the objective of the comparison in a contingency table:

If higher distances from homogeneity than the observed gives the statistic, then the probability associated to observation is obtained by cumulating the probabilities for x = 0, x = 10, x = 11 and x = 12 (red and blue dots on Figures 3 and 4).

Figure 4 present graphically the probability of the observation (calculated from Equation (15)).

Table 10 presents the values of three probabilities: the probability of χ² distribution (p_x2), the probability to observe a higher distance from homogeneity in the direction of the observed value (p_O2) and the probability to observe a higher distance from the homogeneity in any direction (p_D2). The probability obtained from the χ² distribution (p_X2) estimates a higher distance from homogeneity in any direction (p_D2).

Table 10 shows how the Chi-square test is in error when the values in the contingency table are far from the imposed conditions for expected counts or frequencies (no more than 20% of the cells in the contingency table should have counts/frequencies lower that 5). Table 10 also shows how in this case the Chi-square test is exposed to type I errors (giving a lower observation probability than the real one; the risk is to accept the alternative hypothesis even if this is not true).

Frank Yates proposed in 1934 [18] a continuity correction in order to correct the statistical significance in a contingency table. If this correction is applied to Equations (1)–(3), a 0.5 value must be subtracted from the absolute difference between observed and expected frequencies in the hypothesis of independence (the middle of the frequency interval). Mantel and Haenszel proposed in 1959 [52] a correction of Chi-square test by dividing its value to df/(df − 1), where df = degree of freedom.