Application of log-linear model in inference on karyotypic evolution in chronic myelocytic leukemia.

Relationships among additional chromosome abnormalities in chronic myelocytic leukemia (CML) with translocation 9;22 [Philadelphia chromosome (Ph1)-positive CML] were analyzed by log-linear models on 709 karyotypes reported in the literature. Additional abnormalities, such as the gain of chromosome 8 (+8), gain of Philadelphia chromosome (+Ph1), isochromosome of the long arm (q) of chromosome 17 [i(17q)], and the gain of chromosome 19 (+19), were frequently observed. A four-way 2 x 2 x 2 x 2 contingency table was considered with respect to the appearance of these four abnormalities, then the hierarchical log-linear models having at least four main effects were fitted to the observed contingency table. Akaike's information criteria of the models reflected the fitness of the model very well. Parameter estimates of the interaction terms indicated that the combinations of two abnormalities, '+8 and +19', '+Ph1 and +19', and '+8 and i(17q)' were positively associated, while '+Ph1 and i(17q)', and '+19 and i(17q)' were negatively associated. Based on the results of the data analysis, an inference was made on the route of karyotypic evolution in Ph1-positive CML; it statistically supports the hypothesis presented by Heim and Mitelman.


Introduction
The translocation 9;22 found in the karyotypes of chronic myelocytic leukemia (CML) patients is strongly associated with CML as a specific chromosome abnormality. It is regarded as a primary abnormality that plays a fundamental role in initiating the malignant process of CML (1). Additional chromosome abnormalities superimposed on the translocation 9;22, such as +Ph1[j+22q-, +del(22)(qll), etc.], +8 and/or i(17q) also occur in most patients with CML in blastic crisis. These abnormalities reflect the karyotypic evolution of malignant cells in vivo (2)(3)(4).
Heim and Mitelman (4) presented a hypothesis of karyotypic evolution in CML patients by analyzing the patterns of additional chromosome abnormalities other than translocation 9;22. The major route of karyotypic evolution by their hypothesis was as follows: + 8, + Ph1, or i(17q) are the main additional changes after a translocation 9;22 occurs; + 19, on the other hand, seems to occur later in karyotypic evolution, most often in combination with both + 8 and + Ph1; i(17q) in combination with + 8 is a quite frequent phenomenon, whereas the combinations 'i(17q) and + 19', '+ Ph' and i(17q)', '+ 8, i(17q), and + 19', and '+ Ph1, i(17q), and + 19' are only seen very rarely. These findings testify to the fact that i(17q) apparently has a restrictive role in the cytogenetic evolution in CML, at least when no extra chromosome 8 is present in the cells. The hypothesis, however, is derived from single-column freq1uencies of combinations of abnormalities, such as '+ Ph and + 8' and '+ 8 and i(17q)', without analysis of associations among additional abnormalities.
In the present study, we analyzed 709 cases of Ph1 positive CML karyotypes with additional abnormalities that were derived from the same database of Mitelman et al. (4) and quantified the relationships among additional abnormalities by means of multivariate analysis of frequency table (log-linear models).

Materials and Methods
The material used in this study consisted of 709 Ph1positive CML karyotypes with additional abnormalities, which were collected from the Catalog of Chromosome Aberrations in Cancer, second edition (5). Single-column frequencies of the additional abnormalities were analyzed by our previously reported computer program (6)(7)(8). A four-way 2 x 2 x 2 x 2 contingency table where the four indices pertain to categorical (none or present) variables A: +8, B: +Ph', C: i(17q), and D: +19 was considered, and then log-linear models (9) were fitted to the observed four-way contingency To evaluate the goodness of the fit for 113 models, Akaike's information criteria (AIC) (10) was calculated for each model: where G2, df represents the likelihood ratio chi-square, the degrees of freedom, of the model, respectively.
Generally, two models Ml and M2 are said to be nested if all of the X effects in Ml are a subset of the X values contained in M2. Conditioning the effects in Ml, the difference in G' between the Ml and the M2 is the test of the additional effects in M2. This difference also has an asymptotic chi-square distribution with degrees of freedom equal to the difference in the number of parameters fitted to the two models. Therefore, to test the significance of twoor three-way interactions, the difference in the G2 between one model and the other model which does not contain the attended interaction was computed. The program package BMDP (11) was used for these analyses.

Results and Discussion
Additional chromosome abnormalities frequently found in the 709 karyotypes from Ph'-positive CML patients were + 8 (253 cases, 35.6%), + Ph' (238 cases, 33.5%), i(17q) (184 cases, 25.9%), + 19 (109 cases, 15.3%), +21 (48 cases, 6.7%), + 17 (44 cases, 6.2%), +10 (29 cases, 4.0%), -7 (28 cases, 3.9%), and -Y (17 cases, 2.3%). All of the other additional abnormalities were less than 2% of 709 cases. Table 1 shows the 2 x 2 x 2 x 2 contingency table with respect to the presence of four main abnormalities (+ 8, + Ph1, i(17q), and + 19). The table contains the raw data of this analysis. Table 2 summarizes the degrees of freedom (dj), likelihood ratio chi-square (LR X ), and the AIC of the 113 hierarchical models at least with four main effects. AICs are graphically plotted on the right-hand side of the table. Of all the 113 AICs, 19 AICs were less than 0. The maximum and minimum values of AIC were 188.49 in Model 1 (the independent model) and -5.68 in Model 103. In the models with one or two terms of two-way interactions (Models 2 through 22), all of the AICs were greater than 50. In the models with three terms of twoway interactions (Models 23 through  indicates that the two-way interaction of AB does not contribute to increase a fitness. In the models with five terms of two-way interactions (Models 62 through 67), [AC,AD,BC,BD,CD] (Model 67) was the best with an AIC value of -2.93. The AIC was especially lower than other models, and it was the tenth lowest value of all the 113 AICs. The AIC of the second-order full model (Model 80) was -3.69, which was the fifth lowest value. In the models with five terms of two-way interactions and one or two of three-way interactions (Models 81 through 98), Models 89, 92, and 98 (which lacked the two-way term AB) indicated especially lower AIC values than others, that is, -1.11, -1.46, and 0.36, re-  11 10 10 10 10 10 10 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8     Parameter estimates of the cells that correspond to 'presence', 'presence and presence', or 'presence and presence and presence' of one, two, or three abnormalities were shown in Table 4 as the values of 1B = exp(M). Concerning the six interactions of a given two factors, if the value of parameter A is greater than 1, the two factors are considered to be positively correlated to each other. On the other hand, if A is smaller than 1, a negative correlation is suggested between the two factors. Then the parameter estimates and the results of significance tests on Model 80 indicate that '+ 8 and i(17q)', '+ 8 and + 19', and '+ Ph' and + 19' were significantly positively associated, while '+ 22qand i(17q)' and '+ 19 and i(17q)' were significantly negatively associated. The parameter estimates of the two-way interactions in the second-order full model (Model 80) was not greatly different from those in the thirdorder full model (Model 113), and then the parameter estimates of the three-way interactions in the thirdorder full model (Model 113) and the best model (Model 103) were nearly 1. Therefore, the four-way 2 x 2 x 2 x 2 contingency table is considered to be fully explained by the second-order full model (Model 80). Furthermore, the second-order model, which only lacks a two-way term AB (Model 67), is also considered to be a reasonable model.