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Astron. Astrophys. 358, 835-840 (2000)

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3. Results of statistical tests

Before presenting the results of the statistical tests, it is immediately clear from Table 1 that radio-intermediate and radio-loud BAL QSOs are not the most polarized objects. Among the five radio-loud BAL QSOs from Brotherton et al. (1998), only one, J1053-0058, which belongs to the LIBAL class, is significantly polarized with [FORMULA]. It is particularly interesting to note that J1053-0058 has the smallest detachment index among the radio-loud BAL QSOs, in good agreement with the anticorrelation between [FORMULA] and DI found in Paper I 2.

Since [FORMULA] and [FORMULA] are often upper limits, the search for possible correlations between [FORMULA] (or [FORMULA]) and other BAL QSO properties must rely on survival analysis. We then use the standard survival analysis tests available in the ASURV Rev. 1.3 package (LaValley et al. 1992). Several sub-samples are considered: LIBAL QSOs, HIBAL QSOs, and BAL QSOs with [FORMULA]. The latter limit corresponds to the redshift at which the [FORMULA] BAL starts to be detected in the visible. Probabilities (P) that the observed statistics occur by chance among indices uncorrelated with [FORMULA] are summarized in Table 3, using the generalized Kendall [FORMULA] and Spearman [FORMULA] rank order correlation coefficients, and the Cox proportional hazard model (LaValley et al. 1992, Isobe et al. 1986). Results from Table 3 indicate that [FORMULA] is essentially uncorrelated with the other quantities ([FORMULA] 0.05), except within the LIBAL QSO sub-sample. The statistical analysis includes the few radio-loud BAL QSOs. Indeed, these objects, although formally radio-loud, are not powerful radio-sources and mostly in the radio-quiet / radio-loud transition region. However, since the original suggestion by Goodrich (1997) only includes formally radio-intermediate objects, we have also considered the sample of BAL QSOs with [FORMULA]. In this case, the results of Table 3 are basically unchanged, P being only slightly higher for the weak correlations detected in the LIBAL QSO sub-sample. Finally the same statistical analysis has been carried out considering [FORMULA] instead of [FORMULA], with the result that [FORMULA] is totally uncorrelated with the other quantities, whatever the sub-sample.


Table 3. Analysis of correlation of various indices with [FORMULA].
This table gives the probabilities that the observed statistics (Cox, Kendall, Spearman) occur by chance among uncorrelated quantities. n is the number of objects considered in the correlation analysis, and m the number of objects with [FORMULA] upper limits. The sign of the correlation is also given, from the sign of the Spearman [FORMULA]. The three less accurate values of [FORMULA] are considered as detections, although the results are unchanged if they are not taken into account. The Cox model cannot be used to test correlations with [FORMULA], since [FORMULA] is also left-censored. Some results, obtained with very few detections and given here for completeness, must be seen with caution

Within the LIBAL QSO sub-sample, the statistical tests suggest the existence of a weakly significant anticorrelation between [FORMULA] and [FORMULA] ([FORMULA] 0.05 for all tests), and a marginally significant anticorrelation between [FORMULA] and [FORMULA] ([FORMULA] 0.05 for only one test). We have therefore plotted [FORMULA] and [FORMULA] against [FORMULA] in Figs. 1 and 2. The anticorrelations do not appear very convincing. And indeed, if we delete only one object from the LIBAL QSO sub-sample (e.g. B2225-0534 in Fig. 1, and J1053-0058 in Fig. 2), the statistical tests indicate that the correlations are not significant any longer. Further, if one restricts the LIBAL QSO sub-sample to high-redshift objects ([FORMULA]), the correlation between [FORMULA] and [FORMULA] disappears ([FORMULA] 0.35 with n/m = 15/9), suggesting a possible bias. Note that due to the limited sample of LIBAL QSOs, no distinction between LIBAL sub-types was made.

[FIGURE] Fig. 1. The BAL QSO polarization, [FORMULA], is plotted here against the radio-to-optical flux ratio, [FORMULA]. Censored data points are indicated. Open squares represent HIBAL QSOs, filled squares LIBAL QSOs, and squares with a cross unclassified BAL QSOs

[FIGURE] Fig. 2. The [FORMULA] BAL maximum velocity, [FORMULA], is plotted here against the radio-to-optical flux ratio, [FORMULA]. Censored data points are indicated. Open squares represent HIBAL QSOs, filled squares LIBAL QSOs, and squares with a cross unclassified BAL QSOs

In view of the apparently different behavior of LIBAL QSOs, we have also compared the distribution of [FORMULA] for the LIBAL and HIBAL QSO sub-samples, again using standard survival analysis tests from the ASURV package. Tests include the Logrank test, and the Gehan, Peto & Peto, and Peto & Prentice generalized Wilcoxon tests (LaValley et al. 1992, Feigelson et al. 1985). The probability that the two samples (25 HIBALs and 21 LIBALs) are drawn from the same parent population is found to range from 0.06 to 0.09, depending on the test. This means that no significant difference in the distribution of [FORMULA] is detected when comparing the LIBAL and HIBAL QSO sub-samples.

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Online publication: June 20, 2000