
| Domain knowledge:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | {1, 3, 6}<{5}<{4}<{2}   |
| |-------------------------|
| All the six variables analyzed. ICA-LiNGAM: {5}<{6}<{3}<{1}<{4}<{2} DirectLiNGAM: {6}<{2}<{1}<{3}<{4}<{5} GroupLiNGAM: δ=0.500 {6}<{2}<{1}<{4}<{5}<{3} δ=0.100 {6}<{2}<{1}<{4, 5}<{3} δ=0.010 {6}<{2}<{1, 3, 4, 5} δ=0.001 {1, 2, 3, 4, 5, 6} x2 omitted. ICA-LiNGAM: {5}<{6}<{3}<{1}<{4} DirectLiNGAM: {6}<{1}<{3}<{4}<{5} GroupLiNGAM: δ=0.500 {6}<{1}<{3}<{5}<{4} δ=0.100 {6}<{1, 3}<{5}<{4} δ=0.050 {6}<{1, 3}<{4, 5} δ=0.010 {6}<{1, 3, 4, 5} δ=0.001 {1, 3, 4, 5, 6} x2 and x6 omitted. ICA-LiNGAM: {5}<{3}<{1}<{4} DirectLiNGAM: {1}<{3}<{4}<{5} GroupLiNGAM: δ=0.50 {3}<{1}<{5}<{4} δ=0.10 {1, 3}<{5}<{4} δ=0.05 {1, 3}<{4, 5} δ=0.01 {1, 3, 4, 5} Table 1. Estimated orders of groups. |                         |

Next, we omitted son's income (x2) and analyzed the other five variables. Omitting x2 **would not create any unobserved confounder since it does not cause any**
other variables according to the domain knowledge. The results are **shown in the**
third top of Tab. 1. DirectLiNGAM and GroupLiNGAM found consistent time orderings between father's variables (x1,x3) and son's variables (x4,x5). Furthermore, GroupLiNGAM found a reasonable ordering between son's variables, **i.e.**,
son's education (x5) could cause son's occupation level (x4**), but not vice versa,**
whereas DirectLiNGAM failed. However, number of siblings (x6**) is the top variable in every estimated ordering by DirectLiNGAM and GroupLiNGAM and could**
cause father's variables (x1,x3**), which is not easy to interpret.**
We further omitted number of siblings (x6) as well as son's income (x2**) and**
analyzed the other four variables (x1,x3,x4,x5). Omitting x6 could create an unobserved confounder since it could relate father's variables (x1,x3**) and son's variables**
(x4,x5**). The bottom of Tab. 1 shows the results. Every ordering estimated by**
GroupLiNGAM is consistent with the domain knowledge. ICA-LiNGAM wrongly estimated that son's education (x5) could cause father's variables (x1,x3**), but not**
vice versa. DirectLiNGAM also gave inconsistent orderings between father's education (x3) and father's occupation (x1) and between son's education (x5**) and son's**
occupation (x4).

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