Are Women Really more Independent when their Husband Dies?

I saw this article about men being 30% more likely to die if their wife dies:

Male widows are more likely to die shortly after losing their wife than vice versa, new research suggests

Researchers from the Rochester Institute of Technology in America found that grieving husbands were 30 per cent more likely to die after being recently widowed, compared with their normal risk of mortality.

Women, however, did not have any increased risk of dying – perhaps due to them being more independent and prepared, the researchers suggested.

Note how the article shows the men in a negative light–they are unprepared–the women easily soldier on. However, note what the researcher finds when a woman loses a child:

He also uncovered a strong connection between the death of a child and the mortality of the mother, regardless of the cause of death, gender of the child, marital status, family size, income or education level of the mother.

If a woman lost a child, her chances of dying increased by as much as 133 per cent, he found.

Looking at results from more than 69,000 mothers aged between 20 and 50 over nine years, Professor Espinosa found those who had lost a child were three times more likely to die in the two years afterwards.

Naturally, a mother losing a child is a public health problem to be addressed:

The evidence of a heightened mortality rate for the mother, particularly in the first two years of the child’s passing, is especially relevant to public health policy and the timing of interventions that aim to improve the adverse health outcomes mothers experience after the death of a child.

So, my question is, if women are so independent, why are they more likely to die after a child dies? Doesn’t this study rather indicate that they don’t care as much if their husband dies and that their children are more important to them? Men often complain that once kids come along, they are tossed to the side like a pair of old socks. Maybe there is some truth in this. Or maybe men care  more about their wives on average than the other way around.