Some of my family members do not abide by Islamic teachings and commit major sins such as backbiting. I often advise them and sometimes in an annoying manner, such as shouting at them. But I only do so with the intention of advising them. Thus, I have a couple of questions:
1- What is the ruling on shouting at them with the intention of giving them advice?
2- Is it enough to remain silent when I hear them committing backbiting or other sins, intending to change that evil in my heart?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

The Daa‘iyah (caller to Allaah) should adopt the attitude of the Prophet, sallallaahu ‘alayhi wa sallam, and show kindness and benevolence.

In a Hadeeth on the authority of ‘Aa’ishah the Prophet, sallallaahu ‘alayhi wa sallam, stated that Allaah the Almighty loves kindness in all matters. [Bukhari and Muslim] In another Hadeeth on the authority of Jareer the Prophet, sallallaahu ‘alayhi wa sallam, informed us that he who fails to be kind will lose much goodness. [Muslim]  Also, in a Hadeeth on the authority of ‘Aa’ishah the Prophet, sallallaahu ‘alayhi wa sallam, mentioned that Allaah the Almighty is Kind and loves kindness and confers rewards for kindness that he does not confer for severity, nor for anything else besides it (kindness). [Muslim]

In this regard Allaah the Almighty ordered Moosaa (Moses) and Haaroon (Aaron), may Allaah exalt their mention, to speak gently with the staunchest tyrant Fir‘awn (Pharaoh). Allaah the Almighty Says (what means): {Go, both of you, to Pharaoh. Indeed, he has transgressed. And speak to him with gentle speech that perhaps he may be reminded or fear [Allaah].} [Quran 20:43-44]

If this is the case, how much more so will it be the case if the advised person is a Muslim relative, not to mention being a father or mother?

Thus, it is not permissible to raise your voice over them, whatever the reason might be.

Nevertheless, it is not permissible to stop giving advice or to remain silent regarding an evildoing. Unless the person fears for himself or his bodily harm due to advising, then it will be allowed for him to detest it with his heart and that will be enough.

And Allaah Knows best.