At the beginning of a sentence in dialogue that sounds incomplete, should you use an apostrophe?

I'm not sure where I picked up this habit, but I think it might be grammatically incorrect. I'm a fiction writer and have just started working on a new novel, so I want to clear this up before I get too far in. Here's an example, since the question sounds a bit confusing:

The speaker starts off with: "... Shame that...-" Instead of: "... It's a shame that...-" This is how I wrote it: "'Shame that...-"

For some reason, I've picked up the habit of adding an apostrophe at the beginning of an incomplete sentence like that, but I'm not sure if it's just a grammar glitch that my brain contrived randomly or if I saw it used somewhere. Is this incorrect?

Yes, it is incorrect. As noted on Wikipedia:

It is used in contractions, such as can't from cannot, it's from it is or it has, and I'll from I will or I shall.

It is used in abbreviations, as gov't for government.

The omission of entire words outside of contractions is not possible.

You could write

'Tis a shame that [...]

though, but that is due to a specific accent.