It's possible that it might affect your rankings by changing the title.

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It is possible but it also affect your page rank

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If it does affect your rank, it should only be minor since backlinks are the major factor for a pages rank.

But lets say your ranking for the keyword 'dogs' - You will want to make sure you have that keyword in your title tag for a higher CTR and to avoid any possible drop in SERPs for lack of relevancy.

Absolute rubbish.

Make sure that you don't write any keyword in title, which is not relevant as per the page's content. If that keyword is already there in description, heading tags, content part etc., then its ok to use that in the title

Title has nothing to do with PR. And PR doesn't matter anyway. I assume he meant page position by "page rank".

You can change the title and a readable sentence without keyword stuffing.

I did this a while back with a site. I changed several of the article titles to optimize for secondary, closely related keywords. I initially failed to re-create the sitemap and re-submit it to WMT. The pages were all ranking well and once I changed the titles they plummeted. I re-submitted a new sitemap to WMT and they all came back stronger within about 12 hours.

Changes often will change your site ranking. Maybe it's good or bad. It's hard to say.