Let’s say you’re always good about citing your sources, checking that you didn’t introduce typos into your quoted material, and marking any changes to the original with clear indicators like ellipsis points and brackets, with an appropriately placed “[sic]” where you made a point not to change it. These are all good things to be aware of. But there are a handful of changes you can make to quotations that actually shouldn’t be bracketed if you’re using APA or Chicago style.* 
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If you have either manual in front of you, we’ll be working with APA section 6.07 and Chicago section 13.7. There are three types of changes that both manuals allow you to change without indication.

That heading sounds like a very library-centric Nancy Drew mystery, but we’re just talking about lowercase and uppercase letters. Many conscientious scholars have noticed this treatment of a quotation in a book or article and thoughtfully applied it to their own work:

In the original, the word “he” was capitalized, and the author quoting it is being careful enough to indicate their change. There are styles that require that the h be bracketed in this case, but neither Chicago nor APA is among them. In the APA’s words, “The first letter of the first word in a quotation may be changed to an uppercase or a lowercase letter.” So if you’re following either of these styles, you should really write:

The quotation marks that appear in a text may be double (“/”) or single (‘/’) depending on the style the original author was following, on whether the quotation itself is within a quotation, and in some cases whether the text is quoted material or words being presented “as words.” When you quote material including quotation marks, you might need to switch from single to double or vice versa. Both APA and Chicago allow this change without any indication that you did it. Here’s an example:

The dialogue was enclosed within double quotation marks in the original, so they appear in single quotation marks in the quoted version, within double quotation marks that surround the entire quoted sentence. The bracketed s indicates a letter changed from the original. That is a type of quotation change you need to make clear.

This punctuation mark should suit your text’s syntax, which may or may not be that used in the original.

Note that Collins’s sentence ends after “HIV/AIDS,” but mine doesn’t. So while she follows the abbreviations with a period, I follow them with a comma to suit my context. That change doesn’t need to be given brackets or any other kind of identifying marks. Next, I made the opposite change: whereas “women and men” appears unpunctuated in the middle of a sentence for her, it’s at the end of the sentence in mine and thus gets a period, also unbracketed.

And yes, the comma and the period both go within the quotation marks, as they always do in APA and Chicago styles, regardless of what the text is. The rules are a little different for other punctuation marks (e.g., semicolons, question marks, parentheses), but those are beyond the scope of this post.

The three above are the only changes APA allows to be implemented without indication, but Chicago has a few more.

• Chicago allows obvious typos in the original to be corrected silently in most cases. 
• Chicago allows reference marks (footnote/endnote numbers or symbols) to be omitted “unless omission would affect the meaning of the quotation.”
• Chicago allows some archaic characters to be replaced with modern equivalents, such as ʃ, which can be replaced with s without comment.

All three of these changes are subject to context and additional guidelines, so when in doubt, consult the manual. The option of fixing typos is the one most people are likely to have use for, so it’s a good one to bear in mind. It means that the common practice of typing “[sic]” after any error in the original or making a conspicuous correction is not always necessary.

So use your best judgment:

• If it’s clearly an error that the author, as best you can speculate, would have fixed if they’d noticed in time, then a silent change may be the best choice.
• If it’s a peculiarity of a dialect, a historical source, or a genre that is true to how the original author intentionally writes or speaks, then it may be best left alone, unchanged and without comment (or changed universally, with a note explaining that you did so).
• If it’s a word you’re changing the tense of, as in the Pullman example above, to fit the syntax of your context, then bracketing the change will do the trick.
• Finally, if the troublesome word or phrase is erroneous in a way that is true to the author’s intent but could also be mistaken for your own, then a “[sic]” may be the best option.

But all of these are subject to context, so again, use your best judgment, and when in doubt, consult your style guide.

* Chicago Manual of Style (16th ed.) or American Psychological Association (6th ed.). These are the two books my particular work niche has me use most often, and if you’re in the humanities or social sciences, chances are good you need to use one or both of these too.
† Charles W. Eliot, introductory note to The Pilgrim’s Progress, by John Bunyan, The Harvard Classics, Vol. 15 (New York: P. F. Collier & Son, 1909), 3.
‡ Philip Pullman, The Amber Spyglass (New York: Alfred A. Knopf, 2007), 447.
§ Patricia Hill Collins, Black Sexual Politics: African Americans, Gender, and the New Racism (New York: Routledge, 2004), 301.

Your advisor and other committee members. It would be a real faux-pas to forget anyone in this group. Your advisor probably has almost as much emotional investment in your work as you have by the time it’s complete, and your committee on the whole is the reason you’re getting this degree.
Study participants. You don’t have to thank amoebas, but if you performed research with living human participants—in interviews, surveys, participant observation, or any other method—it’s courteous to acknowledge them. You’re not thanking them by name unless their names were used in the dissertation (e.g., experts who were interviewed for their expertise), but a blanket thank-you to the group of them will be a nice professional touch. You couldn’t have done this research without them.
Coresearchers. Although you’re writing your dissertation on your own, you might have had coresearchers who worked with you on data collection and the like. Don’t forget to thank them!
Your editor and/or proofreader(s), whether paid or unpaid. Although I admit I love seeing my name in an Acknowledgments section, that’s not the main reason I include the category I belong to in this discussion. At best, it comes a distant third to etiquette and ethics.
    With respect to professional etiquette, it’s possible that your editor will have spent more time interacting with your dissertation than anyone but you, and depending on how heavy the edit was, that includes your advisor too. You don’t want to slight someone who spent 60 hours or more on this document. Take a look at any published book on your shelves and see how many published authors do and don’t thank their editors.
    The primary reason editors and proofreaders are included, though, is ethics. It is critical to be transparent about how much outside assistance you received on the content of a document you’re submitting for academic credit. Editors Canada recommends that editors include a requirement in their contracts or agreement forms that the thesis/dissertation author acknowledge the editor, and these days I do. Some graduate schools require the dissertation author to do so as well. The last thing you want is for committee members or outside readers to suspect you of hiding assistance of this kind.

Funders. If you received grants of any kind for completing your research, even if it was not for travel or equipment but merely to allow you to work less or not at all during the process, your funding sources deserve an acknowledgment here.
Permission holders. If you reproduced previously copyrighted materials, you’d have had to request permission from the copyright holders to do so. Even if you included them in your (properly formatted APA-style) figure captions, it’s nice — and at some universities required — to thank them here as well.
Transcriptionists and typists. People who performed professional services for you deserve a place in your acknowledgments. Think about how much time they saved you when they did this work and you didn’t have to.