Since the df are different between t-tests and emmeans ones (cf. ), how then should I interpreted the results?
Let us look at some sample data for 5 hypothetical subjects.  library(emmeans) library(lme4)  # generate some sample data # condition (Placebo, Treatment) # type (some factor, e.g. two different medications) # value (outcome of interest) # subject  data &lt;- data.frame(condition = as.factor(rep(c(0,1), times = 10)), value = rnorm(20), type = rep(c("A","B"), each = 10), subject = as.factor(rep(rep(1:5, each = 2), times = 2)))   Now we like to conduct a mixed model and post-hoc tests with emmeans, where we test conditions per type in a pairwise fashion:  # caluculate a mixed model fit_data &lt;- lmer(value ~ condition*type + (1|subject), data = data)  # emmeans post-hoc test emmeans(fit_data, pairwise ~ condition | type)   So for each type (A or B), I have paired data (condition 0 has 5 data points and condition 1 has 5 data points). If I would do a paired t-test like so:  # paired t-test t.test(data[data$condition==0 &amp; data$type=="A","value"], data[data$condition==1 &amp; data$type=="A","value"], paired = TRUE) t.test(data[data$condition==0 &amp; data$type=="B","value"], data[data$condition==1 &amp; data$type=="B","value"], paired = TRUE)   I would get degrees of freedom of 4 for the paired t-test, but emmeans says the degrees of freedom are 12. Why is there this huge difference? If the emmeans package also would use df = 4, then the p-values would also be more comparable.
Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? Let's say I have a document that has many pages and some pages have different formats. For example, most of the pages are fine with \documentclass{article} but there may be pages in the middle that I want to be \documentclass{letter}. I want my letter pages to be numbered inline with the other pages and have them show up in the Table of Contents.   Is it possible to change the class used for an individual or range of pages while still keeping things like page numbering consistent across them?   EDIT I'm trying to create a "welcome to the company" sort of document that has a cover page, table of contents, welcome letter, then various sections of information. I've never tried this in anything other than a word processor where it's easy to set "master pages" or "template pages" for individual pages in the document. So I could use a letterhead template page for the welcome letter part and a standard article-type template for the rest and still have the letter numbered/referenced in the ToC.   So the idea was to take advantage of the classes written, if possible, rather than have to format a single page by hand differently from the rest. For starters I am using this GREAT answer to achieve custom post types as well as a custom way to insert those post types into other pages, you can view that answer here:   I am running into a problem however when wordpress auto saves itself, it clears whatever metadata i have put into the forms.  EXAMPLE:  I select the 3 drop downs I want from my metabox and hit update.  The page reloads and the 3 choices are still selected.  In phpMyAdmin i can look up the metadata and see the array in place as it should be.  I now wait for wordpress to auto save.  As soon as this happens i can either refresh the page (not update, just refresh) and the metadata is lost, or i can look up the metadata in phpMyAdmin and it will display nothing (ie: its gone).  I'm guessing that this has something to do with needing to tie into another hook (some sorta auto update hook) or something, but even then it doesn't make sense that it would actually delete your metadata (rather than just not auto updating it).  Last note: If you copy and paste Mikes code into functions.php, you should be able to recreate my error (in wordpress 3.1) very easy. How can I prove that the Cartesian product of two countable sets is also countable? Good books on complex numbers I searched on this site but I found posts asking for books on complex analysis which I guess is not what I want.   I want a book on problems concerning with roots of unity, argand plane, exponential form, inequalities etc. Some examples : , , .  I don't think these types of problems are covered in a analysis book but I don't know. I think I want a book on algebra of complex numbers.   Are these topics covered in analysis book ? If not, please suggests some books on this topic. I guess my question is bit vague, sorry for that. :).      As suggested in the comments I would like to have books similar to . UK Visa in an expired Nigerian passport - what to use? I am in Nigeria, my passport will expire in 4 months and I have a document to apply for a UK visiting Visa with that passport number.  Should I use it like that or go for another passport. If I choose to get another passport, how will I use it because of the passport number on the about to expired one in the document I want to use for Visa. I am coding a small Python module composed of two parts:   some functions defining a public interface, an implementation class used by the above functions, but which is not meaningful outside the module.   At first, I decided to "hide" this implementation class by defining it inside the function using it, but this hampers readability and cannot be used if multiple functions reuse the same class.  So, in addition to comments and docstrings, is there a mechanism to mark a class as "private" or "internal"? I am aware of the underscore mechanism, but as I understand it it only applies to variables, function and methods name. Where do the regions of Westeros appear to align as of the season 6 finale? Based on , my understanding is that there are 9 regions of Westeros subject to the rule of the Iron Throne, from Dorne joining up to Cersei killing Robert.  With whom does [insert region] align in what appears to be Daenerys vs Cersei?   Kingdom of the North -- Daenerys probably but currently busy with White Walkers Kingdom of the Mountain and the Vale -- Daenerys probably but currently busy with White Walkers (supposedly; depends on what Littlefinger has in mind) Kingdom of the Isles and Rivers (Iron Islands) -- Daenerys Kingdom of the Isles and Rivers (Riverlands) -- Cersei because the Freys are with the Lannisters? Kingdom of the Rock -- Cersei because she is a Lannister Kingdom of the Stormlands -- Cersei? Kingdom of the Reach -- Daenerys after invitation of Varys and Ellaria Principality of Dorne -- Daenerys after invitation of Varys presumably Crownlands -- Cersei since Cersei is currently occupying the Iron Throne   Am I understanding this right? Anything wrong? Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:   Are matrices and second rank tensors the same thing? If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)? I hope this isn't a duplicate.  Similar questions that I found are over a year old.  I have seen cases where shortly after an upvote (perhaps on an old answer?), the summary in my user activity dropdown shows inconsistent counts:    Particularly, the Slid a div offscreen using jQuery answer shows +40, when it was only updated once yesterday.  Now, if this is a monthly total for this question, then it is correct, but when I refreshed later I got:    Now it shows +10, and other entries completely disappear.  So, what is this supposed to show me?  At the very least I am confused about what I am supposed to see here.  Here is another example:    This shows +20 for the same question as before, after I received a single upvote last night.  What gives?  You can see from my reputation that there was only a sinle upvote recently: I see many posts on both this forum and elsewhere the discuss the use of a wide angle on a crop sensor.  For example, they discuss how at 14mm lens is 14mm on a 35mm film or full frame, but with a 1.6x crop sensor it is an effective 28mm.  My question is, is a 14mm on a crop sensor the same image as a 28mm on a full frame.  Or does "effective" include some other conotation.  I read that a 14mm can create distortion on a full frame and some vignette, so people seem to say using it on a crop takes out some of those edge problems. My confusion is how the lens is actually taking in the extra image.  Reading kenrockwells blog I learned that wide angle keeps lines straight, while fish eye actually skews the lines.    So with this as my understanding, I am still a bit confused how a lens that is curved in order to get a greater angle of view, will be the same as a cropped version of a less curved lens.  Or does "effective" only refer to the objects to the sides but not to the distortion of the image.  Or, if it is different, is it so small that for photography it does not matter?  A technically description would be welcome as long as a basic understanding of the situation.  Thanks!  Edit: To be more clear after the comments,  I am specifically talking about the end resulting 2d projection from 3d space.  I have read both those previous answers.  the first one is closer to what I am talking about.  However, it is still confusing for me.  I have experience with 3d modeling and projection matricies, which might be confusing me with this.  But for example, i dont understand the images having a 50 mm lens being drawn Starting from the sensor, so it shows a different fov for crop and full frame . If the lens is the same, it takes the same amount of information from the world, which makes a ray from a certain degree off the last lens projected into a smaller space, there is no projection from the sensor, so drawing the line from the sensor doesnt make sense to me.  The distance of the sensor to the rear of the lens must have some impact on the light projected but is not represented in the images.  Further, again from that first answer it says cropping is the same as zoom, but from what i understand from how perspective works, it is different, since a wide angle will project lines differently than a small fov to the same size, so cropping the center of a wide angle and zooming into something is very different.  This is easily recreated in an opengl application with varying fov projection matrix, i imagine in painting class people learn this as well.  A fisheye would be a good example, since if cropping was the same as zooming, a fish eye lens would keep the center exactly like a normal lens and then a gradient weighted towards the outside would rapidly create a warped perspective, but from what I see, it is even.  To me those images just look like its comparing cropped with full frame in regards to orthogonal projections. I want to have an index entry typeset as $a_\text{T}$, so I issue the command \index{at@$a_\text{T}$}. Each use of this index command creates a separate line in the index. How can I get \makeindex to recognize they all reference the same topic?  My index entries look like this: Accessing the index in 'for' loops? How do I remove log out, reboot, and shut down from GNOME shell application list? I’m new to Ubuntu. I don't know where the log out, reboot and shut down icons in my application list came from. I just need to remove this from here. For some reason, I can run halt, poweroff, shutdown, and reboot as a regular user without root. How can I fix this? Is it something to do with user groups, or is it a seting somewhere? How can I make a message appears on my PC when my phone is ringing? When I'm at work, I usually have a headset on my head and I am listening to music. My phone is always on vibrate to respect co-workers.  However, I don't keep the phone in my pocket because of its size, so I don't feel it vibrating.  I miss a lot of calls.  I was wondering if it's possible to forward "phone is ringing" notifications to the PC, ideally via a Bluetooth connection.  I would not mind installing a 3rd-party app on the phone or software on the PC. I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much. Can't convert string into float/double Is it possible to use multiple here-docs in bash? Can one use multiple here-docs to provide input to a command in bash?  $ cat &lt;&lt;&lt;foo &lt;&lt;&lt;bar bar $ cat &lt;&lt;EOF1 &lt;&lt;EOF2 &gt; foo &gt; EOF1 &gt; bar &gt; EOF2 bar   Obviously, in both cases, the second here-doc is used as stdin, and replaces the first reference. Is the solution to use echos instead?  $ cat &lt;(echo -n foo) &lt;(echo bar) foobar   Also, for some reason, using a combination didn't work for me. Why would that be?  $ cat &lt;&lt;&lt;foo &lt;(echo bar) bar $ cat &lt;(echo -n foo) &lt;&lt;&lt;bar foo How to show that any separable space is CCC