Does it mean anything when all items load negatively on one factor? note: A few people have marked this question as duplicate - But my question here is not answered in the other question. Though a fine distinction, what I was asking about here was not whether the signs of a factor matters in the absolute, but whether there was any significance to the relationship between factors (i.e. whether they were largely the same sign, or largely different signs).  So I am fairly familiar with factor analysis, and am aware of answers  and  that tangentially address my question. I believe I am right in my answer to the question I'm asking, but I wanted to make sure.   Essentially, I have, for each of several hundred participants, answers to two sets of almost identical questions. For examples' sake, let's say these are ratings of their current mood state on 10 items at the start (A) and the end (B) of a questionnaire survey. I expect that these mood rating items should load onto two main factors (positive and negative) and that these loadings should be consistent (i.e. both sets A and B should break down similarly into two factors).  When I run the factor analysis, I find pretty much what I expected, except that for rating set A, items 1:5 load positively on factor 1, and items 6:10 load positively on factor 2, while for rating set B, items 1:5 load positively on factor 1, while items 6:10 load negatively on factor 2. Within each factor, the direction of the loadings are consistent across all items. This analysis was performed using varimax rotation.  Now, my understanding is that the difference between the two results is meaningless. Only the relative loading within each factor matters, and the reversal in sign is simply due to the factor analysis randomly converging on a factor that points in the opposite direction, but is essentially the same (except conceptually reversed). But I have had a colleague try to interpret the difference between the two analyses as meaning something with regards to the difference between the two factors.   Is my understanding right, or is there actually something to the difference?  A visual representation that might be more helpful - this is essentially what my output looks like:
Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign? I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?  Consider this:  set.seed(999) prcomp(data.frame(1:10,rnorm(10)))$x              PC1        PC2  [1,] -4.508620 -0.2567655  [2,] -3.373772 -1.1369417  [3,] -2.679669  1.0903445  [4,] -1.615837  0.7108631  [5,] -0.548879  0.3093389  [6,]  0.481756  0.1639112  [7,]  1.656178 -0.9952875  [8,]  2.560345 -0.2490548  [9,]  3.508442  0.1874520 [10,]  4.520055  0.1761397  set.seed(999) princomp(data.frame(1:10,rnorm(10)))$scores          Comp.1     Comp.2  [1,]  4.508620  0.2567655  [2,]  3.373772  1.1369417  [3,]  2.679669 -1.0903445  [4,]  1.615837 -0.7108631  [5,]  0.548879 -0.3093389  [6,] -0.481756 -0.1639112  [7,] -1.656178  0.9952875  [8,] -2.560345  0.2490548  [9,] -3.508442 -0.1874520 [10,] -4.520055 -0.1761397   Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?    In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)?
Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals.    This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as possible. A professor told me that there are many. So, I invite everyone who has seen proofs of this fact to share them with the community. I think it is a result worth knowing how to prove in many different ways and having a post that combines as many of them as possible will, no doubt, be quite useful. After two days, I will place a bounty on this question to attract as many people as possible. Of course, any comments, corrections, suggestions, links to papers/notes etc. are more than welcome. select all rows with a minimum value Can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? Does the Force wane as you age? @Valorum said     ...not to mention the sheer strain that comes from continually using the Force at an advanced age.   To my feeble mind, this doesn't make sense. Approaching 900 years old, Yoda is still as strong as any whippersnapper in terms of Force power and usage. Despite his old age, Obi Wan shouldn't have any problems with using his Force powers.  This got me to thinking, does the Force begin to leave you as you age? Is it a flow of Force power that goes down based on (I hate to say this) a decrease in midichlorian count as you approach your expiration date? Are "white" and "black" ever capitalized when referring to people? For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. At the end of my animation, the animation pauses a bit then loops again, how can I stop the slow pause, and let the loop work fluently? The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ is _____ I tried by taking the odd $n$ values as in that case, the integral I suppose was easier to calculate. So, denote $I_n=n\int_{0}^{1}e^{x^2}x^ndx$, then we have : $I_1=\frac{e-1}{2},\\I_3=\frac{3}{2},\\I_5=5(\frac{e}{2}-1),\\I_7=7(3-e)$ Then I tried calculating (random) values using calculator as integration was getting cumbersome. $I_{31}=2.488$, $I_{51}=2.57$. I do not see any kind of recurrence so that I can find a general term for odd $n$. I also tried the method given . The idea was that since integral doesn't depend on $n$ considering $I(n)=\int_{0}^{1}e^{x^2}x^ndx$, then $I'(n)=\int_{0}^{1}x^n\ln(x)e^{x^2}dx$, but this also doesn't lead me to any conclusion as well. I believe I couldn't get to a proper approach to tackle this question. Can someone please help with the idea that involves in solving these type of questions? Easiest way to prove that $2^{\aleph_0} = c$ $\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$.  I know that $|P(A)| = 2^{|A|}$. This means that the cardinality of the power set of a set is 2 raised to the power of the cardinality of that set.  This basically means that to prove  $2^{\aleph_0} = c$, I need to prove $c = |P(N)|$. I've seen the Cantor diagonal argument, although I have no idea on how to use it to prove this.   Could anyone suggest ideas of the simplest way to prove that $2^{\aleph_0} = c$? Check if partition is encrypted Is there an official list of Disney Princesses? I'm trying to match SHA1's in generic text with a regular expression.  Ideally I want to avoid matching words.   It's safe to say that full SHA1's have a distinctive pattern (they're long and a consistent length) - so I can match these reliably - but what about abbreviated SHA1's?  Can I rely on the presence of numbers?  Looking at the SHA1's in my commit log - numbers always appear in the first 3 characters. But is this too short? How many characters of SHA1 do I need to consider before I can assume a number would have appeared?    This does not have to be 100% accurate - I just need to match an abbreviated SHA1 99% of the time. Trouble with batch conversion of .png to .pdf using convert Thunderbird - remove unused dictionaries from menu How to disable the "Get Windows 10" icon shown in the notification area (tray)? This icon showed up in my taskbar notification area today and I cannot seem to get rid of it:    Clicking on it displays the following screen:    So how do I disable or remove the "Get Windows 10" icon? MacBook Pro locked with Find My Mac and won't let me boot Ubuntu 14.04: very slow UI in Oracle VirtualBox Short summary: 14.04 is much slower than 12.04 even with more RAM assigned.  I'm using latest VirtualBox 4.3.10 on Mavericks 10.9.2. How can I make it faster? Can I use Divine Intervention or Wish to regain Wish? After reading this question on , I started wondering if there are ways for them to regain that ability.  I figured a Divine Intervention is probably strong enough as that is a god's power, as well as a second caster casting wish to undo the first wish (and thereby undoing its effects).  Are these valid ways of regaining the ability to cast wish? This looks a little bit like a variation on  but specific to the tags view.  Reproduction (app version 1.0.71):   Select any site (say Meta) from the sidebar Switch to the tags view using the top bar Choose a tag, any tag ( is nice)   optionaly refine your search with more terms  Go to any question found, go back   you end on the unfiltered tags view  Go back again   lucky you, your search is here I connected an external screen to my laptop, but my desktop didn't automagically extend to it, as it used to do about a year ago (older version of Ubuntu).  I went to the display settings. The screen was listed there (correct resolution, brand name, etc.). But when i put the switch on 'on' and click 'apply', i get this error:   Failed to apply configuration: %s  GDBus.Error:org.freedesktop.DBus.Error.UnknownMethod: No such interface `org.gnome.SettingsDaemon.XRANDR_2' on object at path /org/gnome/SettingsDaemon/XRANDR   It's been like this for a few months now, but i'd really like to be able to use an external screen/beamer again.  Any ideas?  Thanks.