("Multivariate" regression in this post means a multidimensional response variable.)  I have been playing with multivariate regression for the past two days, and I have noticed something that did not make such an impression on me the first time I studied multivariate regression: it seems to be exactly the same as doing one regression per response variable.  If I do the multivariate regression, I get a matrix of parameters. If I do one regression per response variable, I get several vectors of parameters that can be concatenated to form the matrix of parameters, so doing the multivariate regression does not affect the parameter estimates.  Okay, the parameter estimates are the same either way. The next place where there could be differences is in standard errors on the parameters. The Rencher and Christensen book I'm consulting does not give a way to calculate standard errors for any individual value in the parameter matrix, but it gives a way to determine if a row of the parameter matrix is significant, which is pretty close and at least analogous to an F-test on one parameter in a vanilla OLS regression. However...  ...nothing in that calculation seems to use the dependence of the response variable!  What's going on? What does multivariate regression get us that one regression per response variable lacks?  (For context, a lot of what I do is two-sample ANCOVA where I want to extend a two-sample t-test to account for sources of variability: the covariates in the regression. I now want to try this with multivariate response variables and test the indicator variable, I suppose a two-sample MANCOVA. But it seems that I do not get any advantage by doing one MANCOVA instead of one ANCOVA per response variable.)  Reference  Rencher, Alvin C., and William E. Christensen. Methods of Multivariate Analysis. 3rd ed., Wiley, 2012.
I understand that multivariate regression models contain more than 1 dependent variable, but what is the difference between running a multivariate regression with dependent a and b to a set of independent variables, rather than running two regression models, one for a and one for b against the same set of independent variables?
I have a model similar to below with a continuous outcome, continuous predictor and a categorical variable. the goal of this analysis is to make inferences on the group variable. The issue is that y is bounded by x in the form of y >= -x. When the model makes predictions you can see in the plot below that the predicted value is outside the possible range. See the last plot.   How can this be remedied? A custom loss function to penalize going beyond the bound and use gradient descent regression? Or is there some other way? Thank you.   x =   rep(seq(-10,10,1),each=5)  y =  rep(0,length(x) ) weights = sample( seq(1,1000,1) ,length(x)) groups = rep( letters[1:5], times =length(x)/5  )   ## the outcome y has the relationship y &gt;= -x for(i in 1:length(x)){   y[i] = sample(  seq(-x[i],-x[i]+ ifelse(x[i]&lt;0 ,                                            sample(seq(1,10,.1),1),                                           sample(seq(5,40,.1),1) ),1)                        ,1) }   dat =data.frame(x = x, y = y, weights = weights, group = groups) model = lm(dat = dat, y ~ x + group + group * x, weights = weights) dat$FIT= predict(model, dat, allow.new.levels = TRUE)  library(ggplot2) ggplot(data = dat, aes(x = x, y = y, color = group))+geom_point( aes(size = weights))+ geom_line(data = dat, aes(x = x, y = FIT, color = group))    #data for new predictions newdata = data.frame( x = seq(-100,100,1) ,                        group = rep("c", length( seq(-100,100,1) )),                       bound = -seq(-100,100,1)                       )  ## when pridicting new data the outcome y has the relationship y &gt;= -x ## but the predicted vales are below the bound. See how red line is below blue to the left of zero new_preds  = predict(model, newdata, allow.new.levels = TRUE) ggplot(data = newdata, aes(x = x, y = new_preds, color = group))+geom_line()  +   geom_line(data = newdata, aes(x = x, y = bound, color = "green")) We are working on a service that is quite new and unknown to people. It's is oriented on small business. The target audience is divided into 3 specific different groups. As for now, we have one landing page, which communicates to 3 groups.   We have very low conversion mainly because people don't understand how they can use the service and benefit from it.  So, we decided to change the logic:  There will be one entrance page, where user will have to choose his group (1 of 3) and then he will be forwarded to the page of the group he chose before, where all the explanations how to use the service are provided.  So instead of one page we will have four pages. Will it improve the communication with users and are there any suggestions how to deal with 3 different target audiences? Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$. How to read and write excel file Can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? Entering Unicode characters in LaTeX How do I enter Unicode characters in LaTeX?  What packages do I need to install and what escape sequence do I type to specify Unicode characters in an ASCII source file? I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ How do you correctly say large numbers '+' and 's' in permission strings If the ls -l command gives me a permission string like  rwsr-s--x   What does the 's' mean? The only sources I found mention that it can be present sometimes but do not elaborate.  What does a '+' instead of a '-' mean? I have found mentions of 'extended permission' but nothing clear. Can't boot into Ubuntu in Windows 10 / Ubuntu dual boot I installed Ubuntu on a Dell XPS 13 laptop that had Windows 10 preinstalled on it. I installed it from a USB flash drive on a partition. When I boot my computer I can only boot into Windows 10 and Ubuntu is nowhere to be seen.  If I boot into my USB then I can see that Ubuntu is installed, but I can't get to it from the BIOS boot menu. I am really confused in Gauss law.  Why do E3 and E2 pointing up? and also E1 pointing down?  The lecture note said infer from symmetry and you will get the following but I dont really understand. Could anyone explain to me? I am new to gauss law. Why is energy in a wave proportional to amplitude squared? I'm a mathematics student trying to grasp some basics about wave propagation. A sentence I find very often in introductive physics textbooks is the following:      In a wave, energy is proportional to amplitude squared.   This is something I would like to understand better in the case of mechanical (linear) waves.  The simplest model is the vibrating string of mass density $\mu$ and tension $T$: here an element of string of rest length $dx$ and vertical displacement $y(x, t)$ possesses a kinetic energy $\frac{1}{2}\mu dx \left( \frac{\partial y}{\partial t}\right)^2$ and a potential elastic energy $\frac{1}{2}T dx \left( \frac{\partial y}{\partial x}\right)^2$. So we have the total energy   $$dE=\frac{1}{2}\mu dx \left( \frac{\partial y}{\partial t}\right)^2+\frac{1}{2}T dx \left( \frac{\partial y}{\partial x}\right)^2,$$  which completely explains the sentence. Can we obtain a similar formula for higher-dimensional waves? How should I modify the above formula for a (simplified model of an) elastic membrane, for example? I would expect something like   $$dE=\text{some constant}\left(\frac{\partial z}{\partial t}\right)^2+\text{some other constant}\left\lvert\nabla z\right\rvert^2,$$  where $z=z(x, y, t)$ is the vertical displacement. Am I right? How to setup OpenVPN Client I'm trying to setup OpenVPN using NetworkManager. The GUI seems buggy and unresponsive. Are there any other tools for setting up a VPN client? I attended a business meeting in the Midwest and one of our clients took our management team and a few other business partners to dinner at an awesome Mexican restaurant. When the appetizers were served I commented that it was a wonderful presentation and that everything looked very fresh, including the avocado slices.   Speaking to the woman seated across from me I said that even though I use a good amount of lemon juice, whenever I prepare avocados they still turned dark pretty quickly. This happens whether they are sliced, used in guacamole, etc. She responded that the secret is to use lime juice. She proceeded to tell me that she was born and raised in Central America and that it is customary there to use fresh limes or juice rather than lemon.  I have tried it and lime seems to work much better than lemon, so it would stand to reason that they are more acidic. However, I never thought there was that much difference. So, on to the questions.  Are limes more acidic than lemons? Are there any other differences in their properties that could make a difference? I am getting error:     The argument 2 of the "JSON_VALUE or JSON_QUERY" must be a string literal   ...while trying to pass variable as parameter to JSON_QUERY function. Please help me to resolve it.  DECLARE @cnt INT = 1; DECLARE @cnt_total INT = 1; DECLARE @json NVARCHAR(MAX); DECLARE @json1 NVARCHAR(MAX); declare @str  VARCHAR(200);  WHILE @cnt &lt;= 10 BEGIN     set @str = '$.Seasons[0].Products['+convert(varchar,@cnt)+'].ProductChannels';     set @str = char(39) +@str+ char(39);     PRINT @STR     select @json = json_query(jfile,@STR) from Import.tstjson; --MORE CODE         SET @cnt = @cnt + 1; END   Sample data:  {   "Seasons": [     {       "Season": "12321231",       "Products": [         {           "ProductId": "211",           "ProductChannels": [             {               "ChannelId": 1,               "WeekQuantities": []             }           ],           "ccc": 3,           "Tttt": 4         }       ]     }   ] } When watching US television news or reading US news articles there is this affectation where instead of saying:     "Spotify music-streaming service to launch in U.S. on Thursday" ()   or      "Leading credit rating agency Moody's said on Wednesday that the U.S. government's impeccable credit..." ()   The on preposition is missed out.  Is this just a peculiar US journalistic "news speak" shortcut or do American non-journalist writers write this way too? Change Legend Right to Left Reading in QGIS I am a new QGIS user and i have a challenge to convert legend from right to left reading in QGIS like ArcGIS. Summation of natural number set with power of $m$ Who knows about the summation of this series: $$\sum\limits_{i=1}^{n}i^m $$  where $m$ is constant and $m\in \mathbb{N}$?  thanks Community Wiki posts don't change usernames if the user does Referencing to "above" or "below"?