Re: linear regression with errors in both variables

*To*: mathgroup at smc.vnet.net*Subject*: [mg96436] Re: linear regression with errors in both variables*From*: Ray Koopman <koopman at sfu.ca>*Date*: Fri, 13 Feb 2009 03:46:19 -0500 (EST)*References*: <gmrmga$a1k$1@smc.vnet.net> <gn11v2$8fl$1@smc.vnet.net>

On Feb 12, 3:42 am, Ray Koopman <koop... at sfu.ca> wrote: > On Feb 10, 2:56 am, Joerg <scha... at biologie.hu-berlin.de> wrote: > >> Hi, >> >> I want to test the hypothesis that my data >> follows a known simple linear relationship, >> y = a + bx. However, I have (known) measurements >> errors in both the y and the x values. >> >> I suppose just a simple linear regression >> does not do here. >> >> Any suggestions how do test this correctly? >> >> Thanks, >> >> joerg > > You don't say whether the errors are the same for all data points. > If they are, and sx & sy are the (known) standard deviations of the > measurement errors in x & y, then > > ClearAll[a,b]; Minimize[Evaluate[Expand[ > #.#&[a + b*x - y]]/(b^2 sx^2 + sy^2)], {a,b}] > > will give what you want. You can save a little time if you eliminate > 'a' by centering the data. And if Mathematica honors the parentheses, > centering will also reduce potential roundoff error problems. > > xbar = Mean@x; ybar = Mean@y; ClearAll[a,b]; > {#[[1]], {a -> ybar - #[[2,1,2]]*xbar, #[[2,1]]}}& @ > Minimize[Evaluate[Expand[ > #.#&[b(x-xbar)-(y-ybar)]]/(b^2 sx^2 + sy^2)], b] > > If the errors differ from point to point then sx & sy will be lists. > > ClearAll[a,b]; > Minimize[Tr[ (a + b*x - y)^2 / (b^2 sx^2 + sy^2) ], {a,b}] > > will work, but it's slow. Pre-evaluating doesn't help. > The following is faster: > > worals[x_,y_,sx_,sy_] := Block[ > {a,b,f,z, u = 1/sx, v = 1/sy, w = (sy/sx)^2}, > {a,b} = (y*v).PseudoInverse@{v,x*v}; f = #.#&[(a+b*x-y)v]; > While[f > (z = (x*w + (y-a)b)/(b^2 + w); > {a,b} = (y*v).PseudoInverse@{v,z*v}; > f = #.#&@Join[(z-x)u,(a+b*z-y)v])]; {f,{a,b}}] > > ('worals' is an acronym for Weighted Orthogonal Regression by > Alternating Least Squares.) If the true relation is linear, and if the errors are independent normal with zero means and standard deviations as given in sx & sy, then the minimized function value returned by each of the four code fragments that I gave will have a Chi-Square distribution with n-2 degrees of freedom. The corresponding p-value is GammaRegularized[(n-2)/2,f/2].