2
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I am new to profile likelihood and do not really understand what advantages it may have. Lets say I have the following results estimating the means of three groups. What can I say about them?

enter image description here

R code:

profile.likelihood <- function(dat, muVals){
  likVals <- sapply(muVals,
                    function(mu){
                      (sum((dat - mu)^2) /
                         sum((dat - 
mean(dat))^2)) ^ (-length(dat)/2)
                    }
  )

return(cbind(muVals, likVals))
}
    
muVals <- seq(0,20, length = 10000)
dat1 <- runif(10, 0, 20)
a <- profile.likelihood(dat1, muVals)
dat2 <- runif(10, 0, 20)
b <- profile.likelihood(dat2, muVals)
dat3 <- runif(10, 8, 20)
c <- profile.likelihood(dat3, muVals)

plot(a, type="l", lwd=4,
ylab="Likelihood", xlab="Score"
)
lines(b, lty=2, lwd=4)
lines(c, lty=3, lwd=4)

EDIT:

Here is data more similar to my actual.

  1. data can take any value from 0 to 20

  2. I know from a relatively large number of previous results that the distribution is not normal and looks like the top plot.

My data is shown in the boxplots in the lower panel. I wish to estimate the means of each group and compare them using likelihoods or know why I should not do this.

enter image description here

dput() of new data:

structure(c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 
2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 13.6986301369863, 
16.1643835616438, 12.0547945205479, 12.1722113502935, 9.74559686888454, 
0.430528375733855, 11.3502935420744, 10.6457925636008, 9.9412915851272, 
10.7240704500978, 10.958904109589, 11.6242661448141, 17.9701232444495, 
15.9326690901071, 7.98247314058244, 14.4031004607677, 13.5198221541941, 
2.82421704847366, 16.114045586437, 19.328767512925, 3.74181577004492, 
17.4085859861225, 19.2017483590171, 8.26946665905416, 10.0207103956491, 
16.1689247898757, 13.9989542039111, 9.80047978740185, 17.8596440777183, 
18.1706223106012, 18.1891529858112, 11.7567204562947), .Dim = c(32L, 
2L))

dput() of prior density

structure(list(x = c(0, 0.0391389432485323, 0.0782778864970646, 
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0.160939101818898, 0.160783676919756, 0.160610951636628, 0.160440813332408, 
0.160292840283904, 0.160202196337883, 0.160205045445256, 0.160307691092289, 
0.160525158759831, 0.160872099587403, 0.161427322298974, 0.162146341126922, 
0.16303494787568, 0.1640978213177, 0.165374536988198, 0.166861439709314, 
0.168517967752217, 0.170337253164731, 0.172312150290867, 0.174476220294316, 
0.17675201562322, 0.17912237546253, 0.181569351447975, 0.184077063209181, 
0.186601937519015, 0.189110996532675, 0.19158078566652, 0.193986444864657, 
0.196248038237016, 0.198366671014338, 0.2003193238609, 0.202083414374645, 
0.203583109911179, 0.204784727823111, 0.20571235693773, 0.206350205706743, 
0.206678507619118, 0.206556713002678, 0.206095969055307, 0.205290936474901, 
0.20413741111321, 0.202556572638439, 0.20056355298721, 0.198228613635869, 
0.195558644146613, 0.192555647948946, 0.189126333152985, 0.185410492636506, 
0.181423717899618, 0.177182210388674, 0.172658716028772, 0.167900183255391, 
0.16296802586402, 0.157882368007903, 0.152661212543965, 0.147304635977954, 
0.141882524743064, 0.136414358489409, 0.130919304274034, 0.125422474679471, 
0.119957697079395, 0.11454052598248, 0.109185666974252, 0.103909993572189, 
0.0987683404859644, 0.093741308688079, 0.0888376270971578, 0.0840654236882268, 
0.0794666168513133, 0.0750427325158831, 0.0707727313029462, 0.0666591512693038, 
0.0627091344513033, 0.0589774733641864, 0.0554044324211085, 0.0519885093076509, 
0.0487279204345587, 0.0456550213639223, 0.0427535393057037, 0.0399945562439461, 
0.0373743649778989, 0.0348937863392637, 0.0325878723180957, 0.0304033659564744, 
0.0283361981437062, 0.0263823330082348, 0.0245619867105529, 0.0228595658836749, 
0.0212541351296348, 0.0197423154392725, 0.0183242669772017, 0.0170220160814398, 
0.0158005734119126, 0.0146575124941681, 0.0135905002875402, 0.0126143097870055, 
0.0117191044664318, 0.0108918081116421, 0.0101309359870987, 0.0094379000631823, 
0.00883019944826195, 0.00828380391399826, 0.00779783511360308, 
0.00737145971429946, 0.00701812393006059, 0.0067311252542151, 
0.00650097052493056, 0.00632717374230014, 0.00621198669319391, 
0.00617143557767508, 0.0061855594050197, 0.00625405591658972, 
0.00637663424018689, 0.00656643085439909, 0.00681747969859399, 
0.00712155184640286, 0.00747843433306984, 0.0078907086989986, 
0.00837340291572697, 0.00890806427672648, 0.009494513737272, 
0.010132574128414, 0.010835143782014, 0.0115961949265227, 0.0124081127363209, 
0.0132707231223347, 0.0141867435373067, 0.0151701717571193, 0.0162034512423126, 
0.0172863373562776, 0.0184185690183941, 0.019612438019485, 0.0208613703536178, 
0.0221579723104729, 0.0235016744919049, 0.0248946535533411, 0.0263479485897112, 
0.0278446638511839, 0.0293835133691365, 0.0309631084740302, 0.0325912348690895, 
0.0342591472113346, 0.0359590659180276, 0.0376882779600787, 0.0394452615206507, 
0.0412293518464991, 0.043027543361404, 0.0448353514472685, 0.0466481043572906, 
0.0484577337141087, 0.0502538712990142, 0.0520305447610213, 0.0537813149370389, 
0.0554964047538021, 0.0571515262745735, 0.0587502027060215, 0.0602849490847391, 
0.0617482500354599, 0.0631066082295335, 0.0643579690671984, 0.0655057363703737, 
0.0665429490020974, 0.0674531640036809, 0.0681929810868339, 0.0687958988058021, 
0.0692572514303207, 0.0695726946039624, 0.069695167044965, 0.0696433847901905, 
0.0694350823030356, 0.0690697581056959, 0.068535116834938, 0.0677961078509799, 
0.0669066143489608, 0.0658704054221769, 0.0646916508125923, 0.0633398550663818, 
0.0618485040466386, 0.0602410483121883, 0.0585250659382392, 0.0567013372657685, 
0.0547654519871085, 0.0527582945347469, 0.0506891518402187, 0.0485673654999008, 
0.0463957642449994, 0.0441983057887947, 0.0419884488508511, 0.0397750260303709, 
0.0375684836047041, 0.0353879440268247, 0.0332384672584751, 0.0311267921300081, 
0.0290593901126105, 0.0270601872571371, 0.0251287992452735, 0.0232632081303122, 
0.0214669543868477, 0.0197503127220639, 0.0181339971366657, 0.0165964380609598, 
0.0151383742283497, 0.013760299683845, 0.012485591080821, 0.0112979221667829, 
0.0101874296294859, 0.00915261833812058, 0.00819857079937494, 
0.00733611052320258, 0.00654025398573884, 0.00580844808959148, 
0.0051380838601794, 0.00454221827421074, 0.00400446040436533, 
0.00351615254678973, 0.0030745243393893, 0.00268072850148871, 
0.00233768602660161, 0.00202984861125548, 0.00175479227131742, 
0.00151014383233181, 0.00130072566668639, 0.00111721604283242, 
0.000955112980316734, 0.000812623159815952, 0.000689590092917738, 
0.0005860912862362, 0.000495639552545513, 0.000417014922709681, 
0.000349049284657548, 0.000292958836410902, 0.00024515801710463, 
0.000204040052154968, 0.000168884867932852, 0.000139474416390484, 
0.000115548025328669, 9.51646202245106e-05, 7.79159203801693e-05, 
6.34177645960676e-05, 5.18748729574288e-05, 4.22967579760247e-05, 
3.42684953287067e-05, 2.75891372479148e-05, 2.21737075433437e-05, 
1.79085413159829e-05, 1.43641470627961e-05, 1.14430817192601e-05, 
9.05508126397165e-06, 7.22026287729582e-06, 5.73639534240461e-06, 
4.52392445545382e-06, 3.54207412160509e-06, 2.77052765004019e-06, 
2.18172878629954e-06, 1.7042057736597e-06, 1.32082991902191e-06, 
1.01596818604684e-06, 7.89867561242482e-07, 6.11506981747829e-07, 
4.69391759719292e-07, 3.57363517027134e-07), bw = 0.715374250961732, 
    n = 35L, call = density.default(x = dat.prior, from = 0, 
        to = 20), data.name = "dat.prior", has.na = FALSE), .Names = c("x", 
"y", "bw", "n", "call", "data.name", "has.na"), class = "density")
$\endgroup$
15
  • $\begingroup$ Are you sure it’s a profile likelihood? Is that your question? What are you trying to do? $\endgroup$
    – Elvis
    Dec 2, 2013 at 19:52
  • $\begingroup$ @Elvis I guess not, but that is what it is called in these lecture notes: see pages 24-25 $\endgroup$
    – Flask
    Dec 2, 2013 at 19:54
  • 1
    $\begingroup$ Your function is not constructed like the profile likelihood in these notes... moreover if you want to make a group comparison, you need to write a model (usually, three groups with same variance and three different means). $\endgroup$
    – Elvis
    Dec 2, 2013 at 20:53
  • 1
    $\begingroup$ @Elvis It is constructed exactly the same except instead of being likelihood of the difference between before/after it is the likelihood of the mean value for each group. $\endgroup$
    – Flask
    Dec 2, 2013 at 21:01
  • 1
    $\begingroup$ To address your question, likelihoods are not meant to be compared for different data sets. In your case it seems to work but think to what would happen if the sample sizes were different! You should write a model for your data, with likelihood $L(\mu_1, \mu_2, \mu_3, \sigma^2_1, \sigma^2_2, \sigma^2_3)$ (the product of your three likelihoods) and profile likelihood $L_P(\mu_1, \mu_2, \mu_3)$ (same thing). Then you can compare different values of $L_P$ for different sets of means. This is for normal distribution. If you don’t want to assume a distribution, you can’t use parametric methods. $\endgroup$
    – Elvis
    Dec 3, 2013 at 13:42

0

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