2

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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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, 
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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")
kjetil b halvorsen
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Flask
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  • Are you sure it’s a profile likelihood? Is that your question? What are you trying to do? – Elvis Dec 02 '13 at 19:52
  • @Elvis I guess not, but that is what it is called in these lecture notes: [see pages 24-25](http://ocw.jhsph.edu/courses/MethodsInBiostatisticsI/PDFs/lecture9.pdf) – Flask Dec 02 '13 at 19:54
  • @Elvis I want to know how people can use these likelihood functions to compare multiple groups. For example I could compare the 1/20th max likelihood intervals. – Flask Dec 02 '13 at 20:41
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    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). – Elvis Dec 02 '13 at 20:53
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    @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. – Flask Dec 02 '13 at 21:01
  • In the lecture notes, page 23, it is written that the variance is estimated by ML at $\mu = \mu_0$ fixed, $\widehat{\sigma^2} = \sum_i (x_i - \mu_0)^2 / n$. This is definitely not what your code is doing (you use the empirical mean instead of $\mu_0$). – Elvis Dec 02 '13 at 21:05
  • The problems brought up by @Elvis in the comments was resolved [here](http://stats.stackexchange.com/questions/77528/what-is-the-relationship-between-profile-likelihood-and-confidence-intervals?lq=1) – Flask Dec 03 '13 at 00:09
  • Yes, I am sorry about `mn`, I find the R code provided in your slide very hard to read – as a R programmer and teacher, I would not recommend ;) However, I still have a remark for you: if you want to make a group comparison, you need to write a model (usually, three groups with same variance and three different means). Think about it! – Elvis Dec 03 '13 at 08:46
  • @Elvis Is it wrong to say the mean of group 3 (short dashes) is ~ 5 times more likely to be 14 than the means of groups 1 and 2? In the real data the groups do not have similar variance or come from normal distributions. I was at first analyzing with a complicated hierarchical model using prior data to create a mixture of normal distributions and mcmc but have decided against this because "if a plan is too complicated it will fail". Now I simply want to estimate the overall means and compare in some simple way. – Flask Dec 03 '13 at 12:44
  • The homoscedasticity is a classical assumption; I suggested to use it for simplicity as I was assuming this was some sort of exercise. If you have a concrete problem in mind, you should open a new question and tell us about your data. – Elvis Dec 03 '13 at 13:35
  • 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. – Elvis Dec 03 '13 at 13:42
  • @Elvis Please see the edit. – Flask Dec 03 '13 at 14:10
  • I don’t understand your edit. What is the 'prior density'? From where do your groups come from? Why 'new data'? What is the assumed relation between the 'top plot' and your three groups? If your new data is assumed to come from the same prior density, then all three samples should have the same mean (up to sampling variation...). If you just want a non-parametric group comparison, search for non-parametric anova. If this doesn’t help, please open a new question and describe your problem with more details. – Elvis Dec 03 '13 at 14:53
  • Note that comparing the means of these three groups has they don’t have the same variances is unlikely to be the right thing to do. What is your aim? Try to be clear about that in your future new question. – Elvis Dec 03 '13 at 14:58
  • @Elvis http://stats.stackexchange.com/questions/78435/how-can-i-use-likelihoods-to-compare-these-three-groups-should-i-want-to-do-thi – Flask Dec 03 '13 at 15:41

0 Answers0