I don't understand the term "loglikelihood"? I'd like to have a practical understanding of this word, and of why this is important. Besides this whenever we calculate some statistic like chisquare or doing logistic regression, why we take -2loglikelihood? What is the significance of $-2$ over here?
If someone can explain in plain simple language, it would be great.
I found this on Stack Overflow, but I could not properly understood it.
Thanks very much in advance,
2 Answers
I think you have this in mind.
I think the 'reason' comes from central limit theorem. The statistic $D$ in the link is asymptotically distributed as Chi squared, but when you take log of a normal density function, you get
$$\log (\sqrt{2\pi}) + \frac{1}{2} x^2$$
Note the constant cancels from subtraction but is there a factor of 1/2 in front of $x^2$, multipling 2 gets rid of it.
The - sign is only a convention. If you did loglikelihood of alternative hypothesis - loglikelihood of null, then there is no minus sign.
Loglikelihood, like its name suggests, is the natural logarithm of the likelihood. It is useful in maximum likelihood estimation because it reduces a product of N likelihoods to a sum of N loglikelihoods, with is easier to optimize analytically, and usually numerically as well.
The 2 in the above formula for hypothesis tests is abased on the asymptotic distribution of the likelihood ratio statistic (as noted by Lost1 above). 2ln($\frac{L(H_a)}{L(H_0)})\dot\sim \chi^2_1$. This was proved by Wilk's and is called the Wilk's likelihood ratio statistic. See this for theoretical basis. The minus sign is a convention based on what is in the numerator and denominator of the likelihood ratio (i.e., you want BIG values to lead to rejection, so adjust the sense as needed).
