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,
$\endgroup$2 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$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).
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