Editing Likelihood-ratio test (section)

==Interpretation==
The likelihood ratio is a function of the data <math>x</math>; therefore, it is a [[statistic]], although unusual in that the statistic's value depends on a parameter, <math>\theta</math>. The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small. How small is too small depends on the significance level of the test, i.e. on what probability of [[Type&nbsp;I error]] is considered tolerable (Type&nbsp;I errors consist of the rejection of a null hypothesis that is true).

The [[numerator]] corresponds to the likelihood of an observed outcome under the [[null hypothesis]]. The [[denominator]] corresponds to the maximum likelihood of an observed outcome, varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator; so, the likelihood ratio is between 0 and 1. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative. High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.

===An example===
The following example is adapted and abridged from {{Harvtxt|Stuart|Ord|Arnold|1999|loc=§22.2}}.

Suppose that we have a random sample, of size {{mvar|n}}, from a population that is normally-distributed. Both the mean, {{mvar|&mu;}}, and the standard deviation, {{mvar|&sigma;}}, of the population are unknown. We want to test whether the mean is equal to a given value, {{math|''&mu;''{{sub|0}} }}.

Thus, our null hypothesis is {{math|''H''{{sub|0}}:&nbsp; ''&mu;'' {{=}} ''&mu;''{{sub|0}}&nbsp;}} and our alternative hypothesis is {{math|''H''{{sub|1}}:&nbsp; ''&mu;'' ≠ ''&mu;''{{sub|0}}&nbsp;}}. The likelihood function is 
:<math>\mathcal{L}(\mu,\sigma \mid x) = \left(2\pi\sigma^2\right)^{-n/2} \exp\left( -\sum_{i=1}^n \frac{(x_i -\mu)^2}{2\sigma^2}\right)\,.</math>

With some calculation (omitted here), it can then be shown that 
:<math>\lambda_{LR} = n \ln\left[ 1 + \frac{t^2}{n-1}\right] </math> 
where {{mvar|t}} is the [[t-statistic|{{mvar|t}}-statistic]] with {{math|''n''&thinsp;&minus;&thinsp;1}} degrees of freedom. Hence we may use the known exact distribution of {{math|''t''{{sub|''n''&minus;1}}}} to draw inferences.