H 0 β = 0 , H a β ≠ 0.The test itself is a generalized likelihood ratio. The test statistic is the ratio LR: L R = L ( β = 0 ) a r g m a x β ∈ R L ( β ) Where L is the likelihood function.Then − 2 l n ( L R ) follows a χ 1 2 .I am trying to calculate the p-value for a specific value of LR. Say I look for at a χ 1 2 table (or online calc) and find out that P ( χ 1 2 ≥ − 2 l n ( L R ) ) = αIs α my final p-value?Or do I need to account for the fact that the test is two-sided and set the p-value = 2 ∗ α?

Question

H    0    β  =  0  ,      H    a    β  ≠  0.The test itself is a generalized likelihood ratio. The test statistic is the ratio LR:  L  R  =            L      (      β      =      0      )              a      r      g      m      a              x                  β          ∈          R                    L      (      β      )      Where   L is the likelihood function.Then   −  2  l  n  (  L  R  ) follows a       χ    1    2  .I am trying to calculate the p-value for a specific value of LR. Say I look for at a       χ    1    2   table (or online calc) and find out that  P  (      χ    1    2    ≥  −  2  l  n  (  L  R  )  )  =  αIs   α my final   p-value?Or do I need to account for the fact that the test is two-sided and set the   p-value   =  2  ∗  α?

Zayden Andrade · Accepted Answer

The   p value is related to the rejection region, so the answer depends on how you divide   α over both sides for the test. If you reject       H    0   when the test statistic is less than   F  (  α  /  2  ) or more than   F  (  1  −  α  /  2  ), then   2  α is the right answer. However, you can use any rejection region   (  −  ∞  ,  a  ]  ∪  [  b  ,  ∞  ) as long as   a and   b satisfy       F          −      1        (  b  )  −      F          −      1        (  a  )  =  1  −  α. For skewed unimodal distributions you can impose   f  (  a  )  =  f  (  b  ), which leads to a different answer.In your example of a generalized likelihood ratio test, the rejection region is an interval   [  b  ,  ∞  ), even though the test is two sided. So   α is the   p-value.

H 0 </msub> β = 0 , H a </msub> β<!-- β

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