- Statistika
- Variabel acak
- Ilmu aktuaria
- Statistika matematika
- Uji t Student
- Eksperimen semu
- Efek pengacau
- Analitik prediktif
- Model generatif
- Distribusi t Student
- Likelihood-ratio test
- Likelihood function
- Monotone likelihood ratio
- G-test
- Wald test
- Likelihood ratios in diagnostic testing
- Score test
- F-test
- Wilks' theorem
- Bayes factor
- intuition - What is a *likelihood ratio test* for a specific ...
- lme4 nlme - Likelihood ratio test - lmer R - Cross Validated
- Why is it the likelihood ratio test under Neyman-Pearson and the ...
- AIC vs. Likelihood ratio test - Cross Validated
- probability - Likelihood ratio for binomial - Cross Validated
- AIC versus Likelihood Ratio Test in Model Variable Selection
- Likelihood Ratio Test Equivalent with - Cross Validated
- r - Likelihood ratio tests using ML vs. REML - Cross Validated
- GLM tests involving deviance and likelihood ratios
- hypothesis testing - Likelihood ratio test for a random variable ...
The Greatest of All Time (2024)
The Remarkable Life of Ibelin (2024)
The Hitman’s Bodyguard (2017)
Reagan (2024)
Oppenheimer (2023)
Likelihood-ratio test GudangMovies21 Rebahinxxi LK21
In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the more constrained model (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.
The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.
Definition
= General
=Suppose that we have a statistical model with parameter space
Θ
{\displaystyle \Theta }
. A null hypothesis is often stated by saying that the parameter
θ
{\displaystyle \theta }
lies in a specified subset
Θ
0
{\displaystyle \Theta _{0}}
of
Θ
{\displaystyle \Theta }
. The alternative hypothesis is thus that
θ
{\displaystyle \theta }
lies in the complement of
Θ
0
{\displaystyle \Theta _{0}}
, i.e. in
Θ
∖
Θ
0
{\displaystyle \Theta ~\backslash ~\Theta _{0}}
, which is denoted by
Θ
0
c
{\displaystyle \Theta _{0}^{\text{c}}}
. The likelihood ratio test statistic for the null hypothesis
H
0
:
θ
∈
Θ
0
{\displaystyle H_{0}\,:\,\theta \in \Theta _{0}}
is given by:
λ
LR
=
−
2
ln
[
sup
θ
∈
Θ
0
L
(
θ
)
sup
θ
∈
Θ
L
(
θ
)
]
{\displaystyle \lambda _{\text{LR}}=-2\ln \left[{\frac {~\sup _{\theta \in \Theta _{0}}{\mathcal {L}}(\theta )~}{~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~}}\right]}
where the quantity inside the brackets is called the likelihood ratio. Here, the
sup
{\displaystyle \sup }
notation refers to the supremum. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one.
Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods
λ
LR
=
−
2
[
ℓ
(
θ
0
)
−
ℓ
(
θ
^
)
]
{\displaystyle \lambda _{\text{LR}}=-2\left[~\ell (\theta _{0})-\ell ({\hat {\theta }})~\right]}
where
ℓ
(
θ
^
)
≡
ln
[
sup
θ
∈
Θ
L
(
θ
)
]
{\displaystyle \ell ({\hat {\theta }})\equiv \ln \left[~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~\right]~}
is the logarithm of the maximized likelihood function
L
{\displaystyle {\mathcal {L}}}
, and
ℓ
(
θ
0
)
{\displaystyle \ell (\theta _{0})}
is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes
L
{\displaystyle {\mathcal {L}}}
for the sampled data) and
θ
0
∈
Θ
0
and
θ
^
∈
Θ
{\displaystyle \theta _{0}\in \Theta _{0}\qquad {\text{ and }}\qquad {\hat {\theta }}\in \Theta ~}
denote the respective arguments of the maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem)
λ
LR
{\displaystyle \lambda _{\text{LR}}}
converges asymptotically to being χ²-distributed if the null hypothesis happens to be true. The finite-sample distributions of likelihood-ratio statistics are generally unknown.
The likelihood-ratio test requires that the models be nested – i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below.
If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood.
= Case of simple hypotheses
=A simple-vs.-simple hypothesis test has completely specified models under both the null hypothesis and the alternative hypothesis, which for convenience are written in terms of fixed values of a notional parameter
θ
{\displaystyle \theta }
:
H
0
:
θ
=
θ
0
,
H
1
:
θ
=
θ
1
.
{\displaystyle {\begin{aligned}H_{0}&:&\theta =\theta _{0},\\H_{1}&:&\theta =\theta _{1}.\end{aligned}}}
In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate. For this case, a variant of the likelihood-ratio test is available:
Λ
(
x
)
=
L
(
θ
0
∣
x
)
L
(
θ
1
∣
x
)
.
{\displaystyle \Lambda (x)={\frac {~{\mathcal {L}}(\theta _{0}\mid x)~}{~{\mathcal {L}}(\theta _{1}\mid x)~}}.}
Some older references may use the reciprocal of the function above as the definition. Thus, the likelihood ratio is small if the alternative model is better than the null model.
The likelihood-ratio test provides the decision rule as follows:
If
Λ
>
c
{\displaystyle ~\Lambda >c~}
, do not reject
H
0
{\displaystyle H_{0}}
;
If
Λ
<
c
{\displaystyle ~\Lambda
, reject
H
0
{\displaystyle H_{0}}
;
If
Λ
=
c
{\displaystyle ~\Lambda =c~}
, reject
H
0
{\displaystyle H_{0}}
with probability
q
{\displaystyle ~q~}
.
The values
c
{\displaystyle c}
and
q
{\displaystyle q}
are usually chosen to obtain a specified significance level
α
{\displaystyle \alpha }
, via the relation
q
{\displaystyle ~q~}
P
(
Λ
=
c
∣
H
0
)
+
P
(
Λ
<
c
∣
H
0
)
=
α
.
{\displaystyle \operatorname {P} (\Lambda =c\mid H_{0})~+~\operatorname {P} (\Lambda
The Neyman–Pearson lemma states that this likelihood-ratio test is the most powerful among all level
α
{\displaystyle \alpha }
tests for this case.
Interpretation
The likelihood ratio is a function of the data
x
{\displaystyle x}
; therefore, it is a statistic, although unusual in that the statistic's value depends on a parameter,
θ
{\displaystyle \theta }
. 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 I error is considered tolerable (Type 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 Stuart, Ord & Arnold (1999, §22.2).
Suppose that we have a random sample, of size n, from a population that is normally-distributed. Both the mean, μ, and the standard deviation, σ, of the population are unknown. We want to test whether the mean is equal to a given value, μ0 .
Thus, our null hypothesis is H0: μ = μ0 and our alternative hypothesis is H1: μ ≠ μ0 . The likelihood function is
L
(
μ
,
σ
∣
x
)
=
(
2
π
σ
2
)
−
n
/
2
exp
(
−
∑
i
=
1
n
(
x
i
−
μ
)
2
2
σ
2
)
.
{\displaystyle {\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)\,.}
With some calculation (omitted here), it can then be shown that
λ
L
R
=
n
ln
[
1
+
t
2
n
−
1
]
{\displaystyle \lambda _{LR}=n\ln \left[1+{\frac {t^{2}}{n-1}}\right]}
where t is the t-statistic with n − 1 degrees of freedom. Hence we may use the known exact distribution of tn−1 to draw inferences.
Asymptotic distribution: Wilks’ theorem
If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine.
Assuming H0 is true, there is a fundamental result by Samuel S. Wilks: As the sample size
n
{\displaystyle n}
approaches
∞
{\displaystyle \infty }
, and if the null hypothesis lies strictly within the interior of the parameter space, the test statistic
λ
LR
{\displaystyle \lambda _{\text{LR}}}
defined above will be asymptotically chi-squared distributed (
χ
2
{\displaystyle \chi ^{2}}
) with degrees of freedom equal to the difference in dimensionality of
Θ
{\displaystyle \Theta }
and
Θ
0
{\displaystyle \Theta _{0}}
. This implies that for a great variety of hypotheses, we can calculate the likelihood ratio
λ
{\displaystyle \lambda }
for the data and then compare the observed
λ
LR
{\displaystyle \lambda _{\text{LR}}}
to the
χ
2
{\displaystyle \chi ^{2}}
value corresponding to a desired statistical significance as an approximate statistical test. Other extensions exist.
See also
Akaike information criterion
Bayes factor
Johansen test
Model selection
Vuong's closeness test
Sup-LR test
Error exponents in hypothesis testing
References
Further reading
Glover, Scott; Dixon, Peter (2004), "Likelihood ratios: A simple and flexible statistic for empirical psychologists", Psychonomic Bulletin & Review, 11 (5): 791–806, doi:10.3758/BF03196706, PMID 15732688
Held, Leonhard; Sabanés Bové, Daniel (2014), Applied Statistical Inference—Likelihood and Bayes, Springer
Kalbfleisch, J. G. (1985), Probability and Statistical Inference, vol. 2, Springer-Verlag
Perlman, Michael D.; Wu, Lang (1999), "The emperor's new tests", Statistical Science, 14 (4): 355–381, doi:10.1214/ss/1009212517
Perneger, Thomas V. (2001), "Sifting the evidence: Likelihood ratios are alternatives to P values", The BMJ, 322 (7295): 1184–5, doi:10.1136/bmj.322.7295.1184, PMC 1120301, PMID 11379590
Pinheiro, José C.; Bates, Douglas M. (2000), Mixed-Effects Models in S and S-PLUS, Springer-Verlag, pp. 82–93
Richard, Mark; Vecer, Jan (2021). "Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis". Risks. 9 (2): 31. doi:10.3390/risks9020031. hdl:10419/258120.
Solomon, Daniel L. (1975), "A note on the non-equivalence of the Neyman-Pearson and generalized likelihood ratio tests for testing a simple null versus a simple alternative hypothesis" (PDF), The American Statistician, 29 (2): 101–102, doi:10.1080/00031305.1975.10477383, hdl:1813/32605
External links
Practical application of likelihood ratio test described
R Package: Wald's Sequential Probability Ratio Test
Richard Lowry's Predictive Values and Likelihood Ratios Online Clinical Calculator
Kata Kunci Pencarian:
Likelihood Ratio Test | PDF | Statistical Theory | Statistics
Likelihood Ratio Test | PDF | Statistical Hypothesis Testing | Statistics
Likelihood Ratio Test Example - ReliaWiki
Likelihood ratio test
Likelihood ratio test
Likelihood ratio test
hypothesis testing - Likelihood-ratio test - Cross Validated
statistics - Likelihood Ratio Test explanation - Mathematics Stack Exchange
Likelihood Ratio Test
Likelihood Ratio Test - What Is It, Examples, Formula, Vs Wald Test
8 The Likelihood Ratio Test
The likelihood ratio test results | Download Scientific Diagram
likelihood ratio test
Daftar Isi
intuition - What is a *likelihood ratio test* for a specific ...
Nov 29, 2023 · A likelihood ratio test is just a particular type of hypothesis test where the test statistic is obtained in a specific way. They arise out of Neyman and Pearson's attempt to find a way to obtain "good" test statistics (in the sense of the resulting tests having high power).
lme4 nlme - Likelihood ratio test - lmer R - Cross Validated
(Ordinary) likelihood ratio test can only be used to compare nested models; We cannot compare mean models under REML. (This is not the case here, see @KarlOveHufthammer's comments below.) In the case of using ML, I am aware of using AIC …
Why is it the likelihood ratio test under Neyman-Pearson and the ...
Mar 8, 2021 · In that setting we have a one-sided test for a one-dimensional parameter and the Neyman-Pearson test of a specific $\theta_0$ happens to be the same regardless of $\theta_1$. And in that setting, the uniformly most powerful test …
AIC vs. Likelihood ratio test - Cross Validated
Sep 10, 2020 · That is actually a likelihood ratio test, substracting the log-likelihoods. The significance level can be determined by the critical value $2k_1-2k_0$ and a $\chi^2$ table. This interpretation of the AIC is only possible when the models are nested.
probability - Likelihood ratio for binomial - Cross Validated
Oct 24, 2018 · A. Finding the form of a likelihood ratio test. The basic procedure for constructing a likelihood ratio test is of the following form: maximize the likelihood under the null $\hat{\mathcal{L}}_0$ (by substituting the MLE under the null into the likelihood for each sample)
AIC versus Likelihood Ratio Test in Model Variable Selection
Feb 27, 2016 · AIC and likelihood ratio test (LRT) have different purposes. AIC tells you whether it pays to have a richer model when your goal is to approximate the underlying data generating process the best you can in terms of Kullback-Leibler distance.
Likelihood Ratio Test Equivalent with - Cross Validated
Sep 2, 2021 · F-Test for Equality of Variances as Likelihood Ratio Test 2 Testing the equality of two multivariate mean vectors $μ_1$ and $μ_2$ based on independent random normal samples
r - Likelihood ratio tests using ML vs. REML - Cross Validated
Mar 31, 2015 · Regarding your second question, parameters on the boundary are a problem for likelihood ratio test in general (not just for mixed-effects models). The asymptotics break down when the parameter(s) in one of the models are on the boundary of the parameter space. So yes, this is a problem for ML estimation (too).
GLM tests involving deviance and likelihood ratios
Mar 21, 2022 · There is the null deviance, which is similar to a likelihood ratio for the difference between the saturated model and the model with only an intercept. There is the residual deviance, which is similar to a likelihood ratio for the difference between the saturated model and the current fitted model. Given these,
hypothesis testing - Likelihood ratio test for a random variable ...
Likelihood ratio test for exponential distribution with scale parameter. 2. Likelihood ratio test for two ...