Let the model A be nested in the model B. Consequently the model B has more parameters than the model A. Let LA and LB be the estimated likelihood of the models A and B, respectively. What can you say about the LA and LB? Choose all that apply.

a) LA is always larger or equal to LB
.

b) LB is always larger or equal to LA
.

c) LA can be larger than LB and can be small than LB
, depending on data.

d) (-2log (LA/LB)) follows a chi-square distribution with 1 degree of freedom.

e) LA/LB follows a F-distribution.


a

b

c

d

e

When comparing two models, where model A is nested within model B (meaning model A is a special case of model B with fewer parameters), several statistical properties come into play. Let's analyze each statement:

- **(a) LA is always larger or equal to LB**:  
  **Incorrect.** The likelihood \( L_B \) of the more complex model (model B) will generally be larger because it has more parameters, allowing it to fit the data better. Thus, \( L_A \leq L_B \), not the other way around.

- **(b) LB is always larger or equal to LA**:  
  **Correct.** This is true. Model B has more parameters, allowing it to better fit the data or at least match model A’s likelihood. Therefore, \( L_B \geq L_A \).

- **(c) LA can be larger than LB and can be smaller than LB, depending on data**:  
  **Incorrect.** This statement is false because the likelihood \( L_B \) for the model with more parameters is always greater than or equal to the likelihood \( L_A \), as explained in (b). It cannot be smaller than \( L_A \).

- **(d) \(-2\log (L_A / L_B)\) follows a chi-square distribution with 1 degree of freedom**:  
  **Correct.** This is true for nested models. Under the null hypothesis (that the simpler model A is correct), the test statistic \(-2\log(L_A / L_B)\) follows a chi-square distribution with degrees of freedom equal to the difference in the number of parameters between the two models. In this case, the degrees of freedom is 1 (assuming one extra parameter in model B).

- **(e) \( L_A / L_B \) follows an F-distribution**:  
  **Incorrect.** The likelihood ratio test statistic follows a chi-square distribution, not an F-distribution, so this statement is false.

### Correct choices:  
- **(b) LB is always larger or equal to LA**
- **(d) \(-2\log (L_A / L_B)\) follows a chi-square distribution with 1 degree of freedom**

Would you like further explanation or details on any part of this?

### Related questions:
1. What does it mean for one model to be nested within another?
2. How does increasing the number of parameters in a model affect the likelihood?
3. What is the likelihood ratio test used for?
4. How do you calculate degrees of freedom when comparing two models?
5. What are some limitations of using a chi-square distribution in likelihood tests?

### Tip:
When comparing nested models, always remember that the likelihood of the more complex model (with more parameters) will be equal to or greater than that of the simpler model.

Let the model A be nested in the model B. Consequently, model B has more parameters than model A. Let LA and LB be the estimated likelihood of the models A and B, respectively. What can you say about the LA and LB? Choose all that apply:

a) LA is always larger or equal to LB.
b) LB is always larger or equal to LA.
c) LA can be larger than LB and can be smaller than LB, depending on data.
d) (-2log (LA/LB)) follows a chi-square distribution with 1 degree of freedom.
e) LA/LB follows an F-distribution.

Explore how likelihoods in nested models behave with key concepts like the likelihood ratio test and chi-square distribution. Learn why LB is always greater or equal to LA and how to apply the test statistic -2log(LA/LB). Suitable for advanced undergraduate or graduate-level statistics students.

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