I have two basic models $M_1$ and $M_2$. They each have a likelihood function; $L_{M_1} = f(\mathbf{X}|\mathbf{\theta_1})$ and $L_{M_2} = f(\mathbf{X}|\mathbf{\theta_2})$ (here $\mathbf{X}$ is the data set, and $\mathbf{\theta_i}$ are the parameters for the model). The models are nested since model $M_1$ parameter's are all present in model $M_2$. Specifically, $\mathbf{\theta_1}$ has 2 parameters and $\mathbf{\theta_2}$ has 3 parameters.
QUESTION 1)The model $M_1$ is termed a 'null model' in this case as it is the more simple in the nested case?
QUESTION 2)I want to do model selection. The more complicated model, $M_2$, has a greater likelihood but I would like it to be statistically justified. Is it correct to assume that the likelihood-ratio-test is the suitable choice to apply?
QUESTION 3)For the likelihood ratio test, I am supposed to use the parameter values which maximize the likelihood. In the case of model 1 $M_1$ I can do this analytically, but for model 2 $M_2$ I must approximate it via sampling. Is it suitable that I use $\mathbf{\theta_1}^{max}$ and $\mathbf{\theta_1}^{approx.max}$ in the same mechanism for comparison?
I applied the likelihood-ratio-test, and get this result (in logs):
LogLikModel1 = -61159.4324
LogLikModel2 = -30577.2917
LogLikRationTest_D = 61164.2813
QUESTION 4)To estimate the significance the chi-squared, $\chi^2$ test must be used?
QUESTION 5)If the answer to the question 4) is yes, then must I use this table (link to chisquared and p-values) for mapping the degrees of freedom and chi values to p-values? Are the numbers that the likelihood-ratio-test produce chi values? How do I use these values in logs with the table for chi-squared p-values because the probabilities are too low to not use logs?