Plausible observations and plausible models. The maximum likelihood method. Likelihood = probability that an observation is predicted by the specified model. MLE.
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Likelihood = probability that an observation is predicted by the specified model
what is the range of plausible observations, given the model, and what are the different models that could plausibly have generated the data?
What is a plausible model?
A model that predicts observations with a probability that exceeds a given minimum.
What is the most plausible model?
A model that most likely predicts observations, i.e. that predicts the observations with the largest probability most likely model, given the data.
Probability that an observation is predicted by N(,2): probability that 120 is predicted by N(100,100):
Probability that 120 is predicted by N(120,100):
Ratio of likelihood of any model to likelihood of ‘best’ model
Log-likelihood ratio ln = - ½ z2
z2 = -2ln
the specified model and
A plausible value of is one for which the likelihood ratio exceeds a critical value (less negative), e.g. -1.9208, which corresponds to a 95% confidence interval, or -1.353 which corresponds to a 90% confidence interval.
Values of for which ln > -1.9208 is
the support range for .
When is outside the support range, we reject the claim that does not differ significantly from b . We accept a risk of 5% of wrongly rejecting the claim (Type I error).
To get support range, find * for which ln = -1.9208 (given that ‘best’ value of is 125 and 2 is fixed):
The observation could come from ANY model in the support range. All models in the ‘support range’ are supported by the data.
Data: leaving parental home and index m
Analysis of young adults who left home and index m
leave out censored cases (conditional analysis)