The maximum likelihood method Likelihood = probability that an observation is predicted by the specified model Plausible observations and plausible models.

Slides:

Advertisements

Similar presentations

CHAPTER 21 Inferential Statistical Analysis. Understanding probability The idea of probability is central to inferential statistics. It means the chance.

Advertisements

Sampling: Final and Initial Sample Size Determination

POINT ESTIMATION AND INTERVAL ESTIMATION

Hypothesis testing Week 10 Lecture 2.

EPIDEMIOLOGY AND BIOSTATISTICS DEPT Esimating Population Value with Hypothesis Testing.

Hypothesis testing Some general concepts: Null hypothesisH 0 A statement we “wish” to refute Alternative hypotesisH 1 The whole or part of the complement.

Descriptive statistics Experiment  Data  Sample Statistics Sample mean Sample variance Normalize sample variance by N-1 Standard deviation goes as square-root.

GRAPHS OF MEANS How is a Graph of Means Constructed? What are Error Bars? How Can a Graph Indicate Statistical Significance?

Descriptive statistics Experiment  Data  Sample Statistics Experiment  Data  Sample Statistics Sample mean Sample mean Sample variance Sample variance.

Maximum Likelihood We have studied the OLS estimator. It only applies under certain assumptions In particular,  ~ N(0, 2 ) But what if the sampling distribution.

Stat 418 – Day 3 Maximum Likelihood Estimation (Ch. 1)

HIM 3200 Normal Distribution Biostatistics Dr. Burton.

Chapter 3 Hypothesis Testing. Curriculum Object Specified the problem based the form of hypothesis Student can arrange for hypothesis step Analyze a problem.

Estimating a Population Proportion

Lecture 7 1 Statistics Statistics: 1. Model 2. Estimation 3. Hypothesis test.

458 Fitting models to data – III (More on Maximum Likelihood Estimation) Fish 458, Lecture 10.

Stat 301 – Day 27 Sign Test. Last Time – Prediction Interval When goal is to predict an individual value in the population (for a quantitative variable)

Log-linear analysis Summary. Focus on data analysis Focus on underlying process Focus on model specification Focus on likelihood approach Focus on ‘complete-data.

Tests of Hypothesis [Motivational Example]. It is claimed that the average grade of all 12 year old children in a country in a particular aptitude test.

AM Recitation 2/10/11.

Confidence Intervals and Hypothesis Testing - II

Statistical inference: confidence intervals and hypothesis testing.

The paired sample experiment The paired t test. Frequently one is interested in comparing the effects of two treatments (drugs, etc…) on a response variable.

Go to Index Analysis of Means Farrokh Alemi, Ph.D. Kashif Haqqi M.D.

The Triangle of Statistical Inference: Likelihoood

Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Chapter 22 Using Inferential Statistics to Test Hypotheses.

Prof. Dr. S. K. Bhattacharjee Department of Statistics University of Rajshahi.

Confidence Intervals for the Mean (σ known) (Large Samples)

Logit model, logistic regression, and log-linear model A comparison.

Maximum Likelihood Estimator of Proportion Let {s 1,s 2,…,s n } be a set of independent outcomes from a Bernoulli experiment with unknown probability.

Production Planning and Control. A correlation is a relationship between two variables. The data can be represented by the ordered pairs (x, y) where.

Estimating a Population Proportion

MEGN 537 – Probabilistic Biomechanics Ch.5 – Determining Distributions and Parameters from Observed Data Anthony J Petrella, PhD.

Statistics in Biology. Histogram Shows continuous data – Data within a particular range.

Chapter 22: Comparing Two Proportions. Yet Another Standard Deviation (YASD) Standard deviation of the sampling distribution The variance of the sum or.

Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Section 6.1 Confidence Intervals for the Mean (  Known)

Normal Distribution.

Selecting Input Probability Distribution. Simulation Machine Simulation can be considered as an Engine with input and output as follows: Simulation Engine.

Section 6.1 Confidence Intervals for the Mean (Large Samples) Larson/Farber 4th ed.

Review - Confidence Interval Most variables used in social science research (e.g., age, officer cynicism) are normally distributed, meaning that their.

: An alternative representation of level of significance. - normal distribution applies. - α level of significance (e.g. 5% in two tails) determines the.

Hypothesis Testing Errors. Hypothesis Testing Suppose we believe the average systolic blood pressure of healthy adults is normally distributed with mean.

Summarizing Risk Analysis Results To quantify the risk of an output variable, 3 properties must be estimated: A measure of central tendency (e.g. µ ) A.

Elementary Statistics

Logistic regression. Recall the simple linear regression model: y =  0 +  1 x +  where we are trying to predict a continuous dependent variable y from.

Stochastic Loss Reserving with the Collective Risk Model Glenn Meyers ISO Innovative Analytics Casualty Loss Reserving Seminar September 18, 2008.

Confidence Intervals Chapter 6. § 6.1 Confidence Intervals for the Mean (Large Samples)

Chapter Outline 6.1 Confidence Intervals for the Mean (Large Samples) 6.2 Confidence Intervals for the Mean (Small Samples) 6.3 Confidence Intervals for.

A short introduction to epidemiology Chapter 6: Precision Neil Pearce Centre for Public Health Research Massey University Wellington, New Zealand.

BIOL 582 Lecture Set 2 Inferential Statistics, Hypotheses, and Resampling.

Hypothesis Testing and Statistical Significance

Statistical Inference for the Mean Objectives: (Chapter 8&9, DeCoursey) -To understand the terms variance and standard error of a sample mean, Null Hypothesis,

MEGN 537 – Probabilistic Biomechanics Ch.5 – Determining Distributions and Parameters from Observed Data Anthony J Petrella, PhD.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Inferences Concerning Means.

Statistics for Decision Making Hypothesis Testing QM Fall 2003 Instructor: John Seydel, Ph.D.

Chapter 6 Confidence Intervals.

More on Inference.

Chapter 6 Confidence Intervals.

Making inferences from collected data involve two possible tasks:

Hypothesis Testing for Proportions

Inferential Statistics:

More on Inference.

Statistical Inference for the Mean Confidence Interval

YOU HAVE REACHED THE FINAL OBJECTIVE OF THE COURSE

Estimation Goal: Use sample data to make predictions regarding unknown population parameters Point Estimate - Single value that is best guess of true parameter.

Chapter 6 Confidence Intervals.

Maximum Likelihood We have studied the OLS estimator. It only applies under certain assumptions In particular,  ~ N(0, 2 ) But what if the sampling distribution.

Statistical Inference for the Mean: t-test

Hypothesis Testing for the mean. The general procedure.

Presentation transcript:

The maximum likelihood method Likelihood = probability that an observation is predicted by the specified model Plausible observations and plausible models

MLE Observations are ‘outcomes of random experiments’: the outcome is represented by a random variable (e.g. Y). A representation of Y is y i (I = 1, 2, …. m) The distribution of possible outcomes is given by probability distribution. The same data (observations) can be generated by different models and the different observations may be generated by the same model.  what is the range of plausible observations, given the model, and what are the different models that could plausibly have generated the data? – Plausible observations and plausible models A probability model predicts an outcome and associates a probability with each outcome.

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.

Observation from a normal distribution N( ,  2 ) 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):

Log-likelihood

Range of plausible models Likelihood ratio Withthe specified model andthe ‘best’model Ratio of likelihood of any model to likelihood of ‘best’ model Log-likelihood ratio ln = - ½ z 2 z 2 = -2 ln

A plausible value of  is one for which the likelihood ratio exceeds a critical value (less negative), e.g , which corresponds to a 95% confidence interval, or which corresponds to a 90% confidence interval. Values of  for which ln > 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 = (given that ‘best’ value of  is 125 and  2 is fixed): Solution: The observation could come from ANY model in the support range. All models in the ‘support range’ are supported by the data.

Observation from a binomial distribution with parameter p and index m Likelihood function: Log-likelihood function:

Data: leaving parental home

Analysis of young adults who left home leave out censored cases (conditional analysis) 20 obervations