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Introduction to the idea of Statistical Modelling

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- 1. © 2003 Max Chipulu Previously… • Randomness • Descriptive Statistics 1 © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance © 2003 Max Chipulu Next Four Weeks • Introduction to Statistical Modelling • Types of Data • Simulation • Discrete Probability Distributions • The Method of Maximum Likelihood • Continuous Probability Distributions • Association and Correlation • Regression Analysis • Module Review 2 © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 2. Socratic Logic and Statistical Hypothesis Testing Objectives 1. To introduce the steps in Statistical Modelling 2. To discuss what are statistical hypotheses and how to test them 3. To discuss the three types of II in Statistical Inference 4. To discuss the concept of Statistical Significance • To introduce hypothesis testing using Significance tests © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Have you come across this picture before? © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 3. It is a famous painting of Socrates about to drink his poison in 399 BC It was his death sentence. © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Socrates was a Wise Man •For example, Plato quotes: ‘I reflected as I walked away, well, I am certainly wiser than this man. It is only too likely that neither of us has any knowledge to boast of, but he thinks that he knows something which he does not know, whereas I am quite conscious of my ignorance. At any rate, it seems that I am wiser than he is to the small extent, that I do not think that I know what I do not know.’ © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 4. Socrates’ Crimes • Socrates was brilliant at argument • He was often to be found in debate with groups of the impressionable idle young men of Athens • But there was war on; and the Athenian establishment was nervous • They said he was ‘corrupting the young’ © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Socrates’ Crimes • They also said he rejected the traditional gods; that he introduced new gods! • This was against the Athenian Law, and Socrates was charged; it was a serious charge, which carried the death sentence • So Socrates had to prove that he was innocent © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 5. The Test of Socrates’ Innocence • In that Athens, it was possible to talk with the Gods through the oracle at Delphi • Socrates’ friend asked the oracle, ‘Is there any man wiser than Socrates?’ • ‘No’, said the Delphic oracle © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance The Decision • Since the Gods were not unhappy with Socrates, then surely he was not guilty of the charges? • Not according to the Jury; it rejected these arguments • And Socrates was condemned to death © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 6. Statistical Reasoning But Socrates’ way of thinking remains very valid in tackling contestable theories in Statistics We can see this by looking at an everyday analogy of Socrates’ Reasoning: Does Tobacco Smoking Cause Cancer? © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Statistical Reasoning Socrates’ Apology Socrates’ The General Problem Everyday Problem Socrates was not 1. State the Maintained Tobacco does not cause Guilty of the Charges Hypothesis or H0 cancer How to demonstrate 2. Design an How to test whether ‘Not guilty’: Consult the guilty’ Experiment to test H0 smoking increases Gods Cancer rates Record What the 3. Collect the Data to Random Sample of Delphi Says test H0 Smokers/Non-smokers? Smokers/Non- © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 7. Socrates’ Apology Socrates’ The General Problem Everyday Problem Are the Gods happy 4. Analyse the data with Is difference in rates with Socrates? a Statistical Test between the 2 samples non-zero? non- ‘Yes’, hence CANNOT Yes’ 5. Decision: Reject or ‘Yes, there is a difference’, difference’ reject H0 Accept Hypothesis, H0 Reject H0 But H0 was rejected: 6. Error type I: If They found Socrates Correct H0 is rejected ✔ ✔ guilty 7. Error type II: If Previous results suggest ✔ ✔ Wrong H0 is accepted H0 © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Must be rejected Statistical Significance It is no surprise at all that the word ‘hypothesis’ derives from Greek © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 8. Errors in Statistical Inference • Given a null hypothesis and an alternative hypothesis, the decision will be (1) to reject the null hypothesis or (2) to fail to reject (i.e. accept) the null hypothesis. Both decisions 1 and 2 could be in error: • Error type I: correct hypothesis is rejected • Error type II: the wrong hypothesis is accepted © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Maybe even more serious, error type III is… •To ask the wrong question, i.e. propose the wrong hypothesis in the first place: • “An approximate solution to the right problem is much better than an exact solution to the wrong problem”, George Box, Statistics Icon, most famous for his work on time series forecasting ‘Box- Jenkins’ method © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 9. Statistical Significance: Weird Good Vs Weird Bad • Scenario 1 • Suppose you wake up one day and you have a headache. There is no apparent cause for this headache, e.g. it is NOT the morning after a party. • Would you: • A: Think ‘this is a bit weird- a random headache’- Basically ignore it, maybe take a couple of pain killers?; Or • B: Think ‘oh my, this could be serious, I must go and consult a doctor immediately’ • Please show your hand for A or B… © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Statistical Significance: Weird Good Vs Weird Bad • Scenario 2 • Now suppose you continue to suffer a headache for several days. Let us say you have it for ten days. Would you: • A: Basically ignore it- continue taking pain killers, or • B: Think, ‘This must be serious, I must go and consult a doctor’? • Please show your hand for A or B © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 10. What is the difference between Scenario 1 and 2? • The difference is believability or rather lack of believability: • In scenario 1, most people would believe that the headache is just ‘random’ and it will pass. Why? Well, because the probability that anyone could suffer a random headache on any given day is not small- it happens frequently. • In scenario 2, most people would NOT believe that this a random headache because the probability of a random headache for 10 consecutive days is very, very small. • This is statistical significance. Scenario 1 is NOT statistically significant; Scenario 2 is. How? Well let us see how this works in terms of hypothesis. © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Weird Good=‘Random’ Weird Bad=‘Not Random’, i.e. Statistically Significant • Suppose that the null hypothesis, H0 = ‘the headache is random’ • Then the alternative hypothesis is Ha = ‘the headache is NOT random’ • So when the result, i.e. headache for one day or headache for 10 days is observed, the question is can we believe this headache is random? If the probability is small and we cannot believe that the headache is random, then we must reject the null hypothesis, i.e. we conclude that the observed headache is statistically significant. • Usually, we will reject the null hypothesis if the probability of observing the result (or something worse) under the null hypothesis is 0.05 (5%) or less. © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 11. Tests of Significance ALL hypothesis tests are based on a test of Significance: In order to reject or accept a null hypothesis, we must work out whether the probability of the observed result is small under the null hypothesis. So, always, we need a test probability distribution that the observed result would follow under the null hypothesis. Such a probability distribution is called a statistical test of significance. © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance Type of Test of Significance Parametric Non-Parametric We assume that the observed result We make NO assumptions about the follows a specific probability type of probability distribution Characteristic distribution function, e.g. the function that the observed result normal distribution might follow Strictly only appropriate if Since no assumptions are made, Application assumptions have not been applications can be flexible violated Usually Categorical data or quantitative data for Quantitative data taken from large applicable small samples when most parametric samples for assumptions are violated More exact than non-parametric Not as exact as parametric tests but Advantages tests, therefore whenever research shows are almost as appropriate use parametric tests powerful Z-test (result assumed to follow the normal distribution) Binomial Test T-test: results assumed to follow t- Chi-square Most distribution, e.g. least squares KS (similar to Chi-square but uses Common regression coefficients. proportions) Examples F-test of variance, e.g. variance Wilcoxon (T-test equivalent for explained by a regression categorical data) © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance
- 12. Further Reading • Rae R. Newton and Kjell Erik Rudestam, 1999. Your Statistical Consultant. Sage Publications Inc. • Ramon E. Henkel, 1976. Tests of Significance. Sage Publications Inc. (In Library box HA 33 QAA). © 2009 Max Chipulu, University of Southampton- Socratic Logic, Hypotheses and Statistical Significance

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