Introduction to the idea of Statistical Modelling

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