The Basics

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Introduction

• The Basics

• Three distributions
• Sampling distributions, the central limit theorem
  confidence intervals
• Testing hypotheses the t test, Type I and II
  errors the p value
• Testing hypotheses the 2 sample t test
• One and two tailed tests

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1. Three basic distributions
Normal, Poisson and Binomial
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Normal Distribution

• Described by two parameters Mean and variance

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Useful properties

• Standardising

• Shape

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Poisson Distribution

• Discrete describes counts

• Described by one parameter the mean

• So mean variance

• Probability of r events occurring per unit time
  or space (r 0, 1, 2..).

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Shape

• Depends upon the size of the mean..

• If mean gt 5, can approximate with a normal
  distribution

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Binomial Distribution

• Discrete describes proportions

• Described by two parameters n (number of events,
  trials, sample size etc) and p (the probability
  of a certain outcome).

• Probability of the proportion r/n occurring

• Mean n p Variance n p q

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Shape

• Depends upon p

• n 10
• p 0.1, 0.5, 0.9

• When p close to 0.5 can be approximated by the
  normal distribution

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• These three probability distributions are the
  basis of describing much continuous and
  categorical data

• Under certain conditions, the Poisson and
  Binomial distributions can be approximated using
  the Normal

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2. Populations and samplesCalculating a
confidence interval
Consider this in the context of the simplest
model estimating a mean
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Population Mean ? Variance ?2
Sample
Mean ?y Variance s2
Take a sample of size n
Our view ?y and s2 are known Need to use these to
guess ? and ?2

• Gods view
• and ?2 are known
• ?y and s2 are unpredictable

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True ?
Possible ?y
Observed ?y
Inferred ?
This is what we are doing by constructing a
confidence interval
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Sample n independent data points
Calculate a mean
Calculate a variance
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Variance
DF number of independent pieces of information
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How can we use ?y and s2 to infer a likely value
for ? ?
Imagine a meta-experiment
Population
n
n
n
n
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Distribution of ?y
Often ?y will be close to ?
Occasionally ?y will be some distance from ?
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True ?
Possible ?y
Observed ?y
Inferred ?
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True ?
Possible ?y
Observed ?y
Inferred ?
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True ?
Possible ?y
Inferred ?
Observed ?y
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True ?
Possible ?y
Inferred ?
Observed ?y
The higher the sample size, the better the
estimate of ?
The lower the value of ?2, the better the
estimate of ?
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Distribution of ?y

• Mean ?
• Variance
• Shape Normal

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Standard error

• Variance

• Standard error

standard deviation of the distribution of ?y
that would be obtained through a meta-experiment
standard deviation of a parameter distribution
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Why do we assume the shape is Normal?

• Because of the central limit theorem

In a psychology experiment, peoples reaction
times to operate a push button in response to a
signal are measured. 10 of the time, the
person missed.
The population ? 148 ms
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Aim estimate ? by sampling from the population

• Simulate five meta-experiments
• n 4 n 8 n 16 n 32 n 128.

• When n 4, the reaction times picked from the
  population could be four hits, four misses or any
  combination in-between.

• Could use the binomial distribution to calculate
  the probabilities of these combinations.

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N 4

• First peak is four hits p 0.66, 115 ms
• Next peak, 3 hits and 1 miss, 200 ms
• And so on

Very peculiar
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N8
Still rather peculiar
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N16
Getting there
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N 32
Normalish
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N 128
Spot on
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Summary of the three distributions
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Put this information together
Distribution of ?Y
?
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96 of the time (in our meta-experiment) ?Y lies
between
and
Rearranging, and making it a 95 interval.
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• Problem we dont know ?.
• We have an estimate of ?, which is s
• How good is this estimate?
• That depends upon the number of independent
  pieces of information we have about s i.e. the
  degrees of freedom
• Which in this case n-1

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t distributions compensate for uncertainty in s
When df 8, the t distribution a Normal
distribution
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Use t tables to find out how many standard errors
you need to encompass 95 of the distribution.
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Giving the final formula for a Confidence Interval
Where tcrit has n-1 degrees of freedom
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The General Formula
Parameter estimate tcrit standard
errorparameter
Where the required df for tcrit comes from the
unexplained variation
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As sample size increases.

• ?Y becomes closer to ?
• (because the variance of the sampling
  distribution is ?2/n)
• s becomes closer to ?
• (with df n-1)
• The shape of the sampling distribution becomes
  more Normal
• (due to the central limit theorem)
• The t distribution becomes very close to the
  Normal distribution

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3. Testing hypotheses

• t tests

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One sample t test

• Can be used to test the hypothesis that ? a
  specific value
• Can be used with paired data, to ask if a mean
  difference is significantly different from zero.

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Null hypothesis H0 ? 0
Our sampling distribution
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Units of the x axis are standard errors
Is our ?y here? Or is it here?
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How many standard errors is our observed value of
4.64 away from our hypothesised value of 0?
Answer
with 7 degrees of freedom
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?
?
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The General Formula
Where the required df for ts comes from the
unexplained variation
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So We reject the Null hypothesis with 0.02 lt p lt
0.05
The p value is the probability of getting that
test statistic, or something more extreme, under
the Null hypothesis (i.e. if the Null hypothesis
is true).
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The p value
When we conclude that we have a significant
result with p value of 0.03, what exactly is that
probability measuring?
NOT

• The null hypothesis is true with probability 0.03
• There is a probability of 0.03 that there is no
  difference

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E all possible experiments
E
A
B
A the set of experiments for which the null
hypothesis is true
B the set of experiments for which the null
hypothesis is rejected
The p value the overlap
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Errors
Type I
Type II

• Failing to reject the null hypothesis when it is
  false
• Influenced by a host of factors including power
  of statistical test used experimental design
  etc.
• Cant be measured absolutely (but can relatively)

• Rejecting the null hypothesis when it is true
• Convention sets this at 0.05 for any one test
• Under our control!

Power 1-Probability of Type II error
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E
A
H0 rejected
H0 true
B

• The null hypothesis is true with probability 0.03
• There is a probability of 0.03 that there is no
  difference

This is equivalent to saying that set A is of
size 0.03 Incorrect
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4. Two sample t test

• Are two groups different?

Sample ?y1, s12 ?y2, s22
Population ?1, ?12 ?2, ?22
Does ?1 ?2 ?
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Null hypothesis ?1 - ?2 0
Sampling distribution is now ?y1-?y2
?1-?2
Variance (A-B) Variance (A) Variance (B)
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Two sample t tests are a simple extension of one
sample t tests.
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and also with confidence intervals
Parameter estimate tcrit standard
errorparameter
Degrees of freedom? Two means have been
estimated n1 n2 - 2
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5. One and two tailed tests
Two tails
Null hypothesis H0 ?1 - ?2 0
Alternative hypothesis HA ?1 - ?2 ? 0
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Two tailed test
Reject H0
Reject H0
Standard Errors
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One tail
Null hypothesis ?1 - ?2 0
Alternative hypothesis ?1 lt ?2
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One tailed test
Reject H0
Standard Errors
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Deciding on HA after you know direction will
double your chances of making a Type I error
Reject H0
Reject H0
Standard Errors
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Relationship between CIs and hypothesis testing
The distribution of ?y1 - ?y2
The null distribution
?y1 - ?y2
0
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Next week

• ANOVA
• Regression
• General Linear Models an introduction
• More than 1 variable
• Interactions