montefiore - probability and statistics · 2011. 11. 11. · stat261 statistical inference notes...
TRANSCRIPT
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CHAPTER 6: HYPOTHESIS TESTING 6 - 0
Probability and Statistics
Kristel Van Steen, PhD2
Montefiore Institute - Systems and Modeling
GIGA - Bioinformatics
ULg
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 1
CHAPTER 6: HYPOTHESIS TESTING
1 Terminology and Notation
1.1 Tests of Hypotheses
1.2 Size and Power of Tests
1.3 Examples
2 One-sided and Two-sided Tests
2.1 Introduction
2.2 Case(a) Alternative is one-sided
2.3 Case (b) Two-sided Alternative
2.4 Two Approaches to Hypothesis Testing
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 2
3 Connection between Hypothesis testing and CI’s
3.1 Two faces of the same coin
3.2 The concept of a p-value
3.3 Three approaches for hypothesis testing
4 One-sample problems
4.1 Testing hypotheses about
4.3 Testing hypotheses about
5 Two-Sample Problems
5.1 Testing equality of normal means
5.2 Testing equality of sample variances
6 Course concluding remarks
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 3
1 Terminology and Notation
1.1 Tests of Hypotheses
Introduction
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 4
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 6
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 7
Terminology and notations
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 8
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 9
Tests of hypotheses
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1.2 Size and Power of Tests
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Are the roles of the hypothesis and alternative hypothesis symmetric?
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Considerations when constructing a test
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1.3 Examples
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Fig: CRUpperTail
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these
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2 One-sided and Two-sided Tests
2.1 Introduction
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2.2 Case(a) Alternative is one-sided
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2.3 Case (b) Two-sided Alternative
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2.4 Two Approaches to Hypothesis Testing
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A
A
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3 Connection between Hypothesis testing and CI’s
3.1 Two faces of the same coin
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http://www.statsoft.com/textbook/distribution-tables/
Dark blue is less than one standard deviation from the mean. For the normal distribution, this
accounts for about 68% of the set, while two standard deviations from the mean (medium and
dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue)
account for about 99.7%.
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The vertical line segments represent 50 realizations
of a confidence interval for μ.
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3.2 The concept of a p-value
• The statistical significance of a result is the probability that the observed
relationship (e.g., between variables) or a difference (e.g., between means)
in a sample occurred by pure chance ("luck of the draw"), and that in the
population from which the sample was drawn, no such relationship or
differences exist.
• Using less technical terms, we could say that the statistical significance of a
result tells us something about the degree to which the result is "true" (in
the sense of being "representative of the population").
• More technically, the value of the p-value represents a decreasing index of
the reliability of a result (see Brownlee, 1960).
o The higher the p-value, the less we can believe that the observed
relation between variables in the sample is a reliable indicator of the
relation between the respective variables in the population.
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• Specifically, the p-value represents the probability of error that is involved
in accepting our observed result as valid, that is, as "representative of the
population."
o If the P value is 0.03, that means that there is a 3% chance of observing
a difference as large as you observed even if the two population means
are identical.
o It is tempting to conclude, therefore, that there is a 97% chance that
the difference you observed reflects a real difference between
populations and a 3% chance that the difference is due to chance.
Wrong.
o What you can say is that random sampling from identical populations
would lead to a difference smaller than you observed in 97% of
experiments and larger than you observed in 3% of experiments.
• When there IS a relationship between the variables in the population, the
probability of replicating the study and finding that relationship is related to
the statistical power of the design.
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3.3 Three approaches for hypothesis testing
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Quantile (TS) Probability Confidence Interval (CI)
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Components of a statistical power analysis
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A note about effect sizes
• The effect size encodes the selected research findings on a numeric scale
• There are many different types of
means, correlations, …) , each suited to different research situations
• Each effect size type may also have multiple methods
• An example of a standardized effect size ES is:
CHAPTER 6: HYPOTHESIS TESTING
encodes the selected research findings on a numeric scale
There are many different types of effect size measures (OR, difference in
, each suited to different research situations
type may also have multiple methods of computation
An example of a standardized effect size ES is:
6 -- 46
encodes the selected research findings on a numeric scale
(OR, difference in
, each suited to different research situations
of computation
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4 One-sample problems
4.1 Testing hypotheses about
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X
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X
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Think about the 3 ways to perform statistical hypothesis testing …
X
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Recall
Population Estimation of Test statistic and
distribution
Known
Unknown
,
unbiased for
Case (i)
Case (ii)
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Unknown mean
According to the table, the crucial distribution is now a t-distribution
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4.2 Testing hypotheses about
Case (ii)
Case (i)
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Case (ii)
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Case (i)
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5 Two-Sample Problems
5.1 Testing equality of sample variances
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Reciprocal of an F-variate
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One-sided test
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5.2 Testing equality of normal means
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 65
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 66
Comment
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 67
Recall
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 68
6 Course concluding remarks
Fallacies of Statistical Testing
1. Failure to reject the null hypothesis leads to its acceptance. (WRONG!
Failure ro reject the null hypothesis implies insufficient evidence for its
rejection.)
2. The p value is the probability that the null hypothesis is incorrect.
(WRONG! The p value is the probability of the current data or data that
is more extreme assuming H0 is true.)
3. α = .05 is a standard with an objective basis. (WRONG! α= .05 is merely a
convention that has taken on unwise mechanical use.)
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 69
Fallacies of Statistical Testing (continued)
4. Small p values indicate large effects. (WRONG! p values tell you next to
nothing about the size of a difference.)
5. Data show a theory to be true or false. (WRONG! Data can at best serve
to bolster or refute a theory or claim.)
6. Statistical significance implies importance. (WRONG! WRONG! WRONG!
Statistical significance says very little about the importance of a relation.)
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 70
The importance of the normal distribution
~ Central Limit Theorem
set.seed(7)
norm1 <- rnorm(1000)
norm2 <- rnorm(2000,mean=3)
mixnorm <- c(norm1,norm2)
mixnorm <-
data.frame(cbind(mixnorm,c(rep(0,1000),rep
(1,2000))))
names(mixnorm) <- c("Y","Population")
library(ggplot2)
qplot(Y, colour=Population, data=mixnorm,
geom="density")
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CHAPTER 6: HYPOTHESIS TESTING 6 - 71
Mixing 2 normal distributions
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 72
Sample distributions for the mean of the 3 considered
populations
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CHAPTER 6: HYPOTHESIS TESTING 6 -- 73
Main reference:
STAT261 Statistical inference notes – School of mathematics, statistics and
computer science. University of New England, Oct 4, 2007