thursday, september 12, 2013 effect size, power, and exam review
TRANSCRIPT
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Thursday, September 12, 2013
Effect Size, Power, and Exam Review
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Last Time:
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Which tail? Above or Below?
• In hypothesis testing, we are interested in extreme values of our test statistics.
• We are interested in the tails where p-values are very low. Probability is always <.50 in the tail (by definition)
• For 2-tailed tests, we reject H0 when we encounter values more extreme than +/-Z (higher than +Z or lower than –Z)
• For 1-tailed tests, we reject H0 when we encounter extreme values consistent with HA
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One-Tailed
H0:
HA:
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One-Tailed
H0:
HA:
If is the control condition (the original, null, blue distribution) and is the treatment condition (the alternative, new, green distribution), HA predicts that a sample mean (M) from population 2 will be smaller than the population mean (μ) from population 1. M- would be negative, so a negative Z is consistent with HA, and would lead to rejection of H0 (if the value is sufficiently extreme).
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One-Tailed
H0:
HA:
If is the control condition (the original, null, blue distribution) and is the treatment condition (the alternative, new, green distribution), H0 predicts that a sample mean (M) from population 2 will be equal to or larger than the population mean (μ) from population 1. M- would be 0 or positive, so a positive Z is consistent with H0, and would not lead to rejection of H0, no matter how extreme.
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One-tailed vs. 2-tailed
• One-tailed – if the results are in the direction your theory predicts, then an extreme value of Z (in the tail) will lead you to reject H0 regardless of whether Z is positive or negative.
• Two-tailed – It does not matter whether Z is positive or negative. You reject H0 if the absolute value of Z is sufficiently extreme (in either tail)
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Finishing up chapter 8 (some loose ends)
• Error types (quick review from last time)• Effect size: Cohen’s d• Statistical Power Analysis
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Performing your statistical test
Real world (‘truth’)
H0 is correct
H0 is wrong
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
So there is only one distribution
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
So there are two distributionsThe original (null) distribution
The new (treatment) distribution
The original (null) distribution
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Performing your statistical test
Real world (‘truth’)
H0 is correct
H0 is wrong
So there is only one distribution
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
So there are two distributionsThe original (null) distribution
The new (treatment) distribution
The original (null) distribution
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Effect Size• Hypothesis test tells
us whether the observed difference is probably due to chance or not
• It does not tell us how big the difference is
H0 is wrong
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
So there are two distributionsThe original (null) distribution
The new (treatment) distribution
– Effect size tells us how much the two populations don’t overlap
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Effect Size
• Figuring effect size
The original (null) distribution
The new (treatment) distribution
– Effect size tells us how much the two populations don’t overlap
But this is tied to the particular units of measurement
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Effect Size
The original (null) distribution
The new (treatment) distribution
• Standardized effect size
– Effect size tells us how much the two populations don’t overlap
– Puts into neutral units for comparison (same logic as z-scores)Cohen’s d
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Effect Size
The original (null) distribution
The new (treatment) distribution
• Effect size conventions– small d = .2– medium d = .5– large d = .8
– Effect size tells us how much the two populations don’t overlap
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Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
I conclude that there is an effect
I can’t detect an effect
There really isn’t an effect
There really isan effect
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Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Type I error (α): concluding that there is a difference between groups (“an effect”) when there really isn’t.
Type II error (β): concluding that
there isn’t an effect, when there really is.
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Statistical Power
• The probability of making a Type II error is related to Statistical Power– Statistical Power: The probability that the
study will produce a statistically significant results if the research hypothesis is true (there is an effect)
• So how do we compute this?
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Statistical Power
H0: is true (is no treatment effect)Real world (‘truth’)
Fail to reject H0Reject H0
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
α = 0.05
The original (null) distribution
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Statistical Power
H0: is false (is a treatment effect)
α = 0.05
Reject H0
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error The original (null) distribution
Real world (‘truth’)
The new (treatment) distributionThe new (treatment) distribution
Fail to reject H0
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Statistical Power
Fail to reject H0Reject H0
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
β = probability of a Type II error
The new (treatment) distribution
H0: is false (is a treatment effect)
The original (null) distribution
Real world (‘truth’)
α = 0.05
Failing to Reject H0, even though there is
a treatment effect
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Statistical Power
Fail to reject H0Reject H0
Real world (‘truth’)
H0 is correct
H0 is wrong
Type I error
Type II error
Power = 1 - b
The new (treatment) distribution
H0: is false (is a treatment effect)
The original (null) distribution
Real world (‘truth’)
β = probability of a Type II error
α = 0.05
Failing to Reject H0, even though there is
a treatment effect
Probability of (correctly)
Rejecting H0
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Statistical Power
– α-level
– Sample size
– Population standard deviation σ
– Effect size
– 1-tail vs. 2-tailed
Factors that affect Power:
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
b
Power = 1 - β
Factors that affect Power: a-levelChange from a = 0.05 to 0.01
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: -levelChange from α = 0.05 to 0.01
α = 0.01
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: α-levelChange from α = 0.05 to 0.01
α = 0.01
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 -β
Factors that affect Power: α-levelChange from α = 0.05 to 0.01
α = 0.01
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: α-levelChange from α = 0.05 to 0.01
α = 0.01
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: α-levelChange from α = 0.05 to 0.01
α = 0.01
So as the α level gets smaller, so does the
Power of the test
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Sample sizeChange from n = 25 to 100 Recall that sample
size is related to the spread of the distribution
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Sample sizeChange from n = 25 to 100
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Sample sizeChange from n = 25 to 100
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Sample sizeChange from n = 25 to 100
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Sample sizeChange from n = 25 to 100 As the sample gets
bigger, the standard error gets smaller and the Power gets larger
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Population standard deviationChange from σ = 25 to 20
Recall that standard error is related tothe spread of the distribution
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Population standard deviationChange from σ = 25 to 20
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Population standard deviationChange from σ = 25 to 20
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Population standard deviationChange from σ = 25 to 20
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power:
As the σ gets smaller, the standard error gets smaller and the Power gets larger
Population standard deviationChange from σ = 25 to 20
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
μtreatment μno treatment
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
μtreatment μno treatment
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 -β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
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Statistical Power
α= 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
![Page 43: Thursday, September 12, 2013 Effect Size, Power, and Exam Review](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649d0d5503460f949e1972/html5/thumbnails/43.jpg)
Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
![Page 44: Thursday, September 12, 2013 Effect Size, Power, and Exam Review](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649d0d5503460f949e1972/html5/thumbnails/44.jpg)
Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
![Page 45: Thursday, September 12, 2013 Effect Size, Power, and Exam Review](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649d0d5503460f949e1972/html5/thumbnails/45.jpg)
Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
![Page 46: Thursday, September 12, 2013 Effect Size, Power, and Exam Review](https://reader036.vdocuments.net/reader036/viewer/2022062515/56649d0d5503460f949e1972/html5/thumbnails/46.jpg)
Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: Effect sizeCompare a small effect (difference) to a big effect
As the effect gets bigger, the Power gets larger
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from α = 0.05 two-tailed to a = 0.05 two-tailed
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Statistical Power
α = 0.05
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from a = 0.05 two-tailed to a = 0.05 two-tailed
p = 0.025
p = 0.025
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Statistical Power
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from a = 0.05 two-tailed to α = 0.05 two-tailed
p = 0.025
p = 0.025
α = 0.05
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Statistical Power
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from a = 0.05 two-tailed to α = 0.05 two-tailed
p = 0.025
p = 0.025
α = 0.05
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Statistical Power
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from a = 0.05 two-tailed to a = 0.05 two-tailed
p = 0.025
p = 0.025
α = 0.05
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Statistical Power
Fail to reject H0Reject H0
β
Power = 1 - β
Factors that affect Power: 1-tail vs. 2-tailedChange from a = 0.05 two-tailed to α = 0.05 two-tailed
p = 0.025
Two tailed functionally cuts the α-level in half, which decreases the power.
p = 0.025
α = 0.05
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Statistical Power
Factors that affect Power:– α-level: So as the a level gets smaller, so does the
Power of the test
– Sample size: As the sample gets bigger, the standard error gets smaller and the Power gets larger
– Population standard deviation: As the population standard deviation gets smaller, the standard error gets smaller and the Power gets larger
– Effect size: As the effect gets bigger, the Power gets larger
– 1-tail vs. 2-tailed: Two tailed functionally cuts the α-level in half, which decreases the power
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Why care about Power?
• Determining your sample size– Using an estimate of effect size, and
population standard deviation, you can determine how many participants need to achieve a particular level of power
• When a result if not statistically significant– Is is because there is no effect, or not enough
power
• When a result is significant– Statistical significance versus practical
significance
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Ways of Increasing Power
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Exam I (Tuesday 9/17)
• Covers first 8 chapters• Closed book (materials will be provided)• Bring calculator• Emphasis on material covered in class• No need to use SPSS during the test
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Exam I (Tuesday 9/17)• Ways to describe distributions (using tables, graphs, numeric summaries of
center and spread)• How to enter data into SPSS• Z-scores• Probability• Characteristics of the normal distribution• Normal approximation of the binomial distribution• Use of the unit normal table• Central Limit Theorem & distribution of sample means• Standard error of the mean (σM)• Hypothesis testing
– Four steps of hypothesis testing– Error Types– One-Sample Z-test– Effect sizes and Power