hypothesis testing: significance

Statistics: Unlocking the Power of Data Lock5

Hypothesis Testing: Significance

STAT 101

Dr. Kari Lock Morgan

9/27/12

SECTION 4.3• Significance level• Statistical conclusions• Type I and II errors


Office HoursMy office hours next week will be Wednesday

1 – 3pm, NOT Monday

(and Thurs 1 – 2:30 as always)


Randomization Distributionsp-values can be calculated by randomization

distributions:

simulate samples, assuming H0 is true calculate the statistic of interest for each sample find the p-value as the proportion of simulated

statistics as extreme as the observed statistic

Let’s do a randomization distribution for a randomized experiment…


• In a randomized experiment on treating cocaine addiction, 48 people were randomly assigned to take either Desipramine (a new drug), or Lithium (an existing drug), and then followed to see who relapsed

• Question of interest: Is Desipramine better than Lithium at treating cocaine addiction?

Cocaine Addiction


•What are the null and alternative hypotheses?

•What are the possible conclusions?

Cocaine Addiction

pD, pL: proportion of cocaine addicts who relapse after taking Desipramine or Lithium, respectively

H0: pD = pLHa: pD < pL

Reject H0; Desipramine is better than LithiumDo not reject H0: We cannot determine from these data whether Desipramine is better than Lithium


R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

Desipramine Lithium

1. Randomly assign units to treatment groups


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R N N

N N N N N N

RR

N N N N N N

R = RelapseN = No Relapse

R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R RR

R R N N N N

RR

N N N N N N

2. Conduct experiment

3. Observe relapse counts in each group

LithiumDesipramine

10 relapse, 14 no relapse 18 relapse, 6 no relapse

1. Randomly assign units to treatment groups

10 1824

ˆ ˆ

24.333

D Lp p


To see if a statistic provides evidence against H0, we need to

see what kind of sample statistics we would observe,

just by random chance, if H0 were true

Measuring Evidence against H0


• “by random chance” means by the random assignment to the two treatment groups

• “if H0 were true” means if the two drugs were equally effective at preventing relapses (equivalently: whether a person relapses or not does not depend on which drug is taken)

• Simulate what would happen just by random chance, if H0 were true…

Cocaine Addiction


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R N N

N N N N N N

RR

N N N N N N



R R R R R R

R R R R N N

N N N N N N

N N N N N N

R R R R R R

R R R R R R

R R R R R R

N N N N N N

R N R N

R R R R R R

R N R R R N

R N N N R R

N N N R

N R R N N N

N R N R R N

R N R R R R

Simulate another randomization

Desipramine Lithium


ˆ ˆ16 1224 240.167

LDp p


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R N R R N N

R R N R N R

RR

R N R N R R

Simulate another randomization

Desipramine Lithium


ˆ ˆ17 1124 240.250

D Lp p


Simulate Your Own SampleIn the experiment, 28 people relapsed and 20 people

did not relapse. Create cards or slips of paper with 28 “R” values and 20 “N” values.

Pool these response values together, and randomly divide them into two groups (representing Desipramine and Lithium)

Calculate your difference in proportions

Plot your statistic on the class dotplot

To create an entire randomization distribution, we simulate this process many more times with technology: StatKey

http://www.lock5stat.com/statkey


www.lock5stat.com/statkey

p-value

http://www.lock5stat.com/statkey


Formal DecisionsIf the p-value is small:

REJECT H0

the sample would be extreme if H0 were true the results are statistically significant we have evidence for Ha

If the p-value is not small: DO NOT REJECT H0

the sample would not be too extreme if H0 were true the results are not statistically significant the test is inconclusive; either H0 or Ha may be true


A formal hypothesis test has only two possible conclusions:

1. The p-value is small: reject the null hypothesis in favor of the alternative

2. The p-value is not small: do not reject the null hypothesis

Formal Decisions

How small?


Significance Level

The significance level, , is the threshold below which the p-value is deemed small enough to reject the null hypothesis

p-value < Reject H0

p-value > Do not Reject H0


Significance LevelIf the p-value is less than , the results are

statistically significant, and we reject the null hypothesis in favor of the alternative

If the p-value is not less than , the results are not statistically significant, and our test is inconclusive

Often = 0.05 by default, unless otherwise specified


• Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents, and has recently been investigated in primates (specifically, the Grey Mouse Lemur).

• A sample of lemurs had various measurements taken before and after receiving resveratrol supplementation for 4 weeks

Red Wine and Weight Loss

BioMed Central (2010, June 22). “Lemurs lose weight with ‘life-extending’ supplement resveratrol. Science Daily.



In the test to see if the mean resting metabolic rate is higher after treatment, the p-value is 0.013.

Using = 0.05, is this difference statistically significant? (should we reject H0: no difference?)

a) Yesb) No The p-value is lower than

= 0.05, so the results are statistically significant and we reject H0.



In the test to see if the mean body mass is lower after treatment, the p-value is 0.007.


a) Yesb) No The p-value is lower than

= 0.05, so the results are statistically significant and we reject H0.



In the test to see if locomotor activity changes after treatment, the p-value is 0.980.


a) Yesb) No

The p-value is not lower than = 0.05, so the results are not statistically significant and we do not reject H0.



In the test to see if mean food intake changes after treatment, the p-value is 0.035.


a) Yesb) No

The p-value is lower than = 0.05, so the results are statistically significant and we reject H0.


H0 : X is an elephantHa : X is not an elephant

Would you conclude, if you get the following data?

• X walks on two legs

• X has four legs

Elephant Example

Reject H0; evidence that X is not an elephant

Although we can never be certain!

Do not reject H0; we do not have sufficient evidence to determine whether X is an elephant


“For the logical fallacy of believing that a hypothesis has been proved to be true, merely because it is not contradicted by the available facts, has no more right to insinuate itself in statistical than in other kinds of scientific reasoning…”

-Sir R. A. Fisher

Never Accept H0

•“Do not reject H0” is not the same as “accept H0”!

• Lack of evidence against H0 is NOT the same as evidence for H0!


Statistical Conclusions

In a hypothesis test of

H0: = 10 vs Ha: < 10

the p-value is 0.002. With α = 0.05, we conclude:

a) Reject H0

b) Do not reject H0

c) Reject Ha

d) Do not reject Ha

The p-value of 0.002 is less than α = 0.05, so we reject H0




H0: = 10 vs Ha: < 10


a) There is evidence that = 10 b) There is evidence that < 10

c) We have insufficient evidence to conclude anything




H0: = 10 vs Ha: < 10


a) Reject H0

b) Do not reject H0

c) Reject Ha

d) Do not reject Ha

The p-value of 0.21 is not less than α = 0.01, so we do not reject H0




H0: = 10 vs Ha: < 10


a) There is evidence that = 10 b) There is evidence that < 10

c) We have insufficient evidence to conclude anything


Informal strength of evidence against H0:

Formal decision of hypothesis test, based on = 0.05 :

statistically significant

not statistically significant



Multiple Sclerosis and Sunlight• It is believed that sunlight offers some protection

against multiple sclerosis, but the reason is unknown

• Researchers randomly assigned mice to one of:• Control (nothing)• Vitamin D Supplements• UV Light

• All mice were injected with proteins known to induce a mouse form of MS, and they observed which mice got MS

Seppa, Nathan. “Sunlight may cut MS risk by itself”, Science News, April 24, 2010 pg 9, reporting on a study appearing March 22, 2010 in the Proceedings of the National Academy of Science.


Multiple Sclerosis and SunlightFor each situation below, write down

Null and alternative hypotheses Informal description of the strength of evidence against H0

Formal decision about H0, using α = 0.05 Conclusion in the context of the question

In testing whether UV light provides protection against MS (UV light vs control group), the p-value is 0.002.

In testing whether Vitamin D provides protection against MS (Vitamin D vs control group), the p-value is 0.47.


Multiple Sclerosis and SunlightIn testing whether UV light provides

protection against MS (UV light vs control group), the p-value is 0.002.

• H0: pUV – pC = 0 Ha: pUV – pC < 0• We have strong evidence against H0

• Reject H0

• We have strong evidence that UV light provides protection against MS, at least in mice.


Multiple Sclerosis and SunlightIn testing whether Vitamin D provides

protection against MS (Vitamin D vs control group), the p-value is 0.47.

• H0: pD – pC = 0 Ha: pD – pC < 0• We have little evidence against H0

• Do not reject H0

• We cannot conclude anything about Vitamin D and MS.


There are four possibilities:Errors

Reject H0 Do not reject H0

H0 true

H0 false TYPE I ERROR

TYPE II ERRORTrut

h

Decision

• A Type I Error is rejecting a true null

• A Type II Error is not rejecting a false null


• In the test to see if resveratrol is associated with food intake, the p-value is 0.035.

o If resveratrol is not associated with food intake, a Type I Error would have been made

• In the test to see if resveratrol is associated with locomotor activity, the p-value is 0.980.

o If resveratrol is associated with locomotor activity, a Type II Error would have been made



A person is innocent until proven guilty.

Evidence must be beyond the shadow of a doubt.

Types of mistakes in a verdict?

Convict an innocent

Release a guilty

Ho Ha

Type I error

Type II error

Analogy to Law

p-value from data


• The probability of making a Type I error (rejecting a true null) is the significance level, α

• α should be chosen depending how bad it is to make a Type I error

Probability of Type I Error


If the null hypothesis is true:• 5% of statistics will be in the most extreme 5% • 5% of statistics will give p-values less than 0.05• 5% of statistics will lead to rejecting H0 at α = 0.05• If α = 0.05, there is a 5% chance of a Type I error

Distribution of statistics, assuming H0 true:



If the null hypothesis is true:• 1% of statistics will be in the most extreme 1% • 1% of statistics will give p-values less than 0.01• 1% of statistics will lead to rejecting H0 at α = 0.01• If α = 0.01, there is a 1% chance of a Type I error

Distribution of statistics, assuming H0 true:



Probability of Type II ErrorThe probability of making a Type II Error (not

rejecting a false null) depends on

Effect size (how far the truth is from the null)

Sample size

Variability

Significance level


Choosing αBy default, usually α = 0.05

If a Type I error (rejecting a true null) is much worse than a Type II error, we may choose a smaller α, like α = 0.01

If a Type II error (not rejecting a false null) is much worse than a Type I error, we may choose a larger α, like α = 0.10


Come up with a hypothesis testing situation in which you may want to…

• Use a smaller significance level, like = 0.01

• Use a larger significance level, like = 0.10

Significance Level


• Results are statistically significant if the p-value is less than the significance level, α• In making formal decisions, reject H0 if the p-value is less than α, otherwise do not reject H0

• Not rejecting H0 is NOT the same as accepting H0

• There are two types of errors: rejecting a true null (Type I) and not rejecting a false null (Type II)

Summary


To DoProject 1 proposal due TODAY at 5pm

Read Section 4.3

Do Homework 4 (due Thursday, 10/4)

http://stat.duke.edu/courses/Fall12/sta101.002/project1proposal.pdf

http://stat.duke.edu/courses/Fall12/sta101.002/HW4.pdf

http://stat.duke.edu/courses/Fall12/sta101.002/HW4.pdf

hypothesis testing: significance

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