P-value and H

[Pages:6]3/5/2015

STAT 250 Dr. Kari Lock Morgan

Hypothesis Testing: Significance

SECTION 4.3 ? Significance level (4.3) ? Statistical conclusions (4.3)

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p-value and H0

If the p-value is small, then a statistic as extreme as that observed would be unlikely if the null hypothesis were true, providing significant evidence against H0

The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative

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p-value and H0

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p-value and H0

Which of the following p-values gives the strongest evidence against H0?

a) 0.005 b) 0.1 c) 0.32 d) 0.56 e) 0.94

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p-value and H0

Which of the following p-values gives the strongest evidence against H0?

a) 0.22 b) 0.45 c) 0.03 d) 0.8 e) 0.71

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p-value and H0

Two different studies obtain two different pvalues. Study A obtained a p-value of 0.002 and Study B obtained a p-value of 0.2. Which study obtained stronger evidence against the null hypothesis?

a) Study A b) Study B

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Formal Decisions

If the p-value is small:

the sample would be extreme if H0 were true the results are statistically significant

REJECT H0 we have evidence for Ha

If the p-value is not small:

the sample would not be too extreme if H0 were true the results are not statistically significant

DO NOT REJECT H0 the test is inconclusive; either H0 or Ha may be true

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Formal Decisions

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

How small?

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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

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Significance Level

If 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



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Statistical Significance

p-value <

p-value

Results would be rare, if the null were true

Reject H0

We have evidence that the alternative is true!

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Results would not be rare, if the null were true

Do not reject H0

We can make no conclusions either way

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Statistical Conclusions

In a hypothesis test of H0: = 10 vs Ha: < 10

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

a) Reject H0 b) Do not reject H0 c) Reject Ha d) Do not reject Ha

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Statistical Conclusions

In a hypothesis test of H0: = 10 vs Ha: < 10

the p-value is 0.002. With = 0.01, we conclude:

a) There is evidence that = 10 b) There is evidence that < 10 c) We have insufficient evidence to conclude anything

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Statistical Conclusions

In a hypothesis test of H0: = 10 vs Ha: < 10

the p-value is 0.21. With = 0.01, we conclude:

a) Reject H0 b) Do not reject H0 c) Reject Ha d) Do not reject Ha

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Statistical Conclusions

In a hypothesis test of H0: = 10 vs Ha: < 10

the p-value is 0.21. With = 0.01, we conclude:

a) There is evidence that = 10 b) There is evidence that < 10 c) We have insufficient evidence to conclude anything

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Elephant Example

H0 : X is an elephant Ha : X is not an elephant Would you conclude, if you get the following data?

? X walks on two legs

? X has four legs

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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!

"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

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Red Wine and Weight Loss

? 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

BioMed Central (2010, June 22). "Lemurs lose weight with `life-extending' supplement resveratrol. Science Daily.

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Red Wine and Weight Loss

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) Yes b) No

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Red Wine and Weight Loss

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

Using = 0.05, what can we conclude?

a) Mean resting metabolic rate is higher after resveratrol supplementation in lemurs b) Mean resting metabolic rate is not higher after resveratrol supplementation in lemurs c) Nothing

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Red Wine and Weight Loss

In the test to see if the mean body mass is lower after treatment, the p-value is 0.007. Using = 0.05, is this difference statistically significant? (should we reject H0: no difference?)

a) Yes b) No

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Red Wine and Weight Loss

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

Using = 0.05, what can we conclude?

a) Mean body mass is higher after resveratrol supplementation in lemurs b) Mean body mass is not higher after resveratrol supplementation in lemurs c) Nothing

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Red Wine and Weight Loss

In the test to see if locomotor activity changes after treatment, the p-value is 0.980. Using = 0.05, is this difference statistically significant? (should we reject H0: no difference?)

a) Yes b) No

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Red Wine and Weight Loss

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

Using = 0.05, what can we conclude?

a) Locomotor activity changes after resveratrol supplementation in lemurs b) Locomotor activity does not change after resveratrol supplementation in lemurs c) Nothing

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Red Wine and Weight Loss

In the test to see if mean food intake changes after treatment, the p-value is 0.035. Using = 0.10, is this difference statistically significant? (should we reject H0: no difference?)

a) Yes b) No

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Red Wine and Weight Loss

In the test to see if mean food intake changes after treatment, the p-value is 0.035. Using = 0.01, is this difference statistically significant? (should we reject H0: no difference?)

a) Yes b) No

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Statistical Conclusions

Formal decision of hypothesis test, based on = 0.05 :

statistically significant not statistically significant Informal strength of evidence against H0:

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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.

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Multiple Sclerosis and Sunlight

For each situation below, write down

Null and alternative hypotheses Informal description of the strength of evidence against H0 Formal decision about H0, using = 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 pvalue is 0.47.

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Multiple Sclerosis and Sunlight

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

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Multiple Sclerosis and Sunlight

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

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Conclusions

p-value <

p-value

H0

Generic conclusion:

Generic conclusion:

Reject H0

Do not reject H0

Ha

Conclusion in context: We have (strong?) evidence

that [fill in alternative hypothesis]

Conclusion in context: We do not have enough evidence to conclude that [fill in alternative hypothesis]

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To Do

Read Section 4.3 (will cover errors next class) Do HW 4.3 (due Friday, 3/20) Enjoy spring break!!!

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