Hypothesis Testing: Conclusions STAT 250 Dr. Kari Lock Morgan Hypothesis Testing: Conclusions SECTION 4.3 Significance level (4.3) Statistical conclusions (4.3) Statistical significance (4.3)

Resveratrol https://www.youtube.com/watch?v=aRpcdbhAf64 A powerful antioxidant found in grapes and red wine with many potential benefits

Does red wine (resveratrol) promote weight loss? Question #1 of the Day Does red wine (resveratrol) promote weight loss? If so, why?

Red Wine and Weight Loss Resveratrol has been shown to promote weight loss in rodents. Today we’ll look at a study on Grey Mouse Lemurs (a primate). A sample of lemurs had various measurements taken before and after receiving resveratrol supplementation for 4 weeks: body mass resting metabolic rate locomotor activity food intake BioMed Central (2010, June 22). “Lemurs lose weight with ‘life-extending’ supplement resveratrol. Science Daily.

Study Design Can we use this study to make conclusions about causality? Yes No

Resveratrol and Weight Loss Which of the following tests gives the strongest evidence against the null hypothesis of no difference? Body mass (p-value = 0.007) Resting metabolic rate (p-value = 0.013) Locomotor activity (p-value = 0.49) Food intake (p-value = 0.035)

Formal Decisions A formal hypothesis test has only two possible conclusions: The p-value is small: reject the null hypothesis in favor of the alternative The p-value is not small: do not reject the null hypothesis

Reject the Null? H0: ⍴ = 0 Ha: ⍴ > 0 r = 0.6 Based on the randomization distribution, do you think we should reject the null? Yes No

Reject the Null? H0: ⍴ = 0 Ha: ⍴ > 0 r = 0.1 Based on the randomization distribution, do you think we should reject the null? Yes No

Reject the Null? H0: ⍴ = 0 Ha: ⍴ > 0 r = 0.336 Based on the randomization distribution, do you think we should reject the null? Yes No

Formal Decisions How small? A formal hypothesis test has only two possible conclusions: The p-value is small: reject the null hypothesis in favor of the alternative The p-value is not small: do not reject the null hypothesis How small?

Significance Level p-value > α => Do not Reject H0 The significance level, α, is the threshold below which the p-value is deemed small enough to reject the null hypothesis p-value < α => Reject H0 p-value > α => Do not Reject H0 Often α = 0.05, unless otherwise specified (Why 0.05?)

p-value < α p-value ≥ α Results would be rare, if the null were true Results would not be rare, if the null were true Reject H0 Do not reject H0 We have evidence that the alternative is true! Our test is inconclusive

Resveratrol and Weight Loss It is hypothesized that resveratrol decreases body mass. For the hypothesis test on body mass, what are the relevant hypotheses? B = before resveratrol supplementation A = after resveratrol supplementation H0: pB = pA; Ha: pB < pA H0: pB = pA; Ha: pB > pA H0: μB = μA; Ha: μB < μA H0: μB = μA; Ha: μB > μA

Resveratrol and Weight Loss In the test to see if the mean body mass is lower after treatment, the p-value is 0.007. Using α = 0.05, we should… Reject H0 Not reject H0 Reject Ha Not reject Ha

Conclusions p-value < α p-value ≥ α H0 Ha Generic conclusion: Reject H0 Generic conclusion: Do not reject H0 Ha Conclusion in context: We have convincing evidence that [fill in alternative hypothesis] Conclusion in context: We do not have convincing evidence that [fill in alternative hypothesis]

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 α = 0.05, what can we conclude? We have convincing evidence that mean body mass is lower after resveratrol supplementation in lemurs We do not have convincing evidence that mean body mass is lower after resveratrol supplementation in lemurs We have convincing evidence that mean body mass is the same before and after resveratrol supplementation in lemurs

Resveratrol and Weight Loss It is hypothesized that resveratrol may decrease body mass by increasing locomotor activity. For the hypothesis test on resting metabolic rate, what are the relevant hypotheses? B = before resveratrol supplementation A = after resveratrol supplementation H0: pB = pA; Ha: pB < pA H0: pB = pA; Ha: pB > pA H0: μB = μA; Ha: μB < μA H0: μB = μA; Ha: μB > μA

Resveratrol and Weight Loss In the test to see if the mean locomotor activity is higher after treatment, the p-value is 0.49. Using α = 0.05, we should… Reject H0 Not reject H0 Reject Ha Not reject Ha

Red Wine and Weight Loss In the test to see if locomotor activity is lower after treatment, the p-value is 0.49. Using α = 0.05, what can we conclude? We have convincing evidence that mean locomotor activity is higher after resveratrol supplementation in lemurs We do not have convincing evidence that mean locomotor activity is higher after resveratrol supplementation in lemurs We have convincing evidence that mean locomotor activity is the same before and after resveratrol supplementation in lemurs

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

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

Statistical Significance When the p-value is less than α, the results are statistically significant. If our sample is statistically significant, we have convincing evidence against H0, in favor of Ha

p-value < α p-value ≥ α Results would be rare, if the null were true Results would not be rare, if the null were true Reject H0 Do not reject H0 We have evidence that the alternative is true! Our test is inconclusive Results are statistically significant Results are not statistically significant

Resveratrol and Weight Loss It is hypothesized that resveratrol may decrease body mass by increasing resting metabolic rate. For the hypothesis test on resting metabolic rate, what are the relevant hypotheses? B = before resveratrol supplementation A = after resveratrol supplementation H0: pB = pA; Ha: pB < pA H0: pB = pA; Ha: pB > pA H0: μB = μA; Ha: μB < μA H0: μB = μA; Ha: μB > μA

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 α = 0.05, is this statistically significant? Yes No

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 α = 0.05, what can we conclude? We have convincing evidence that mean resting metabolic rate is higher after resveratrol supplementation in lemurs We do not have convincing evidence that mean resting metabolic rate is higher after resveratrol supplementation in lemurs We have convincing evidence that mean resting metabolic rate is the same before and after resveratrol supplementation in lemurs

Resveratrol and Weight Loss It is hypothesized that resveratrol may decrease body mass by decreasing food intake. For the hypothesis test on food intake, what are the relevant hypotheses? B = before resveratrol supplementation A = after resveratrol supplementation H0: pB = pA; Ha: pB < pA H0: pB = pA; Ha: pB > pA H0: μB = μA; Ha: μB < μA H0: μB = μA; Ha: μB > μA

Red Wine and Weight Loss In the test to see if mean food intake is lower after treatment, the p-value is 0.035. Using α = 0.10, is this statistically significant? Yes No

Red Wine and Weight Loss In the test to see if mean food intake is lower after treatment, the p-value is 0.035. Using α = 0.01, is this difference statistically significant? Yes No

Statistical Conclusions Formal decision of hypothesis test, based on  = 0.05 : Statistically significant Not statistically significant Informal strength of evidence against H0:

Resveratrol and Weight Loss Body mass (p-value = 0.007) evidence that resveratrol decreases body mass Resting metabolic rate (p-value = 0.013) evidence that resveratrol decreases metabolic rate Food intake (p-value = 0.035) evidence that resveratrol decreases food intake Locomotor activity (p-value = 0.49) evidence that resveratrol increases activity

Conclusions p-value < α p-value ≥ α H0 Ha Generic conclusion: Reject H0 Generic conclusion: Do not reject H0 Ha Conclusion in context: We have (strong?) evidence that [fill in alternative hypothesis] Conclusion in context: We do not have convincing evidence that [fill in alternative hypothesis]

What aspect of sunlight might help protect against multiple sclerosis? Question #2 of the Day What aspect of sunlight might help protect against multiple sclerosis?

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.

Study Design Can this study be used to make conclusions about causality, at least in mice? Yes No

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 α = 0.05 Conclusion in the context of the question In testing whether Vitamin D provides protection against MS (Vitamin D vs control group), the p- value is 0.47. In testing whether UV light provides protection against MS (UV light vs control group), the p-value is 0.002.

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

Multiple Sclerosis and Sunlight In testing whether UV light provides protection against MS (UV light vs control group), the p- value is 0.002. https://www.youtube.com/watch?v=KbJqyhTkYSY

To Do Do HW 4.1, 4.2 (due Friday, 3/3) Do HW 4.3 (due Wednesday, 3/15) Enjoy spring break!!!