Statistics: Unlocking the Power of Data Lock 5 Hypothesis Testing: Conclusions STAT 250 Dr. Kari Lock Morgan SECTION 4.3 Significance level (4.3) Statistical conclusions (4.3) Statistical significance (4.3)

Statistics: Unlocking the Power of Data Lock 5 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?

Statistics: Unlocking the Power of Data Lock 5 Significance Level The significance level, α, is the threshold below which the p-value is deemed small enough to reject the null hypothesis p-value Reject H 0 p-value > α=> Do not Reject H 0 Often α = 0.05, unless otherwise specified

Statistics: Unlocking the Power of Data Lock 5 p-value < α Results would be rare, if the null were true Reject H 0 We have evidence that the alternative is true! p-value ≥ α Results would not be rare, if the null were true Do not reject H 0 Our test is inconclusive

Statistics: Unlocking the Power of Data Lock 5 Statistical Conclusions In a hypothesis test of H 0 : μ = 10 vs H a : μ < 10 the p-value is With α = 0.05, we conclude: a) Reject H 0 b) Do not reject H 0 c) Reject H a d) Do not reject H a

Statistics: Unlocking the Power of Data Lock 5 Statistical Conclusions In a hypothesis test of H 0 : μ = 10 vs H a : μ < 10 the p-value is With α = 0.01, we conclude: a) There is evidence that μ = 10 b) There is evidence that μ < 10 c) We have insufficient evidence to conclude anything

Statistics: Unlocking the Power of Data Lock 5 Statistical Conclusions In a hypothesis test of H 0 : μ = 10 vs H a : μ < 10 the p-value is With α = 0.01, we conclude: a) Reject H 0 b) Do not reject H 0 c) Reject H a d) Do not reject H a

Statistics: Unlocking the Power of Data Lock 5 Statistical Conclusions In a hypothesis test of H 0 : μ = 10 vs H a : μ < 10 the p-value is With α = 0.01, we conclude: a) There is evidence that μ = 10 b) There is evidence that μ < 10 c) We have insufficient evidence to conclude anything

Statistics: Unlocking the Power of Data Lock 5 H 0 : X is an elephant H a : 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

Statistics: Unlocking the Power of Data Lock 5 “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 H 0 “Do not reject H 0 ” is not the same as “accept H 0 ”! Lack of evidence against H 0 is NOT the same as evidence for H 0 !

Statistics: Unlocking the Power of Data Lock 5 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 H 0, in favor of H a

Statistics: Unlocking the Power of Data Lock 5 p-value < α Results would be rare, if the null were true Reject H 0 We have evidence that the alternative is true! Results are statistically significant p-value ≥ α Results would not be rare, if the null were true Do not reject H 0 Our test is inconclusive Results are not statistically significant

Statistics: Unlocking the Power of Data Lock 5 Question #1 of the Day Does red wine (resveratrol) promote weight loss? If so, why?

Statistics: Unlocking the Power of Data Lock 5 Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents. 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 Red Wine and Weight Loss BioMed Central (2010, June 22). “Lemurs lose weight with ‘life-extending’ supplement resveratrol. Science Daily.

Statistics: Unlocking the Power of Data Lock 5 Study Design This is a a) Experiment b) Observational study

Statistics: Unlocking the Power of Data Lock 5 Study Design Can we use this study to make conclusions about causality? a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if the mean resting metabolic rate is higher after treatment, the p-value is Using α = 0.05, is this difference statistically significant? (should we reject H 0 : no difference?) a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if the mean resting metabolic rate is higher after treatment, the p-value is Using α = 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

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if the mean body mass is lower after treatment, the p-value is Using α = 0.05, is this difference statistically significant? (should we reject H 0 : no difference?) a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if the mean body mass is lower after treatment, the p-value is Using α = 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

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if locomotor activity changes after treatment, the p-value is Using α = 0.05, is this difference statistically significant? (should we reject H 0 : no difference?) a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if locomotor activity changes after treatment, the p-value is Using α = 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

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if mean food intake changes after treatment, the p-value is Using α = 0.10, is this difference statistically significant? (should we reject H 0 : no difference?) a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Red Wine and Weight Loss In the test to see if mean food intake changes after treatment, the p-value is Using α = 0.01, is this difference statistically significant? (should we reject H 0 : no difference?) a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Informal strength of evidence against H 0 : Formal decision of hypothesis test, based on  = 0.05 : Statistical Conclusions

Statistics: Unlocking the Power of Data Lock 5 Question #2 of the Day What aspect of sunlight might help protect against multiple sclerosis?

Statistics: Unlocking the Power of Data Lock 5 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.

Statistics: Unlocking the Power of Data Lock 5 Study Design Can this study be used to make conclusions about causality, at least in mice? a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Multiple Sclerosis and Sunlight For each situation below, write down  Null and alternative hypotheses  Informal description of the strength of evidence against H 0  Formal decision about H 0, 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 In testing whether Vitamin D provides protection against MS (Vitamin D vs control group), the p- value is 0.47.

Statistics: Unlocking the Power of Data Lock 5 Multiple Sclerosis and Sunlight In testing whether UV light provides protection against MS (UV light vs control group), the p-value is

Statistics: Unlocking the Power of Data Lock 5 Multiple Sclerosis and Sunlight In testing whether Vitamin D provides protection against MS (Vitamin D vs control group), the p-value is 0.47.

Statistics: Unlocking the Power of Data Lock 5 Conclusions p-value < α Generic conclusion: Reject H 0 Conclusion in context: We have (strong?) evidence that [fill in alternative hypothesis] p-value ≥ α Generic conclusion: Do not reject H 0 Conclusion in context: We do not have enough evidence to conclude that [fill in alternative hypothesis] H0H0 HaHa

Statistics: Unlocking the Power of Data Lock 5 To Do Read Section 4.3 Do HW 4.2, 4.3 (due Friday, 10/23)