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![Power Analysis
• Example: power
• Remember, power = 1 -
= Pr[Type II error]
• Typical goal is power = 0.80
• For a two-sample t-test, the sample size
needed for a power of 80% to detect a
difference of D is:
n = 16
D
2](https://image.slidesharecdn.com/16exdesign-250706074654-3653c9a4/75/design-of-experiments-25-desigfnExDesign-ppt-52-2048.jpg)
![Power Analysis
• Example: power
• Remember, power = 1 -
= Pr[Type II error]
• Typical goal is power = 0.80
• For a two-sample t-test, the sample size
needed for a power of 80% to detect a
difference of D is:
n = 16
D
2](https://crownmelresort.com/image.slidesharecdn.com/16exdesign-250706074654-3653c9a4/75/design-of-experiments-25-desigfnExDesign-ppt-52-2048.jpg)
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Similar to design of experiments 25 desigfnExDesign.ppt
design of experiments 25 desigfnExDesign.ppt
- 1. lab exam • when:Nov 27 - Dec 1 • length = 1 hour – each lab section divided in two • register for the exam in your section so there is a computer reserved for you • If you write in the 1st hour, you can’t leave early! If you write in the second hour, you can’t arrive late!
- 2. lab exam • format: –open book! – similar to questions in lab manual – last section in the lab manual has review questions – show all your work: hypotheses, tests of assumptions, test statistics, p-values and conclusions
- 3.
- 4. Experimental Design • Experimentaldesign is the part of statistics that happens before you carry out an experiment • Proper planning can save many headaches • You should design your experiments with a particular statistical test in mind
- 5. Why do experiments? •Contrast: observational study vs. experiments • Example: – Observational studies show a positive association between ice cream sales and levels of violent crime – What does this mean?
- 6. Why do experiments? •Contrast: observational study vs. experiments • Example: – Observational studies show a positive association between ice cream sales and levels of violent crime – What does this mean?
- 7. Alternative explanation Hot weather Icecream Violent crime
- 8. Alternative explanation Hot weather Icecream Violent crime Correlation is not causation
- 9. Why do experiments? •Observational studies are prone to confounding variables: Variables that mask or distort the association between measured variables in a study – Example: hot weather • In an experiment, you can use random assignments of treatments to individuals to avoid confounding variables
- 10. Goals of ExperimentalDesign • Avoid experimental artifacts • Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding • Reduce sampling error 1. Replication 2. Balance 3. Blocking
- 11. Goals of ExperimentalDesign • Avoid experimental artifacts • Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding • Reduce sampling error 1. Replication 2. Balance 3. Blocking
- 12. Experimental Artifacts • Experimentalartifacts: a bias in a measurement produced by unintended consequences of experimental procedures • Conduct your experiments under as natural of conditions as possible to avoid artifacts
- 13. Experimental Artifacts • Example:diving birds
- 14. Goals of ExperimentalDesign • Avoid experimental artifacts • Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding • Reduce sampling error 1. Replication 2. Balance 3. Blocking
- 15. Control Group • Acontrol group is a group of subjects left untreated for the treatment of interest but otherwise experiencing the same conditions as the treated subjects • Example: one group of patients is given an inert placebo
- 16. The Placebo Effect •Patients treated with placebos, including sugar pills, often report improvement • Example: up to 40% of patients with chronic back pain report improvement when treated with a placebo • Even “sham surgeries” can have a positive effect • This is why you need a control group!
- 17. Randomization • Randomization isthe random assignment of treatments to units in an experimental study • Breaks the association between potential confounding variables and the explanatory variables
- 18.
- 19.
- 20. Experimental units Confounding variable Treatments Without randomization, theconfounding variable differs among treatments
- 21.
- 22. Experimental units Confounding variable Treatments With randomization, theconfounding variable does not differ among treatments
- 23. Blinding • Blinding isthe concealment of information from the participants and/or researchers about which subjects are receiving which treatments • Single blind: subjects are unaware of treatments • Double blind: subjects and researchers are unaware of treatments
- 24. Blinding • Example: testingheart medication • Two treatments: drug and placebo • Single blind: the patients don’t know which group they are in, but the doctors do • Double blind: neither the patients nor the doctors administering the drug know which group the patients are in
- 25. Goals of ExperimentalDesign • Avoid experimental artifacts • Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding • Reduce sampling error 1. Replication 2. Balance 3. Blocking
- 26. Replication • Experimental unit:the individual unit to which treatments are assigned Experiment 1 Experiment 2 Experiment 3 Tank 1 Tank 2 All separate tanks
- 27. Replication • Experimental unit:the individual unit to which treatments are assigned Experiment 1 Experiment 2 Experiment 3 Tank 1 Tank 2 All separate tanks 2 Experimental Units 2 Experimental Units 8 Experimental Units
- 28. Replication • Experimental unit:the individual unit to which treatments are assigned Experiment 1 Experiment 2 Experiment 3 Tank 1 Tank 2 All separate tanks 2 Experimental Units 2 Experimental Units 8 Experimental Units Pseudoreplication
- 29. Why is pseudoreplicationbad? • problem with confounding and replication! • Imagine that something strange happened, by chance, to tank 2 but not to tank 1 • Example: light burns out • All four lizards in tank 2 would be smaller • You might then think that the difference was due to the treatment, but it’s actually just random chance Experiment 2 Tank 1 Tank 2
- 30. Why is replicationgood? • Consider the formula for standard error of the mean: SEY s n Larger n Smaller SE
- 31. Balance • In abalanced experimental design, all treatments have equal sample size Better than Balanced Unbalanced
- 32. Balance • In abalanced experimental design, all treatments have equal sample size • This maximizes power • Also makes tests more robust to violating assumptions
- 33. Blocking • Blocking isthe grouping of experimental units that have similar properties • Within each block, treatments are randomly assigned to experimental treatments • Randomized block design
- 34.
- 35. Randomized Block Design •Example: cattle tanks in a field
- 37. Very sunny Not SoSunny
- 38. Block 1 Block 4 Block2 Block 3
- 39. What good isblocking? • Blocking allows you to remove extraneous variation from the data • Like replicating the whole experiment multiple times, once in each block • Paired design is an example of blocking
- 40. Experiments with 2Factors • Factorial design – investigates all treatment combinations of two or more variables • Factorial design allows us to test for interactions between treatment variables
- 41. Factorial Design 5.5 6.57.5 25 n=2 n=2 n=2 30 n=2 n=2 n=2 35 n=2 n=2 n=2 40 n=2 n=2 n=2 Temperature pH
- 42. Interaction Effects • Aninteraction between two (or more) explanatory variables means that the effect of one variable depends upon the state of the other variable
- 43. Interpretations of 2-wayANOVA Terms 0 10 20 30 40 50 60 70 25 30 35 40 Temperature Growth Rate pH 5.5 pH 6.5 pH 7.5 Effect of pH and Temperature, No interaction
- 44. 0 5 10 15 20 25 30 35 40 45 25 30 3540 Temperature Growth Rate pH 5.5 pH 6.5 pH 7.5 Interpretations of 2-way ANOVA Terms Effect of pH and Temperature, with interaction
- 45. Goals of ExperimentalDesign • Avoid experimental artifacts • Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding • Reduce sampling error 1. Replication 2. Balance 3. Blocking
- 46. What if youcan’t do experiments? • Sometimes you can’t do experiments • One strategy: – Matching – Every individual in the treatment group is matched to a control individual having the same or closely similar values for known confounding variables
- 47. What if youcan’t do experiments? • Example: Do species on islands change their body size compared to species in mainland habitats? • For each island species, identify a closely related species living on a nearby mainland area
- 48. Power Analysis • Beforecarrying out an experiment you must choose a sample size • Too small: no chance to detect treatment effect • Too large: too expensive • We can use power analysis to choose our sample size
- 49. Power Analysis • Example:confidence interval • For a two-sample t-test, the approximate width of a 95% confidence interval for the difference in means is: (assuming that the data are a random sample from a normal distribution) precision = 4 2 n
- 50. Power Analysis • Example:confidence interval • The sample size needed for a particular level of precision is: n = 32 Precision 2
- 51. Power Analysis • Assumethat the standard deviation of exam scores for a class is 10. I want to compare scores between two lab sections. A. How many exams do I need to mark to obtain a confidence limit for the difference in mean exam scores between two classes that has a width (precision) of 5? n = 32 Precision 2 n = 32 10 5 2 =128
- 52. Power Analysis • Example:power • Remember, power = 1 - = Pr[Type II error] • Typical goal is power = 0.80 • For a two-sample t-test, the sample size needed for a power of 80% to detect a difference of D is: n = 16 D 2
- 53. Power Analysis • Assumethat the standard deviation of exam scores for a class is 10. I want to compare scores between two lab sections. B. How many exams do I need to mark to have sufficient power (80%) to detect a mean difference of 10 points between the sections? n = 16 D 2 n = 16 10 10 2 = 16