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! format: lab exam 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 Experimental Design Experimental Design Experimental design 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
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? 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? Alternative explanation Alternative explanation Ice cream Ice cream Correlation is not causation Hot weather Violent crime Hot weather Violent crime
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 Goals of Experimental Design Avoid experimental artifacts Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding Reduce sampling error 1. Replication 2. Balance 3. Blocking Goals of Experimental Design Avoid experimental artifacts Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding Reduce sampling error 1. Replication 2. Balance 3. Blocking Experimental Artifacts Experimental artifacts: 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
Experimental Artifacts Example: diving birds Goals of Experimental Design Avoid experimental artifacts Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding Reduce sampling error 1. Replication 2. Balance 3. Blocking Control Group A control 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 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!
Randomization Experimental units Randomization is the random assignment of treatments to units in an experimental study Breaks the association between potential confounding variables and the explanatory variables Confounding variable Experimental units Treatments Experimental units Treatments Confounding variable Confounding variable Without randomization, the confounding variable differs among treatments
Experimental units Treatments Experimental units Treatments Confounding variable Confounding variable With randomization, the confounding variable does not differ among treatments Blinding Blinding is the 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 Blinding Example: testing heart 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
Goals of Experimental Design Avoid experimental artifacts Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding Reduce sampling error 1. Replication 2. Balance 3. Blocking Replication Experimental unit: the individual unit to which treatments are assigned Tank 1 Tank 2 All separate tanks Experiment 1 Experiment 2 Experiment 3 Replication Experimental unit: the individual unit to which treatments are assigned Replication Experimental unit: the individual unit to which treatments are assigned 2 Experimental Units Experiment 1 2 Experimental Units Experiment 1 2 Experimental Units Experiment 2 2 Experimental Units Pseudoreplication Experiment 2 Tank 1 Tank 2 Tank 1 Tank 2 8 Experimental Units All separate tanks Experiment 3 8 Experimental Units All separate tanks Experiment 3
Why is pseudoreplication bad? Tank 1 Tank 2 problem with confounding and replication! Imagine that something strange happened, by chance, to tank 2 but not to tank 1 Example: light burns out Experiment 2 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 Why is replication good? Consider the formula for standard error of the mean: Larger n SE Y = s n Smaller SE Balance In a balanced experimental design, all treatments have equal sample size Better than Balance In a balanced experimental design, all treatments have equal sample size This maximizes power Also makes tests more robust to violating assumptions Balanced Unbalanced
Blocking Blocking is the grouping of experimental units that have similar properties Within each block, treatments are randomly assigned to experimental treatments Randomized block design Randomized Block Design Example: cattle tanks in a field Randomized Block Design
Very sunny Block 1 Block 2 Block 3 Not So Sunny What good is blocking? 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 Block 4 Experiments with 2 Factors Factorial design investigates all treatment combinations of two or more variables Factorial design allows us to test for interactions between treatment variables
Factorial Design Interaction Effects Temperature 25 30 35 5.5 ph 6.5 7.5 An interaction between two (or more) explanatory variables means that the effect of one variable depends upon the state of the other variable 40 Interpretations of 2-way ANOVA Terms Interpretations of 2-way ANOVA Terms Effect of ph and Temperature, No interaction Effect of ph and Temperature, with interaction
Goals of Experimental Design Avoid experimental artifacts Eliminate bias 1. Use a simultaneous control group 2. Randomization 3. Blinding Reduce sampling error 1. Replication 2. Balance 3. Blocking What if you can 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 What if you can 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 Power Analysis Before carrying 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
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: precision = 4 " #2 #n Power Analysis Example: confidence interval The sample size needed for a particular level of precision is: n = 32 " Precision 2 (assuming that the data are a random sample from a normal distribution) Power Analysis Assume that 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 n = 32 " Precision 10 5 2 2 =128 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
Power Analysis Assume that 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