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!
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
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 Hot weather Ice cream Violent crime
Alternative explanation Hot weather Ice cream Violent crime Correlation is not causation
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 Randomization is the random assignment of treatments to units in an experimental study Breaks the association between potential confounding variables and the explanatory variables
Experimental units Confounding variable
Experimental units Confounding variable Treatments
Experimental units Confounding variable Treatments Without randomization, the confounding variable differs among treatments
Experimental units Confounding variable Treatments
Experimental units Confounding variable Treatments 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 Experiment 1 Experiment 2 Experiment 3 Tank 1Tank 2 All separate tanks
Replication Experimental unit: the individual unit to which treatments are assigned Experiment 1 Experiment 2 Experiment 3 Tank 1Tank 2 All separate tanks 2 Experimental Units 2 Experimental Units 8 Experimental Units
Replication Experimental unit: the individual unit to which treatments are assigned Experiment 1 Experiment 2 Experiment 3 Tank 1Tank 2 All separate tanks 2 Experimental Units 2 Experimental Units 8 Experimental Units Pseudoreplication
Why is pseudoreplication bad? 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 1Tank 2
Why is replication good? Consider the formula for standard error of the mean: Larger n Smaller SE
Balance In a balanced experimental design, all treatments have equal sample size Better than BalancedUnbalanced
Balance In a balanced experimental design, all treatments have equal sample size This maximizes power Also makes tests more robust to violating assumptions
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
Very sunny Not So Sunny
Block 1 Block 4 Block 2 Block 3
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
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 n=2 30n=2 35n=2 40n=2 Temperature pH
Interaction Effects An interaction between two (or more) explanatory variables means that the effect of one variable depends upon the state of the other variable
Interpretations of 2-way ANOVA Terms Effect of pH and Temperature, No interaction
Interpretations of 2-way ANOVA Terms 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: (assuming that the data are a random sample from a normal distribution) precision = 4 2n2n
Power Analysis Example: confidence interval The sample size needed for a particular level of precision is: n = 32 Precision 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. 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 = =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 DD 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 DD = 16