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STAB22 Statistics I Lecture 13 1 Principles of Experimental Design 1. 2. Control: control sources of variation, besides factors, by making conditions as similar as possible for all treatment groups Randomize: helps equalize effects of unknown/uncontrollable sources of variation Note: Randomization does not eliminate the effects of these sources, but tries to spread them out across the treatment levels so that we can see past them 2 Principles of Experimental Design 3. 4. Replicate: get several measurements of response for each treatment Blocking: for variables we can identify but cannot control and which affect response, divide subjects into groups of same variable values (a.k.a. blocks) and randomize within each block Removes much of the variability due to the difference among the blocks. 3 Experimental Designs Completely Randomized Design (CRD): All experimental units are allocated at random among all treatments Randomized Block Design (RBD): Random assignment of units to treatments is carried out separately within each block. 4 Blocking Assume 36 adult & 6 child participants in previous insomnia experiment If you just randomize, you could get all children in one treatment How would you block age? 5 Experiments Placebo: “fake” treatment designed to look like a real one, used when just knowledge of receiving any treatment can affect response E.g. Used to test effectiveness of pain medication For comparing results, often use current standard treatment as baseline Subjects getting placebo/standard treatment called control group 6 Blinding Knowledge of assigned treatment can often influence the assessment of the response Two classes of individuals can affect experiment Those who influence results (e.g. subjects, nurses) Those who evaluate results (e.g. physicians, judges) Blinding avoids bias from knowing treatment Single-blind: every individual in either one of the two classes doesn’t know treatments Double-blind: every individual in both of the two classes doesn’t know treatments 7 Experiments Golden standard for experiments: randomized double-blind ● comparative ● placebo-controlled. Even so, could have confounding problems Confounding variable: variable associated with both factor & response Cannot tell whether effect on response is caused by factor or confounding variable E.g. Subject’s weight might be is associated with both insomnia & diet (veg/non-veg) 8 Probability Many things in life are random (uncertain): roll of a die, tomorrow’s weather, etc Don’t know for certain what will happen, but Could know how likely is something to happen Probability describes how likely is “something” (event) to happen. Probabilities take values between 0 and 1: A value of 1 means event is sure to happen A value of 0 means event is not going to happen 9 Probability Terminology Random phenomenon: process whose result is not known beforehand Trial: particular realization of random phenomenon E.g. Particular roll of a standard die Outcome: basic result of a trial that cannot be broken down to simpler results E.g. Rolling a standard (6-sided) die E.g. “Rolling a 1” is an outcome Sample Space: collection of all possible outcomes Sample space usually denoted by S E.g. for rolling standard die, S={1, 2, 3, 4, 5, 6} 10 Events Event: collection of one or more outcomes Usually denoted by capital letters (A, B, etc) E.g. A = {2, 4, 6} = “Rolling an even number” Events can contain just single outcome, e.g. A = {1} Probabilities are assigned to events Example: Student’s course grade is random phenomenon w/ sample space S={A, B, C, D, F} Are the following events of outcomes? “Getting a passing grade” “Getting a failing grade” 11 Empirical Probability In short term, unpredictable In long run, predictable Limiting rel. frequency value called empirical probability of event A, denoted by P(A) 0.8 0.6 0.4 Rel. Freq. {Heads} 0.2 Relative Frequency of # times A occurs event A: total # of trials 1.0 Observe sequence of coin tosses (trials) & count # of times of event A={Heads} 0.0 0 200 400 600 800 # of trials 1000 12 Law of Large Numbers (LLN) LLN guarantees relative frequency will eventually settle on specific value LLN holds if trials are independent (i.e. outcome of one trial does not influence that of another) LLN does not apply in the short-run E.g. Assume you flip fair coin, i.e. P(Heads)=1/2, and first 10 outcomes are “Tails”. Is next flip more likely to be “Heads”? NO LLN only applies in the (infinitely) long run 13 Other Ways to Assign Probabilities Theoretical Probability: Sometimes P(A) can be deduced from mathematical model For equally likely outcomes (e.g. rolling fair die) # of outcomes in A probability is: P A total # outcomes Personal / Subjective Probability: Assign P(A) based on personal beliefs Typically expert’s views (e.g. sports analyst’s chances of Maple Leafs winning Stanley Cup) Most general case (can always be used) 14 Example Consider rolling 2 fair dice: find (theoretical) probability of their sum being 4 15