Download Stats in Data Analysis: Pseudoreplication, Multiple Testing, Meta-Analysis, & Bayesian and more Study notes Environmental Science in PDF only on Docsity! 1 Module 4: Drawing Conclusions From Data 4.3 Pseudoreplication, Multiple Testing, Meta-Analysis, and Bayesian Methods 4/12/2002 Module 2 Pseudoreplication Pseudoreplication is an issue when data points are correlated in space or time or when the data collected are not representative of the entire population In these cases, you may have less information than you think You should adjust your degrees of freedom downward 2 4/12/2002 Module 3 Pseudoreplication Example: • In a study I was involved with, a graduate student was to collect data on lead contamination in homes in the Bunker Hill Superfund area • The plan was to drive to a neighborhood and go door-to-door asking if they would participate. If yes, then information and samples were collected and they went next door and continued. • At the end of the week, the data collection would end. 4/12/2002 Module 4 Pseudoreplication Example: • But, homes in a neighborhood tend to be alike in age, value, condition • People who live there also tend to be alike • Also, some neighborhoods would be expected to be more contaminated than others • So, data points collected in this way are correlated. • Also, some neighborhoods would be well covered and others may be skipped 5 4/12/2002 Module 9 Bayesian Methods There are two types of statisticians in the world, Frequentists and Bayesians Frequentists view probability as completely objective. They look at all statistical methods from a standpoint of what would happen in the long run if a sample were taken over and over Statistics is often taught by them beginning with coin flips, pulling balls from an urn, or using a deck of cards 4/12/2002 Module 10 Bayesian Methods Bayesians, on the other hand, view probability as subjective Probability can be expressed as a degree of belief that an event will occur That belief could be based purely on the data in hand or could involve past experience, other data, expert judgement, theory, etc. 6 4/12/2002 Module 11 Bayesian Methods Bayesians express their degree of belief about a parameter as a prior probability distribution Then they incorporate new data into the analysis using a likelihood function, the likelihood that those data occurred given a particular parameter value The result is another probability distribution called a posterior distribution 4/12/2002 Module 12 Bayesian Methods The Reverend Thomas Bayes created this approach, called Bayes Theorem Θ = the parameter being estimated For simplicity assume that Θ can take on a small number of values Θ1 Θ 2 ...Θ n You, as the investigator, may have some guess as to the probabilities of these being the best estimate of Θ If you don’t, then they are equally likely 7 4/12/2002 Module 13 Bayesian Methods These probabilities are your priors: P(Θ1), P(Θ 2), …, P(Θ n) (Note: these probabilities must sum to 1) Then you collect some new data You can determine the probability that those data occurred given that Θ = Θi That’s called a likelihood and denoted by P(data| Θi) 4/12/2002 Module 14 Bayesian Methods Then your prior beliefs and the data are combined to give a new, posterior, set of probabilities using Bayes Theorem P data P data P data P i i k k k n( | ) ( | ) ( | ) ( ) Θ Θ Θ Θ = = ∑ 1
