274 CH A P T E R 11: Null Hypothesis Significance Testing More fundamentally, the argument, that if any intention leads to nearly the same critical values then it's okay to use intentions, still fully admits that exper- imenter intentions influence the interpretation of data. It's like arguing that we shouldn't worry about the butchers putting their fingers on the scale, because no matter which butcher does it, the cheating is about the same. Admitting that experimenter intention influences the interpretation of data contradicts a basic premise of the data collection, that experimenter intentions have no influence on the data. 11.1.4 Bayesian Analysis The Bayesian interpretation of data does not depend on the covert intentions of the data collector. In general, for data that are independent across trials, the probability of the conjoint set of data is simply the product of the prob- abilities of the individual outcomes. Thus, for z = N y i heads in N flips, i=1 the likelihood is N y i (1 - ) 1-y i = z (1 - ) N-z , regardless of the exper- i=1 imenter's private reasons for collecting those data. The likelihood function captures everything we assume to influence the data. In the case of the coin, we assume that the bias of the coin is the only influence on its outcome, and that the flips are independent. The Bernoulli likelihood function completely captures those assumptions. In summary, the NHST analysis and conclusion depend on the covert inten- tions of the experimenter, because those intentions define the space of all possible (unobserved) data. This dependence of the analysis on the exper- imenter's intentions conflicts with the opposite assumption that the experi- menter's intentions have no effect on the observed data. The Bayesian analysis operates only with the actual data obtained and does not depend on the space of possible unobserved data. 11.2 PRIOR KNOWLEDGE ABOUT THE COIN Suppose that we are not flipping a coin, but we are flipping a flat-headed nail. In a social science setting, this is like asking a survey ques- tion about left- or right-handedness of the respondent, which we know is far from 50/50, as opposed to asking a survey question about male or female sex of the respondent, which we know is close to 50/50. When we flip the nail, it can land with its point touching the ground (which I'll call tails) or it can land balanced on its head with its point sticking up (which I'll call heads). We believe, just by looking at the nail and our previous experience with nails, that it will not come up heads and tails equally often. Indeed, with its nar- row head, the nail will very probably come to rest with its point touching the ground (i.e., "tails"). In other words, we have a strong prior belief that the nail