Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
9/14/2016 Hypotheses • What are they? Framing and Testing Hypotheses – Explanations that account for our observations • Inferences – We draw conclusions about causes based on our data – ‘DATA’ do not make conclusions – Previous discussions about ‘P‐values’ Hypothesis • Manipulative Studies – Well‐designed manipulative experiments allow us to be confident in our inferences Hypotheses • Scientific hypothesis must be testable – Additional observations or results could cause us to modify or reject our hypothesis • Observational or Correlative Studies – Less confidence in inferences drawn from poorly designed experiments or studies in which variables of interest were not directly manipulated • Good hypotheses should also generate novel predictions Scientific Method • Deduction – Proceeds from the general case to the specific case • All hares in Forests for the World (FFW) are Snowshoe Hares • I sampled this Hare from FFW • This hare is a Snowshoe Hare – The conclusion must be logically true if the first two premises are correct ... Scientific Method • Induction – Proceeds from the specific case to the general case • All 25 hare observed were Snowshoe Hare • All of these observations were made in FFW • All hare in Forests for the World are Snowshoe Hare – Although the conclusion is likely to be true, it may be false – confidence with n 1 9/14/2016 Statistics • Statistics is an inductive process – Always trying to draw general conclusions based on a specific, limited set of observations Scientific Reasoning • Both induction and deduction are used in all models of scientific reasoning – Have different emphases • Begin with observation that needs explanation • Through induction develop a single hypothesis to explain it • Once formed, the hypothesis (through deduction) generates further predictions Scientific Reasoning Scientific Reasoning • Test predictions with additional observations – Hypothesis supported or rejected • Hypothesis prediction observation repeated ... Hypothetico‐Deductive Method Hypothetico‐Deductive Method • Begins with an initial observation that you are trying to observe • But uses multiple working hypotheses • Goal is to falsify the hypotheses • Falsification eliminates some alternatives 2 9/14/2016 Hypothetico‐Deductive Method • Advantages Hypothetico‐Deductive Method • Disadvantages – Consideration of multiple working hypotheses – Highlights key predictive differences between hypotheses – Emphasis on falsification leads to simple, testable hypotheses Bayesian Inference – Multiple working hypotheses may not always be available – Must have the ‘correct’ hypothesis as one of the alternatives Bayesian Inference • The null hypothesis is the starting point for hypothesis testing • Reject null hypothesis then move on to more complex hypotheses • But where to start? Bayesian Inference • Do we start from a completely ‘null’ hypothesis? • “There is no relationship between light and photosynthesis?” Bayesian Inference • Using a bit of knowledge of plant physiology we can formulate a more realistic initial hypothesis: No Relationship • The relationship should be non‐linear representing an upper limit on photosynthesis No Relationship 3 9/14/2016 Bayesian Inference • Data could then be used to test the support of this hypothesis Information‐Theoretic Approach Bayesian Inference • Bayesian approach – Use prior knowledge or information to generate and test hypotheses – Could also be based on published literature – Subjective? – Compared to no prior information? Approach? • Simultaneously assess support for multiple, plausible hypotheses … Testing Statistical Hypotheses • The statistical null hypothesis H0 is usually one of ‘no pattern’ – No difference between groups – No effect of one variable on another Statistical Significance and P‐values • “The control and treatment groups differed significantly from one another (P = 0.01)” • What does this mean? • The alternative hypothesis HA: the pattern exists – Difference in means measured between groups 4 9/14/2016 Statistical Significance and P‐values Statistical Null Hypothesis (H0) • All statistical tests produce a test statistic, which is the numerical result of the test, and a probability value associated with that test • “differences between groups are no greater than we would expect due to random variation” • Statistical Null Hypothesis: the effect does not operate Statistical Alternative Hypothesis (HA) Statistical Alternative Hypothesis (HA) • With H0 stated, can propose one or more alternative hypotheses • Strength of the tests and inference depend on how well the experiment is designed • In most tests, HA is not explicitly stated because there are many alternate explanations – Confounding factors? The P‐value The P‐value • In many statistical analyses, we ask whether H0 can be rejected • The P‐value measures the probability that the observed or more extreme differences would be found if the null hypothesis is true • Specific to the test statistic (and therefore the test used) • Therefore, small P‐values indicate that the result is improbable given the null hypothesis • A large P‐value indicates that the observed result did occur given H0 • Thus the importance of meeting the assumptions of our test(s) 5 9/14/2016 The P‐value • For comparing 2 groups, P‐values determined by – The number of observations (n) – The difference between the means of the samples What is a significant P‐value? • The operational critical value is P ≤ 0.05 • “Significant” • Should we be rigid in this interpretation? – The among of variation among individuals (s2) What is a significant P‐value? • Sets a standard or 1 in 20 (high) standard for testing • This set point (α) determines the Type I error of a test • Still not a safeguard against a poorly designed experiment 6