NONPARAMETRIC STATISTICS EDRM 712
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This 27 page Class Notes was uploaded by Edwin Considine on Monday October 26, 2015. The Class Notes belongs to EDRM 712 at University of South Carolina - Columbia taught by M. Seaman in Fall. Since its upload, it has received 24 views. For similar materials see /class/229492/edrm-712-university-of-south-carolina-columbia in OTHER at University of South Carolina - Columbia.
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Date Created: 10/26/15
Introduction to Inference Probability Risk and Approximating Truth Descriptive Statistics I Used to describe the characteristics of empirical sample distributions I Provides research outcomes for a sample I Reveals relationships among variables in the sample Size of a correlation Size of a difference Specific variable effects Inferential Statistics Uses of Inferential Statistics Used to infer if observed effects are real Used to infer the value of parameters I Forms of Inferential Statistics Point estimate Hypothesis test outcome Confidence interval Types of Inferences Population Inference lnfer from the sample statistic to a population parameter The sample statistic is compared to all possible chance statistics Causal Inference lnfer nonchance effects from one randomization The sample outcome is compared to all possible chance outcomes Both types of inference rely on the observation of a subset taken from a set of all possibilities Causal inference uses one randomization Population inference uses one sample Point Estimates A point estimate is a descriptive statistic that is used to estimate a population parameter S ESTIMATES 0 X ESTIMATES U 19 ESTIMATES p I ESTIMATES IO Hypothesis Testing State an alternative hypothesis I State a null hypothesis Test the null hypothesis Estimate the size of the effect Determine the conditional probability a Make a decision I Make inferences based on test results The Null Hypothesis Postulate a null hypothesis I Determine if the sample statistic is improbable given the truth of the null hypothesis If improbable reject the null hypothesis If plausible retain the null hypothesis Hoz zh Hypothesis Test A hypothesis test is a way to determine how likely it is that the observed data would occur if speculation about a parameter is true OBSERVED VALUE OF STATISTIC HYPOTHESIZED VALUE OF The data were unlikely to be observed If PARAMETER speculation about the parameter is true Falsi oation I A theory must be stated so that it can be falsified by a finite set of observations I A scientific theory can only be falsified never proved correct I If a hypothesis does not receive support the theory may be incorrect in its present form I If a theory is repeatedly not supported it should be thrown out or revised I If a hypothesis is supported it does not prove the theory correct Characteristics of Hypotheses Hypotheses can be made about any parameter of interest I Hypotheses can be one or twosided Onesided hypotheses are used to confirm a theoretical expectation Twosided hypotheses are used to explore potential values of a parameter I A statistical hypothesis should parallel a research hypothesis Type 1 Errors Incorrectly rejecting the null hypothesis is called a Type I error I The consequence of a Type I error is a false conclusion about an effect or parameter I The researcher can set the maximum probability of a Type I error symbolized as 0 Type II Errors Incorrectly retaining a null hypothesis is called a Type II error I The consequence of a Type II error is a failure to reach a conclusion about an effect or parameter I The researcher can set the maximum probability of a Type II error symbolized as B I The power of a hypothesis test is the probability of rejecting a false null hypothesis 1 B Power 05 H 04 03 02 01 0 I 03 02 01 REJECTION REGION Statistic The power of a test is the actual probability of rejecting H0 The Type I error rate of a test is the probability of rejecting H0 when H0 is true Possible Statistical Outcomes Actual State of H0 H0 is True H0 is False Reject HO Type Error Correct Decision Retain HO Correct Type II Error Decision Researcher Controls I The maximum Type I error rate can be set by the researcher I Power can be set by the researcher I Sample size can be set by the researcher I The limitation is that only two of these three controls can be set at any one time PValues A Pvalue is a measure of inconsistency with the null hypothesis P equals the probability of obtaining a value of a statistic that is at least as inconsistent with the null hypothesis as the observed value of the statistic Decisions about HO are based on the Pvalue and the maximum Type I error rate Reject H0 when P s or Retain H0 when P gt or Con dence Intervals A confidence interval bounds a parameter within a range with a specified level of con dence I The confidence interval is the set of all retained hypotheses if we test all possible hypotheses I The level of confidence 10 is inversely related to hypothesis testing risk 0 Con dence Intervals A confidence interval consists of all values for the parameter that are not rejected given the observed data when we test all hypotheses I I We would expect that I I four out of five 80 confidence intervals I I will capture the parameter Cautions About Inference Statistical inference is only as good as the design used to collect the data Statistical significance does not imply practical significance I Failure to reject the null hypothesis does not prove that the null hypothesis is true I Conducting multiple hypotheses tests can lead to a compounding of errors Inference in Educational Behavioral and Social Research Adapted from Draper D 1995 Inference and hierarchical modeling in the social sciences Journal of Educational and Behavioral Statistics 20 115147 I Calibration Inference I Sampling Inference I Specific Causal Inference I General Causal Inference Calibration Inference I Uncertain exchangeability UE samples samples of convenience I Observational design no treatment assignment I Low external and internal validity Example Chicago study people are quite good at identifying interesting patterns in dataso good in fact that they are capable of finding them even when they are not really there to be found we need some form of calibration inference to restrain our enthusiasm in the search fOr scientific relationships Specific Causal Inference I UE sample I Control of potential confounding factors PCFs treatment allocation or strong ignorability of the allocation mechanism I Low external and high internal validity Example Cognitive Strategies in Writing Project In permutation inference you condition on the observed data and consider all possible ways in which the observations might be rearranged a p value may then be calculated by asking how often differences as large as the one observed or larger would occur Sampling Inference I Representative sample from the target population I Observational design I High external and low internal validity Example Thailand study In hypothetical sampling inference the data values or sets of values before us are interpreted as a random sample from a hypothetical infinite population of such values as may have arisen in the same circumstances The distribution of this population will be capable of some kind of mathematical specification involving 39a certain number usually few of parameters R A Fisher 1925 General Causal Inference I Representative sample I Control of PCFs I High external and internal validity Example None found in HLM educational literature many examples where general causal inference is implied eg Miami Beach study In specific causal inference there is something causal going on for people in the experiment In general causal inference there is something causal going on for everybody in the target population Strong Ignorability of Treatment Assignment Exchangeability of Sampled and Unsampled Units in Target Population Difficult to Justify Justifiable Difficult to Justify JustIfIable Calibration SCPGCifif Inference aUSa Inference Sampnng generall Inference ausa Inference The Role of Subsets Inference depends on the observation of a subset taken from a set of all possibilities Specific causal inference depends on one randomization Sampling inference depends on one sample