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But is is clear that the answers to the questions are inextricable interrelated. Other issues implied by the question include the statistical model that is used, the robustness of the model, missing data, and an assessment of the many sources of variability. The ordering reflects the process of science.
Multiplication rule for independent events ( ) ( ) ( ) General multiplication rules ( ) ( ) ( , ) A P not A P A P A and B P A P B P A and B P A P B given A = • = • ( ) ( ) ( , ) Addition rule for mutually exclusive events ( ) ( ) + ( ) General addition rule ( ) ( ) + ( ) ( ) P A and B P A P A given B PAorB PA PB P A or B P A P B P A and B ...
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Calculate a test statistic and a p-value. Make a decision, and interpret it in context of the research question. Recap: Hypothesis testing for a population mean. Set the hypotheses. H0 : = null value HA : < or > or 6= null value. Check conditions. Ind: random sampling/assignment. < 10% (when sampling without replacement)
- Three Modes of Statistical Inference
- Statistics for Social Scientists
- | {z } | {z } | {z }
- x) for all x
- f (X)
- k pk(1 p)n
- Simple Model-Based Inference
- Estimated Probability of Obama Victory in 2008
- Key Points
- The Key Assumptions
- Yi(0) j Ti = 1)
- Design Considerations
- Tradeoff between internal and external validity
- Identification vs. Estimation
- Joint distribution of yz = (Yi(0) = y; Z = z) is identified:
- Key Points
- Type II error
- Power Analysis
- Statistical Control of False Discovery
- Key Points
Descriptive Inference: summarizing and exploring data Inferring “ideal points” from rollcall votes Inferring “topics” from texts and speeches Inferring “social networks” from surveys Predictive Inference: forecasting out-of-sample data points Inferring future state failures from past failures Inferring population average turnout from a sample of vo...
Quantitative social science research: Find a substantive question Construct theory and hypothesis Design an empirical study and collect data Use statistics to analyze data and test hypothesis Report the results No study in the social sciences is perfect Use best available methods and data, but be aware of limitations Many wrong answers but no singl...
subjective objective subjective Statistical methods are no substitute for good research design
x) being continuous at every x If Xn ! d X, then for any continuous function f ( ), d f (Xn) !
Implication: Justifies asymptotic (normal) approximation
Sir Francis Galton’s Quincunx, Boston Museum of Science, or just check out YouTube
Setup: njk respondents of poll j from state k Model for # of Obama supporters in poll j and state k: indep:
Estimate pk for each state Simulate M elections using pk ^ and its standard error: for state k, sample Obama’s voteshare from N(^ \ pk; V(^ pk)) collect all electoral votes from winning states Plot M draws of total electoral votes Distribution of Obama's Predicted Electoral Votes Electoral Votes
Random sampling enables statistical inference Design-based vs. Model-based inference Design-based: random sampling as basis for inference Model-based: probability model as basis for inference Sampling weights: inverse probability weighting Challenges of survey research: cluster sampling, multi-stage sampling =) loss of efficiency stratified samplin...
The notation implies three assumptions: No simultaneity (different from endogeneity) No interference between units: Yi(T1; T2; : : : ; Tn) Same version of the treatment = Yi(Ti) Stable Unit Treatment Value Assumption (SUTVA) Potential violations: feedback effects spill-over effects, carry-over effects different treatment administration Potential ou...
Treatment effect heterogeneity: Zero ATE doesn’t mean zero effect for everyone! =) Conditional ATE Other quantities: Quantile treatment effects etc.
Randomized experiments Laboratory experiments Survey experiments Field experiments Observational studies
Endogeneity: selection bias Generalizability: sample selection, Hawthorne effects, realism “Designing” observational studies Natural experiments (haphazard treatment assignment) Examples: birthdays, weather, close elections, arbitrary administrative rules Generalizing experimental results: possible extrapolation Bottom line: No study is perfect, st...
Observational studies =) No randomization of treatment Difference in means between two populations can still be estimated without bias Valid inference for ATE requires additional assumptions Law of Decreasing Credibility (Manski): The credibility of inference decreases with the strength of the assumptions maintained Identification: How much can you...
i assumptions are valid, yz should be positive for all y and z Suppose that a negative value of ^yz is observed. Did this happen by chance? Statistical hypothesis test (next topic)
Causal inference is all about predicting counter-factuals Association (comparison between treated and control groups) is not causation (comparison between factuals and counterfactuals) Randomization of treatment eliminates both observed and unobserved confounders Design-based vs. model-based inference Observational studies =) identification problem...
Hypothesis tests control the probability of Type I error They do not control the probability of Type II error Tradeoff between the two types of error Size (level) of test: probability that the null is rejected when it is true Power of test: probability that a test rejects the null Typically, we want a most powerful test with the proper size
Null hypotheses are often uninteresting But, hypothesis testing may indicate the strength of evidence for or against your theory Power analysis: What sample size do I need in order to detect a certain departure from the null? Power = 1 Pr(Type II error) Four steps: Specify the null hypothesis to be tested and the significance level Choose a true va...
Pre-registration system: reduces dishonesty but cannot eliminate multiple testing problem Family-wise error rate (FWER): Pr(making at least one Type I error) Bonferroni procedure: reject the jth null hypothesis Hj if pj < m where m is the total number of tests Very conservative: some improvements by Holm and Hochberg
Stochastic proof by contradiction Assume what you want to disprove (null hypothesis) Derive the reference distribution of test statistic Compare the observed value with the reference distribution Interpretation of hypothesis test Statistical significance 6= Pay attention to effect size scientific significance Power analysis Failure to reject null 6...
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Example (General Addition Rule) Rolling a pair of fair dice, what is the probability of getting a total of 10 or a double? Sol. The two events are A = {Total of 10}= {(4,6),(6,4),(5,5)}and B = {Double}= {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)} Their intersection is A∩B = {(5,5)}. P(Total of 10 ∪double) = P(Total of 10)+ P(Double)−P(Total of ...
Definition of Expectation. = E(X ) Def: Mean aka. Expected value. Let X be a random variable with p(d)f f (x). The mean, or expected value of X, denoted E(X), is defined as follows. discrete: X E(X) = xf(x) All x. assuming the sum exists. continuous: 1. E(X) = xf(x) dx. 1. assuming the integral exists.
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Statistics is a branch of mathematics used to summarize, analyze, and interpret a group of numbers or observations. We begin by introducing two general types of statistics: • Descriptive statistics: statistics that summarize observations. • Inferential statistics: statistics used to interpret the meaning of descriptive statistics.
