Intro to Research Methods SPH-X590  Summer 2015 Data Analysis: Methodological Big Picture Statistical Analysis Probability, Inference & Estimation Notation

Presentation Outline Review Statistical Analysis Structure of Research Dimensions of Research Research Process Study Designs Statistical Analysis Types of Statistics: Descriptive and Inferential Notation

The Structure of Research: Deduction The “Hourglass" Notion of Research Begin with broad questions narrow down, focus in. Operationalize OBSERVE Analyze Data Reach Conclusions Generalize back to Questions

The Scientific Method Problem/Question Observation/Research Formulate a Hypothesis Experiment Collect and Analyze Results Conclusion Communicate the Results

The Empirical Research Process: D EDUCT ION T H E OR Y Step 1 Identification of Area of Study: Problem Formulation Step 2 Literature Review: Context Step 3 Research Objectives to Hypotheses: Content to Methodology Concepts to Variables Step 4 Study Design I: Data Collection Methods Research Design: experimental, quasi-experimental, or non-experimental Time & Unit of Analysis Step 5 Procedures: Sampling, Assignment, Recruitment, & Ethics Step 6 Collection: Instruments, Materials, & Management Step 7 Study Design II: Analysis Statistical Approaches & Analytical Techniques Sample Size & Power Step 8 Results: Dissemination Publication, Presentation, & New Application

The Dimensions of Empirical Research: A movement from the theoretical to analytical Theories Postulates Propositions Hypotheses Deductive Reasoning SCIENTIFIC METHOD Analysis Data Collection Constructs Variables Concepts Measurement

Describe Characteristics Data Analysis: In the Big Picture of Methodology Question to Answer Hypothesis to Test Theory Note: Results of empirical scientific studies always begin with the Descriptive Statistics, whether results conclude with Inferential Statistics depends of the Research Objectives/ Aims Study Design: Data Collection Method & Analysis Inferential Statistics Causal Inference Test Hypothesis, Conclusions, Interpretation, & Identification Relationships Collect Data: Measurements, Observations Data Extraction Data Storage Descriptive Statistics Describe Characteristics Organize, Summarize, & Condense the Numbers Decision: Statistics?

Data Analysis: Types of Statistics Descriptive Statistics Summarization & Organization of variable values/scores for the sample Inferential Statistics Inferences made from the Sample Statistic to the Population Parameter. Able to Estimate Causation or make Causal Inference Isolate the effect of the Experimental (Independent) Variable on the Outcome (Dependent) Variable

Data Analysis: Descriptive Statistics Descriptive Statistics are procedures used for organizing and summarizing scores in a sample so that the researchers can describe or communicate the variables of interest. Note: Descriptive Statistics apply only to the sample: says nothing about how accurately the data may reflect the reality in the population Use Sample Statistics to “infer” something about relationships in the entire population: assumes sample is representative of population. Descriptive Statistics summarize 1 variable: aka Univariate Statistics Mean, Median, Mode, Range, Frequency Distribution, Variance and Standard Deviation are the Descriptive Statistics: Univariates

Data Analysis: Inferential Statistics Inferential Statistics are procedures designed to test the likelihood of finding the same results from one sample with another sample drawn from the same population: in fact, mathematically tests whether the sample results would be obtained if all possible samples from the population were tested. Attempts to rule out chance as an explanation for the results: that results reflect real relationships that exist in the population and are not just random or only by chance. Before you can describe or evaluate a relationship using statistics, you must design your study so that your research question can be addressed. This is Methodology: where theory meets Data Collection Methods & Data Analysis.

Data Analysis: Statistics Notation Capitalization In general, capital letters refer to population attributes (i.e., parameters); and lower-case letters refer to sample attributes (i.e., statistics). For example, P refers to a population proportion; and p, to a sample proportion. X refers to a set of population elements; and x, to a set of sample elements. N refers to population size; and n, to sample size. Greek vs. Roman Letters Like capital letters, Greek letters refer to population attributes. Their sample counterparts, however, are usually Roman letters. For example, μ refers to a population mean; and x, to a sample mean. σ refers to the standard deviation of a population; and s, to the standard deviation of a sample.

Data Analysis: Statistics Notation Population Parameters By convention, specific symbols represent certain population parameters. Notation μ refers to a population mean. σ refers to the standard deviation of a population. σ2 refers to the variance of a population. P refers to the proportion of population elements that have a particular attribute. Q refers to the proportion of population elements that do not have a particular attribute, so Q = 1 - P. ρ is the population correlation coefficient, based on all of the elements from a population. N is the number of elements in a population. Sample Statistics By convention, specific symbols represent certain sample statistics. Notation x refers to a sample mean. s refers to the standard deviation of a sample. s2 refers to the variance of a sample. p refers to the proportion of sample elements that have a particular attribute. q refers to the proportion of sample elements that do not have a particular attribute, so q = 1 - p. r is the sample correlation coefficient, based on all of the elements from a sample. n is the number of elements in a sample.

Data Analysis: Summation/ Sigma Notation Summation Notation is shorthand that relies on Greek alphabet and mathematical symbols to indicate how to process values: aka formulae.  = summation X = Variable What do each of these mean? X Add up the values of X X + 2 versus (X + 2) Add up the values of X and add 2 to the Sum, Add 2 to each value of X and then Sum the values X2 versus (X)2 Square each value of X and then Sum Sum the values of X and then Square the Sum (X + 2)2 versus (X2 + 2) Add 2 to each value of X, square the value, then Sum the squared values Square each value of X, add 2 to the value, then Sum the values

Data Analysis: Summation/ Sigma Notation X : Independent Variable, typically Y: Dependent Variable, typically N= Size of the Population n= Size of the Sample ≤ ≥ ≠ = : Equalities or Inequalities ± × ÷ + - : Mathematical Operators α: alpha, refers to constant/ intercept µ: mu, sample mean β: beta coefficient/ standardized δ: sigma, sample standard deviation δ2: sigma squared, sample variance