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Parametric Statistics
Parametric statistics are the most common type of inferential statistics. Inferential statistics are calculated with the purpose of generalizing the findings of a sample to the population it represents, and they can be classified as either parametric or non-parametric. Parametric tests make assumptions about the parameters of a population, whereas nonparametric tests do not include such assumptions or include fewer. For instance, parametric tests assume that the sample has been randomly selected from the population it represents and that the distribution of data in the population has a known underlying distribution. The most common distribution assumption is that the distribution is normal. Other distributions include the binomial distribution (logistic regression) and the Poisson distribution (Poisson regression). Additionally, parametric statistics require that the data are measured using an interval or ratio scale, whereas nonparametric statistics use data that are measured with a nominal or ordinal scale. Some frequently used parametric tests that include the assumption of a normal distribution include the following: Student's t test, analysis of variance, Pearson's r correlation, and linear regression. Selection of the correct statistical test is important because if the wrong test is selected, the researcher increases the chances of coming to the wrong conclusion.
This entry first reviews inferential statistics methodologies. Next, it discusses the underlying assumptions for parametric statistics and parametric methods, and corrections for assumption violations. Finally, this entry discusses the advantages of parametric statistics.
Inferential Statistics
There are two types of methodologies used in inferential statistics: hypothesis testing and estimation of population parameters. Each of these methodologies includes parametric and nonparametric tests.
Hypothesis testing uses sample data to test a prediction about a population or the relationship between two or more populations. The predictions are stated as two statistical hypotheses. The null hypothesis, H0, states that there is no effect or no difference. The alternative hypothesis, Ha, indicates the presence of an effect or difference. This is usually the hypothesis that the researcher expects to be supported. Within hypothesis testing, it is possible for a researcher to commit two types of errors, Type I and Type II. A Type I error occurs when a true null hypothesis is rejected; that is, the conclusion is made that the alternative hypothesis is true when it is not. The likelihood of committing such an error is specified by the alpha level. For example, the likelihood of committing a Type I error if α = .05 is 5%. A Type II error occurs when a false null hypothesis is not rejected; that is, the conclusion is made that an alternative hypothesis is not true when it is. The likelihood of committing such an error is specified by beta, β. This is related to the power of a statistical test. Power refers to the probability that a null hypothesis will be rejected when it is false—in other words, the probability of finding a statistically significant result. Power depends on the significance level (α), the sample size, and the population effect size.
Estimation of population parameters includes point estimation and interval estimation. Point estimation involves estimating the parameter value from the computed sample statistics. It is the “best guess” for an unknown population parameter. Statistics refer to characteristics of an observed sample and are measured using measures of central tendency (i.e., mean, median, mode) and measures of variability (i.e., variance and standard deviation). Parameters refer to characteristics of a population. Interval estimation, the more commonly used, involves using sample data to compute a range of values that includes the unknown population parameter. The confidence interval is the most common form of interval estimation.
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- Descriptive Statistics
- Distributions
- Graphical Displays of Data
- Hypothesis Testing
- Alternative Hypotheses
- Beta
- Critical Value
- Decision Rule
- Hypothesis
- Nondirectional Hypotheses
- Nonsignificance
- Null Hypothesis
- One-Tailed Test
- p Value
- Power
- Power Analysis
- Significance Level, Concept of
- Significance Level, Interpretation and Construction
- Significance, Statistical
- Two-Tailed Test
- Type I Error
- Type II Error
- Type III Error
- Important Publications
- “Coefficient Alpha and the Internal Structure of Tests”
- “Convergent and Discriminant Validation by the Multitrait-Multimethod Matrix”
- “Meta-Analysis of Psychotherapy Outcome Studies”
- “On the Theory of Scales of Measurement”
- “Probable Error of a Mean, The”
- “Psychometric Experiments”
- “Sequential Tests of Statistical Hypotheses”
- “Technique for the Measurement of Attitudes, A”
- “Validity”
- Aptitudes and Instructional Methods
- Doctrine of Chances, The
- Logic of Scientific Discovery, The
- Nonparametric Statistics for the Behavioral Sciences
- Probabilistic Models for Some Intelligence and Attainment Tests
- Statistical Power Analysis for the Behavioral Sciences
- Teoria Statistica Delle Classi e Calcolo Delle Probabilità
- Inferential Statistics
- Association, Measures of
- Coefficient of Concordance
- Coefficient of Variation
- Coefficients of Correlation, Alienation, and Determination
- Confidence Intervals
- Margin of Error
- Nonparametric Statistics
- Odds Ratio
- Parameters
- Parametric Statistics
- Partial Correlation
- Pearson Product-Moment Correlation Coefficient
- Polychoric Correlation Coefficient
- Q-Statistic
- R2
- Randomization Tests
- Regression Coefficient
- Semipartial Correlation Coefficient
- Spearman Rank Order Correlation
- Standard Error of Estimate
- Standard Error of the Mean
- Student's t Test
- Unbiased Estimator
- Weights
- Item Response Theory
- Mathematical Concepts
- Measurement Concepts
- Organizations
- Publishing
- Qualitative Research
- Reliability of Scores
- Research Design Concepts
- Aptitude-Treatment Interaction
- Cause and Effect
- Concomitant Variable
- Confounding
- Control Group
- Interaction
- Internet-Based Research Method
- Intervention
- Matching
- Natural Experiments
- Network Analysis
- Placebo
- Replication
- Research
- Research Design Principles
- Treatment(s)
- Triangulation
- Unit of Analysis
- Yoked Control Procedure
- Research Designs
- A Priori Monte Carlo Simulation
- Action Research
- Adaptive Designs in Clinical Trials
- Applied Research
- Behavior Analysis Design
- Block Design
- Case-Only Design
- Causal-Comparative Design
- Cohort Design
- Completely Randomized Design
- Cross-Sectional Design
- Crossover Design
- Double-Blind Procedure
- Ex Post Facto Study
- Experimental Design
- Factorial Design
- Field Study
- Group-Sequential Designs in Clinical Trials
- Laboratory Experiments
- Latin Square Design
- Longitudinal Design
- Meta-Analysis
- Mixed Methods Design
- Mixed Model Design
- Monte Carlo Simulation
- Nested Factor Design
- Nonexperimental Design
- Observational Research
- Panel Design
- Partially Randomized Preference Trial Design
- Pilot Study
- Pragmatic Study
- Pre-Experimental Designs
- Pretest-Posttest Design
- Prospective Study
- Quantitative Research
- Quasi-Experimental Design
- Randomized Block Design
- Repeated Measures Design
- Response Surface Design
- Retrospective Study
- Sequential Design
- Single-Blind Study
- Single-Subject Design
- Split-Plot Factorial Design
- Thought Experiments
- Time Studies
- Time-Lag Study
- Time-Series Study
- Triple-Blind Study
- True Experimental Design
- Wennberg Design
- Within-Subjects Design
- Zelen's Randomized Consent Design
- Research Ethics
- Research Process
- Clinical Significance
- Clinical Trial
- Cross-Validation
- Data Cleaning
- Delphi Technique
- Evidence-Based Decision Making
- Exploratory Data Analysis
- Follow-Up
- Inference: Deductive and Inductive
- Last Observation Carried Forward
- Planning Research
- Primary Data Source
- Protocol
- Q Methodology
- Research Hypothesis
- Research Question
- Scientific Method
- Secondary Data Source
- Standardization
- Statistical Control
- Type III Error
- Wave
- Research Validity Issues
- Bias
- Critical Thinking
- Ecological Validity
- Experimenter Expectancy Effect
- External Validity
- File Drawer Problem
- Hawthorne Effect
- Heisenberg Effect
- Internal Validity
- John Henry Effect
- Mortality
- Multiple Treatment Interference
- Multivalued Treatment Effects
- Nonclassical Experimenter Effects
- Order Effects
- Placebo Effect
- Pretest Sensitization
- Random Assignment
- Reactive Arrangements
- Regression to the Mean
- Selection
- Sequence Effects
- Threats to Validity
- Validity of Research Conclusions
- Volunteer Bias
- White Noise
- Sampling
- Cluster Sampling
- Convenience Sampling
- Demographics
- Error
- Exclusion Criteria
- Experience Sampling Method
- Nonprobability Sampling
- Population
- Probability Sampling
- Proportional Sampling
- Quota Sampling
- Random Sampling
- Random Selection
- Sample
- Sample Size
- Sample Size Planning
- Sampling
- Sampling and Retention of Underrepresented Groups
- Sampling Error
- Stratified Sampling
- Systematic Sampling
- Scaling
- Software Applications
- Statistical Assumptions
- Statistical Concepts
- Autocorrelation
- Biased Estimator
- Cohen's Kappa
- Collinearity
- Correlation
- Criterion Problem
- Critical Difference
- Data Mining
- Data Snooping
- Degrees of Freedom
- Directional Hypothesis
- Disturbance Terms
- Error Rates
- Expected Value
- Fixed-Effects Models
- Inclusion Criteria
- Influence Statistics
- Influential Data Points
- Intraclass Correlation
- Latent Variable
- Likelihood Ratio Statistic
- Loglinear Models
- Main Effects
- Markov Chains
- Method Variance
- Mixed- and Random-Effects Models
- Models
- Multilevel Modeling
- Odds
- Omega Squared
- Orthogonal Comparisons
- Outlier
- Overfitting
- Pooled Variance
- Precision
- Quality Effects Model
- Random-Effects Models
- Regression Artifacts
- Regression Discontinuity
- Residuals
- Restriction of Range
- Robust
- Root Mean Square Error
- Rosenthal Effect
- Serial Correlation
- Shrinkage
- Simple Main Effects
- Simpson's Paradox
- Sums of Squares
- Statistical Procedures
- Accuracy in Parameter Estimation
- Analysis of Covariance (ANCOVA)
- Analysis of Variance (ANOVA)
- Barycentric Discriminant Analysis
- Bivariate Regression
- Bonferroni Procedure
- Bootstrapping
- Canonical Correlation Analysis
- Categorical Data Analysis
- Confirmatory Factor Analysis
- Contrast Analysis
- Descriptive Discriminant Analysis
- Discriminant Analysis
- Dummy Coding
- Effect Coding
- Estimation
- Exploratory Factor Analysis
- Greenhouse-Geisser Correction
- Hierarchical Linear Modeling
- Holm's Sequential Bonferroni Procedure
- Jackknife
- Latent Growth Modeling
- Least Squares, Methods of
- Logistic Regression
- Mean Comparisons
- Missing Data, Imputation of
- Multiple Regression
- Multivariate Analysis of Variance (MANOVA)
- Pairwise Comparisons
- Path Analysis
- Post Hoc Analysis
- Post Hoc Comparisons
- Principal Components Analysis
- Propensity Score Analysis
- Sequential Analysis
- Stepwise Regression
- Structural Equation Modeling
- Survival Analysis
- Trend Analysis
- Yates's Correction
- Statistical Tests
- Bartlett's Test
- Behrens-Fisher t′ Statistic
- Chi-Square Test
- Duncan's Multiple Range Test
- Dunnett's Test
- F Test
- Fisher's Least Significant Difference Test
- Friedman Test
- Honestly Significant Difference (HSD) Test
- Kolmogorov-Smirnov Test
- Kruskal-Wallis Test
- Mann-Whitney U Test
- Mauchly Test
- McNemar's Test
- Multiple Comparison Tests
- Newman-Keuls Test and Tukey Test
- Omnibus Tests
- Scheffé Test
- Sign Test
- t Test, Independent Samples
- t Test, One Sample
- t Test, Paired Samples
- Tukey's Honestly Significant Difference (HSD)
- Welch's t Test
- Wilcoxon Rank Sum Test
- z Test
- Theories, Laws, and Principles
- Bayes's Theorem
- Central Limit Theorem
- Classical Test Theory
- Correspondence Principle
- Critical Theory
- Falsifiability
- Game Theory
- Gauss-Markov Theorem
- Generalizability Theory
- Grounded Theory
- Item Response Theory
- Occam's Razor
- Paradigm
- Positivism
- Probability, Laws of
- Theory
- Theory of Attitude Measurement
- Weber-Fechner Law
- Types of Variables
- Validity of Scores
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