Model Specification

In statistics, a model is a (joint) probability distribution.

Parametric Probabilistic Models

Parametric and nonparametric estimators do not have essential difference between or superiority to each other. Both take random samples as the sole input. Both are collections of models. Parametric estimators, such as MLE, are algorithms selecting a unique probability model from a subspace of probability model space, indexed by model parameter. Nonparametric methods are algorithms selecting a unique probability model from an other subspace of probability model space, only without an index.

Generally, non-parametric methods are non-mechanistic methods, which are basically statistical in the essence.

Sample

Random Sample: Statistic, Sampling Distributions from Gaussian Population, Order Statistic.

Notation: Use \( X \) to denote random variables; use \( x \) to denote samples.

Traditional statistics assumes "large n, small p" (n for observations, p for parameters measured.) While in modern statistics, the problem typically is "small n, large p".

Asymptotic Analysis:

Statistical Inference

Principles of Statistical Inference: Sufficiency principle, Conditionality principle, Likelihood principle

Point Estimation: methods of finding and evaluating estimators, UMVU estimators

Interval Estimation: Confidence Interval, Tolerance Interval

Hypothesis Testing

Likelihood Ratio Test (LRT), Uniformly Most Powerful (UMP) Test

False discovery rate (FDR)

Model Selection

Linear Regression: simple linear regression, ordinary least square estimator

Logistic Regression

Information criteria.

Causal Inference

Causal inference in statistics is hard.

Reference

  1. Statistical Tables: Normal, t, F, Chi-Squared
  2. Statistical Tables: Binomial
  3. Statistical Tables: Poisson
  4. Univariate Distribution Relationships

Notes on Intuitive Biostatistics {Motulsky}

Table 1: Statistical Techniques

Purpose Continuous Data Count or Ranked Data Arrival Time Binary Data
(Height) (Number of headaches in a week; Self-report score) (Life expectancy of a patient; Minutes until REM sleep begins Recurrence of infection)
Describe one sample Frequency distribution; Sample mean; Quantiles; Sample standard deviation Frequency distribution; Quantiles; Kaplan-Meier survival curve; Median survival curve; Five-year survival percentage Proportion
Distributional Test Normality tests; Outlier tests N/A N/A N/A
Infer about one population One-sample t test Wilcoxon’s rank-sum test Confidence bands around survival curve; CI of median survival CI of proportion; Binomial test to compare observed distribution with a theoretical (expected) distribution
Compare two unpaired groups Unpaired t test Mann-Whitney test Log-rank test; Gehan-Breslow test; CI of ratio of median survival times; CI of hazard ratio Fisher’s exact test;
Compare two paired groups Paired t test Wilcoxon’s matched paires test Conditional proportional hazards regression McNemar’s test
Compare three or more unpaired groups One-way ANOVA followed by multiple comparison tests Kruskal-Wallis test; Dunn’s posttest Log-rank test; Gehan-Breslow test Chi-squared test (for trend)
Compare three or more paired groups Repeated-measures ANOVA followed by multiple comparison tests Friedman’s test; Dunn’s posttest Conditional proportional hazards regression Cochran’s Q
Quantify association between two variables Pearson’s correlation Spearman’s correlation N/A N/A
Predict one variable from one or several others linear/nonlinear regression N/A Cox’s proportional hazards regression Logistic regression

[Based on Motulsky. Intuitive Biostatistics, Appendix 7.]