Applied Statistics course notes
Preface
I Preparing Data for Analysis
1
Workflow and Data Cleaning
1.1
Reproducible workflows
1.1.1
Literate programming
1.2
Import data
1.3
Data Management
1.4
Renaming variable names for sanity sake
1.5
Identifying Variable Types
1.6
Convert number to factor
1.7
Identifying Outliers
1.8
Changing numeric values
1.9
Creating secondary variables
1.10
Wrangling Factors
1.10.1
Collapsing categorical variables into fewer categories
1.11
Binning a continuous variable into categorical ranges.
1.12
Dichotomizing a measure into 2 categories
1.13
Sum or Average values across multiple variables
1.14
Transformations for Normality
1.15
Export and Save
1.16
Wide vs. Long data
1.17
Missing data
1.17.1
Identifying missing data
1.17.2
Model predictions
1.17.3
Regression
1.17.4
Factor Analysis and Principle Components
2
Visualizing Data
2.1
The syntax of
ggplot
2.1.1
Required arguments
2.1.2
Optional but helpful arguments
2.2
The Data
2.3
Univariate Visualizations
2.3.1
Categorical variables
2.3.2
Continuous Measures
2.4
Bivariate Visualizations
2.4.1
Categorical v. Categorical
2.4.2
Grouped bar charts with percentages
2.4.3
Continuous v. Continuous
2.4.4
Continuous v. Categorical
2.4.5
Joy plots / Ridgelines
2.5
Faceting / paneling
2.6
Multiple plots per window
2.7
Multivariate (3+ variables)
2.7.1
Three continuous
2.7.2
Scatterplot matrix
2.7.3
Two categorical and one continuous
2.7.4
Two continuous and one categorical
2.8
Paneling on two variables
2.9
Troubleshooting
2.10
But what about…
2.10.1
Other plots not mentioned
2.11
Additional Resources
3
Selecting Appropriate Analyses
II Statistical Inference
4
Foundations for Inference
5
Bivariate Analysis
5.1
Assumption of Independent Observations
5.2
Choosing appropriate bivariate analysis
5.3
(Q~B) Two means: T-Test
5.3.1
Assumptions
5.3.2
Sampling Distribution for the difference
5.3.3
Example: Smoking and BMI
5.4
(Q~C) Multiple means: ANOVA
5.4.1
Terminology
5.4.2
Formulation of the One-way ANOVA model
5.4.3
Analysis of Variance Table*:
5.4.4
The F-distribution
5.4.5
Assumptions
5.4.6
Example: A comparison of plant species under low water conditions
5.4.7
Coefficient of determination
\(R^{2}\)
5.4.8
Multiple Comparisons
5.5
(C~C) Multiple Proportions:
\(\chi^{2}\)
5.5.1
Conditions for the sampling distribution to be normal.
5.5.2
Example: Are Mammograms effective?
5.5.3
Example: Smoking and General Health
5.5.4
Multiple Comparisons
5.6
(Q~Q) Correlation
5.6.1
Strength of the correlation
5.6.2
Example: Federal spending per capita and poverty rate
III Regression Modeling
6
Introduction
6.1
Opening Remarks
7
Simple Linear Regression
7.1
Example
7.1.1
Caution on out of range predictions
7.2
Mathematical Model
7.2.1
Unifying model framework
7.3
Parameter Estimates
7.3.1
Sum of Squares
7.4
Assumptions
7.5
Example
7.6
Model Diagnostics
7.7
Prediction
7.7.1
Predict the
average
value of Y (
\(\hat{y}\)
) based on the model
7.7.2
Confidence intervals for the predicted mean
7.7.3
Predict the
average
value of Y
\((\hat{y_{i}})\)
for a certain value of
\(x^{*}\)
7.7.4
Predict a
new
value of Y
\(\hat{y_{i}}\)
for a certain value of
\(x^{*}\)
7.8
ANOVA for regression
7.9
Correlation Coefficient
8
Moderation and Stratification
8.1
Moderation
8.1.1
Motivating Example - Admissions at UC Berkeley
8.1.2
Motivating Example: Association of flower parts
8.2
Stratification
8.3
Identifying a moderator
8.4
What to look for in each type of analysis
8.5
Ex: Correlation
8.6
Ex: Regression
8.7
Ex: ANOVA
8.8
Ex: 4 Chi-Squared
9
Multiple Linear Regression
9.1
Mathematical Model
9.2
Parameter Estimation
9.3
Fitting the model
9.4
Interpreting Coefficients
9.4.1
Continuous predictors
9.4.2
Binary predictors
9.4.3
Categorical Predictors
9.5
Presenting results
9.6
Confounding
9.6.1
Example: Does smoking affect pulse rate?
9.7
What to watch out for
10
Model Building
10.1
Interactions (PMA6 8.8)
10.1.1
Fitting interaction models & interpreting coefficients
10.1.2
Categorical Interaction variables
10.1.3
Example 2
10.1.4
Example 3:
10.2
Simultaneous test of multiple variables (PMA6 9.5)
10.2.1
Example: Modeling depression score
10.2.2
Testing for a moderation effect in a multiple regression model.
10.3
Multicollinearity (PMA6 8.9)
10.4
Variable Selection Process
10.4.1
Automated selection procedures (PMA6 9.6)
10.4.2
LASSO Regression (PMA6 9.7)
10.5
Comparing between models (PMA6 9.4)
10.5.1
RSS: Residual Sum of Squares
10.5.2
General F Test
10.5.3
Likelihood function
10.5.4
Multiple
\(R^{2}\)
10.5.5
Adjusted
\(R^{2}\)
10.5.6
Mallows
\(C_{p}\)
10.5.7
Akaike Information Criterion (AIC)
10.5.8
Bayesian Information Criterion (BIC)
10.5.9
AIC vs BIC
10.6
Model Diagnostics
10.6.1
Linearity
10.6.2
Normality of residuals.
10.6.3
Homogeneity of variance
10.6.4
Posterior Predictions
10.6.5
All at once
10.7
General Advice (PMA6 9.9)
10.8
What to watch out for (PMA6 9.10)
11
Generalized Linear Models
11.1
Fitting GLMs
11.2
Log-linear models
11.2.1
Example
11.3
Binary outcome data
11.4
Logistic Regression
11.4.1
Interpreting Odds Ratios
11.4.2
Confidence Intervals
11.4.3
Example: The effect of gender on Depression
11.4.4
Multiple Logistic Regression
11.4.5
Effect of a k unit change
11.4.6
Example: Predictors of smoking status
11.4.7
Model Fit
11.5
Count outcome data
11.6
Categorical outcome data
12
Classification of Binary outcomes
12.1
Predicted Probabilities
12.2
Distribution of Predicted probabilities
12.2.1
Plotting predictions against covariates
12.3
Predicted Class (outcome)
12.4
Confusion Matrix
12.5
Vocabulary terms
12.6
ROC Curves
12.7
Model Performance
IV Multivariate Analysis
13
Introduction
14
Principal Component Analysis
14.1
When is Principal Components Analysis (PCA) used?
14.2
Basic Idea - change of coordinates
14.3
More Generally
14.4
R commands
14.4.1
Generating PC’s
14.4.2
Viewing the amount of variance contained by each PC
14.4.3
Vizualize Loadings
14.5
Data Reduction
14.5.1
Choosing
\(m\)
14.6
Standardizing
14.7
Example
14.8
Use in Multiple Regression
14.8.1
Example: Modeling acute illness
14.9
Things to watch out for
14.10
Additional References
15
Factor Analysis
15.1
Introduction
15.1.1
Latent Constructs
15.1.2
Comparison with PCA
15.1.3
EFA vs CFA
15.2
Factor Model
15.2.1
Components of Variance
15.2.2
Two big steps
15.3
Example data setup
15.4
Factor Extraction Methods
15.4.1
Principal components (PC Factor model)
15.4.2
Iterated components
15.4.3
Maximum Likelihood
15.4.4
Uniqueness
15.4.5
Resulting factors
15.5
Rotating Factors
15.5.1
Varimax Rotation
15.5.2
Oblique rotation
15.6
Factor Scores
15.7
What to watch out for
15.8
Additional Resources
V Multi-level Modeling
16
Introduction
16.1
Example School Data
16.2
Multi-level models
17
Random Intercept Models
17.1
Pooling
17.2
Mathematical Models
17.2.1
Complete Pooling
17.2.2
No Pooling
17.2.3
Partial Pooling (RI)
17.3
Components of Variance
17.4
Fitting models in R
17.4.1
Comparison of estimates
17.5
Estimation Methods
17.6
Including Covariates
17.7
More Random Effects
17.8
Centering terms
17.8.1
A generic
dplyr
approach to centering.
17.9
Specifying Correlation Structures
17.9.1
Changing covariance structures in R
17.10
Additional References
17.10.1
Lecture notes from other classes found on the interwebs
17.10.2
Package Vignettes
VI Other Topics
18
Missing Data
18.1
Identifying missing data
18.1.1
Visualize missing patterns
18.2
Effects of Nonresponse
18.3
Missing Data Mechanisms
18.3.1
Demonstration via Simulation
18.4
General strategies
18.4.1
Complete cases analysis
18.4.2
Available-case analysis
18.4.3
Imputation
18.5
Imputation Methods
18.6
Multiple Imputation (MI)
18.6.1
Goals
18.6.2
Technique
18.6.3
MI as a paradigm
18.6.4
Inference on MI
18.7
Multiple Imputation using Chained Equations (MICE)
18.7.1
Overview
18.7.2
Process / Algorithm
18.7.3
Convergence
18.7.4
Imputation Methods
18.8
Diagnostics
18.9
Example: Prescribed amount of missing.
18.9.1
Multiply impute the missing data using
mice()
18.9.2
Check the imputation method used on each variable.
18.9.3
Check Convergence
18.9.4
Look at the values generated for imputation
18.9.5
Create a complete data set by filling in the missing data using the imputations
18.9.6
Visualize Imputations
18.9.7
Calculating bias
18.10
Post MICE data management
18.11
Final thoughts
18.12
Additional References
VII APPENDIX
19
Setup R & RStudio
19.1
⚠️ Before you begin
19.2
Download and install R
🔽 Download R v 4.1+
✏️ Install
🎦 Video Tutorials for both R and R Studio.
19.3
Download and install R Studio
19.4
Navigating RStudio
19.5
Setting preferences
19.6
Installing packages
19.6.1
Common packages used in this notebook
19.7
Organization using R Projects
19.8
Literate programming with Quarto
19.8.1
Creating PDF’s
19.9
Seeking Help
19.9.1
Advice on asking for help
19.9.2
Help from inside R Studio
19.9.3
Other Online
19.9.4
Written
19.10
Saving and closing your work.
19.10.1
Restart R
19.11
Acknowledgements
View source on Github
Applied Statistics
10.8
What to watch out for (PMA6 9.10)
Multicollinearity
Missing Data
Use previous research as a guide
Variables not included can bias the results
Significance levels are only a guide
Perform model diagnostics after selection to check model fit.