Essential Statistics with R: Agenda

1. NHANES data – overview, import
2. Descriptive statistics & Exploratory data analysis (EDA)
3. Continuous variable statistics
   • T-tests
   • Wilcoxon / Mann-Whitney U tests
   • ANOVA
   • Linear models & multiple regression
4. Discrete variable statistics
   • Contingency tables
   • Chi-square & Fisher exact tests
   • Logistic regression
5. If time allows: Power & sample size analysis
6. Bonus: Model tidying and extraction

Getting started

Go to bioconnector.org. Hit the data link. Download nhanes.csv and nhanes_dd.csv. Save them both somewhere on your computer you can easily find. E.g., create a new folder on your desktop called rstats and save it there.

Open RStudio. File – New file – R script. Save this file as nhanes_analysis.R in the same rstats folder above.

Quit RStudio, then double-click the R script to open R running in the rstats folder. You can also do this through the menu with Session – Set working directory – To Source File Location.

Load the required libraries and data:

library(dplyr)
library(readr)
nh <- read_csv("nhanes.csv")

(If you don’t have these packages installed, use install.packages("dplyr") and install.packages("dplyr")).

T-tests vs ANOVA vs linear models

• T-test: difference in means between two groups
• ANOVA: difference in means between 2+ groups
• Linear model: How does a continuous outcome change with one or more predictor variables?

T-tests are specific cases of ANOVA, which is a specific case of linear regression.

Linear regression: “Do levels of a categorical variable affect the response?”

ANOVA/t-tests: “Does the mean response differ between levels of a categorical variable?”

Linear regression

Simple linear regression, single predictor variable X:

\[Y = \beta_{0} + \beta_{1}X\]

Multiple regression:

\[Y = \beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \epsilon\]

• \(Y\) is the response
• \(X_{1}\) and \(X_{2}\) are the predictors
• \(\beta_{0}\) is the intercept, and \(\beta_{1}\), \(\beta_{2}\) etc are coefficients that describe what 1-unit changes in \(X_{1}\) and \(X_{2}\) do to the outcome variable \(Y\).
• \(\epsilon\) is random error. Our model will not perfectly predict \(Y\). It will be off by some random amount. We assume this amount is a random draw from a Normal distribution with mean 0 and standard deviation \(\sigma\).

Logistic regression: model

What if we wanted to model the discrete outcome, e.g., whether someone is insured, against several other variables, similar to how we did with multiple linear regression? We can’t use linear regression because the outcome isn’t continuous – it’s binary, either Yes or No. For this we’ll use logistic regression to model the log odds of binary response. That is, instead of modeling the outcome variable, \(Y\), directly against the inputs, we’ll model the log odds of the outcome variable.

If \(p\) is the probability that the individual is insured, then \(\frac{p}{1-p}\) is the odds that person is insured. The generalized linear model is expressed as:

\[log(\frac{p}{1-p}) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k\]

Where \(\beta_0\) is the intercept, \(\beta_1\) is the increase in the log odds of the outcome for every unit increase in \(x_1\). Rearranging the formula a little bit will show that \(e^{\beta_1}\) is the odds ratio for \(x_1\).

Logistic regression: implementation

Logistic regression is a type of generalized linear model (GLM). We fit GLM models in R using the glm() function. It works like the lm() function except we specify which GLM to fit using the family argument. Logistic regression requires family=binomial.

mod <- glm(y ~ x, data=yourdata, family='binomial')
summary(mod)

Power & sample size

Power (a.k.a. “sensitivity”): probability that your test correctly rejects the null hypothesis when the alternative hypothesis is true. That is, if there really is an effect (difference in means, association between categorical variables, etc.), how likely are you to be able to detect that effect at a given statistical significance level, given certain assumptions. Know two, estimate the third:

1. Power: How likely are you to detect the effect? (Usually want 80%).
2. N: What is the sample size you have (or require)?
3. Effect size: How big is the difference in means, odds ratio, etc?

(Other assumptions include desired \(\alpha\) and \(\sigma\)).

# Power for a t-test
power.t.test(power, n, delta, sig.level=0.05, sd=1)
# Power for a two-sample test of proportions
power.prop.test(power, n, p1, p2, sig.level=0.05)