---
title: "Statistical Foundations"
author: "Prof. Eric A. Suess"
date: "1/26/2026"
format: 
  revealjs:
    self-contained: true
---

## Statistical Foundations

The authors of our book make many important observations about Data Science at the beginning of Chapter 9.

- "The objective of Data Science is to extract meaning from data."
- "Visualization is good for seeing patterns in noisy data."
- "It is important to be able to see when the patterns we see are strong enough that they are not mere accidents."

## Statistical Foundations

- **"Statistical methods quantify patterns and their strengths."**
- "Some people think that **Big Data** has made statistics obsolete.  The argument is that with lots of data, the data can speak clearly for themselves.  This is wrong, as we shall see."
- "This chapter will illustrate a Data Science **workflow**."

## Samples and Populations

- Recall the Law of Large Numbers
- Recall the Central Limit Theorem

## LLN

Sample mean $\bar{x}$ converges to the population mean $\mu$.

Simulation assuming the populations parameters are known.

```{r}
#| echo: true
mu <- 65; sigma <- 3;  # population parameters assumed
B <- 1000
x.sample <- rnorm(B, mu, sigma)
x.running.mean <- numeric(B)
for(i in 1: B){
  x.running.mean[i] <- sum(x.sample[1:i])/i
}
```

## LLN

```{r}
#| echo: true
plot(x.running.mean, type="l")
abline(h=mu, col="red")
```


## CLT

The *sampling distribution* of $\bar{x}$ is approximately $N(\mu, \frac{\sigma^2}{n})$.

Simulation assuming the populations parameters are known.

```{r}
#| echo: true
mu <- 65; sigma <- 3; n <- 25
B <- 1000
x.mean <- numeric(B)
for(i in 1: B){
  x.mean[i] <- mean(rnorm(n, mu, sigma))
}
```

## CLT

```{r}
#| echo: true
hist(x.mean)
```

## CLT in the Tidyverse

Simulation assuming the populations parameters are known.  See Chapter 7.

```{r}
#| echo: true
library(pacman)
p_load(tidyverse)

n <- 25
B <- 10000

Trials_n <- 1:B |> 
  map_dbl(
     ~ rnorm(n, mu, sigma) |> mean(.x)
  ) |>  
  tibble() |>  
  rename(mean = starts_with("map") )

head(Trials_n, 2)  # Show only the first 2 rows
```

## CLT in the Tidyverse

```{r}
#| echo: true
Trials_n |> ggplot(aes(mean)) +
  geom_histogram(bins=10)
```

## ModernDive Chapter 7 Sampling

For a [ModernDive (v2)](https://moderndive.com/v2/) into sampling and the Central Limit Theorem try the code in [Chapter 7 Sampling](https://moderndive.com/v2/sampling.html).

## Bootstrap

The *bootstrap* is a statistical method that allows us to approximate the *sampling distribution* even without access to the population.

Bootstrapping is a *resampling* method.

Bootstrapping uses *sampling with replacement.*

## Sketch the difference

Note that the main difference between the CLT and the Bootstrap is that for the CLT the sample size $n$ goes to **infinity** and with the Bootstrap the sample size remains fixed and the number of samples $B$ goes to **infinity**.

## ModernDive Chapter 8 Bootstrapping and Confidence Intervals.

For a [ModernDive (v2)](https://moderndive.com/v2/) into Bootstrapping and Confidence Intervals try the code in [Chapter 8 Bootstrapping and Confidence Intervals](https://moderndive.com/v2/confidence-intervals.html).

In this chapter the [infer](http://infer.tidymodels.org/) R package is ued.  This package has a number of modern functions to easily simulate resampling from a dataframe.

For further modern code, check out the [tidymodels](https://www.tidymodels.org/) R package [rsample](https://rsample.tidymodels.org/).

## Outliers

Outliers should never be dropped unless there is a clear rationale.  If outliers are dropped this should be clearly reported.

```{r, eval=FALSE}
#| echo: true
p_load(anomalize, tibbletime)

tidyverse_cran_downloads |>
    time_decompose(count, merge = TRUE) |>
    anomalize(remainder) |>
    time_recompose() |>
    plot_anomalies(ncol = 3, alpha_dots = 0.25) 
```

## Outliers

```{r}
#| echo: false
p_load(anomalize, tibbletime)

tidyverse_cran_downloads |>
    time_decompose(count, merge = TRUE) |>
    anomalize(remainder) |>
    time_recompose() |>
    plot_anomalies(ncol = 3, alpha_dots = 0.25) 
```

## Statistical Models

Statistical models are used to explain variation between *response variables* and *explanatory variables*.

Linear Regression models are commonly used to build models.  They are fit using the *least squares* **algorithm**.  This algorithm leads to unbiased estimators that have minimum variance.

- We know that the estimators of the parameters in the model are computed using **optimization**.
- We know that the estimators are **unbiased**.

## Confounding Variables

What does the correlation coefficient measure?  **Answer:**  ???

Recall "Correlation does not imply causation."

The gold standard is a *controlled experiment*.  The authors describe the idea of *A/B testing*.

Most data collected today is observational.  So no designed *experiment* has been used.

Recall **Simpson's Paradox**.

## Problems with **p-values**

Using *many many many* p-values all assuming $\alpha = 0.05$.

This causes much higher overall error rates.

When using *multiple comparisons* and overall error rate should be addressed.  

## Appendix E.

Be sure to read Appendix E at the end of the book.  

It includes a very nice summary of fitting Multiple Linear Regression.


## Confounding variables

A [confounding variable](https://en.wikipedia.org/wiki/Confounding) is another variable that influences
the other variables.

[Simpson's Paradox](http://en.wikipedia.org/wiki/Simpson%27s_paradox)

[Edward Simpson: Bayes at Bletchley Park](http://onlinelibrary.wiley.com/doi/10.1111/j.1740-9713.2010.00424.x/abstract)


## Example of Simpson's Paradox 

```{r}
#| echo: true
### synthetic data

# Consider book price (y) by number of pages (x)

z = c("hardcover","hardcover",
      "hardcover","hardcover",
      "paperback", "paperback","paperback", 
      "paperback")

x1 = c( 150, 225, 342, 185)
y1 = c( 27.43, 48.76, 50.25, 32.01 )

x2 = c( 475, 834, 1020, 790)
y2 = c( 10.00, 15.73, 20.00, 17.89 )

x = c(x1, x2)
y = c(y1, y2)
```

## Example of Simpson's Paradox 


```{r}
#| echo: true
plot(x,y)
```

## Example of Simpson's Paradox 

```{r}
#| echo: true
# correlation

cor(y, x)

cor(y1, x1)
cor(y2, x2)
```

## Example of Simpson's Paradox 

```{r}
#| echo: true
# linear regression

lm(y ~ x)

```


## Example of Simpson's Paradox 

```{r}
#| echo: true
# linear regression

lm(y1 ~ x1)
```


## Example of Simpson's Paradox 

```{r}
#| echo: true
# linear regression

lm(y2 ~ x2)

```

## Example of Simpson's Paradox 

**Summary:** Simpson's Paradox is the changing of the direction 
of a relationship with the introduction of another variable.

The relationship between Price and Number of pages in a book 
changes with the introduction of the variable Type of Book
(Hardcover, Paperback).

See the R Markdown document [SimpsonsParadox](http://rpubs.com/esuess/SimpsonsParadox) available on [RPubs.com/esuess](http://rpubs.com/esuess).