---
title: "Covariates, Confounding, Association, and Causation"
subtitle: "SD01 — Statistical Thinking"
author: "Samer El Kababji"
output:
  html_document:
    toc: true
format:
  html:
    code-links:
      - text: "Download R Markdown"
        icon: file-earmark-arrow-down
        href: SD01-02-covariates-confounding-and-causation.Rmd?download=1
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = TRUE)
```

**Course:** Statistics and Probability for Data Science (DS14270).  
**Source deck:** Samer El Kababji, *SD01 — Statistical Thinking*.

This lesson uses a small fictional example to translate the ideas in the deck
into R. It uses group averages rather than a regression model.

## Response, treatment, and covariate

Suppose we want to know whether a new medication reduces systolic blood
pressure after eight weeks.

- The **response** is blood pressure after eight weeks.
- The **treatment** is the medication received: new or standard.
- A **covariate** is another recorded characteristic, such as age group.

The table below is constructed for teaching. Each row describes a group of
patients rather than an individual patient.

```{r teaching-data}
blood_pressure <- data.frame(
  medication = c("Standard", "Standard", "New", "New"),
  age_group = c("Younger", "Older", "Younger", "Older"),
  patients = c(20, 80, 90, 10),
  mean_bp = c(130, 140, 125, 135)
)
blood_pressure
```

Age group is a covariate because it is an additional variable that may help us
understand differences in blood pressure.

## Comparing the two medication groups

The overall mean blood pressure for each medication combines its younger and
older patients. Each age group contributes according to its patient count:

$$
\bar y=\frac{\sum_j n_j\bar y_j}{\sum_j n_j}.
$$

Here $j$ runs over the age groups within one medication, $n_j$ is the patient
count, and $\bar y_j$ is that age group's mean blood pressure. Apply the
equation separately for New and Standard.

**Worked example — comparing medications.** For New, combine 90 younger
patients with a mean of 125 mmHg and 10 older patients with a mean of 135 mmHg:

$$
\bar y_{\text{New}}
=\frac{90\times125+10\times135}{90+10}
=\frac{12600}{100}
=126\text{ mmHg}.
$$

For Standard, combine 20 younger patients with a mean of 130 mmHg and 80 older
patients with a mean of 140 mmHg:

$$
\bar y_{\text{Standard}}
=\frac{20\times130+80\times140}{20+80}
=\frac{13800}{100}
=138\text{ mmHg}.
$$

These calculations produce the table below: `weighted_bp` is the numerator,
`patients` is the denominator, and `mean_bp` is their quotient. Younger and
older patients are combined within each medication, giving one row for New
and one for Standard.

```{r overall-comparison}
overall_means <- aggregate(
  cbind(weighted_bp = blood_pressure$mean_bp * blood_pressure$patients,
        patients = blood_pressure$patients),
  by = list(medication = blood_pressure$medication),
  FUN = sum
)
overall_means$mean_bp <- overall_means$weighted_bp / overall_means$patients
overall_means

overall_difference <- with(
  overall_means,
  mean_bp[medication == "New"] - mean_bp[medication == "Standard"]
)
overall_difference
```

The new-medication group has an average blood pressure 12 mmHg lower than the
standard-medication group. This is an **association** in the observed data. It
does not yet tell us how much of the difference was caused by the medication.

## How age can confuse the comparison

The treatment groups have very different age compositions.

```{r age-composition}
age_counts <- xtabs(patients ~ medication + age_group, data = blood_pressure)
age_counts
prop.table(age_counts, margin = 1)
```

Most patients receiving the standard medication are older, while most
patients receiving the new medication are younger. Older patients also have
higher blood pressure in both medication groups. Age is therefore connected
to both:

1. which medication patients received; and
2. their blood pressure.

In this example, age is a **confounder**. It mixes the medication comparison
with an age comparison.

## Compare similar patients

We can compare medications separately within each age group:

$$
\text{difference within an age group}
=\bar y_{\text{New}}-\bar y_{\text{Standard}}.
$$

**Worked example — compare within each age group.** For younger patients,
$125-130=-5$ mmHg; for older patients, $135-140=-5$ mmHg. A negative
difference means the new-medication group has lower average blood pressure.

```{r within-age-comparisons}
younger <- subset(blood_pressure, age_group == "Younger")
older <- subset(blood_pressure, age_group == "Older")

within_age <- data.frame(
  age_group = c("Younger", "Older"),
  difference = c(
    younger$mean_bp[younger$medication == "New"] -
      younger$mean_bp[younger$medication == "Standard"],
    older$mean_bp[older$medication == "New"] -
      older$mean_bp[older$medication == "Standard"]
  )
)
within_age
```

Within both age groups, the difference is only −5 mmHg. The larger overall
difference of −12 mmHg partly reflected the unequal age composition.

```{r visualize-comparisons}
bp_matrix <- xtabs(mean_bp ~ medication + age_group, data = blood_pressure)
barplot(bp_matrix, beside = TRUE, ylim = c(0, 150),
        col = c("steelblue", "tomato"),
        ylab = "Mean systolic blood pressure (mmHg)",
        legend.text = rownames(bp_matrix),
        args.legend = list(x = "topleft"))
```

This simple separation by age is useful for explaining confounding. In a real
study, many covariates may matter at the same time, and some age groups may
contain very few patients. More advanced statistical methods are then needed.

## Association and causation

Two variables are **associated** when their values tend to differ together.
An association can be measured from observational data, but it does not by
itself show that changing one variable will change the other.

A **causal effect** asks a different question: what would happen to the same
kind of patients under the new medication compared with the standard
medication? We cannot give both medications to the same person under identical
conditions, so study design matters.

In a randomized experiment, chance assigns the treatment. This helps make the
treatment groups similar in age and other characteristics before treatment.
An observational study can reveal useful associations, but differences between
the groups may provide other explanations for the result.

The lesson from this example is:

- compare similar groups when possible;
- look for variables connected to both the treatment and response; and
- do not describe an observed association as a causal effect without support
  from the study design and assumptions.

**Try it:** change the number of older and younger patients so both medication
groups have the same age composition. Recalculate the overall difference.