Scatterplots and Correlation

Introduction

• In this module, we begin discussing regression analysis.

• Regression is a very, very wide area.

  • We are barely scratching the surface.

• SDS II will deepen your understanding on statistical modeling techniques.

R Set Up

library(tidyverse)
library(ssstats)
training <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/7-1-training.csv")
wellness <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/7-2-wellness.csv")
reports <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/7-3-reports.csv")
break_room <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSI/refs/heads/main/files/data/lectures/7-4-break_room.csv")

Scatterplots

• A scatterplot is a graph that shows the relationship between two quantitative variables measured on the same individual.

• Every individual in the dataset is represented by a point on the scatterplot.

• Note! The explanatory variable is on the x-axis and the response variable is on the y-axis.

• Like we have seen before, plotting the data is a first step in identifying issues.

Scatterplots

• Positive relationship: As x increases, y also increases.

Scatterplots

• Negative relationship: As x increases, y decreases.

Scatterplots

• No relationship: As x increases, we cannot predict what happens to y.

Scatterplots

• Hmmmm…

Scatterplots

• Hmmmm…

Scatterplot (R)

• To construct a scatterplot in R, the base code uses ggplot() from the ggplot2 package (called in with tidyverse).

dataset_name |> ggplot(aes(x = x_variable, y = y_variable)) +
  geom_point() +
  theme_minimal() +
  labs(x = "Explanatory variable (x)",
       y = "Response variable (y)")

Example 1

• Okonkwo stops by your desk: “Training division keeps encouraging recruits to spend more time on practice exercises. Before we build extra practice time into the schedule, I want to know if recruits who practice more tend to score higher?”

• We will examine the following question using the training dataset:

  • Is there a relationship between weekly practice hours (x) and exam score (y)?

Example 1

• Constructing the scatterplot,

Example 2

• Okonkwo raises an eyebrow: “Someone in records claims our taller officers write more accurate incident reports. … That can’t be true. Is there actually a relationship?”

• We will examine the following question using the reports dataset:

  • Is there a relationship between officer height (x) and report accuracy (y)?

Example 2

• Constructing the scatterplot,

Correlation

• Correlation: A measure of the strength and direction of the linear relationship between two quantitative variables.

  • \rho represents the population correlation coefficient.

  • r represents the sample correlation coefficient.

• Note!! Correlation is bounded to [-1, 1].

  • r=-1 represents perfect negative correlation.
  • r=1 represents perfect positive correlation.
  • r=0 represents no correlation.

• The closer |r| is to 1, the stronger the relationship between x and y.

Pearson’s Correlation Coefficient

• Pearson’s correlation coefficient:

r = \frac{\sum_{i=1}^n \left( \frac{x_i - \bar{x}}{s_x} \right)\left( \frac{y_i - \bar{y}}{s_y} \right)}{n-1},

• where

  • x_i and y_i make up the the i^th observation
  • \bar{x} and \bar{y} are the sample means
  • s_x and s_y are the sample standard deviations
  • n is the total sample size

Pearson’s Correlation Coefficient

• We can also determine if a linear relationship exists.

• Hypotheses:

  • H_0: \rho = 0
  • H_1: \rho \ne 0

• Test Statistic:

t_0 = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \sim t_{n-2}

Pearson’s Correlation (R)

• We will use the correlation() function from library(ssstats).

• Generic syntax:

1dataset_name |> correlation(x = 1st_continuous_variable,
2                            y = 2nd_continuous_variable)

Example 1

• Finding the correlation for the training dataset,

  • Is there a relationship between weekly practice hours (x) and exam score (y)?

training |> correlation(x = practice_hours, y = exam_score)

• Note that the choice (order) of x and y are arbitrary.

training |> correlation(y = practice_hours, x = exam_score)

Example 1

• Hypotheses:

  • H_0: \rho = 0
  • H_1: \rho \ne 0

• Test Statistic and p-Value:

  • t_0 = 8.837
  • p < 0.001

• Rejection Region:

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation:

  • Reject H_0. There is sufficient evidence to suggest that there is a linear relationship between the practice time and exam score.

Example 2

• Finding the correlation for the reports dataset,

  • Is there a relationship between officer height (x) and report accuracy (y)?

reports |> correlation(height_cm, report_accuracy)

Example 2

• Hypotheses:

  • H_0: \rho = 0
  • H_1: \rho \ne 0

• Test Statistic and p-Value:

  • t_0 = 0.285
  • p = 0.777

• Rejection Region:

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation:

  • Fail to reject H_0. There is not sufficient evidence to suggest that there is a linear relationship between height and report accuracy.

Assumptions on Pearson’s Correlation (R)

• An assumption for Pearson’s correlation is that both x and y are normally distributed.

• We will check this using a q-q plot as we have in the past.

• We will use the correlation() function from library(ssstats).

• Generic syntax:

1dataset_name |> correlation_qq(x = 1st_continuous_variable,
2                               y = 2nd_continuous_variable)

Example 1

• Checking the assumption,

training |> correlation_qq(practice_hours, exam_score)

Example 2

• Checking the assumption,

reports |> correlation_qq(height_cm, report_accuracy)

Spearman’s Correlation Coefficient

• Spearman’s correlation coefficient is a nonparametric alternative to Pearson’s correlation.

  • Spearman’s correlation only assumes that the data is ordinal.

  • It does not work for nominal data.

• To compute Spearman’s, the following algorithm is used:

  1. Rank the x and, separately, the y values.

     • (x, y) \to (R_x, R_y)

  2. Find Pearson’s correlation for the ranked data.

• Spearman’s correlation is interpreted the same as Pearson’s correlation.

Pearson’s Correlation Coefficient

• We can also determine if a monotonic relationship exists.

• Hypotheses:

  • H_0: \rho_S = 0
  • H_1: \rho_S \ne 0

• Test Statistic:

S = \frac{n(n^2 - 1)}{6}(1 - r_s)

Spearman’s Correlation (R)

• We will again use the correlation() function from library(ssstats).

• Generic syntax:

1dataset_name |> correlation(x = 1st_continuous_variable,
2                            y = 2nd_continuous_variable,
3                            method = "spearman")

Example 2

• Finding the Spearman’s correlation for the reports dataset,

  • Is there a relationship between officer height (x) and report accuracy (y)?

reports |> correlation(height_cm, report_accuracy,
                       method = "spearman")

Example 2

• Hypotheses:

  • H_0: \rho_S = 0
  • H_1: \rho_S \ne 0

• Test Statistic and p-Value:

  • S_0 = 10396.93
  • p = 0.880

• Rejection Region:

  • Reject H_0 if p < \alpha; \alpha = 0.05

• Conclusion and Interpretation:

  • Fail to reject H_0. There is not sufficient evidence to suggest that there is a linear relationship between height and report accuracy.

Wrap Up

• guessthecorrelation.com!

• In the next lectures, we will learn how to quantify the relationship between x and y using regression analysis.