Module 3 Review

Putting It All Together

• Relationships between continuous variables.

• Correlation:

  • Unitless measure of linear (Pearson’s) or monotonic (Spearman’s) pairwise relationships.
    • Does not adjust for other variables.
  • Quantifies strength and direction of relationship.
    • Allows us to compare strength of relationships across different pairs of variables.

• Linear regression:

  • Models relationship between continuous outcome and one or more predictors.
    • Units are attached, so the relationship can be stated in context.
  • Provides estimates of association adjusted for other variables in the model.
  • Allows for prediction of outcome based on predictor values.

Example

• During a puzzle-heavy session in Ponyville, Twilight wants to understand what helps small teams solve logic puzzles faster. For each puzzle, she records:

  • MindPractice: hours of individual “mind” drills the day before.
  • SleepHours: hours slept the night before.
  • FocusMinutes: minutes of calm focus/box-breathing right before starting.
  • PuzzleComplexity: a continuous difficulty score Twilight assigns from 1–10.
  • SolveTime — minutes to solve the puzzle (lower is better).

• Our task is to determine how MindPractice, SleepHours, FocusMinutes, and PuzzleComplexity relate to SolveTime.

Example

• Let’s first look at pairwise scatterplots.

Example

• Next, let’s determine what method of correlation we should use. Checking normality,

Example

• Because we met the normality assumption, we can use Pearson’s correlation to quantify pairwise relationships.

Example

• Because we met the normality assumption, we can use Pearson’s correlation to quantify pairwise relationships.

• Mind practice is significantly correlated with solve time (p < 0.001).
  • The correlation is negative and modest (r \approx -0.42), indicating that increased mind practice is associated with decreased solve times (i.e., they are faster to solve).

Example

• Because we met the normality assumption, we can use Pearson’s correlation to quantify pairwise relationships.

• Sleep duration is significantly correlated with solve time (p < 0.001).
  • The correlation is negative and weak (r \approx -0.27), indicating that increased sleep duration is associated with decreased solve times (i.e., they are faster to solve).

Example

• Because we met the normality assumption, we can use Pearson’s correlation to quantify pairwise relationships.

• Focus is not significantly correlated with solve time (p = 0.219).
  • The correlation is negative and weak (r \approx -0.10), indicating that increased focus is associated with decreased solve times (i.e., they are faster to solve).

Example

• Because we met the normality assumption, we can use Pearson’s correlation to quantify pairwise relationships.

• Puzzle complexity is significantly correlated with solve time (p < 0.001).
  • The correlation is positive and modest (r \approx 0.64), indicating that increased complexity is associated with increased solve times (i.e., they are slower to solve).

Example

• Let’s now examine the multiple regression model,

• This gives the regression model,

\hat{\text{solve time}} = 37.54 - 4.12 \text{ practice} - 1.82 \text{ sleep} - 0.02 \text{ focus} + 3.10 \text{ complexity}

Example

• If I were reporting the table of results, I would submit

Example

• Before we make inference on each predictor, let’s examine the significance of the regression model.

Likelihood Ratio Test for Significant Regression Line:
Null: H₀: β₁ = β₂ = ... = βₖ = 0
Alternative: H₁: At least one βᵢ ≠ 0
Test statistic: χ²(4) = 8661.511
p-value: p < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.05)

Example

• Hypotheses
  • H_0: \beta_{\text{mind}} = \beta_{\text{sleep}} = \beta_{\text{focus}} = \beta_{\text{complex}} = 0
  • H_1: at least one \beta_i \ne 0
• Test Statistic and p-Value
  • \chi^2_0 = 8661.51; p < 0.001
• Rejection Region
  • Reject H_0 if p<\alpha; \alpha=0.05
• Conclusion and Intepretation
  • Reject H_0. There is sufficient evidence to conclude that at least one of the predictors is significantly associated with solve time, after adjusting for the other predictors.

Example

• Now, let’s provide some interpretations and comment on significance.

• Mind practice is significantly associated with solve time (p < 0.001). After adjusting for sleep, focus, and complexity, each additional hour of mind practice is associated with a decrease in solve time of approximately 4.12 minutes (95% CI: -5.02, -3.22).

Example

• Now, let’s provide some interpretations and comment on significance.

• Sleep hours is significantly associated with solve time (p < 0.001). After adjusting for mind practice, focus, and complexity, each additional hour of sleep is associated with a decrease in solve time of approximately 1.82 minutes (95% CI: -2.58, -1.06).

Example

• Now, let’s provide some interpretations and comment on significance.

• Focus minutes is not significantly associated with solve time (p = 0.857). After adjusting for mind practice, sleep, and complexity, each additional minute of focus is associated with a decrease in solve time of approximately 0.02 minutes (95% CI: -0.21, 0.17).

Example

• Now, let’s provide some interpretations and comment on significance.

• Puzzle complexity is significantly associated with solve time (p < 0.001). After adjusting for mind practice, sleep, and focus, each additional unit increase in puzzle complexity is associated with an increase in solve time of approximately 3.10 minutes (95% CI: 2.63, 3.57).

Example

• Let’s create a visualization of the model.

  • We know that solve time will be on the y-axis.
  • We can choose one predictor to put on the x-axis – let’s choose sleep duration.
  • The other predictors will be set to their median values.

# A tibble: 5 × 3
  variable         mean_sd    median_iqr 
  <chr>            <chr>      <chr>      
1 FocusMinutes     10.6 (4.7) 10.7 (6.8) 
2 MindPractice     2.9 (1.0)  2.8 (1.4)  
3 PuzzleComplexity 6.0 (1.9)  5.9 (3.0)  
4 SleepHours       7.1 (1.2)  7.1 (1.6)  
5 SolveTime        30.9 (9.4) 31.1 (12.5)

Example

• Our first step is to create predicted values using the found regression model.
  • Let sleep duration vary.
  • Plug in medians for other predictors.

\hat{\text{solve time}} = 37.54 - 4.12 \text{ practice} - 1.82 \text{ sleep} - 0.02 \text{ focus} + 3.10 \text{ complexity}

puzzles <- puzzles %>%
  mutate(predicted = 37.54 - 4.12 * median(MindPractice)
                           - 1.82 * SleepHours
                           - 0.02 * median(FocusMinutes)
                           + 3.10 * median(PuzzleComplexity))

Example

• Then we can graph.
  • Scatterplot: x = SleepHours, y = SolveTime
  • Line: x = SleepHours, y = predicted

puzzles %>% ggplot(aes(x = SleepHours, y = SolveTime)) + # specify x and y
  geom_point(size = 2.5, color = "gray50") + # plot points
  geom_line(aes(y = predicted), linewidth = 1, color = "black") + # plot line
  labs(x = "Sleep Duration (hours)", # edit x-axis label
       y = "Time to Complete Puzzle (min)") + # edit y-axis label
  theme_bw() # change theme of graph

Example

• The resulting graph,

Wrap Up

• This module covers the basics of linear regression.

• As a reminder, this is just the beginning. There are many more advanced topics to explore, including:

  • Interaction terms
  • Nonlinear relationships
  • Model selection
  • Diagnostics and remedial measures

• STA4231 - Statistics for Data Science II dives deeper into regression topics.