Linear regression with multiple predictors
Lecture 17
October 26, 2026
While you wait: Participate 📱💻
Which of the following is true about linear regression models?
- The regression line always passes through the origin (x = 0, y = 0).
- The slope indicates the average change in the outcome for a one-unit increase in the predictor.
- The intercept indicates the average value of the predictor when the outcome is 0.
- Least squares regression line minimizes the sum of the vertical distances between the observed and predicted values of the outcome.
Go to wooclap.com and use the code YJMSXJE.
Boxplot
Violin plot
Density plot
Code
ggplot(
nc_places,
aes(
x = mental_distress,
color = development_tier,
fill = development_tier
)
) +
geom_density(alpha = 0.5) +
scale_x_continuous(labels = label_percent(scale = 1)) +
scale_color_viridis_d(end = 0.85) +
scale_fill_viridis_d(end = 0.85) +
labs(
x = "Adults experiencing frequent mental distress (%)",
y = "Density",
color = "Development\ntier",
fill = "Development\ntier",
title = "Mental distress by county development tier"
)
Multiple geoms
Code
ggplot(
nc_places,
aes(x = development_tier, y = mental_distress)
) +
geom_boxplot(outlier.shape = NA) +
geom_jitter(width = 0.12, height = 0, size = 2, alpha = 0.5) +
scale_y_continuous(labels = label_percent(scale = 1)) +
labs(
x = "Development tier",
y = "Adults experiencing\nfrequent mental distress (%)",
title = "Mental distress by county development tier"
)
Participate 📱💻
Go to wooclap.com and use the code YJMSXJE.
Participate 📱💻
What is the baseline level of development_tier in the following model?
# A tibble: 3 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 18.4 0.136 136. 1.75e-112
2 development_tierTier 1 1.00 0.192 5.22 1.04e- 6
3 development_tierTier 3 -1.57 0.235 -6.70 1.40e- 9- Tier 1
- Tier 2
- Tier 3
- Cannot tell from the output
Go to wooclap.com and use the code YJMSXJE.
Visualize
Code
ggplot(
nc_places,
aes(x = loneliness, y = mental_distress, fill = development_tier)
) +
geom_point(
alpha = 0.7,
size = 3,
shape = "circle filled",
color = "#674574"
) +
scale_x_continuous(labels = label_percent(scale = 1)) +
scale_y_continuous(labels = label_percent(scale = 1)) +
scale_fill_viridis_d(end = 0.85) +
labs(
x = "Adults experiencing loneliness (%)",
y = "Adults experiencing\nfrequent mental distress (%)",
fill = "Development tier",
) +
theme(
legend.position = "inside",
legend.position.inside = c(0.8, 0.22)
)
Participate 📱💻
What do the following look like on a scatterplot?
Tier 1: \(\widehat{\text{mental distress %}} = 6.04 + 0.379 \times \text{loneliness %}\)
Tier 2: \(\widehat{\text{mental distress %}} = 5.66 + 0.379 \times \text{loneliness %}\)
Tier 3: \(\widehat{\text{mental distress %}} = 4.63 + 0.379 \times \text{loneliness %}\)
- Parallel lines with different intercepts
- Parallel lines with the same intercept
- Lines with different slopes and different intercepts
- Lines with different slopes and the same intercept
Go to wooclap.com and use the code YJMSXJE.
Model - visualize
Code
md_tier_lone_aug <- augment(
md_tier_lone_fit,
new_data = nc_places
)
ggplot(
md_tier_lone_aug,
aes(x = loneliness, y = mental_distress, fill = development_tier)
) +
geom_line(
aes(y = .pred, color = development_tier),
linewidth = 1
) +
geom_point(
alpha = 0.7,
size = 3,
shape = "circle filled",
color = "#674574"
) +
scale_x_continuous(labels = label_percent(scale = 1)) +
scale_y_continuous(labels = label_percent(scale = 1)) +
scale_color_viridis_d(end = 0.85, guide = "none") +
scale_fill_viridis_d(end = 0.85) +
labs(
x = "Adults experiencing loneliness (%)",
y = "Adults experiencing\nfrequent mental distress (%)",
fill = "Development tier",
title = "Main effects models",
subtitle = "Linear model for each tier: same slope, different intercepts"
) +
theme(
legend.position = "inside",
legend.position.inside = c(0.8, 0.22)
)
Other explanations?
Current model: Mental distress prevalence is higher by the same amount for each additional percentage point in loneliness prevalence, regardless of development tier.
Could there be a different reality?
Code
md_tier_lone_aug <- augment(
md_tier_lone_fit,
new_data = nc_places
)
ggplot(
md_tier_lone_aug,
aes(x = loneliness, y = mental_distress, fill = development_tier)
) +
geom_line(
aes(y = .pred, color = development_tier),
linewidth = 1
) +
geom_point(
alpha = 0.7,
size = 3,
shape = "circle filled",
color = "#674574"
) +
scale_x_continuous(labels = label_percent(scale = 1)) +
scale_y_continuous(labels = label_percent(scale = 1)) +
scale_color_viridis_d(end = 0.85, guide = "none") +
scale_fill_viridis_d(end = 0.85) +
labs(
x = "Adults experiencing loneliness (%)",
y = "Adults experiencing\nfrequent mental distress (%)",
fill = "Development tier",
) +
theme(
legend.position = "inside",
legend.position.inside = c(0.8, 0.22)
)
