Introduction to programming for data analysis
Session 4 — Statistical Tests and Regressions
Computing the statistics “by hand” (4/4)
qt(p, df)is the quantile function of the Student’s \(t\)-distribution: it returns the value \(t^\star\) such that \(\mathbb{P}[T_{\nu} \le t^\star] = p\).- To get the critical value \(t_{1-\alpha/2,\,\nu}\) for a two-sided test at level \(\alpha\):
- Since \(|T| > t_{1-\alpha/2,\,\nu}\), we reject \(\text{H}_0\) at the 5% level: happy and unhappy countries do not have the same average GDP.
Code
alpha <- 0.05
t_crit <- qt(1 - alpha / 2, df = df_obs)
# x-range for the graph
x_max <- max(abs(t_obs), t_crit) + 1.5
curve_data <- tibble(x = seq(-x_max, x_max, by = .01)) |>
mutate(y = dt(x, df = df_obs))
# Rejection area
reject_data <- curve_data |>
filter(abs(x) >= t_crit)
ggplot(curve_data, aes(x = x, y = y)) +
# rejection regions (both tails), shaded
geom_area(
data = reject_data,
mapping = aes(group = x > 0),
fill = "#D55E00", alpha = .25
) +
geom_line(linewidth = 0.8) + # density curve
# critical values \pm t_crit
geom_vline(
xintercept = c(-t_crit, t_crit),
colour = "#D55E00", linetype = "dotted", linewidth = .7
) +
# observed statistic
geom_vline(
xintercept = t_obs,
colour = "#0072B2", linetype = "dashed", linewidth = .9
) +
# labels
annotate(
geom = "label",
x = t_obs, y = max(curve_data$y) * .95,
label = str_c("t[obs] == ", round(t_obs, 2)),
parse = TRUE, colour = "#0072B2", fill = "white"
) +
annotate(
geom = "text",
x = c(-t_crit, t_crit), y = -max(curve_data$y) * .03,
label = str_c(c("-", "+"), round(t_crit, 2)),
colour = "#D55E00", size = 3.5
) +
labs(
title = "Student's t-distribution under H0",
subtitle = str_c(
"df = ", round(df_obs, 1),
", shaded areas: rejection region at α = ", alpha
),
x = "t",
y = "Density"
) +
theme_minimal(base_size = 14) +
theme(plot.title.position = "plot")
Example
ggplot(
data = tb |> filter(year == 2018),
mapping = aes(x = gdp, y = life_satisf)
) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) +
labs(
x = "Real per capita GDP in 2013 (2020 Euro)",
y = "Life satisfaction"
) +
scale_x_continuous(
labels = scales::label_number(
suffix = "", scale = 1, big.mark = ","
)
)
Predictions
Residuals
Residuals: Quantile-Quantile Plot
Residuals: Relationship with \(X\)?
Bivariate Graph for Residuals
Interaction in a Model
We may believe the effect of a variable (\(x_1\)) depends on the value of another (\(x_2\)).
Visualize the Slopes in a Graph
A Quick Glance at the Residuals
Interaction Between Two Continuous Variables
