Figure 4.6 data

This article uses the figure4_6 dataset to illustrate residual and influence diagnostics in linear regression, as described in RMB2e Chapter 4.

1 Introduction

Residual diagnostics are an essential part of the five-step RMB regression workflow: after fitting a model, examining residuals against fitted values, normal quantile plots, and Cook’s distance identifies violations of linearity, homoscedasticity, and normality that can invalidate inference. The figure4_6 dataset provides 100 observations of a single predictor x that serves as the basis for a pedagogical illustration of these diagnostic tools (RMB2e Ch. 4).

data(rmb_datasets, package = "rmb")
rmb_datasets$study_design[rmb_datasets$object == "figure4_6"]
#> [1] "Illustrative teaching dataset used for residual and influence diagnostics in Chapter 4."

Do the standard regression diagnostic plots reveal any violations of linear model assumptions for data simulated from a linear relationship with normally distributed errors?

1.1 Causal assumptions

set.seed(42)
dag <- ggdag::dagify(
  y ~ x,
  labels = c(y = "Outcome y", x = "Predictor x"),
  exposure = "x",
  outcome = "y"
)
ggdag::ggdag(dag, use_labels = "label", text = FALSE) +
  ggdag::theme_dag_blank() +
  ggplot2::labs(title = "Figure 4.6: Causal DAG")

Figure 1: Directed acyclic graph for the simulated linear-regression example.

2 Methods

2.1 Study sample

data(figure4_6, package = "rmb")
dat <- figure4_6
dim(dat)
#> [1] 100   1
summary(dat)
#>        x       
#>  Min.   :0.00  
#>  1st Qu.:0.25  
#>  Median :0.50  
#>  Mean   :0.50  
#>  3rd Qu.:0.75  
#>  Max.   :1.00

2.2 Statistical analysis

A simple linear regression of a simulated outcome y (constructed from x with normally distributed errors) is fitted, and the full suite of diagnostic plots from RMB2e Chapter 4 is produced: residuals vs fitted, normal Q-Q, scale-location, and Cook’s distance (RMB2e Ch. 4).

set.seed(42)
y <- 2 + 1.5 * dat$x + stats::rnorm(100)
demo_df <- data.frame(x = dat$x, y = y)

formula_main <- y ~ x
formula_main
#> y ~ x

3 Results

3.1 Descriptive statistics

summary(demo_df)
#>        x              y          
#>  Min.   :0.00   Min.   :-0.3989  
#>  1st Qu.:0.25   1st Qu.: 2.0596  
#>  Median :0.50   Median : 2.9090  
#>  Mean   :0.50   Mean   : 2.7825  
#>  3rd Qu.:0.75   3rd Qu.: 3.5636  
#>  Max.   :1.00   Max.   : 4.8002
with(demo_df, cor(x, y))
#> [1] 0.3730699

ggplot2::ggplot(demo_df, ggplot2::aes(x = x, y = y)) +
  ggplot2::geom_point(alpha = 0.4, size = 1.2) +
  ggplot2::geom_smooth(method = "lm", se = FALSE, color = "#d95f02", linewidth = 1) +
  ggplot2::labs(
    title = "Figure 4.6: Simulated data for diagnostic illustration",
    x = "x",
    y = "y (simulated)"
  ) +
  ggplot2::theme_minimal()

Figure 2: Scatterplot with fitted linear trend for the simulated outcome data.

3.2 Model estimates

fit <- stats::lm(formula_main, data = demo_df)
summary(fit)
#> 
#> Call:
#> stats::lm(formula = formula_main, data = demo_df)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.0195 -0.6618  0.0809  0.6527  2.2264 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   2.0682     0.2077   9.956  < 2e-16 ***
#> x             1.4286     0.3589   3.981 0.000132 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.046 on 98 degrees of freedom
#> Multiple R-squared:  0.1392, Adjusted R-squared:  0.1304 
#> F-statistic: 15.85 on 1 and 98 DF,  p-value: 0.000132

3.3 Model diagnostics

if (!requireNamespace("ggfortify", quietly = TRUE)) {
  stop("Package 'ggfortify' is required for model diagnostic autoplots.")
}
ggplot2::autoplot(
  fit,
  which = 1:4,
  ncol = 2
) +
  ggplot2::theme_minimal()

Figure 3: Standard 2x2 linear-model diagnostic panel from autoplot.lm.

3.4 Inference

ci <- stats::confint(fit)
coefs <- summary(fit)$coefficients
knitr::kable(data.frame(
  term = rownames(coefs),
  estimate = coefs[, "Estimate"],
  conf_low = ci[, 1],
  conf_high = ci[, 2],
  p_value = coefs[, "Pr(>|t|)"]
), digits = 3)

Table 1: Estimated regression coefficients with 95% confidence intervals and p-values.

	term	estimate	conf_low	conf_high	p_value
(Intercept)	(Intercept)	2.068	1.656	2.480	0
x	x	1.429	0.716	2.141	0

4 Discussion

When data truly follow a linear model with homoscedastic normal errors, the four standard diagnostic plots should show: no pattern in residuals vs fitted, points falling close to the diagonal in the Q-Q plot, roughly constant spread in the scale-location plot, and no observations with large Cook’s distances (RMB2e Ch. 4). This simulated dataset with correct model specification provides a baseline reference for what “good” diagnostics look like, against which practitioners can compare plots from real datasets where assumptions may be violated.

5 Source

• UCSF Regression Methods companion data: figure 4.6 illustrative dataset (RMB2e Chapter 4).
• Book: Vittinghoff E, Glidden DV, Shiboski SC, McCulloch CE (2012). Regression Methods in Biostatistics (2nd edition).