Author: Zhuoran (Angela) Yu
Date: 2025-11-12

# Linear_Model_Example

#### Zhuoran (Angela) Yu

#### 2025-11-12

Due to runtime constraints, some code chunks in this vignette are not executed during the build process. Users may run these examples locally or refer to the package website for the full outputs.

The example here is to use simulated data to construct the simultaneous outer and inner confidence regions (SCRs) from simultaneous confidence bands (SCB) using linear regression. `SCoRES::SCB_linear_outcome()` function uses a non-parametric bootstrap algorithm to construct the SCB in linear regression. The argument `df_fit` specifies a data frame containing the training design matrix used to fit the linear model, while `grid_df` defines the mean outcome for which simultaneous confidence bands (SCB) are constructed. Use argument `model` to specify the formula used for fitting the linear model.

## 1 SCBs for Mean Outcome of Linear Model

In the following example, we use a linear regression to estimate the simultaneous confidence band (SCB) for the mean outcome, with modeling equation as follows:

\[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i1}^2 + \beta_3 x_{i1}^3 + \beta_4 x_{i2} + \beta_5 x_{i2}^2 + \beta_6 x_{i2}^3 + \epsilon_i \]

We establish simultaneous confidence bands for the expected response surface \(E[Y|X_1,X_2]\), with the independent variables \(X_1\) and \(X_2\) discretized into 100 equidistant points over the domain \([-1, 1]\).

    library(SCoRES)
    set.seed(262)
    # generate simulated data
    x1 <- rnorm(100)
    x2 <- rnorm(100)
    epsilon <- rnorm(100,0,sqrt(2))
    y <- -1 + x1 + 0.5 * x1^2 - 1.1 * x1^3 - 0.5 * x2 + 0.8 * x2^2 - 1.1 * x2^3 + epsilon
    df <- data.frame(x1 = x1, x2 = x2, y = y)
    grid <- data.frame(x1 = seq(-1, 1, length.out = 100), x2 = seq(-1, 1, length.out = 100))
    # fit the linear regression model and obtain the SCB for y
    model <- "y ~ x1 + I(x1^2) + I(x1^3) + x2 + I(x2^2) + I(x2^3)"
    results <- SCB_linear_outcome(df_fit = df, model = model, grid_df = grid)

The `levels = c(-0.3, 0, 0.3)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plots the estimated inverse regions, the inner confidence regions, and the outer confidence regions.

    results <- tibble::as_tibble(results)
    suppressWarnings(plot_cs(results,
            levels = c(-0.3, 0, 0.3),
            x = seq(-1, 1, length.out = 100),
            mu_hat = results$Mean,
            xlab = "x1",
            ylab = "y",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black"))

In the following example, we establish simultaneous confidence bands for the expected response surface \(E[Y|X_1, X_2 =0]\), with the independent variable \(X_1\) discretized into 100 equidistant points over the domain \([-1, 1]\).

    x1 <- rnorm(100)
    x2 <- rnorm(100)
    epsilon <- rnorm(100,0,sqrt(2))
    y <- -1 + x1 + 0.5 * x1^2 - 1.1 * x1^3 - 0.5 * x2 + 0.8 * x2^2 - 1.1 * x2^3 + epsilon
    df <- data.frame(x1 = x1, x2 = x2, y = y)
    # fit the linear regression model and obtain the SCB for y
    model <- "y ~ x1 + I(x1^2) + I(x1^3) + x2 + I(x2^2) + I(x2^3)"
    grid <- data.frame(x2 = seq(-1, 1, length.out = 100))
    results <- SCB_linear_outcome(df_fit = df, model = model, grid_df = grid)

The `levels = c(-1.5, -1.0, -0.5)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plots the estimated inverse regions, the inner confidence regions, and the outer confidence regions.

    results <- tibble::as_tibble(results)
    suppressWarnings(plot_cs(results,
            levels = c(-1.5, -1.0, -0.5),
            x = seq(-1, 1, length.out = 100),
            mu_hat = results$Mean,
            xlab = "x2",
            ylab = "y",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black"))

## 2 SCBs for Mean Outcome of Logistic Model

In addition to linear regression, `SCoRES` also provides`SCoRES::SCB_logistic_outcome()` for estimating the SCB for outcome of logistic regression.

Here, we use a similar model structure with logistic regression to estimate the simultaneous confidence band (SCB) for the outcome probability, with modeling equation as follows:

\[ Y_i \sim Bernoulli(p_i) \] \[ \log\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i1}^2 + \beta_3 x_{i1}^3 + \beta_4 x_{i2} + \beta_5 x_{i2}^2 + \beta_6 x_{i2}^3 \]

We establish simultaneous confidence bands for the expected response surface \(\text{logit}(E[Y|X_1,X_2])\), with the independent variables \(X_1\) and \(X_2\) discretized into 100 equidistant points over the domain \([-1, 1]\).

    # generate simulated data
    x1 <- rnorm(100)
    x2 <- rnorm(100)
    mu <- -1 + x1 + 0.5 * x1^2 - 1.1 * x1^3 - 0.5 * x2 + 0.8 * x2^2 - 1.1 * x2^3
    p <- expit(mu)
    y <- rbinom(100, size = 1, prob = p)
    df <- data.frame(x1 = x1, x2 = x2, y = y)
    grid <- data.frame(x1 = seq(-1, 1, length.out = 100), x2 = seq(-1, 1, length.out = 100))
    # fit the logistic regression model and obtain the SCB for y
    model <- "y ~ x1 + I(x1^2) + I(x1^3) + x2 + I(x2^2) + I(x2^3)"
    results <- SCB_logistic_outcome(df_fit = df, model = model, grid_df = grid)

Likewise, the `levels = c(0.3, 0.4, 0.5)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plot the estimated inverse region, the inner confidence region, and the outer confidence region.

    results <- tibble::as_tibble(results)
    plot_cs(results,
            levels = c(0.3, 0.4, 0.5),
            x = seq(-1, 1, length.out = 100),
            mu_hat = results$Mean,
            xlab = "x1",
            ylab = "y",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black")

## 3 SCBs for Coefficients of Linear/Logistic Model

Besides, `SCoRES::SCB_regression_coef` can estimate the SCB for every coefficient in the linear/logistic model.

Here, we use a linear regression model with 50 coefficients in total to estimate the simultaneous confidence band (SCB) for all coefficients and the intercept, with modeling equation as follows: \[ Y_i = \beta_0 + \sum^{50}_{j=1}X_{ij}\beta_j + \epsilon_i, \] where \(Cov(X_i, X_j)=0.4^{|i-j|}\).

    library(MASS)
    # generate simulated data
    M <- 50
    rho <- 0.4
    n <- 500
    beta <- rnorm(M, mean = 0, sd = 1)
    Sigma <- outer(1:M, 1:M, function(i, j) rho^abs(i - j))
    X <- MASS::mvrnorm(n = n, mu = rep(0, M), Sigma = Sigma)
    epsilon <- rnorm(n, mean = 0, sd = 1)
    y <- X %*% beta + epsilon
    df <- as.data.frame(X)
    names(df) <- paste0(1:M)
    df$y <- as.vector(y)
    # fit the linear regression model and obtain the SCB for all betas
    model <- "y ~ ."
    results <- SCB_regression_coef(df, model)

Likewise, the `levels = c(-2, -0.5, -0.1, 0.2)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plots the estimated inverse regions, the inner confidence regions, and the outer confidence regions.

    results <- tibble::as_tibble(results)
    plot_cs(results,
            levels = c(-2, -0.5, -0.1, 0.2),
            x = c("intercept", as.character(1:50)),
            mu_hat = results$Mean,
            xlab = "Coefficients",
            ylab = "SCBs for Coefficients",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black")

Users can also compute the SCB for any linear combination of these coefficients by specifying a weight vector of the same length as the coefficient vector. This can be done by using `SCoRES::SCB_regression_outcome()`, and specifying the linear combination through `w`. For example, suppose we want to calculate the SCB for the linear combination \(\beta_1+\beta_2+\beta_3\) for model: \(y=\beta_0+\beta_1 X_1 +\beta_2X^2_1+\beta_3X^3_1+\beta_4X_2+\beta_5X^2_2+\beta_6X^2_3\):

    x1 <- rnorm(100)
    x2 <- rnorm(100)
    epsilon <- rnorm(100,0,sqrt(2))
    y <- -1 + x1 + 0.5 * x1^2 - 1.1 * x1^3 - 0.5 * x2 + 0.8 * x2^2 - 1.1 * x2^3 + epsilon
    df <- data.frame(x1 = x1, x2 = x2, y = y)
    grid <- data.frame(x1 = seq(-1, 1, length.out = 100), x2 = seq(-1, 1, length.out = 100))
    # fit the linear regression model and obtain the SCB for y
    model <- "y ~ x1 + I(x1^2) + I(x1^3) + x2 + I(x2^2) + I(x2^3)"
    w <- matrix(c(0, 1, 1, 1, 0, 0, 0), ncol = 7)

    SCB_regression_outcome(df_fit = df, model = model,
                           grid_df = grid, n_boot = 50, alpha = 0.1,
                           fitted = FALSE, w = w)
    #>            scb_low      Mean   scb_up
    #> scb_low -0.3723075 0.5968281 1.565964

In the following example, we load `ipad` dataset, and establish simultaneous confidence bands for all the coefficients and the intercept in the model fitted. Here, we set `log(t_mmr1)` as the outcome, and include all variables from `Pupillography`, `Tablet (task metrics)` and `Cardiovascular` as predictors.

    library(SCoRES)
    library(dplyr)
    library(ggplot2)
    set.seed(262)
    data(ipad)
    df <- ipad %>%
      filter(t_mmr1 > 0) %>%
      select(p_fpc1, p_fpc2, p_fpc3, p_fpc4, p_fpc5, p_fpc6,
             p_PMC_pctChg, p_auc, t_mmr1,
             i_prop_false_timeout, i_prop_failed1, i_prop_failed2,
             i_judgement_time1, i_judgement_time2, i_time_outside_reticle,
             i_time_on_edge, i_prop_hit, i_correct_reaction2,
             i_reaction_time_max2, i_reaction_time2, i_rep_shapes12,
             i_rep_shapes34, i_memory_time12, i_memory_time34,
             i_composite_score, h_hr, h_dbp, h_sbp, recent_smoke) %>%
      mutate(log_tmmr1 = log(t_mmr1))
    df_lin <- df %>%
      select(-recent_smoke, -t_mmr1)
    # fit the linear regression model and obtain the SCB for y
    model_lin <- "log_tmmr1 ~ ."
    results <- SCB_regression_coef(df_fit = df_lin, model = model_lin)

The `levels = c(0.2, 0.3, 0.4)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plot the estimated inverse region, the inner confidence region, and the outer confidence region. Here, for illustration, we filter out the SCB for intercept.

    results <- tibble::as_tibble(results[-1, ])
    suppressWarnings(plot_cs(results,
            levels = c(0.2, 0.3, 0.4),
            x = c(names(df_lin)[1:(length(names(df_lin))-1)]),
            mu_hat = results$Mean,
            xlab = "",
            ylab = "",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black"))

We fit a logistic model with `recent_smoke` as outcome with the same predictors in linear regression model, and obtain SCBs for all coefficients and the intercept.

    df_log <- df %>%
      select(-t_mmr1, -log_tmmr1)
    # fit the logistic regression model and obtain the SCB for y
    model_log <- "recent_smoke ~ ."
    results <- SCB_regression_coef(df_fit = df_log, model = model_log, type = "logistic")

The `levels = c(2e-07, 3e-07, 4e-07)` argument specifies a set of thresholds, and `SCoRES::plot_cs()` function estimates multiple inverse upper excursion sets corresponding to these thresholds, and plots the estimated inverse regions, the inner confidence regions, and the outer confidence regions. Here, for illustration, we filter out the SCB for intercept.

    results <- tibble::as_tibble(results[-1, ])
    suppressWarnings(plot_cs(results,
            levels = c(2e-07, 3e-07, 4e-07),
            x = c(names(df_log)[1:(length(names(df_log))-1)]),
            mu_hat = results$Mean,
            xlab = "",
            ylab = "",
            level_label = T,
            min.size = 40,
            palette = "Spectral",
            color_level_label = "black"))