README.Rmd

---
output: github_document
bibliography: references.bib
---

<!-- README.md is generated from README.Rmd. Please edit that file -->

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)
```

# RegrCoeffsExplorer

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_Always present effect sizes for primary outcomes_ [@Wilkinson1999Statistical].

_The size of the regression weight depends on the other predictor variables included in the equation and is, therefore, prone to change across situations involving different combinations of predictors_ [@Dunlap1998Interpretations].

_Any interpretations of weights must be considered context-specific_ [@Thompson1998]. 

_Dissemination of complex information derived from sophisticated statistical models requires diligence_ [@Chen2003].

The **goal** of `RegrCoeffsExplorer` is to **enhance the interpretation** of 
regression results by providing **visualizations** that integrate empirical data 
distributions. This approach facilitates a more thorough understanding of the 
impacts of changes exceeding one unit in the independent variables on the
dependent variable for models fitted within Linear Models (LM), 
Generalized Linear Models (GLM), and 
Elastic-Net Regularized Generalized Linear Models (GLMNET) frameworks.

## Installation

You can install the current version of `RegrCoeffsExplorer` from 
[GitHub](https://github.com/vadimtyuryaev/RegrCoeffsExplorer) with:

```{r, eval=F, echo=T, results="hide", warning=F, message=F }
library(devtools)

devtools::install_github("vadimtyuryaev/RegrCoeffsExplorer",    
                         ref = "main",
                         dependencies = TRUE,
                         build_vignettes = TRUE)
```
Alternatively, you can use the `remotes` library with the following command:
`remotes::install_github()`.

## A Very Brief Recap on Logistic Regression

Logistic regression [@mccullagh1989] is a statistical method used for binary classification.
Unlike linear regression, which predicts continuous outcomes, logistic 
regression predicts the probability of a binary outcome (1 or 0, Yes or No, 
True or False). The core function in logistic regression is the 
logistic function, also known as the **sigmoid function**, which maps any input 
into the range (0, 1), making it interpretable as a **probability**. In logistic
regression, we model the **log odds** (logarithm of Odds Ratio) of the probability 
as a linear function of the input features.

Sigmoid function is defined as:

$$\sigma(z)=\frac{1}{1+\exp(-z)}$$

Probability of **success** is calculated in the following manner:

$$P(Y=1|\mathbf{X}) = \pi(\mathbf{X}) = \frac{\exp(\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{n}X_{n})}{1+\exp(\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{n}X_{n})} =\frac{\exp(\mathbf{X}\beta)}{1+\exp(\mathbf{X}\beta)} = \frac{1}{1+\exp(-\mathbf{X}\beta)}$$

Odds Ratio is: 

$$OR = \frac{\pi(\mathbf{X})}{1-\pi(\mathbf{X})} =\frac{\frac{\exp(\mathbf{X}\beta)}{1+\exp(\mathbf{X}\beta)}}{1- \frac{\exp(\mathbf{X}\beta)}{1+\exp(\mathbf{X}\beta)}} =\exp(\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{n}X_{n})$$

Log Odds or the Logistic Transformation of the probability of success is:

$$\log{OR}=\log{\frac{\pi(\mathbf{X})}{1-\pi(\mathbf{X})}} =\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{n}X_{n}$$

Change in log odds when one predictor variable ($X_{1}$) increases by one unit, 
while **all other variables remain unchanged**:

$$\log{\frac{P(Y=1|\mathbf{X_{X_1=X_1+1}})}{P(Y=0|\mathbf{X_{X_1=X_1+1}})}} -\log{\frac{P(Y=1|\mathbf{X_{X_1=X_1}})}{P(Y=0|\mathbf{X_{X_1=X_1}})}} \overset{1}{=}$$

$$\overset{1}{=} \beta_{0}+\beta_{1}(X_{1}+1)+\ldots+\beta_{n}X_{n} - (\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{n}X_{n}) =\beta_{1}$$

Therefore, coefficient $\beta_{1}$ shows expected change in the Log Odds for a
one unit increase in $X_1$. Thus, expected change in the Odds Ratio is 
$\exp(\beta_{1})$. Finally, expected change in the Odds Ratio if $X_1$ changes 
by k units whilst all other variables remain unchanged is $[\exp(\beta_{1})]^k$ 
or $\exp(\beta_{1} \times k)$.

## Examples

### Consider implications of a change exceeding one unit on the Odds Ratio  

To generate a dataset for logistic regression analysis, we simulate four
continuous predictor variables and one categorical predictor variable. 
The continuous predictors are sampled from a normal distribution, each with
distinct means and standard deviations. The categorical predictor is generated 
as a dummy variable with two levels. The binary response variable is calculated 
by applying a logistic function to a linear combination of the predictors, 
thereby transforming the latent variable to a probability scale, which serves 
as the basis for generating binary outcomes through a binomial sampling process.

```{r example, fig.width=12, fig.height=6}

library(RegrCoeffsExplorer)
library(gridExtra)

# Set seed for reproducibility
set.seed(1945)

# Set the number of observations
n = 1000

# Random means and SDs
r_means = sample(1:5, 4, replace = TRUE)
r_sd = sample(1:2, 4, replace = TRUE)

# Generate predictor variables
X1 = rnorm(n, mean = r_means[1], sd = r_sd[1])
X2 = rnorm(n, mean = r_means[2], sd = r_sd[2])
X3 = rnorm(n, mean = r_means[3], sd = r_sd[3])
X4 = rnorm(n, mean = r_means[4], sd = r_sd[4])

# Create a dummy variable
F_dummy=sample(1:2, n, replace = TRUE) - 1

# Convert to factor
Factor_var=factor(F_dummy)

# Define coefficients for each predictor
beta_0 = -0.45
beta_1 = -0.35
beta_2 = 1.05
beta_3 = -0.7
beta_4 = 0.55  
beta_5 = 1.25

# Generate the latent variable
latent_variable = beta_0 + beta_1*X1 + beta_2*X2 + beta_3*X3 + beta_4*X4 +beta_5*F_dummy

# Convert the latent variable to probabilities using the logistic function
p = exp(latent_variable) / (1 + exp(latent_variable))

# Generate binomial outcomes based on these probabilities
y = rbinom(n, size = 1, prob = p)

# Fit a GLM with a logistic link, including the factor variable
glm_model = glm(y ~ X1 + X2 + X3 + X4 + Factor_var, 
                family = binomial(link = "logit"),
                data = data.frame(y, X1, X2, X3, X4, Factor_var))


grid.arrange(vis_reg(glm_model, CI = TRUE, intercept = TRUE,
        palette = c("dodgerblue", "gold"))$"SidebySide")


```

Note that upon consideration of the empirical distribution of
data, particularly concerning the influence on the response variable, `y`, 
attributable to the interquartile change (Q3-Q1) in the 
dependent variables, there is a discernible enlargement in the magnitudes of
coefficients `X2` and `X4`. 

Let us delve further into the underlying reasons for this phenomenon.

### Check estimated coefficients

```{r glm-summary}
summary(glm_model)
```

### Obtain Odds Ratio (OR)

```{r Odds-Ratio}
exp(summary(glm_model)$coefficients[,1])
```

The coefficients for `X1` through `X4` represent the change in the OR
for a **one-unit shift** in each respective coefficient, while the coefficient 
for `Factor_var1` signifies the variation in OR resulting from a transition from
the reference level of 0 to a level 1 in the factor variable. At first glance, 
it may seem that the factor variable exerts the most significant impact on the
odds ratio.Yet, this interpretation can often be deceptive, as it fails to take 
into account the distribution of empirical data.

### Real data differences

```{r real-data-differences}

# Calculate all possible differences (1000 choose 2)
all_diffs <- combn(X2, 2, function(x) abs(x[1] - x[2]))

# Count differences that are exactly 1 units
num_diffs_exactly_one = sum(abs(all_diffs) == 1)

# Count the proportion of differences that more or equal to 2 units
num_diffs_2_or_more = sum(abs(all_diffs)>=2)/sum(abs(all_diffs))

print("Number of differences of exactly 1 unit:")
num_diffs_exactly_one
print("Proportion of differences of two or more units:")
num_diffs_2_or_more

```

**None** of the differences observed within the values of the variable `X2` equate
to a single unit. Furthermore, **in excess of 15 percent** of these differences 
are equal or surpass a magnitude of two units.Therefore, when analyzing standard
regression output displaying per-unit interpretations, we, in a sense, comment
on difference that might not exist in the real data.Consequently, when engaging 
in the analysis of standard regression outputs that provide interpretations on a
per-unit basis, there is an implicit commentary on disparities that may not be 
present within the actual data. A more realistic approach is to utilize an 
actual observable difference, for example `Q3`-`Q1`, to calculate the OR. 

### Plot changes in OR and empirical data distribution for the `X2` variable 

```{r plot-Odds-Ratio,fig.width=12, fig.height=8}

plot_OR(glm_model, 
        data.frame(y, X1, X2, X3, X4, Factor_var), 
        var_name="X2",
        color_filling=c("#008000", "#FFFF00","#FFA500","#FF0000"))$"SidebySide"

```

The top plot delineates the variations in the OR corresponding to data 
differentials spanning from the minimum to the first quartile (Q1), 
the median (Q2), the third quartile (Q3), and the maximum.The bottom plot 
depicts a boxplot with a notch to display a confidence interval around the 
median and jitters to add random noise to data points preventing overlap and
revealing the underlying data distribution more clearly. Substantial changes in
the OR progressing alone the empirical data are clearly observed. 

### Customize plots - 1/2

```{r, warning=F, message=F}
require(ggplot2)

vis_reg(glm_model, CI = TRUE, intercept = TRUE,
        palette = c("dodgerblue", "gold"))$PerUnitVis+
  ggtitle("Visualization of Log Odds Model Results (per unit change)")+
  ylim(0,6)+
  xlab("Predictors")+
  ylab("Estimated OR")+
  theme_bw()+
  scale_fill_manual(values = c("red","whitesmoke" ))+                            
  theme(plot.title = element_text(hjust = 0.5))   

```
As observed, when returning individual plots, the resulting entities are `ggplot` 
objects. Consequently, any operation that is compatible with ggplot can be
applied to these plots using the `+` operator.

### Customize plots - 2/2

```{r, warning=F, message=F}
vis_reg(glm_model, CI = TRUE, intercept = TRUE,
        palette = c("dodgerblue", "gold"))$RealizedEffectVis+
  scale_fill_manual(values = c("#DC143C","#DCDCDC" ))+
  geom_hline(yintercept=exp(summary(glm_model)$coefficients[,1][3]*IQR(X2)), # note the calculation
             linetype="dashed", color = "orange", size=1)
  
```

## Vignettes

Would you like to know more? Please, check out the in-depth vignettes below. 

```{r vignettes, eval=F, echo=T, results="hide", warning=F, message=F}

vignette("BetaVisualizer", 
         package = "RegrCoeffsExplorer")  # To visualize realized effect sizes 

vignette("OddsRatioVisualizer",           # To visualize Odds Ratios
         package = "RegrCoeffsExplorer")

```

## A note on estimation of Confidence Intervals for objects fitted using regularized regression

It is imperative to to gain a comprehensive understanding  of the post-selection 
inference [@Hastie2015] rationale and methodologies  prior to the generation and
graphical representation of confidence intervals for objects fitted via the
Elastic-Net Regularized Generalized Linear Models. Please, kindly
consult the hyperlinks below containing the designated literature
and the `BetaVisualizer` vignette. 

1. [Statistical Learning with Sparsity](https://www.ime.unicamp.br/~dias/SLS.pdf)
2. [Recent Advances in Post-Selection Statistical Inference](https://www.math.wustl.edu/~kuffner/TibshiraniSlides.pdf)
3. [Tools for Post-Selection Inference](https://CRAN.R-project.org/package=selectiveInference)

## A cautionary note on intepretation of interaction effects in Generalized Linear Models (GLM)

A frequently misrepresented and misunderstood concept is that coefficients 
for interaction terms in GLMs **do not** have straightforward slope 
interpretations. This implies, among other considerations,
that in models including interaction terms, the ORs derived from 
coefficients might not be meaningful [@Chen2003]. Many situations demand 
recalculation of correct ORs, and the interpretation of interaction terms depends
on other predictors in the model due to inherent non-linearity. Consequently,
researchers should exercise caution when including and interpreting interaction
terms in GLM models.

In the following, we adopt the approach by @McCabe2021 to demonstrate 
computationally how interactions may depend on all predictors in the model and 
how these interactions can be estimated and interpreted on the probability scale.

### Theoretical considerations regarding the interpretation of interaction terms

Consider a Linear Model with two continuous predictors and an interaction term:

$$E[Y|\textbf{X}] = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2$$

Define the **marginal effect** by taking the partial derivative with respect to 
$x_2$:

$$\gamma_2 = \frac{\partial E[Y|\textbf{X}]}{\partial x_2} = \beta_2$$

Therefore, $\beta_2$ is sufficient to quantify how much $E[Y|\textbf{X}]$ changes
with respect to every one unit increase in $\beta_2$, holding all other variables
constant. 

Now, take the second order cross-partial derivative of $E[Y|\textbf{X}]$ with 
respect to both $x_1$ and $x_2$:

$$\gamma_{12}^2 = \frac{\partial^2 E[Y| \textbf{X}]}{\partial x_1 \partial x_2} = \beta_{12}$$

Similar intuition as above holds. The interaction term $\beta_{12}$ shows 
how effect of $x_1$ on $E[Y|\textbf{X}]$ changes for every one unit increase in
$x_2$ and vice versa. 

Now consider a logistic regression model with a non-linear link function $g(\cdot)$,
two continuous predictors and an interaction term: 

$$g(E[Y|\textbf{X}])=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2$$

Converting GLM to a natural scale using the inverse link function: 

$$E[Y|\textbf{X}]=g^{-1}(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2)$$

Note that the relationship is **no longer linear**. 

As an example, consider logistic regression: 

$$\log(\frac{E[Y|\textbf{X}]}{1-E[Y|\textbf{X}]})=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2=\eta$$

Transformation leads to:

$$E[Y|\textbf{X}]=\frac{1}{1+exp(-\{\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2\})}=\frac{1}{1+exp(-\eta)}=\frac{exp(\eta)}{1+exp(\eta)}$$

Let's take the second order cross-partial derivative.

Using the chain rule:

$$\gamma_{12}^2 = \frac{\partial^2 E[Y|\textbf{X}]}{\partial x_1 \partial x_2} = \frac{\partial^2 g^{-1}(\eta)}{\partial x_1 \partial x_2} = \frac{\partial }{\partial x_1} \left[ \frac{\partial g^{-1}(\eta)}{\partial x_2} \right] \overset{2}{=}$$

$$\overset{2}{=} \frac{\partial}{\partial x_1} \left[ \frac{\partial g^{-1}(\eta)}{\partial \eta} \frac{\partial \eta}{ \partial x_2} \right] = \frac{\partial}{\partial x_1} [(\beta_2+\beta_{12} x_1)\dot{g}^{-1}(\eta)]$$

Utilizing the product rule followed by the chain rule:

$$\frac{\partial}{\partial x_1} \left[(\beta_2+\beta_{12} x_1)\dot{g}^{-1}(\eta) \right] =\frac{\partial}{\partial x_1} [(\beta_2+\beta_{12} x_1)]\dot{g}^{-1}(\eta) +  [(\beta_2+\beta_{12} x_1)]\frac{\partial}{\partial x_1}[\dot{g}^{-1}(\eta)] \overset{3}{=}$$

$$\overset{3}{=} \beta_{12} \dot{g}^{-1}(\eta)+(\beta_2+\beta_{12}x_1)(\beta_1+\beta_{12}x_2)\ddot{g}^{-1}(\eta)$$

First and second derivative of the inverse link function are:

$$\dot{g}^{-1}(\eta)=\frac{exp(\eta)}{(1+exp(\eta))^2}$$

$$\ddot{g}^{-1}(\eta)=\frac{exp(\eta)(1-exp(\eta))}{(1+exp(\eta))^3}$$

Therefore: 

$$\gamma_{12}^2=\beta_{12} \frac{e^{\eta}}{(1+e^{\eta})^2}+(\beta_1+\beta_{12}x_2)(\beta_2+\beta_{12}x_1)\frac{e^{\eta}(1-e^{\eta})}{(1+e^{\eta})^3}$$

Calculation above show that  an interaction term in GLMs depends on all predictors 
within the model.This implies that the coefficient $\beta_{12}$ alone does not 
adequately describe how the effect of variable $x_1$ on $E[Y|\textbf{X}]$ changes for 
each one-unit increase in the variable $x_2$, and vice versa.

### Computational perspectives on interaction terms in Generalized Linear Models

We sample two moderately correlated predictors `X1b` and `X2b` from a standard
bivariate normal distribution and use them to simulate a logistic regression
model. By fitting the model, we obtain the estimated coefficients and 
calculate the values of $\hat{\gamma}^2_{12}$. Subsequently, we visualize
the slopes of $\hat{E}[Y|\mathbf{X}]$ calculated for several combinations of
`X1b` and `X2b`.

#### Sample two predictors `X1b` and `X2b` from a standard bivariate normal distribution: 

$$\left( \begin{array}{c}
X_1 \\
X_2 
\end{array} \right)
\sim \mathcal{N}\left( 
\begin{array}{c}
\begin{pmatrix}
\mu_1 \\
\mu_2 
\end{pmatrix}
\end{array}, 
\begin{pmatrix}
\sigma_1^2 & \rho \sigma_1 \sigma_2 \\
\rho \sigma_1 \sigma_2 & \sigma_2^2
\end{pmatrix}
\right)$$

Recall that the the pdf of the bivariate normal distribution has the following
form [@PSUStat505Bivariate]:

$$\phi(x_1, x_2) = \frac{1}{2\pi \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \exp\left(-\frac{1}{2(1 - \rho^2)} \left[ \left(\frac{x_1 - \mu_1}{\sigma_1}\right)^2 - 2\rho \left(\frac{x_1 - \mu_1}{\sigma_1}\right) \left(\frac{x_2 - \mu_2}{\sigma_2}\right) + \left(\frac{x_2 - \mu_2}{\sigma_2}\right)^2 \right] \right)$$

```{r}

# Load necessary library
library(MASS)          # for sampling from a multivariate normal distribution
library(ggplot2)
library(reshape2)      # for melting data frames

# Set parameters
n_samples   = 1000     # Number of samples
mean_vector = c(0, 0)  # Mean vector for X1 and X2
std_devs    = c(1, 1)  # Standard deviations for X1 and X2
correlation = 0.6      # Correlation between X1 and X2

# Generate the covariance matrix
cov_matrix = matrix(c(std_devs[1]^2, 
                       std_devs[1]*std_devs[2]*correlation, 
                       std_devs[1]*std_devs[2]*correlation, 
                       std_devs[2]^2), 
                    nrow = 2)

# Generate samples from the bivariate normal distribution
set.seed(2020)  
samples = mvrnorm(n = n_samples, mu = mean_vector, Sigma = cov_matrix)

# Convert samples to a data frame
bivariate_sample_df = data.frame(X1 = samples[, 1], X2 = samples[, 2])

X1b = bivariate_sample_df$X1
X2b = bivariate_sample_df$X2

```

#### Fit logistic model with an interaction term:

```{r}

# Set parameters
n_samples =  1000
beta_0    = -1
beta_1    =  2
beta_2    = -1.5
beta_3    =  0.5   # Coefficient for the interaction term

# Calculate probabilities including interaction term
log_odds = beta_0 + beta_1 * X1b + beta_2 * X2b + beta_3 * X1b * X2b
prob     = 1 / (1 + exp(-log_odds))

# Generate response variable
Y = rbinom(n_samples, size = 1, prob = prob)

# Fit logistic regression
data  = data.frame(Y = Y, X1 = X1b, X2 = X2b, X1X2 = X1b * X2b)
model = glm(Y ~ X1 + X2 + X1X2, family = binomial(link = "logit"), data = data)

# Print estimated coefficients
summary(model)

```
#### Check distribution of interaction term values at different values of `X1b` and`X2b`

```{r, fig.height=6, fig.width=6, dpi=1200}

# Define x1 and x2 ranges
x1_range     = seq(-3, 3, length.out = 100)
x2_quantiles = quantile(X2b, probs = c(0.25, 0.50, 0.75))

# Define the function to calculate gamma squared d[E[Y|X]]/dx1dx2
second_derivative = function(x1, x2, beta) {
  z = beta[1] + beta[2]*x1 + beta[3]*x2 + beta[4]*x1*x2
  g = exp(z) / (1 + exp(z))
  term1 = beta[4] * exp(z) / (1 + exp(z))^2
  term2 = (beta[2] + beta[4]*x2) * (beta[3] + beta[4]*x1) * exp(z) * (1 - exp(z)) / (1 + exp(z))^3
  gamma_squared = term1 + term2
  return(gamma_squared)
}

# Calculate gamma squared for each combination
gamma_squared_values = outer(x1_range, x2_quantiles, Vectorize(function(x1, x2) {
  second_derivative(x1, x2, coef(model))
}))

# Create a data frame from the matrix
gamma_df = as.data.frame(gamma_squared_values)

# Add X1 values as a column for identification
gamma_df$X1 = x1_range

# Melt the data frame for use with ggplot
long_gamma_df = melt(gamma_df, id.vars = "X1", 
                     variable.name = "X2_Quantile",
                     value.name = "GammaSquared")

# Convert X2_Quantile to a factor
levels(long_gamma_df$X2_Quantile) = c("25%", "50%", "75%")

# Plot the boxplot
ggplot(long_gamma_df, aes(x = X2_Quantile, y = GammaSquared)) +
  geom_boxplot() +
  labs(title = "Boxplot of Gamma Squared Values",
       x = "X2b Percentile",
       y = "Gamma Squared") +
  theme_minimal()+
  theme(plot.title = element_text(hjust = 0.5))

```

Note that the estimate of the interaction term is positive ($0.68345$). Yet, 
significant number of the gamma squared values are negative. Moreover, 
the magnitude and sign of $\hat{\gamma}_{12}^2$ are contingent upon the specific
combination of `X1b` and `X2b` variables utilized in the analysis.

#### Visualize changes in $\hat{E}[Y|\textbf{x}]$ associated with one unit increase in `X1b` at different quantiles of `X2b`. 

```{r,fig.height=8, fig.width=8, dpi=1200}

# Generate a sequence of x1 values for plotting
x1_values = seq(-1.5, 1.5, length.out = 100)

# Calculate predicted probabilities for each combination of x1 and quantile x2
predictions = lapply(x2_quantiles, function(x2) {
  predicted_probs = 1 / (1 + exp(-(coef(model)[1] + coef(model)[2] * x1_values + coef(model)[3] * x2 + coef(model)[4] * x1_values * x2)))
  return(predicted_probs)
})

# Calculate values for plotting slopes
x1_values_lines_1    = c(-1,0)
x1_values_lines_2    = c(0.5,1.5)
x2_quantiles_lines = quantile(X2b)[c(2,4)]

predictions_lines_1 = lapply(x2_quantiles_lines, function(x2) {
  predicted_probs = 1 / (1 + exp(-(coef(model)[1] + coef(model)[2] * x1_values_lines_1 + coef(model)[3] * x2 + coef(model)[4] * x1_values_lines_1 * x2)))
  return(predicted_probs)
})

predictions_lines_2 = lapply(x2_quantiles_lines, function(x2) {
  predicted_probs = 1 / (1 + exp(-(coef(model)[1] + coef(model)[2] * x1_values_lines_2 + coef(model)[3] * x2 + coef(model)[4] * x1_values_lines_2 * x2)))
  return(predicted_probs) 
})

# Plot the results
plot(x1_values, predictions[[1]], type = 'l', lwd = 2, ylim = c(0, 1), 
     ylab = "Estimated Probability", xlab = "X1b", main = "Interaction Effects")
segments(-1, predictions_lines_1[[2]][1],0, predictions_lines_1[[2]][2], 
         col="red4", lwd=2.5,lty="dotdash")
segments(-1, predictions_lines_1[[2]][1],0, predictions_lines_1[[2]][1], 
         col="red4", lwd=2.5,lty="dotdash")
segments( 0, predictions_lines_1[[2]][1],0, predictions_lines_1[[2]][2], 
          col="red4", lwd=2.5,lty="dotdash")
segments(-1, predictions_lines_1[[1]][1],0, predictions_lines_1[[1]][2], 
         col="red4", lwd=2.5,lty="dotdash")
segments(-1, predictions_lines_1[[1]][1],0, predictions_lines_1[[1]][1], 
         col="red4", lwd=2.5,lty="dotdash")
segments( 0, predictions_lines_1[[1]][1],0, predictions_lines_1[[1]][2], 
          col="red4", lwd=2.5,lty="dotdash")
segments(0.5, predictions_lines_2[[2]][1],1.5, predictions_lines_2[[2]][2],     
         col="springgreen4", lwd=3,lty="twodash")
segments(0.5, predictions_lines_2[[2]][1],1.5, predictions_lines_2[[2]][1], 
         col="springgreen4", lwd=3,lty="twodash")
segments(1.5, predictions_lines_2[[2]][1],1.5, predictions_lines_2[[2]][2], 
         col="springgreen4", lwd=3,lty="twodash")
segments(0.5, predictions_lines_2[[1]][1],1.5, predictions_lines_2[[1]][2], 
         col="springgreen4", lwd=3,lty="solid")
segments(0.5, predictions_lines_2[[1]][1],1.5, predictions_lines_2[[1]][1], 
         col="springgreen4", lwd=3,lty="solid")
segments(1.5, predictions_lines_2[[1]][1],1.5, predictions_lines_2[[1]][2], 
         col="springgreen4", lwd=3,lty="solid")
lines(x1_values, predictions[[2]], lty = 2, lwd = 2)
lines(x1_values, predictions[[3]], lty = 3, lwd = 2)
legend("topleft", legend = c("25% X2b", "50% X2b", "75% X2b"), lty = 1:3, lwd = 2)

```

The alterations in $\hat{E}[Y|\textbf{x}]$ associated with
one-unit increments in `X1b` at the first and third quartiles of `X2b` demonstrate 
significant disparities. Observe the variations in $\hat{E}[Y|\textbf{x}]$ for a 
unit change in `X1b` from $-1$ to $0$ (red) compared to the changes in
$\hat{E}[Y|\textbf{x}]$ for a unit change in X1b from $0.5$ to $1.5$ (green). 
Observe how the magnitudes of changes in the estimated probability associated
with a unit increment in `X1b` vary depending on the position along the number 
line where the increment occurs. This observation substantiates the preliminary
assertion regarding the inadequacy of $\hat{\beta}_{12}$ in capturing the
interaction effects on the probability scale within the GLM framework, which
incorporates two continuous predictors and an interaction term. Consequently,
meticulous attention is required when incorporating interaction terms and 
interpreting the outcomes of models that include such terms.

# References