feat: new article

Author: DiogoRibeiro7

_posts/2020-01-30-cox_proportional_hazards_model.md

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 ---
 author_profile: false
 categories:
-- Medical Statistics
 - Data Science
 classes: wide
 date: '2020-01-30'
@@ -18,6 +17,10 @@ keywords:
 - Survival Analysis
 - Medical Statistics
 - Clinical Trials
+- Time-to-Event Data
+- Censored Data
+- Hazard Ratios
+- Proportional Hazards Assumption
 seo_description: Explore the Cox Proportional Hazards Model and its application in survival analysis, with examples from clinical trials and medical research.
 seo_title: Understanding Cox Proportional Hazards Model for Medical Survival Analysis
 seo_type: article
@@ -26,6 +29,144 @@ tags:
 - Cox Proportional Hazards Model
 - Survival Analysis
 - Medical Studies
+- Clinical Trials
+- Time-to-Event Data
+- Censored Data
 title: 'Cox Proportional Hazards Model: A Guide to Survival Analysis in Medical Studies'
 ---
 
+## Overview of the Cox Proportional Hazards Model
+
+In medical research, understanding how different factors impact patient survival is critical for guiding treatment decisions, improving healthcare outcomes, and evaluating the effectiveness of interventions. The **Cox Proportional Hazards Model** is one of the most widely used methods for analyzing **time-to-event data**, which records the time until a particular event of interest occurs, such as death, disease recurrence, or recovery.
+
+The Cox model, introduced by Sir David Cox in 1972, has become an essential tool in survival analysis because of its flexibility, particularly its ability to handle **censored data**. In survival studies, not all patients experience the event during the study period; some patients are lost to follow-up or their study period ends before the event occurs. The Cox model can accommodate this partial information, enabling researchers to still derive meaningful conclusions from incomplete data.
+
+### Why Use the Cox Proportional Hazards Model?
+
+The main reasons for the widespread use of the Cox model in medical studies include:
+
+- **Flexibility**: Unlike parametric models (e.g., exponential or Weibull models), the Cox model does not require a specific distributional form for survival times. Instead, it leaves the **baseline hazard** unspecified, making it a **semi-parametric model**. This allows it to be used in a wide variety of scenarios without strong assumptions about the underlying survival mechanism.
+  
+- **Handling of Censored Data**: The Cox model is particularly suited for survival data, where **censoring** is common. Censored observations occur when the event of interest has not yet been observed for some individuals by the end of the study or when a subject withdraws from the study before the event happens.
+
+- **Multiple Covariates**: The model allows researchers to examine the effect of multiple predictor variables (covariates) on survival simultaneously. This is crucial in medical studies where various factors—age, gender, treatment type, disease severity—may all influence patient outcomes.
+
+- **Hazard Ratios**: One of the strengths of the Cox model is its ability to compute **hazard ratios** for each covariate, which are easily interpretable as the relative risk of the event occurring for different levels of the covariates. For example, a hazard ratio of 2 for a certain covariate indicates that individuals with that characteristic have twice the risk of experiencing the event compared to those without it.
+
+Given its wide applicability, the Cox model is used extensively in medical research, from clinical trials evaluating new therapies to epidemiological studies investigating risk factors for chronic diseases.
+
+---
+
+## Understanding the Key Concepts
+
+To fully grasp the Cox Proportional Hazards Model, it's essential to understand the key statistical concepts that underpin it. This section explores the most important ideas in survival analysis and how they are applied in the Cox model.
+
+### Hazard Function
+
+The **hazard function**, denoted as $h(t)$, represents the **instantaneous rate of occurrence** of the event at time $t$, given that the individual has survived up until that point. In practical terms, the hazard function tells us how likely it is that an event (e.g., death or disease progression) will occur in the next moment, assuming that the individual has not experienced the event before time $t$.
+
+Mathematically, the hazard function can be expressed as:
+
+\[
+h(t) = \lim_{\Delta t \to 0} \frac{\Pr(t \leq T < t + \Delta t \mid T \geq t)}{\Delta t}
+\]
+
+Here, $T$ represents the time-to-event, and the hazard function captures the conditional probability of the event happening shortly after time $t$, given survival up to time $t$. The hazard function is closely related to the **survival function**, $S(t)$, which represents the probability of surviving beyond time $t$.
+
+The relationship between the hazard function and the survival function is:
+
+\[
+S(t) = \exp\left(-\int_0^t h(u) du \right)
+\]
+
+This shows that survival probabilities are directly influenced by the cumulative hazard over time.
+
+### Proportional Hazards Assumption
+
+The Cox model is built on the **proportional hazards assumption**, which states that the hazard ratio between any two individuals remains **constant over time**. This assumption simplifies the modeling process and makes the interpretation of covariates easier. In mathematical terms, the Cox model specifies that:
+
+\[
+h(t \mid X_i) = h_0(t) \cdot \exp(\beta_1 X_{i1} + \beta_2 X_{i2} + \dots + \beta_p X_{ip})
+\]
+
+Where:
+- $h_0(t)$ is the **baseline hazard**, representing the hazard function for an individual with baseline (or zero) values for all covariates.
+- $X_i$ is a vector of covariates for individual $i$.
+- $\beta_1, \dots, \beta_p$ are the regression coefficients corresponding to the covariates.
+
+The **exponentiated coefficients** $\exp(\beta_j)$ represent the **hazard ratio** associated with a one-unit increase in the covariate $X_j$. The proportional hazards assumption implies that while the baseline hazard function $h_0(t)$ may vary with time, the effect of the covariates on the hazard is multiplicative and **remains constant** over time.
+
+#### Testing the Proportional Hazards Assumption
+
+In practice, the proportional hazards assumption does not always hold. Violations of this assumption can lead to biased estimates and incorrect conclusions. To assess whether the assumption holds, researchers use several diagnostic techniques, including:
+
+- **Schoenfeld Residuals**: These residuals are used to test the proportional hazards assumption by examining whether the residuals for each covariate are independent of time. If a covariate’s residuals show a time-dependent pattern, this suggests that the proportional hazards assumption may be violated for that covariate.
+- **Graphical Methods**: Plotting **log-log survival curves** or **scaled Schoenfeld residuals** against time can provide a visual check for proportionality.
+
+If the proportional hazards assumption is violated, alternative models, such as **time-varying covariate models** or **stratified Cox models**, may be more appropriate.
+
+### Censored Data
+
+In survival analysis, not all subjects experience the event of interest during the study period. For these individuals, we only know that they have survived beyond a certain time, but we don't know when (or if) the event will occur. Such observations are referred to as **censored data**. Censoring can occur in several ways:
+
+- **Right Censoring**: This is the most common type of censoring, where the subject's event time is unknown but is known to be greater than the censoring time. For example, in a clinical trial, a patient may not have died by the time the study ends, so their survival time is censored.
+  
+- **Left Censoring**: Occurs when the event of interest has already happened before the subject enters the study, but the exact time of the event is unknown. For example, a patient may have already developed a disease before entering the study, but the exact onset time is unknown.
+
+- **Interval Censoring**: Happens when the exact time of the event is unknown, but it is known to occur within a specific time interval. For example, patients may be followed up at regular intervals, and the exact time of disease progression may fall between two follow-up visits.
+
+Handling censored data correctly is one of the strengths of the Cox Proportional Hazards Model. By incorporating censored data into the likelihood function, the model makes efficient use of all available information, even for subjects who do not experience the event during the study period.
+
+---
+
+## Mathematical Foundations of the Cox Model
+
+At the core of the Cox Proportional Hazards Model is its mathematical formulation, which allows for the flexible analysis of survival data without needing to specify a distribution for survival times. The Cox model is a **semi-parametric model**, meaning that it estimates the effects of covariates on the hazard function while leaving the baseline hazard function unspecified.
+
+### The Cox Proportional Hazards Function
+
+The Cox model expresses the **hazard at time $t$**, for an individual with covariate values $X = (X_1, X_2, \dots, X_p)$, as:
+
+\[
+h(t \mid X) = h_0(t) \cdot \exp(\beta_1 X_1 + \beta_2 X_2 + \dots + \beta_p X_p)
+\]
+
+Where:
+- $h(t \mid X)$ is the hazard function at time $t$ given the covariate values.
+- $h_0(t)$ is the **baseline hazard function**, representing the hazard for an individual with all covariates set to zero.
+- $\beta_1, \dots, \beta_p$ are the **regression coefficients** that quantify the relationship between the covariates and the hazard.
+
+The **baseline hazard function** $h_0(t)$ is left unspecified, which gives the Cox model its semi-parametric flexibility. However, the model does assume that the effects of the covariates on the hazard are **multiplicative** and constant over time.
+
+### Partial Likelihood and Parameter Estimation
+
+Unlike parametric models, the Cox model does not attempt to estimate the baseline hazard function directly. Instead, it uses the **partial likelihood method** to estimate the **regression coefficients** $\beta_1, \dots, \beta_p$. The partial likelihood focuses only on the ordering of event times, rather than their exact values, making the model more robust to the unknown baseline hazard.
+
+For a dataset with $n$ individuals, let $T_i$ denote the survival time for individual $i$, and let $\delta_i$ be an indicator variable that equals 1 if the event was observed for individual $i$, and 0 if the observation is censored. The **partial likelihood** for the Cox model is given by:
+
+\[
+L(\beta) = \prod_{i:\delta_i = 1} \frac{\exp(\beta' X_i)}{\sum_{j \in R(T_i)} \exp(\beta' X_j)}
+\]
+
+Here, $R(T_i)$ is the **risk set** at time $T_i$, representing the set of individuals who are still at risk of experiencing the event at time $T_i$. The partial likelihood is constructed by considering only the times when an event occurs and comparing the covariates of the individual who experienced the event to those of the individuals still at risk at that time.
+
+By maximizing the partial likelihood, we can estimate the **regression coefficients** $\beta_1, \dots, \beta_p$. These coefficients represent the **log-hazard ratios** for the covariates, and their **exponentiated values**, $\exp(\beta_j)$, represent the hazard ratios, which quantify the relative risk associated with each covariate.
+
+### Confidence Intervals and Hypothesis Testing
+
+Once the regression coefficients are estimated, we can compute **confidence intervals** for the hazard ratios to assess the precision of the estimates. A common method for constructing confidence intervals is based on the **Wald test**, which uses the estimated standard errors of the regression coefficients to compute confidence intervals.
+
+For each covariate $X_j$, the **Wald statistic** is given by:
+
+\[
+W_j = \frac{\hat{\beta}_j}{\text{SE}(\hat{\beta}_j)}
+\]
+
+Where $\hat{\beta}_j$ is the estimated coefficient, and $\text{SE}(\hat{\beta}_j)$ is its standard error. The Wald statistic follows a standard normal distribution under the null hypothesis that $\beta_j = 0$ (i.e., that the covariate has no effect on the hazard).
+
+Hypothesis testing in the Cox model often involves comparing nested models using the **likelihood ratio test** or examining individual covariates using the **Wald test**. These tests provide insights into the statistical significance of the covariates and help guide model selection.
+
+---
+
+
+

_posts/2024-06-13-Stepwise_regression.md

@@ -1,9 +1,6 @@
 ---
 author_profile: false
 categories:
-- Mathematics
-- Statistics
-- Data Science
 - Machine Learning
 classes: wide
 date: '2024-06-13'
@@ -28,6 +25,9 @@ tags:
 - python
 - r
 - julia
+- - Mathematics
+- Statistics
+- Data Science
 title: 'Stepwise Regression: Methodology, Applications, and Concerns'
 ---
 

_posts/2024-06-14-matthew_correlation.md

@@ -1,9 +1,6 @@
 ---
 author_profile: false
 categories:
-- Mathematics
-- Statistics
-- Data Science
 - Machine Learning
 classes: wide
 date: '2024-06-14'
@@ -27,6 +24,9 @@ keywords:
 - fortran
 - sh
 - c
+- Mathematics
+- Statistics
+- Data Science
 seo_description: Learn about Matthew’s Correlation Coefficient (MCC), an essential metric for evaluating binary classification models, particularly in imbalanced datasets, and how it improves upon traditional metrics.
 seo_title: 'Matthew’s Correlation Coefficient (MCC): A Guide to Binary Classification'
 seo_type: article