MishenMed: Statistical Consulting for Clinical Trials

 The Cox Proportional Hazards (Cox PH) model , introduced by Sir David Cox in 1972 , is a semi-parametric survival analysis model used to assess the effect of covariates on the time-to-event outcome, such as progression-free survival (PFS) or overall survival (OS) in clinical trials. 1. Model Specification The hazard function at time t t  is modeled as: h ( t ∣ X ) = h 0 ( t ) exp ⁡ ( β 1 X 1 + β 2 X 2 + . . . + β p X p ) h(t | X) = h_0(t) \exp(\beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p) where: h ( t ∣ X ) h(t | X) = hazard function (risk of the event occurring at time t t ) h 0 ( t ) h_0(t) = baseline hazard (when all covariates are zero) X 1 , X 2 , . . . , X p X_1, X_2, ..., X_p = covariates (e.g., treatment group, age, biomarker levels) β 1 , β 2 , . . . , β p \beta_1, \beta_2, ..., \beta_p = regression coefficients exp ⁡ ( β ) \exp(\beta)  = hazard ratio (HR) for each covariate 2. Key Interpretations a. Hazard Ratio (HR) HR = exp ⁡ ( β ) \exp(\beta)   ...

Try to interpret the following survival analysis results from an oncology clinical trial: Hazard Ratio (HR) for PFS = 0.58 (95% Confidence Interval [CI]: 0.49–0.70 ) Median Progression-Free Survival (PFS) = 16.4 months These values describe the efficacy of a treatment in delaying disease progression compared to a control group. 1. Interpretation of the Hazard Ratio (HR) for PFS = 0.58 Definition : The hazard ratio (HR) quantifies the relative risk of disease progression or death between two groups (e.g., treatment vs. control). HR = 0.58 means: The treatment group has a 42% lower risk (1 - 0.58 = 0.42 or 42% reduction ) of disease progression or death compared to the control group. Patients receiving the treatment are progressing more slowly than those in the control group. 95% Confidence Interval (0.49–0.70) : The true hazard ratio is likely between 0.49 and 0.70 in 95% of similar trials . Since the entire CI is below 1 , the result is statistically significant (indicating ...

  The Kaplan-Meier (K-M) survival plot is a statistical tool used in clinical research to estimate and visualize time-to-event data , such as progression-free survival (PFS) or overall survival (OS). It helps compare the probability of survival (or remaining event-free) over time between different treatment groups. Key Components of a K-M Survival Plot X-Axis (Time in Months/Years) Represents time since treatment initiation or study enrollment . Could be in months, years, or another unit depending on the study. Y-Axis (Survival Probability or PFS Probability) Represents the proportion of patients still event-free (e.g., alive, progression-free) at a given time point. Starts at 1 (100%) when all patients are event-free and decreases over time. Survival Curves (One for Each Group) Each curve shows the estimated survival probability over time for a specific group (e.g., treatment vs. control ). A steeper drop indicates faster progression or death in that group. If curves remain ...

  In oncology clinical trials, various endpoints are used to evaluate treatment efficacy. These can be classified into tumor response-based , time-to-event , and patient-reported endpoints. 1. Tumor Response-Based Endpoints a. Overall Response Rate (ORR) Definition : The proportion of patients with a confirmed complete response (CR) or partial response (PR) based on RECIST criteria . Formula : O R R = CR + PR Total Patients Assessed × 100 Significance : A key efficacy measure, often used in early-phase trials . b. Best Overall Response (BOR) Definition : The best response (CR, PR, stable disease [SD], or progressive disease [PD]) observed in a patient at any time during the trial. Interpretation : Shows the maximum effect of the treatment. c. Duration of Response (DOR) Definition : The time from first confirmed response (CR or PR) until disease progression or death . Significance : Measures how long a patient benefits from the therapy. Example : If DOR = 8.5 months ,...

 Target: Clustering | Anomaly detection  in contrast: clustering Applications of clustering: Grouping similar news Market segmentation DNA analysis Astronomical data analysis K-means intuition: Repeat the steps above to achieve the following result. Continue until no further changes occur, indicating convergence. K-means algorithm: If a cluster has zero points, then k=k-1 clusters. or reinitialize k cluster centroids. K-means Optimize the distortion cost function: The distortion cost function should go down. How to initialize k-means? Always use multiple random initializations, as this significantly improves K-means' ability to minimize the distortion cost function and select better cluster centroids. Choosing the number of clusters: Elbow method is not frequently used You can use K-means to determine t-shirt sizes by clustering data. Using three clusters may categorize sizes as small, medium, and large, while five clusters could define extra small, small, medium, large, and e...

 TensorFlow implementation How data is represented in Numpy and Tensorflow ? FYI: TensorFlow: An Overview TensorFlow is an open-source machine learning library developed by Google that is widely used for deep learning, neural networks, and large-scale machine learning tasks. It provides tools for building, training, and deploying machine learning models efficiently. 1. Key Features of TensorFlow Scalability : Supports CPUs, GPUs, and TPUs (Tensor Processing Units). Flexibility : Works for both simple models (linear regression) and complex deep learning architectures (CNNs, RNNs, Transformers). Eager Execution & Graph Computation : Allows dynamic computation (eager mode) and optimized execution (graph mode). TensorFlow Extended (TFX) : Tools for deploying ML models in production. Integration with Keras : TensorFlow supports Keras as a high-level API for building deep learning models. Building a neural network in TensorFlow  simplify the code above  Most machine lear...

 Neural Networks inpput layer----hidden layer (activation values)----output layer The nice properties of a neural network are that when you train it from data, you don't need to go in to explicitly decide what other features, such as affordability and so on, the neural network should compute instead or figure out all by itself what are the features it wants to use in this hidden layer.  computer vision application  neural network More complex neural networks 4 layers = hidden layers + output layer (not counting the input layer ) Foward propagation: making predictions Since this computation moves from left to right, you start with a1, followed  by a 2 a_2 , and then a 3 a_3 . This process is known as forward propagation because it propagates neuron activations forward through the network. It contrasts with backpropagation , which is used to adjust weights during training. Additionally, a neural network architecture where the number of hidden units is larger in earlie...