Word Count : 666

---

Let’s solve this step by step, manually, using the Newton-Raphson method for logistic regression and then apply the Hosmer-Lemeshow test. We are given: * Initial parameter estimates: (\beta_0 = -2.148), (\beta_1 = 0.36) * Observations: (x_i) (concentration), (y_i) (recovered), (n_i) (total subjects) We want one iteration of Newton-Raphson. --- Step 1: Logistic regression basics For a logistic regression: [ \pi_i = \frac{e^{\beta_0 + \beta_1 x_i}}{1 + e^{\beta_0 + \beta_1 x_i}} ] The Newton-Raphson update is: [ \boldsymbol{\beta}^{(new)} = \boldsymbol{\beta}^{(old)} + (\mathbf{X}^\top \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\top (\mathbf{y} - \boldsymbol{\mu}) ] Where: * (\mathbf{X} = \begin{bmatrix} 1 & x_1 \ \vdots & \vdots \ 1 & x_n \end{bmatrix}) * (\mathbf{W} = \text{diag}(n_i \pi_i (1-\pi_i))) * (\boldsymbol{\mu} = n_i \pi_i) --- Step 2: Compute (\pi_i) with initial estimates [ \pi_i = \frac{e^{\beta_0 + \beta_1 x_i}}{1 + e^{\beta_0 + \beta_1 x_i}} ] * For (x_1 = 2): [ \eta_1 = -2.148 + 0.36 \cdot 2 = -2.148 + 0.72 = -1.428 ] [ \pi_1 = \frac{e^{-1.428}}{1 + e^{-1.428}} \approx \frac{0.240}{1 + ____ ________ __________ ___ _________ _________.
_____ ______ _________ __________ _______ _________.
___ ________ ____ ______ ______.
____ _______ _______ ______ ________ _________.
________ _______ ____ ___ _______ _______ _____ ____.
_________ ______ ________ _______ _________ ________ __________ ________ __________ ____ _____.
________ _________ _______ ________ __________ ___ ____ __________ ________ ________.
__________ _____ _________ __________ _____.
_____ ___ ______ ______ ___ ____ ______ ______ ___.
__________ _____ ____ ____ ___ ____ _______.
____ _______ ____ _________ __________ ________ ____ ______ ______ _________.
___ __________ _________ ________ ______ _______ ________ _______ ________.
_________ _______ ________ _______ _______.
________ _______ _____ ______ _____ ______ _____ ________ _____ ___.
__________ ______ _________ _________ ___ _____ _____ ____ ___.
______ _________ _______ ___ ___ _____ _______ ______ _________.
________ ___ __________ ______ ___ ______ _____ ________.
_______ _____ ____ ___ _______ _____ _______.
__________ ____ ______ ____ _________ ______ ____ ___ _____ ______ ________.
______ ______ _____ ________ _______ ______ ____ ______ _______ ___ _______.
____ ______ _________ _________ ___.
_______ __________ _______ _________ _______ ________.
____ ____ _____ ________ ___.
___ _____ _____ _______ __________ _______ _______ _________ _________ ________ __________ ____.
_______ ______ ____ ____ _________ ______ ________ ______.
_____ ________ __________ ___ ___ ____ ________.
________ _______ ________ ____ ________ __________ ____ ________ _______ ___ ___ ______.
____ ___ _________ ______ ____.
_________ ___ _________ ___ ___ _____ ______ ____ ________ ______.
________ __________ ______ ___ _______ ____ __________ ________ ______ ______.
__________ ____ ____ _____ ______ _______.
_____ ____ __________ _______ __________ ______ ___ ______.
___ __________ __________ ___ _________ __________ ___ _______ _____.
_______ _________ _______ _________ ____ ______ _________ _____.
__________ _________ ______ ________ ___ _____ _________ ________ ___ _________ ___ _______.
__________ ______ _____ ______ ______ _________.
______ __________ ____ ___ ___ ______ ______ __________ ____ _____.
________ _____ ____ ________ __________ __________ ___ __________ ____ _____ ___.
________ _________ ________ _________ _______ ______ _____.
_______ ______ ____ ____ ____.
__________ _______ ____ ______ ________ _____ __________ _________.
_______ ______ __________ ____ ______ __________ ____ ___ __________ _______ ____ ______.
________ ___ ______ _____ __________ _____ ____ __________.
__________ _____ _____ _________ ___ _______.
___ __________ ___ ____ ___ ___.
____ ___ _____ ___ _______ ___ ________ ______ _______ _________ ______ _______.
__________ ___ _________ ___ _________ _____.
________ ____ _________ _________ ____ ______ ______ __________ ___ ___ __________ _________.
____ ___ _____ ________ __________ _____ __________ _______ ______ __________ __________ _______.
_______ ________ ___ ___ ___.
________ ________ ____ ____ ____ ______ _______.
________ ________ ___ __________ ___ ___ _____.
_________ ____ _________ ________ __________ ___ _______ _________.
_______ _________ ______ ___ __________ ______ ________ __________ ________ _________.
___ ____ __________ ______ ___.
_______ _______ ______ ___ _________ ________ ____ ___.
______ ____ _________ __________ ______.
_____ ______ ____ ______ _________ __________ ________.
_________ ____ _____ _________ __________ _______ _____.
___ ___ __________ ________ _________ ___ ____ ______.
_______ ___ ____ ___ ______ ___.
___ ________ ___ ____ _______ _______.
____ ______ _________ _____ __________.
Get Full Answer on WhatsApp