In huangrh/rstarating: CMS Hospital Compare Star Rating Replicate

1. Latent Variable Model

${X_m = \mu_m + \ell_m u + \epsilon_m}$

$\epsilon_m \sim \mathcal{N}(0,\sigma^2)$

$u \sim \mathcal{N}(0,1) \hspace{71pt}$

• $X_m$ : The standardized measure vriable
• $\mu_m$ : The offset to the mean
• $\ell_m$ : The loading of the latent variable ${u}$
• $\mathcal{u}$: The latent variable
• $\epsilon_m$: Normal distributed error
• Example: suppose we have a set of M measure variables ($m = 1,2,...,m,...,M$)
  ${X_1 = \mu_1 + \ell_1 u + \epsilon_1}$
  ${X_2 = \mu_2 + \ell_2 u + \epsilon_2}$
  $...$
  ${X_m = \mu_m + \ell_m u + \epsilon_m}$
  $...$
  ${X_M = \mu_M + \ell_M u + \epsilon_M}$

2. Weighted likelihood ($\mathcal{L}$) & log likelihood($\mathcal{LL}$)

For each hospital $h$:
$\mathcal{L}^{(h)} = \prod_{m=1}^M \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m})^{w_m^{(h)}} \hspace{35pt}$

or

$\mathcal{LL}^{(h)} = \sum_{m=1}^M (w_m^{(h)} * \log( \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m}))) \hspace{35pt}$

For all hospitals $(h= 1,2,3,...,H)$:

$\mathcal{L}^{(total)} = \prod_{h=1}^H \prod_{m=1}^M \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m})^{w_m^{(h)}} \hspace{35pt}$

or

$\mathcal{LL}^{(total)} = \sum_{h=1}^H \sum_{m=1}^M (w_m^{(h)} * \log( \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m}))) \hspace{35pt}$

• $\mathcal{L}$: normal likelihood function

3. Random Effect Log Likelihood

$Random^{(h)} = \log(\mathcal{L}({u}^{(h)}, mean = 0, sd = 1))$

4. The Random-Effect Latent Variable Model (Objective Function)

The joint log likelihood for one hospital ${h}$

• Joint likelihood
  $Joint{_\mathcal{L}}^{(h)} = \mathcal{L}^{(h)} \mathcal{L}({u}^{(h)}, mean = 0, sd = 1)$
  or
  $Joint_\mathcal{L}^{(h)} = \prod_{m=1}^M \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m})^{w_m^{(h)}} \mathcal{L}({u}^{(h)}, mean = 0, sd = 1)$

or

• Joint log likelihood
  $Joint{_\mathcal{LL}}^{(h)} = \mathcal{LL}^{(h)} + Random^{(h)}$
  or

$Joint_{\mathcal{LL}}^{(h)} = \sum_{m=1}^M (w_m^{(h)} * \log( \mathcal{L}( X_{m}^{(h)}, \; mean=\mu_{m} + \ell_{m} * {u}^{(h)}, sd = \sigma_{m}))) + \log(\mathcal{L}({u}^{(h)}, mean = 0, sd = 1))$

The joint log likelihood for all hospitals $(h = 1,2,3,...,H)$

$Joint^{(Total)} = \sum_{h=1}^{H}(\mathcal{LL}^{(h)} + Random^{(h)}) \hspace{40pt}$

• $H$: The total number of hospitals in the model.

5. The marginal distribution in a mixed model:

$m^{(h)}(\theta) = \int \mathcal{L}^{h}\mathcal{L}({u}^{(h)},mean=0,sd=1)d{u}^{(h)}$

6. The objective function

$f(\theta) = -\log\prod_{h=1}^{H}m^{(h)}(\theta)$

7. Quadrature approximation:

7.1 Gaussian-Hermite quadrature rule:

$\int_{}^{} \exp(-x^2)f(x)\mathrm{dx} \approx \sum_{i=1}^{N}w_{i}f(x_{i})$

Consider a function $h(y)$, where the variable $y$ is Normally districubted: $y \sim N(\mu,\sigma^2)$. The expectation of $h(y)$ corresponds to the follwoing integral:

$\int \cfrac{1}{\sigma \sqrt{2 \pi} } \exp(- \cfrac{(a-\hat{u})^2}{2 \sigma ^2}) h(a) \mathrm{da} \hspace{20pt} \$ # $substitute \; z = \cfrac{a-\hat{u}}{\sqrt{2} \sigma} \to a = \hat{u} + \sqrt{2} \sigma z$

$= \int \cfrac{1}{\sigma \sqrt{2 \pi} } \exp(- \cfrac{(a-\hat{u})^2}{2 \sigma ^2}) h(\hat{u} + \sqrt{2} \sigma z) \mathrm{d}(\hat{u} + \sqrt{2} \sigma z)$

$= \int \cfrac{1}{\sqrt{\pi}}\exp(-z^2)h(\sqrt{2}\sigma z + \mu) \mathrm{dz}\$

$\approx \cfrac{1}{\sqrt{\pi}} \sum_{i=1}^{N}w_i\cdot h(\sqrt{2}\sigma z_i + \mu)$

Ref: Gauss Hermite_quadrature

7.2 Marginal probability of mixed effect with Gauss-Hermite quadrature approximation for hospital $h$:

$\int p(y^h)q(u)\mathrm{du}$

$= \int p(y^h)q(a)\mathrm{da} \hspace{85pt}$ # Substitute variable $u \rightarrow a$

$= \int p(y^h)q(a) \exp(z^2) \exp(-z^2) \mathrm{da} \hspace{12pt}$ # Add $\exp(z^2)\exp(-z^2) \hspace{15pt} (where\;z = \cfrac{a-\hat{u}}{\sqrt{2} \sigma})$

$= \int p(y^h)q(a) \exp(z^2) \exp(-z^2) \mathrm{d}(\hat{u} + \sqrt{2} \sigma z)$ # Substitute $a$ with $\hat{u} + \sqrt{2} \sigma z$

$= \sqrt{2} \sigma \int p(y^h)q(a) \exp(z^2) \exp(-z^2) \mathrm{dz}$ # Simpflify

$\approx \sqrt{2} \sigma \sum_{i=1}^{N} w_i \cdot p(y^h) q(a_i) \exp(z_i^2)$

• $a_i = \hat{u} + \sqrt{2} \sigma z_i$
• $\sigma = \cfrac{1}{\sqrt{f^{''}(- \log (p(y^h)q(u)))}}$, $f^{''}$ is the second derivative of $f(u) = - \log (p(y^h)q(u)))$

• $z_i,w_i (i = 1,...N): \text{the standard Gauss-Hermite abscissas and weights}$

• $f^{'} = \cfrac{\partial}{\partial{u}} (-\log[p(y^h)q(u)])$ $= u_h - \sum w_m^h\cfrac{y_m^h-(\mu_m + l _m \cdot u^h)}{\sigma _m^2} \cdot \ell_m$

• $f^{''} = \cfrac{\partial{^{2}}}{\partial{u{^{2}}}} (-\log[p(y^h)q(u^h)])$ $= 1 + \sum w_m^h\cdot \cfrac{l _m ^2}{\sigma _m ^2}$

8.

8.1 Likelihood function definition:

The likelihood of a parameter value (or vector of parmeter values) $\theta$, given outcomes $\mathcal{x}$, is equal to the probability (density) assumed for those observed outcomes given those parameter values, that is:

$\mathcal{L}(\theta | \mathcal{x}) = \mathrm{P}(\mathcal{x}|\theta)$

Example, a likelihood function for a normal distribution is below:
$\mathcal{L(\mu,\sigma^2|x) = f(x|\mu,\sigma^2) = \cfrac{1}{\sqrt{2 \pi \sigma ^2}}\exp{(- \cfrac{(x-\mu)^2}{2 \sigma ^2})}}$

8.2 The Objective function:

$\mathcal{ -\log [\prod_{h=1}^\mathrm{H}m^{h}(\theta)] = - \sum_{h=1}^\mathrm{H}\log m^{h}(\theta)}$

• $\mathcal{m^{h}(\theta)}$ is the marginal likelihood function for a hospital $\mathcal{h}$.

8.3 The marginal likelihood function for a hospital $\mathcal{h}$:

1. $\mathcal{ m^h(\theta) = \int \prod_{m=1}^{\mathrm{M}}[\cfrac{1}{\sqrt{2 \pi \sigma_m^2}}\exp[-\cfrac{1}{2}\cfrac{(x_m - mean)^2}{\sigma_m^2}]]^{\mathrm{w}_m^h} \cdot \cfrac{1}{\sqrt{2 \pi}}\exp[- \cfrac{u^2}{2}]} \mathrm{du}$

2. where mean =$\mu_{m} + \ell_{m} * \mathcal{u^{h}}$

3. $\mathcal{ m^h(\theta) = \int \prod_{m=1}^{\mathrm{M}}[\cfrac{1}{\sqrt{2 \pi \sigma_m^2}}\exp[-\cfrac{1}{2}\cfrac{(x_m - \mu_{m} - \ell_{m} * \mathcal{u})^2}{\sigma_m^2}]] ^{\mathrm{w}_m^h} \cdot \cfrac{1}{\sqrt{2 \pi}}\exp(- \cfrac{u^2}{2})} \mathrm{du}$

4. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \prod_{m=1}^{\mathrm{M}} \exp[-\cfrac{1}{2}\cfrac{(x_m - \mu_{m} - \ell_{m} * \mathcal{u})^2}{\sigma_m^2}] ^{\mathrm{w}_m^h} \cdot \cfrac{1}{\sqrt{2 \pi}}\exp(- \cfrac{u^2}{2})} \mathrm{du}$

5. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \prod_{m=1}^{\mathrm{M}} \exp[-\cfrac{w_m^h}{2}\cfrac{(x_m - \mu_{m} - \ell_{m} * \mathcal{u})^2}{\sigma_m^2}] \cdot \cfrac{1}{\sqrt{2 \pi}}\exp(- \cfrac{u^2}{2})} \mathrm{du}$

6. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \prod_{m=1}^{\mathrm{M}} \exp[-\cfrac{w_m^h}{2 \sigma_m^2} (x_m - \mu_{m} - \ell_{m} * \mathcal{u})^2] \cdot \cfrac{1}{\sqrt{2 \pi}} \exp(- \cfrac{u^2}{2})} \mathrm{du}$

7. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \prod_{m=1}^{\mathrm{M}} \exp[-\cfrac{w_m^h}{2 \sigma_m^2} [(x_m - \mu_{m})^2 - 2 (x_m - \mu_{m}) (\ell_{m} * \mathcal{u})+ (\ell_{m} * \mathcal{u})^2]] }$ ${\cdot \cfrac{1}{\sqrt{2 \pi}} \exp(- \cfrac{u^2}{2})} \mathrm{du}$

8. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \exp\sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2} [(x_m - \mu_{m})^2 - 2 (x_m - \mu_{m}) (\ell_{m} * \mathcal{u})+ (\ell_{m} * \mathcal{u})^2]] }$ ${\cdot \cfrac{1}{\sqrt{2 \pi}} \exp(- \cfrac{u^2}{2})} \mathrm{du}$

9. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \exp {\sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2}(x_m - \mu_{m})^2] +
   \sum_{m=1}^{\mathrm{M}} [\cfrac{w_m^h}{ \sigma_m^2} (x_m - \mu_{m}) (\ell_{m} * \mathcal{u})]
   }$ ${+ \sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2} (\ell_{m} * \mathcal{u})^2]}
   \cdot \cfrac{1}{\sqrt{2 \pi}} \exp(- \cfrac{u^2}{2})} \mathrm{du}$

10. $\mathcal{ m^h(\theta) = \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \exp {\sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2}(x_m - \mu_{m})^2] +
    \sum_{m=1}^{\mathrm{M}} [\cfrac{w_m^h}{ \sigma_m^2} (x_m - \mu_{m}) (\ell_{m} * \mathcal{u})] }$ ${+ \sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2} (\ell_{m} * \mathcal{u})^2]}
    \cdot \cfrac{1}{\sqrt{2 \pi}} \exp(- \cfrac{u^2}{2})} \mathrm{ du }$

11. $\mathcal{ m^h(\theta) = \cfrac{1}{\sqrt{2\pi}} \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}} \int \exp {\sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2}(x_m - \mu_{m})^2] +
    \sum_{m=1}^{\mathrm{M}} [\cfrac{w_m^h}{ \sigma_m^2} (x_m - \mu_{m}) \ell_{m} * \mathcal{u}] }$ ${- \frac{1}{2} [1 + \sum_{m=1}^{\mathrm{M}} (\cfrac{w_m^h}{\sigma_m^2} \ell_{m}^2)] * \mathcal{u}^2}} \mathrm{ du }$
    
    

12. Set:

13. a = $- \frac{1}{2} [1 + \sum_{m=1}^{\mathrm{M}} \cfrac{w_m^h}{\sigma_m^2} (\ell_{m})^2]$
14. b = $\sum_{m=1}^{\mathrm{M}} [\cfrac{w_m^h}{ \sigma_m^2} (x_m - \mu_{m}) \ell_{m}]$
15. c = $\sum_{m=1}^{\mathrm{M}} [-\cfrac{w_m^h}{2 \sigma_m^2}(x_m - \mu_{m})^2]$

16. f = $\cfrac{1}{\sqrt{2 \pi}} \prod_{m=1}^{\mathrm{M}} (2 \pi \sigma_m^2)^{- \cfrac{w_m^h}{2}}$

17. $\mathcal{ m^h(\theta) = f \cdot \int \exp (a \cdot u^2 + b \cdot u + c) \mathrm{du}}$

18. $\mathcal{ m^h(\theta) = f \cdot \int \exp [a \cdot (u^2 + \cfrac{b}{a} \cdot u) + c] \mathrm{du}}$

19. $\mathcal{ m^h(\theta) = f \cdot \int \exp [a \cdot (u^2 + 2 \cfrac{b}{2a} u + (\cfrac{b}{2a})^2) + c-a (\cfrac{b}{2a})^2] \mathrm{du}}$

20. $\mathcal{ m^h(\theta) = f \cdot \exp (c-\cfrac{b^2}{4a}) \cdot \int \exp [a \cdot (u^2 + 2 \cfrac{b}{2a} u + (\cfrac{b}{2a})^2)] \mathrm{du}}$

21. $\mathcal{ m^h(\theta) = f \cdot \exp (c-\cfrac{b^2}{4a}) \cdot \int \exp [a \cdot (u + \cfrac{b}{2a})^2] \mathrm{du}}$

22. $\mathcal{ m^h(\theta) = f \cdot \exp (c-\cfrac{b^2}{4a}) \cdot \int \exp [-\cfrac{1}{2} \cdot \cfrac{(u + \cfrac{b}{2a})^2}{-\cfrac{1}{2a}}] \mathrm{du}}$

23. $\mathcal{ m^h(\theta) = f \cdot \exp (c-\cfrac{b^2}{4a}) \cdot \sqrt{2 \pi (-\cfrac{1}{2a})}\cdot {\cfrac{1}{\sqrt{2 \pi (-\cfrac{1}{2a})}} \cdot \int \exp [-\cfrac{1}{2} \cdot \cfrac{(u + \cfrac{b}{2a})^2}{-\cfrac{1}{2a}}] \mathrm{du}}}$

24. $\mathcal{\cfrac{1}{\sqrt{2 \pi (-\cfrac{1}{2a})}} \int \exp [-\cfrac{1}{2} \cdot \cfrac{(u + \cfrac{b}{2a})^2}{-\cfrac{1}{2a}}] \mathrm{du}} = 1$ because the intergation over a probability density function equals one.

25. $\mathcal{ m^h(\theta) = f \cdot \exp (c-\cfrac{b^2}{4a}) \cdot \sqrt{2 \pi (-\cfrac{1}{2a})}}$

9. Why Gaussian_quadrature approach failed to approximate the marginal likelihood function.

• An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1, ..., n. Ref: wikipedia

• To intergrate a polynomials of degree 2n-1 exactly use only n nodes.

• ${\exp(x) = \sum_{k=0}^{\infty}\cfrac{x^k}{k!}=1 + x + \cfrac{x^2}{2!}+\cfrac{x^3}{3!}+\cfrac{x^4}{4!}+ \cdots}$

• The degree of an exponential function, $exp(x)$ is ${\infty}$. Ref: wikipedia

• deg f = $\lim\limits_{x \to \infty}\cfrac{\log|f(x)|}{log(x)}$

10. The corresponding SAS code

proc nlmixed data=input tech=dbldog qpoints=30 noad;  
    parms mu1-mu7=0;

    array x{*}      x1-x7;  
    array mu{*}     mu1-mu7;  
    array fl{*}     fl1-fl7;  
    array loglik{*} loglik1-loglik7;  
    array w{*}      w1-w7;  
    array err{*}    err1-err7;  
    totloglik = 0;

    do i=1 to 5;  
        if x{i} = . then loglik{i}= 0;  
           else loglik{i} = w{i} * log(pdf('normal', x{i}, mu{i}+fl{i}*lv, err{i})); 
        totloglik = loglik{i} + totloglik;  
    end;

    model   id ~ general(totloglik);   
    random  lv ~ normal(0, 1) subject= id;

    ods output ParameterEstimates=ParEst;  
    predict lv out=Pred;  
run;

huangrh/rstarating documentation built on March 28, 2022, 6:44 p.m.