• IUR for SMR and SHR
• C statistics (C index) for SMR and SHR
• Variable selection for SHR

The SMR-type measure is calculated by \[ SMR_j = \frac{\sum_i^{n_j} O_{ji}}{\sum_i^{n_j} E_{ji}}, \] where

• \( j \): \( 1,\cdots,J \), facility index
• \( n_j \): the number of patients in facility \( j \)
• \( i \): patient index
• \( O_{ji} \): observed value for patient \( i \) in facility \( j \)
• \( E_{ji} \): expected value for patient \( i \) in facility \( j \)
• \( SMR_j \): SMR for facility \( j \).

• \( b \): bootstrap index
• \( B \): total number of bootstraps
• \( O_{ji}^{(b)} \): observed value for patient \( i \) in facility \( j \) for \( b \)-th bootstrap data
• \( E_{ji}^{(b)} \): expected value for patient \( i \) in facility \( j \) for \( b \)-th bootstrap data
• \( SMR_j^{(b)} \): SMR for facility \( j \) for \( b \)-th bootstrap data, \[ SMR_j^{(b)} = \frac{\sum_i^{n_j} O_{ji}^{(b)}}{\sum_i^{n_j} E_{ji}^{(b)}}. \]

• For \( b = 1,\cdots, B \), sample data from original data using facility-stratified sampling.
• Calculate \( SMR^{(b)} \) for each bootstrap data.
• Calculate within-facility variance: \[ \sigma^2_w = \frac{\sum_j \sum_b (SMR_j^{(b)}-\overline{SMR}_j)^2}{\sum_j(n_j-1)}, \] where \( \overline{SMR}_j= \sum_b SMR_j^{(b)}/n_j \).
• \[ n' = \frac{\sum_j n_j - \sum_j n_j^2/\sum_j n_j}{\sum_j n_j-1}. \]

• Calculate total variance: \[ \sigma^2_t = \frac{\sum_j n_j(SMR_j-\overline{SMR})^2}{n'(\sum_j n_j -1)}, \] where \( \overline{SMR} = \sum_j n_jSMR_j/\sum_j n_j \).
• Between facility variance \( \sigma^2_b = \sigma^2_t- \sigma^2_w \).
• \[ IUR = \frac{\sigma^2_b}{\sigma^2_t}. \]
• Facility IUR, \[ IUR_j = \frac{\sigma^2_b}{\sigma^2_b+\sigma^2_w/n_j}. \]

SMR

rm(list=ls())
uniqname = 'yuanyang'
output_path = 'K:/Users/kecc-yuanyang/IUR0321'
data_path = 'K:/Projects/Dialysis_Reports_Shared/Data/SMR/special_request'
raw_data_name = 'SMRSHR_2014to2017.sas7bdat'

obs_death = "dial_drd" #observed death
exp_death = "expectda" #expeceted death
facility = "provfs" #facility id
death_yar = "DIAL_yar" #death year
year = "year" #data year

IUR for 2014-2017 SMR data:

rm(list=ls())
uniqname = 'yuanyang'
output_path = 'K:/Users/kecc-yuanyang/IUR0321'
data_path = 'K:/Projects/Dialysis_Reports_Shared/Data/SMR/special_request'
raw_data_name = 'SMRSHR_2014to2017.sas7bdat'

obs_admission = "h_admits" #observed admission
exp_admission = "expectta" #expeceted admission
hosp_yar = "h_dy_yar" #hospital YAR
facility = "provfs" #facility id
year = "year" #data year

IUR for 2017 SHR data:

• C statistic is to measure the proportion of concordance pairs of observed and expected outcomes to all possible pairs.

• Normally, C statistic is defined as \[ C_{stat}=\frac{\rm number \ of\ concordances}{\rm number\ of\ pairs}= \frac{ \sum_{i < j} I(y_i < y_j\ \&\ \hat{y}_i < \hat{y}_j) }{\sum_{i < j}1}. \] where

  • \( y_i \): the obeserved outcome;
  • \( \hat{y}_i \): the fitted value.

• \( 0 \le C \le 1 \). Higher C indicates higher concordance between y and y_hat, and thus better model fitting.

• Data:

  ID	y	y_hat
  1	1.1	4.5
  2	2.3	3.4
  3	4.3	5.3

  • Numerator = \( \sum_{i < j} I(y_i < y_j\ \&\ \hat{y}_i < \hat{y}_j) \)= 2
  • Denominator = \( \sum_{i < j}1 \)=3
  • Therefore, C=2/3.

Data preparation:

• \( m_i \)(h_admits): observed number of events.
• \( T_i \)(h_dy_yar): year-at-risk.
• \( {\rm year}_i \): patient hospitalization year. For example, pyf4_mod1 data is from 2010 to 2013.
• \( e_i \)(expectta): expected number of events.
• \( r_i=m_i/T_i \): observed event rate.
• \( \hat{r}_i = e_i/T_i \): expected event rate.
• \( SHR_i \)(shrty1_f, shrty2_f, shrty3_f, shrty4_f): SHR from 2010 to 2013 in pyf4_mod1 data.
• \( \hat{r}_{2i}=\hat{r}_i*SHR_i \): SHR adjusted expected rate.

1. Define category by \( {\rm year}_i \).
2. Sort the data by \( r_i \).
3. Within category \( k \), \( \rm number \ of\ concordances = N_k = \sum_i \sum_{j:j > i} I((r_i < r_j)\ \&\ (\hat{r}_{2i} < \hat{r}_{2j})) \), \( \rm number \ of\ pairs = D_k = \sum_i \sum_{j:j > i} 1 \).
4. Then \( C_{stat} = \frac{\sum_k N_k}{\sum_k D_k} \).

rm(list=ls())
uniqname = 'yuanyang'
output_path = 'K:/Users/kecc-yuanyang/C_statistics0214'
data_path = 'K:/Projects/Dialysis_Reports_Shared/Data/SMR/special_request'
raw_data_name = 'SMRSHR_2014to2017.sas7bdat'

SHR = 'shrty'
obs_admission = "h_admits" #observed admission
exp_admission = "expectta" #expeceted admission
hosp_yar = "h_dy_yar" #hospital YAR
facility = "provfs" #facility id
year = "year" #data year

Sort data by event time, then the C statistic for Survival model is defined as \[ C_{stat} = \frac{ \rm number \ of\ concordances}{\rm number\ of\ pairs} =\frac{\sum_i \sum_{j:j > i} I(T_i < T_j\ \&\ ( X \hat{\beta})_i>( X \hat{\beta})_j)}{\sum_i \sum_{j:j > i}\delta_i}, \] where

• \( T_i \): the obeserved survival time
• \( \delta_i \): 1= event; 0 = censored; \( \Delta \): \( \{ i:\delta_i=1 \} \)
• \( (X\hat{\beta})_i=X_i \hat{\beta} \)
  

Data:

• \( T_i (\rm DIAL\_yar) \): observed survival time.
• \( (X\hat{\beta})_i (pred\_xbeta) \): \( X \) is the covariates and \( \hat{\beta} \) is from the model.
• \( k (\rm provfs) \): facility number.
• \( \delta_i (\rm dial\_drd) \): event indicator.

• if \( \delta_i = 1 \) and \( \delta_j = 1 \),

  • if \( |T_i-T_j|< 0.0001 \) (tie), then concordance = 0.5;
  • else if \( |(X\hat{\beta})_i-(X\hat{\beta})_j|<0.0001 \), then concordance=0.5; else if \( (X\hat{\beta})_i>(X\hat{\beta})_j \), concordance = 1.

• if \( \delta_i = 1 \) and \( \delta_j=0 \),

  • if \( |(X\hat{\beta})_i-(X\hat{\beta})_j|<0.0001 \), then concordance=0.5;
  • else if \( (X\hat{\beta})_i>(X\hat{\beta})_j \), then concordance = 1.

1. Sort data by \( T_i \) (increasing order).
2. Within facility \( k \), \[ \rm number \ of\ concordances = N_k = \sum_i \sum_{j:j>i} concordance(i,j), \] \[ \rm number \ of\ pairs = D_k = \sum_i \sum_{j:j>i} \delta_i. \]
3. Then \( C_{stat} = \frac{\sum_k N_k}{\sum_k D_k} \).

• On a toy data with 1000 obervations:
  
  
  
• About 1000 times faster.

rm(list=ls())
uniqname = 'yuanyang'
output_path = 'K:/Users/kecc-yuanyang/C_statistics0214'
data_path = 'K:/Projects/Dialysis_Reports_Shared/Data/SMR/special_request'
raw_data_name = 'SMRSHR_2014to2017.sas7bdat'


obs_death = "dial_drd" #observed death
exp_xbeta = "xbeta_smr" #predicted Xbeta
death_yar = "DIAL_yar" #dialysis YAR
facility = "provfs" #facility id

Following Liu 2012, we first introduce some notation:

• \( k \): facility index, \( 1,\dots,K \).
• \( l \): interval index, \( 1, \dots, L \). The intervals are defined by \( a_0,\cdots,a_L \), e.g. \( [a_0,a_1],\cdots, [a_{L-1},a_{L}] \). In ED_SHR data, we use T_start and T_stop to define the intervals.
• \( i \): patient index, \( 1,\dots, n_{kl} \).
• \( \mathbf{Z}_{kli} \): covarites of \( i \)-th patient in \( k \)-th facility \( l \)-th interval. \( p*1 \) vector.

The partial likelihood for SHR model is \[ PL_f = \prod_{k=1}^{K} \prod_{l=1}^{L} \prod_{i=1}^{n_{kl}} \Big[ \frac{e^{ \{\mathbf{\beta}^T\mathbf{Z}_{kli}+o_{kli}\}}}{\sum_{j=1}^{n_{kl}}t_{kli}e^{\{\mathbf{\beta}^T\mathbf{Z}_{kli}\}}} \Big]^{d_{kli}}. \]

Then the log-partial likelihood is \[ l(\mathbf{\beta})=\sum_{j=1}^{K} \sum_{l=1}^{L} \sum_{i=1}^{n_{kl}} \Big[ \mathbf{\beta}^T\mathbf{Z}_{kli}+o_{kli} - log\{ \sum_{j=1}^{n_{kl}}t_{kli}e^{\{\mathbf{\beta}^T\mathbf{Z}_{kli}\}} \} \Big] {d_{kli}}. \]

1. For \( j=1,\dots,p \), we calculate the first and second derivative of \( l(\mathbf{\beta}) \) with respect to \( \beta_j \), denoted as \( U(\mathbf{\beta})_j \) and \( G(\mathbf{\beta})_j \), respectively.
2. Find \( j^* \), where \( U(\mathbf{\beta})_{j^*} = \max\{U(\mathbf{\beta})_{j},j=1,\cdots,p\} \).
3. Update \( \beta_{j^*}=\beta_{j^*} - \frac{U(\mathbf{\beta})_j}{G(\mathbf{\beta})_j} \),
   

The goal is to select important comorbidity groupers. At each step, we update the coefficients for all adjustment variables and one comorbidity grouper using Boosting Algorithm. The groupers should be normalized before boosting.

• The goal of this task is to reduce the number of false positives.
• Fdr is an abbreviation of empirical Bayes false discovery rate, which is defined as the probability we have made a false discovery given this variable is selected.
• Selection rule: for a prespecified q (0<q<1), we select variables with their Fdr <= q. This procedure will ensure that at most q proportion of the selected variables would be false positives.
  • For example, q=0.1, if 10 variables are selected, at most 1 of them is false positive.

• \( S_0=\{ j:\beta_j \ne 0\} \): the set of indexes of true variables
• \( \hat{S}=\{ j:\hat{\beta}_j \ne 0\} \): the set of indexes of selected variables
• \( N_+ = |\hat{S}| \), \( e_+ = {\rm E}N_+ \)
• \( N_0=|\hat{S}\setminus S_0| \), \( e_0={\rm E}N_0 \)
• Empirical FDR \( \overline{Fdr} = \frac{e_0}{N_+} \)

• On the original dataset, we bootstrap B times. Then we calculate \( \hat{\pi}_j=1/B \sum_{b=1}^{B}I\{ \hat{\beta}_j^{(b)} \ne 0\} \). \( \hat{\pi}_{(1)}\ge \hat{\pi}_{(2)} \ge \dots \ge \hat{\pi}_{(n)} \) are the ordered \( \hat{\pi}_i \)'s.
• We permute orginal data M times.
  • For \( m \)-th data, bootstrap B times; calculate \( \tilde{\pi}_j^m=\frac{1}{B} \sum_{b=1}^{B}I\{ \tilde{\beta}_j^{(m,b)} \ne 0\} \); get ordered \( \tilde{\pi}_{(i)}^m \)'s.
  • \( \tilde{\pi}_{(j)}^* = \frac{1}{M} \sum_{m=1}^{M} \tilde{\pi}_{(j)}^{m} \).

• Transform \( \pi \) to be a Normal variable \( z \).
  • \( \triangle_{(j)} = \hat{z}_{(j)} - \tilde{z}_{(j)}^* \) and \( c_{(j)}=min \{ \hat{\pi}_{(l)}: \triangle_{(l)} \ge \triangle_{(j)} ,\quad l=1,\dots,p\} \).
  • \( \overline{Fdr}_{(j)} = \underset{1\le j' \le j}{max} \frac{\sum_{m=1}^{M} \sum_{l=1}^{p} I(\tilde{\pi}_{(l)}^m>c_{(j')})/M}{\sum_{l=1}^{p} I(\hat{\pi}_{(l)}^0>c_{(j')})} \).