59228 - The Bayesian LASSO

Sample 59228: The Bayesian LASSO

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Overview

The least absolute shrinkage and selection operator (LASSO) was developed by Tibshirani (1996) as an alternative to the ordinary least squares (OLS) method with two objectives in mind. The first was to improve prediction accuracy, and the second was to improve model interpretation by determining a smaller subset of regressors that exhibit the strongest effects. This example presents a fully Bayesian interpretation and implementation of the LASSO that provides estimates for the regression parameters and their variances and provides Bayesian credible intervals for the regression parameters that can guide variable selection.

Analysis

The LASSO is commonly used to estimate the parameters in the linear regression model

$\mb {y} = \mu \mb {1}_ n + \mb {X}\bbeta + \bepsilon$

where $\mb {y}$ is the $n \times 1$ vector of responses, $\mu$ is the overall mean, $\mb {X}$ is the $n \times p$ matrix of standardized regressors, and $\bepsilon$ is the $n \times 1$ vector of independent and identically distributed normal errors with mean 0 and unknown variance $\sigma ^2$ . The LASSO estimates of Tibshirani (1996) are the solution to the minimization problem

$\min _{\bbeta } {(\mb {\tilde{y}} - \mb {X}\bbeta )}^\prime (\mb {\tilde{y}} - \mb {X}\bbeta ) + \lambda \sum _{j=1}^{p} |\beta _ j|$

for some $\lambda \ge 0$ , where $\mb {\tilde{y}} = \mb {y} - \bar{y}\mb {1}_ n$ .

Tibshirani (1996) suggested that the LASSO estimates can be interpreted as posterior mode estimates when the regression parameters have independent and identical Laplace priors. Park and Casella (2008) consider a fully Bayesian analysis by using a conditional Laplace prior specification of the form

$\pi (\bbeta |\sigma ^2) = \prod _{j=1}^{p} \frac{\lambda }{2\sqrt {\sigma ^2}}\exp {\left( \frac{-\lambda |\beta _ j|}{\sqrt {\sigma ^2}}\right) }$

and the noninformative scale-invariant marginal prior $\pi (\sigma ^2) = 1/\sigma ^2$ on $\sigma ^2$ . Conditioning of $\sigma ^2$ is important because it guarantees a unimodal full posterior. Park and Casella (2008) also note that any inverse-gamma prior for $\sigma ^2$ maintains conjugacy.

Exploiting the fact that the Laplace distribution can be represented as a scale mixture of normal densities with an exponential mixing density, Park and Casella (2008) propose the following hierarchical representation of the full model:

$\begin{eqnarray*} \mb {y}|\mu ,\mb {X},\bbeta ,\sigma ^2 & \sim & \mb {N}_ n(\mu \mb {1}_ n + \mb {X}\bbeta , \sigma ^2\mb {I}_ n) \\ \bbeta |\sigma ^2,\tau _1^2, \dots ,\tau _ p^2 & \sim & \mb {N}_ p(\mb {0}_ p,\sigma ^2\mb {D}_\tau ) \\ \mb {D}_\tau & = & \mr {diag}(\tau ^2_1, \dots ,\tau ^2_ p) \\ \sigma ^2,\tau _1^2, \dots ,\tau _ p^2 & \sim & \pi (\sigma ^2)d\sigma ^2 \prod _{j=1}^{p}\frac{\lambda ^2}{2} \exp { \left( \frac{-\lambda ^2\tau ^2_ j}{2} \right)} d\tau ^2_ j \\ \sigma ^2,\tau _1^2, \dots ,\tau _ p^2 & > & 0 \end{eqnarray*}$

The parameter $\mu$ can be given an independent, flat prior. After you integrate out $\tau _1^2, \dots ,\tau _ p^2$ , the conditional prior on $\bbeta$ has the desired conditional Laplace distribution.

The Bayesian LASSO parameter $\lambda$ can be chosen by using marginal maximum likelihood or an appropriate hyperprior. The example in the next section demonstrates the latter and considers, as suggested by Park and Casella (2008), the class of gamma priors on $\lambda ^2$ ,

$\pi (\lambda ^2) = \frac{\delta ^ r}{\Gamma {(r)}}{(\lambda ^2)}^{r-1}{\Mathtext{e}}^{-\delta \lambda ^2}$

Example

This example from Park and Casella (2008) fits a Bayesian LASSO model to the diabetes data from Efron et al. (2004). In the original study, statisticians were asked to construct a model that predicted the response variable, Y, a quantitative measure of disease progression one year after baseline, from 10 covariates: Age, Sex, BMI, MAP, TC, LDL, HDL, TCH, LTG, and GLU. It was hoped that the model would produce accurate baseline predictions of response for future patients and that the form of the model would suggest which covariates were important factors in disease progression. The following SAS statements read the data and create the SAS data set Diabetes:


data diabetes;
  input age sex bmi map tc ldl hdl tch ltg glu y;
  sex=ifn(sex=2,1,0);
  datalines;
59 2 32.1 101.00 157 93.2 38.0 4.00 4.8598 87.000 151
48 1 21.6 87.00 183 103.2 70.0 3.00 3.8918 69.000 75
72 2 30.5 93.00 156 93.6 41.0 4.00 4.6728 85.000 141

   ... more lines ...

60 2 24.9 99.67 162 106.6 43.0 3.77 4.1271 95.000 132
36 1 30.0 95.00 201 125.2 42.0 4.79 5.1299 85.000 220
36 1 19.6 71.00 250 133.2 97.0 3.00 4.5951 92.000 57
;

Before specifying the model in the MCMC procedure, you need to standardize the model’s regressors, excluding the indicator variable Sex. You can use the STDIZE procedure as follows to perform this task:


proc stdize data=diabetes out=std_diabetes;
  var age bmi map tc ldl hdl tch ltg glu;
run;

The following statements specify the Bayesian LASSO in PROC MCMC:

ods graphics on;
ods output postintervals=intervals;
proc mcmc data=std_diabetes seed=45678 nmc=50000 propcov=quanew
          monitor=(b0 beta1-beta10 tau1-tau10 sigma2 lasso)
          outpost=posterior;

  array D[10,10];
  array beta[10] beta1-beta10;
  array mu0[10];
  array data[10] age sex bmi map tc ldl hdl tch ltg glu;

  begincnst;
   call identity(D);
   call zeromatrix(mu0);
  endcnst;

  beginnodata;
    lasso=sqrt(lambda);
    b=lambda/2;

    %macro loop;
      %do k = 1 %to 10;
         tau&k = exp(omega&k);
         D[&k,&k]=sigma2*tau&k;
         %end;
    %mend loop;
    %loop;
  endnodata;

  call mult(beta, data,xb);

  parms lambda ;
  prior lambda ~ gamma(1,scale=.1);
  parms omega1-omega10;
  prior omega: ~ expexpon(iscale=b);
  parms sigma2 1;
  prior sigma2 ~ igamma(shape = .1, iscale = .1);
  parms b0 0;
  prior b0 ~ general(0);
  parms beta;
  prior beta ~ mvn(mu0,D);

  model y ~ normal(b0 + xb,var=sigma2);

run;

The ODS OUTPUT statement saves the posterior credible intervals in the SAS data set Intervals. The NMC= option in the PROC MCMC statement requests 50,000 MCMC iterations, excluding the burn-in iterations. A large sample is used because the posterior samples are highly autocorrelated. The PROPCOV= option in the PROC MCMC statement requests that the quasi-Newton method be used in constructing the initial covariance matrix for the Metropolis-Hastings algorithm. The OUTPOST= option saves the posterior samples in the data set Posterior.

The next four statements create arrays that are used in the model. The array D is the covariance matrix for the regression parameters Beta1–Beta10. The array Beta is the vector of the regression parameters Beta1–Beta10. The array Mu0 is the mean vector for the prior distribution of the regression parameters Beta1–Beta10. The array Data is the matrix of regressors, excluding the intercept.

The BEGINCNST and ENDCNST statements define a statement block within which PROC MCMC processes the programming statements only during the setup stage of the simulation. You can use the BEGINCNST and ENDCNST statement block to initialize the matrices D and Mu0. D is initially set to an identity matrix, and Mu0 is initialized as a zero vector.

The BEGINNODATA and ENDNODATA statements define a block within which PROC MCMC processes the programming statements without stepping through the entire data set. The programming statements are executed only twice: at the first and last observations of the data set. Within this statement block, the parameters Lasso and b are defined. The macro %LOOP repopulates the matrix D. The purpose of the parameters Omega1–Omega10 and their relationship with the parameters Tau1–Tau10 are explained later.

The next statement uses the MULT CALL routine to define the matrix XB, which contains the product of the regressors and the regression parameters Beta1–Beta10. That is, it contains the linear predictor, excluding the intercept.

The following block of statements declares the model parameters and assigns prior distribution to them. The parameter Lambda, which represents $\lambda ^2$ , is specified to have a gamma distribution. The parameters Omega1–Omega10 are specified to have exponential exponential distributions. The parameters $\tau ^2_1, \dots , \tau ^2_{10}$ have exponential distributions, but modeling these parameters directly can cause convergence problems. Instead, the parameters Omega1–Omega10 are modeled directly, and within the macro %LOOP the parameters Tau1–Tau10, which represent $\tau ^2_1, \dots , \tau ^2_{10}$ , are defined as being the exponential of Omega1–Omega10, respectively. The parameter Sigma2, which represents $\sigma ^2$ , is specified to have an inverse-gamma distribution. The parameter B0, which represent $\mu$ , is specified to have an improper uniform distribution. The parameter vector Beta, which represents $\bbeta$ , is specified to have a multivariate normal distribution with mean equal to 0 and variance matrix equal to D.

Finally, the MODEL statement specifies that the response variable Y have a normal distribution.

Output 1 shows that the Monte Carlo standard errors (MCSE) of each parameter are small relative to the posterior standard deviations (SD). A small MCSE/SD ratio indicates that the Markov chain has stabilized and that the mean estimates do not vary much over time.

Output 1: Monte Carlo Standard Errors

The MCMC Procedure

Monte Carlo Standard Errors
Parameter	MCSE	Standard Deviation	MCSE/SD
b0	0.1105	3.7685	0.0293
beta1	0.0769	2.8187	0.0273
beta2	0.2072	5.8461	0.0354
beta3	0.1029	3.1602	0.0326
beta4	0.1093	3.3479	0.0326
beta5	0.3238	11.1592	0.0290
beta6	0.3241	11.4013	0.0284
beta7	0.1018	3.5868	0.0284
beta8	0.2234	7.4085	0.0302
beta9	0.1765	6.2293	0.0283
beta10	0.0992	3.3421	0.0297
tau1	0.0875	3.7978	0.0230
tau2	0.0925	4.0235	0.0230
tau3	0.1082	3.9813	0.0272
tau4	0.1124	3.8682	0.0291
tau5	0.0934	3.7605	0.0248
tau6	0.1107	3.8139	0.0290
tau7	0.0922	3.6053	0.0256
tau8	0.1004	3.6653	0.0274
tau9	0.1105	4.1032	0.0269
tau10	0.0969	3.7154	0.0261
sigma2	1.1459	202.7	0.00565
lasso	0.00400	0.1500	0.0266

Output 2 shows the “Effective Sample Sizes” table. The autocorrelation times for the parameters range from 1.59 to 62.83, and most of the efficiency rates are low. These results account for the relatively small effective sample sizes, given a nominal sample size of 50,000.

Output 2: Effective Sample Sizes

Effective Sample Sizes
Parameter	ESS	Autocorrelation Time	Efficiency
b0	1164.0	42.9555	0.0233
beta1	1343.4	37.2178	0.0269
beta2	795.7	62.8345	0.0159
beta3	942.3	53.0600	0.0188
beta4	938.5	53.2742	0.0188
beta5	1188.0	42.0892	0.0238
beta6	1237.5	40.4033	0.0248
beta7	1241.1	40.2856	0.0248
beta8	1100.0	45.4557	0.0220
beta9	1245.4	40.1475	0.0249
beta10	1134.0	44.0920	0.0227
tau1	1883.1	26.5514	0.0377
tau2	1892.7	26.4173	0.0379
tau3	1353.3	36.9456	0.0271
tau4	1183.6	42.2445	0.0237
tau5	1619.6	30.8725	0.0324
tau6	1186.1	42.1539	0.0237
tau7	1529.5	32.6909	0.0306
tau8	1333.9	37.4850	0.0267
tau9	1378.2	36.2783	0.0276
tau10	1468.9	34.0388	0.0294
sigma2	31287.3	1.5981	0.6257
lasso	1409.3	35.4777	0.0282

The following SAS statements use the OUTPOST data set Posterior and the ODS OUTPUT data set Intervals to generate a table of the Bayesian LASSO parameter estimates, which are the modes of the posterior samples for B0 and Beta1–Beta10, and their respective 95% HPD intervals:


proc means data=posterior mode;
  var b0 beta1-beta10;
  output out=parameters(drop=_TYPE_ _FREQ_) mode(b0 beta1-beta10)=b0 beta1-beta10;
run;

proc transpose data=parameters out=parameters;
run;


data parameters;
  length parameter $ 6;
  set parameters(rename=(col1=mode _NAME_=Parameter));
  label Parameter=;
  index=_N_;
run;


proc sort data=parameters out=parameters;
  by parameter;
run;

proc sort data=intervals out=intervals;
  by parameter;
run;


data parameters(where=(~missing(mode)));
  merge parameters intervals;
  by parameter;
  label parameter="Parameter" mode="Mode";
run;


proc sort data=parameters out=parameters;
  by index;
run;


proc print data=parameters noobs label;
  var parameter mode hpdlower hpdupper;
run;

Output 3 shows that the HPD intervals for the parameters Beta1, Beta5, Beta6, Beta8, and Beta10 all contain 0. Unlike what happens in the frequentist version of the LASSO, regression parameters are not set to 0, so the inclusion of 0 in the HPD interval is the only indication that a variable is a candidate for exclusion from the model. Based on this criterion, the variables Age, TC, LDL, TCH, and GLU are the leading candidates for exclusion from the model.

Output 3: Bayesian LASSO Parameter Estimates and 95% HPD Intervals

Parameter	Mode	HPDLower	HPDUpper
b0	162.2	155.2	170.1
beta1	-1.4406	-5.8311	5.2856
beta2	-21.2417	-33.7811	-11.0211
beta3	26.5303	18.3849	30.6794
beta4	13.6935	9.4601	22.4973
beta5	-10.1466	-33.9135	9.7807
beta6	6.9991	-17.1198	27.7032
beta7	-13.3526	-17.2501	-3.3104
beta8	1.7061	-14.5072	14.3417
beta9	25.8343	15.8578	40.0418
beta10	0.3158	-2.5716	10.4408

References

Efron, B., Hastie, T. J., Johnstone, I. M., and Tibshirani, R. (2004), “Least Angle Regression (with Discussion),” Annals of Statistics, 32, 407–499.
Kyung, M., Gill, J., Ghosh, M., and Casella, G. (2010), “Penalized Regression, Standard Errors, and Bayesian Lassos,” Bayesian Analysis, 5, 369–412.
Park, T. and Casella, G. (2008), “The Bayesian Lasso,” Journal of the American Statistical Association, 103, 681–686.
Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.

These sample files and code examples are provided by SAS Institute Inc. "as is" without warranty of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Recipients acknowledge and agree that SAS Institute shall not be liable for any damages whatsoever arising out of their use of this material. In addition, SAS Institute will provide no support for the materials contained herein.

data diabetes;
  input age sex bmi map tc ldl hdl tch ltg glu y;
  sex=ifn(sex=2,1,0);
  datalines;
59 2 32.1 101.00 157 93.2 38.0 4.00 4.8598 87.000 151 
48 1 21.6 87.00 183 103.2 70.0 3.00 3.8918 69.000 75 
72 2 30.5 93.00 156 93.6 41.0 4.00 4.6728 85.000 141 
24 1 25.3 84.00 198 131.4 40.0 5.00 4.8903 89.000 206 
50 1 23.0 101.00 192 125.4 52.0 4.00 4.2905 80.000 135 
23 1 22.6 89.00 139 64.8 61.0 2.00 4.1897 68.000 97 
36 2 22.0 90.00 160 99.6 50.0 3.00 3.9512 82.000 138 
66 2 26.2 114.00 255 185.0 56.0 4.55 4.2485 92.000 63 
60 2 32.1 83.00 179 119.4 42.0 4.00 4.4773 94.000 110 
29 1 30.0 85.00 180 93.4 43.0 4.00 5.3845 88.000 310 
22 1 18.6 97.00 114 57.6 46.0 2.00 3.9512 83.000 101 
56 2 28.0 85.00 184 144.8 32.0 6.00 3.5835 77.000 69 
53 1 23.7 92.00 186 109.2 62.0 3.00 4.3041 81.000 179 
50 2 26.2 97.00 186 105.4 49.0 4.00 5.0626 88.000 185 
61 1 24.0 91.00 202 115.4 72.0 3.00 4.2905 73.000 118 
34 2 24.7 118.00 254 184.2 39.0 7.00 5.0370 81.000 171 
47 1 30.3 109.00 207 100.2 70.0 3.00 5.2149 98.000 166 
68 2 27.5 111.00 214 147.0 39.0 5.00 4.9416 91.000 144 
38 1 25.4 84.00 162 103.0 42.0 4.00 4.4427 87.000 97 
41 1 24.7 83.00 187 108.2 60.0 3.00 4.5433 78.000 168 
35 1 21.1 82.00 156 87.8 50.0 3.00 4.5109 95.000 68 
25 2 24.3 95.00 162 98.6 54.0 3.00 3.8501 87.000 49 
25 1 26.0 92.00 187 120.4 56.0 3.00 3.9703 88.000 68 
61 2 32.0 103.67 210 85.2 35.0 6.00 6.1070 124.000 245 
31 1 29.7 88.00 167 103.4 48.0 4.00 4.3567 78.000 184 
30 2 25.2 83.00 178 118.4 34.0 5.00 4.8520 83.000 202 
19 1 19.2 87.00 124 54.0 57.0 2.00 4.1744 90.000 137 
42 1 31.9 83.00 158 87.6 53.0 3.00 4.4659 101.000 85 
63 1 24.4 73.00 160 91.4 48.0 3.00 4.6347 78.000 131 
67 2 25.8 113.00 158 54.2 64.0 2.00 5.2933 104.000 283 
32 1 30.5 89.00 182 110.6 56.0 3.00 4.3438 89.000 129 
42 1 20.3 71.00 161 81.2 66.0 2.00 4.2341 81.000 59 
58 2 38.0 103.00 150 107.2 22.0 7.00 4.6444 98.000 341 
57 1 21.7 94.00 157 58.0 82.0 2.00 4.4427 92.000 87 
53 1 20.5 78.00 147 84.2 52.0 3.00 3.9890 75.000 65 
62 2 23.5 80.33 225 112.8 86.0 2.62 4.8752 96.000 102 
52 1 28.5 110.00 195 97.2 60.0 3.00 5.2417 85.000 265 
46 1 27.4 78.00 171 88.0 58.0 3.00 4.8283 90.000 276 
48 2 33.0 1.00 23 253.0 163.6 44.00 6.0000 5.425 97 
48 2 27.7 73.00 191 119.4 46.0 4.00 4.8520 92.000 90 
50 2 25.6 101.00 229 162.2 43.0 5.00 4.7791 114.000 100 
21 1 20.1 63.00 135 69.0 54.0 3.00 4.0943 89.000 55 
32 2 25.4 90.33 153 100.4 34.0 4.50 4.5326 83.000 61 
54 1 24.2 74.00 204 109.0 82.0 2.00 4.1744 109.000 92 
61 2 32.7 97.00 177 118.4 29.0 6.00 4.9972 87.000 259 
56 2 23.1 104.00 181 116.4 47.0 4.00 4.4773 79.000 53 
33 1 25.3 85.00 155 85.0 51.0 3.00 4.5539 70.000 190 
27 1 19.6 78.00 128 68.0 43.0 3.00 4.4427 71.000 142 
67 2 22.5 98.00 191 119.2 61.0 3.00 3.9890 86.000 75 
37 2 27.7 93.00 180 119.4 30.0 6.00 5.0304 88.000 142 
58 1 25.7 99.00 157 91.6 49.0 3.00 4.4067 93.000 155 
65 2 27.9 103.00 159 96.8 42.0 4.00 4.6151 86.000 225 
34 1 25.5 93.00 218 144.0 57.0 4.00 4.4427 88.000 59 
46 1 24.9 115.00 198 129.6 54.0 4.00 4.2767 103.000 104 
35 1 28.7 97.00 204 126.8 64.0 3.00 4.1897 93.000 182 
37 1 21.8 84.00 184 101.0 73.0 3.00 3.9120 93.000 128 
37 1 30.2 87.00 166 96.0 40.0 4.15 5.0106 87.000 52 
41 1 20.5 80.00 124 48.8 64.0 2.00 4.0254 75.000 37 
60 1 20.4 105.00 198 78.4 99.0 2.00 4.6347 79.000 170 
66 2 24.0 98.00 236 146.4 58.0 4.00 5.0626 96.000 170 
29 1 26.0 83.00 141 65.2 64.0 2.00 4.0775 83.000 61 
37 2 26.8 79.00 157 98.0 28.0 6.00 5.0434 96.000 144 
41 2 25.7 83.00 181 106.6 66.0 3.00 3.7377 85.000 52 
39 1 22.9 77.00 204 143.2 46.0 4.00 4.3041 74.000 128 
67 2 24.0 83.00 143 77.2 49.0 3.00 4.4308 94.000 71 
36 2 24.1 112.00 193 125.0 35.0 6.00 5.1059 95.000 163 
46 2 24.7 85.00 174 123.2 30.0 6.00 4.6444 96.000 150 
60 2 25.0 89.67 185 120.8 46.0 4.02 4.5109 92.000 97 
59 2 23.6 83.00 165 100.0 47.0 4.00 4.4998 92.000 160 
53 1 22.1 93.00 134 76.2 46.0 3.00 4.0775 96.000 178 
48 1 19.9 91.00 189 109.6 69.0 3.00 3.9512 101.000 48 
48 1 29.5 131.00 207 132.2 47.0 4.00 4.9345 106.000 270 
66 2 26.0 91.00 264 146.6 65.0 4.00 5.5683 87.000 202 
52 2 24.5 94.00 217 149.4 48.0 5.00 4.5850 89.000 111 
52 2 26.6 111.00 209 126.4 61.0 3.00 4.6821 109.000 85 
46 2 23.5 87.00 181 114.8 44.0 4.00 4.7095 98.000 42 
40 2 29.0 115.00 97 47.2 35.0 2.77 4.3041 95.000 170 
22 1 23.0 73.00 161 97.8 54.0 3.00 3.8286 91.000 200 
50 1 21.0 88.00 140 71.8 35.0 4.00 5.1120 71.000 252 
20 1 22.9 87.00 191 128.2 53.0 4.00 3.8918 85.000 113 
68 1 27.5 107.00 241 149.6 64.0 4.00 4.9200 90.000 143 
52 2 24.3 86.00 197 133.6 44.0 5.00 4.5747 91.000 51 
44 1 23.1 87.00 213 126.4 77.0 3.00 3.8712 72.000 52 
38 1 27.3 81.00 146 81.6 47.0 3.00 4.4659 81.000 210 
49 1 22.7 65.33 168 96.2 62.0 2.71 3.8918 60.000 65 
61 1 33.0 95.00 182 114.8 54.0 3.00 4.1897 74.000 141 
29 2 19.4 83.00 152 105.8 39.0 4.00 3.5835 83.000 55 
61 1 25.8 98.00 235 125.8 76.0 3.00 5.1120 82.000 134 
34 2 22.6 75.00 166 91.8 60.0 3.00 4.2627 108.000 42 
36 1 21.9 89.00 189 105.2 68.0 3.00 4.3694 96.000 111 
52 1 24.0 83.00 167 86.6 71.0 2.00 3.8501 94.000 98 
61 1 31.2 79.00 235 156.8 47.0 5.00 5.0499 96.000 164 
43 1 26.8 123.00 193 102.2 67.0 3.00 4.7791 94.000 48 
35 1 20.4 65.00 187 105.6 67.0 2.79 4.2767 78.000 96 
27 1 24.8 91.00 189 106.8 69.0 3.00 4.1897 69.000 90 
29 1 21.0 71.00 156 97.0 38.0 4.00 4.6540 90.000 162 
64 2 27.3 109.00 186 107.6 38.0 5.00 5.3083 99.000 150 
41 1 34.6 87.33 205 142.6 41.0 5.00 4.6728 110.000 279 
49 2 25.9 91.00 178 106.6 52.0 3.00 4.5747 75.000 92 
48 1 20.4 98.00 209 139.4 46.0 5.00 4.7707 78.000 83 
53 1 28.0 88.00 233 143.8 58.0 4.00 5.0499 91.000 128 
53 2 22.2 113.00 197 115.2 67.0 3.00 4.3041 100.000 102 
23 1 29.0 90.00 216 131.4 65.0 3.00 4.5850 91.000 302 
65 2 30.2 98.00 219 160.6 40.0 5.00 4.5218 84.000 198 
41 1 32.4 94.00 171 104.4 56.0 3.00 3.9703 76.000 95 
55 2 23.4 83.00 166 101.6 46.0 4.00 4.5218 96.000 53 
22 1 19.3 82.00 156 93.2 52.0 3.00 3.9890 71.000 134 
56 1 31.0 78.67 187 141.4 34.0 5.50 4.0604 90.000 144 
54 2 30.6 103.33 144 79.8 30.0 4.80 5.1417 101.000 232 
59 2 25.5 95.33 190 139.4 35.0 5.43 4.3567 117.000 81 
60 2 23.4 88.00 153 89.8 58.0 3.00 3.2581 95.000 104 
54 1 26.8 87.00 206 122.0 68.0 3.00 4.3820 80.000 59 
25 1 28.3 87.00 193 128.0 49.0 4.00 4.3820 92.000 246 
54 2 27.7 113.00 200 128.4 37.0 5.00 5.1533 113.000 297 
55 1 36.6 113.00 199 94.4 43.0 4.63 5.7301 97.000 258 
40 2 26.5 93.00 236 147.0 37.0 7.00 5.5607 92.000 229 
62 2 31.8 115.00 199 128.6 44.0 5.00 4.8828 98.000 275 
65 1 24.4 120.00 222 135.6 37.0 6.00 5.5094 124.000 281 
33 2 25.4 102.00 206 141.0 39.0 5.00 4.8675 105.000 179 
53 1 22.0 94.00 175 88.0 59.0 3.00 4.9416 98.000 200 
35 1 26.8 98.00 162 103.6 45.0 4.00 4.2047 86.000 200 
66 1 28.0 101.00 195 129.2 40.0 5.00 4.8598 94.000 173 
62 2 33.9 101.00 221 156.4 35.0 6.00 4.9972 103.000 180 
50 2 29.6 94.33 300 242.4 33.0 9.09 4.8122 109.000 84 
47 1 28.6 97.00 164 90.6 56.0 3.00 4.4659 88.000 121 
47 2 25.6 94.00 165 74.8 40.0 4.00 5.5255 93.000 161 
24 1 20.7 87.00 149 80.6 61.0 2.00 3.6109 78.000 99 
58 2 26.2 91.00 217 124.2 71.0 3.00 4.6913 68.000 109 
34 1 20.6 87.00 185 112.2 58.0 3.00 4.3041 74.000 115 
51 1 27.9 96.00 196 122.2 42.0 5.00 5.0689 120.000 268 
31 2 35.3 125.00 187 112.4 48.0 4.00 4.8903 109.000 274 
22 1 19.9 75.00 175 108.6 54.0 3.00 4.1271 72.000 158 
53 2 24.4 92.00 214 146.0 50.0 4.00 4.4998 97.000 107 
37 2 21.4 83.00 128 69.6 49.0 3.00 3.8501 84.000 83 
28 1 30.4 85.00 198 115.6 67.0 3.00 4.3438 80.000 103 
47 1 31.6 84.00 154 88.0 30.0 5.10 5.1985 105.000 272 
23 1 18.8 78.00 145 72.0 63.0 2.00 3.9120 86.000 85 
50 1 31.0 123.00 178 105.0 48.0 4.00 4.8283 88.000 280 
58 2 36.7 117.00 166 93.8 44.0 4.00 4.9488 109.000 336 
55 1 32.1 110.00 164 84.2 42.0 4.00 5.2417 90.000 281 
60 2 27.7 107.00 167 114.6 38.0 4.00 4.2767 95.000 118 
41 1 30.8 81.00 214 152.0 28.0 7.60 5.1358 123.000 317 
60 2 27.5 106.00 229 143.8 51.0 4.00 5.1417 91.000 235 
40 1 26.9 92.00 203 119.8 70.0 3.00 4.1897 81.000 60 
57 2 30.7 90.00 204 147.8 34.0 6.00 4.7095 93.000 174 
37 1 38.3 113.00 165 94.6 53.0 3.00 4.4659 79.000 259 
40 2 31.9 95.00 198 135.6 38.0 5.00 4.8040 93.000 178 
33 1 35.0 89.00 200 130.4 42.0 4.76 4.9273 101.000 128 
32 2 27.8 89.00 216 146.2 55.0 4.00 4.3041 91.000 96 
35 2 25.9 81.00 174 102.4 31.0 6.00 5.3132 82.000 126 
55 1 32.9 102.00 164 106.2 41.0 4.00 4.4308 89.000 288 
49 1 26.0 93.00 183 100.2 64.0 3.00 4.5433 88.000 88 
39 2 26.3 115.00 218 158.2 32.0 7.00 4.9345 109.000 292 
60 2 22.3 113.00 186 125.8 46.0 4.00 4.2627 94.000 71 
67 2 28.3 93.00 204 132.2 49.0 4.00 4.7362 92.000 197 
41 2 32.0 109.00 251 170.6 49.0 5.00 5.0562 103.000 186 
44 1 25.4 95.00 162 92.6 53.0 3.00 4.4067 83.000 25 
48 2 23.3 89.33 212 142.8 46.0 4.61 4.7536 98.000 84 
45 1 20.3 74.33 190 126.2 49.0 3.88 4.3041 79.000 96 
47 1 30.4 120.00 199 120.0 46.0 4.00 5.1059 87.000 195 
46 1 20.6 73.00 172 107.0 51.0 3.00 4.2485 80.000 53 
36 2 32.3 115.00 286 199.4 39.0 7.00 5.4723 112.000 217 
34 1 29.2 73.00 172 108.2 49.0 4.00 4.3041 91.000 172 
53 2 33.1 117.00 183 119.0 48.0 4.00 4.3820 106.000 131 
61 1 24.6 101.00 209 106.8 77.0 3.00 4.8363 88.000 214 
37 1 20.2 81.00 162 87.8 63.0 3.00 4.0254 88.000 59 
33 2 20.8 84.00 125 70.2 46.0 3.00 3.7842 66.000 70 
68 1 32.8 105.67 205 116.4 40.0 5.13 5.4931 117.000 220 
49 2 31.9 94.00 234 155.8 34.0 7.00 5.3982 122.000 268 
48 1 23.9 109.00 232 105.2 37.0 6.00 6.1070 96.000 152 
55 2 24.5 84.00 179 105.8 66.0 3.00 3.5835 87.000 47 
43 1 22.1 66.00 134 77.2 45.0 3.00 4.0775 80.000 74 
60 2 33.0 97.00 217 125.6 45.0 5.00 5.4467 112.000 295 
31 2 19.0 93.00 137 73.0 47.0 3.00 4.4427 78.000 101 
53 2 27.3 82.00 119 55.0 39.0 3.00 4.8283 93.000 151 
67 1 22.8 87.00 166 98.6 52.0 3.00 4.3438 92.000 127 
61 2 28.2 106.00 204 132.0 52.0 4.00 4.6052 96.000 237 
62 1 28.9 87.33 206 127.2 33.0 6.24 5.4337 99.000 225 
60 1 25.6 87.00 207 125.8 69.0 3.00 4.1109 84.000 81 
42 1 24.9 91.00 204 141.8 38.0 5.00 4.7958 89.000 151 
38 2 26.8 105.00 181 119.2 37.0 5.00 4.8203 91.000 107 
62 1 22.4 79.00 222 147.4 59.0 4.00 4.3567 76.000 64 
61 2 26.9 111.00 236 172.4 39.0 6.00 4.8122 89.000 138 
61 2 23.1 113.00 186 114.4 47.0 4.00 4.8122 105.000 185 
53 1 28.6 88.00 171 98.8 41.0 4.00 5.0499 99.000 265 
28 2 24.7 97.00 175 99.6 32.0 5.00 5.3799 87.000 101 
26 2 30.3 89.00 218 152.2 31.0 7.00 5.1591 82.000 137 
30 1 21.3 87.00 134 63.0 63.0 2.00 3.6889 66.000 143 
50 1 26.1 109.00 243 160.6 62.0 4.00 4.6250 89.000 141 
48 1 20.2 95.00 187 117.4 53.0 4.00 4.4188 85.000 79 
51 1 25.2 103.00 176 112.2 37.0 5.00 4.8978 90.000 292 
47 2 22.5 82.00 131 66.8 41.0 3.00 4.7536 89.000 178 
64 2 23.5 97.00 203 129.0 59.0 3.00 4.3175 77.000 91 
51 2 25.9 76.00 240 169.0 39.0 6.00 5.0752 96.000 116 
30 1 20.9 104.00 152 83.8 47.0 3.00 4.6634 97.000 86 
56 2 28.7 99.00 208 146.4 39.0 5.00 4.7274 97.000 122 
42 1 22.1 85.00 213 138.6 60.0 4.00 4.2767 94.000 72 
62 2 26.7 115.00 183 124.0 35.0 5.00 4.7875 100.000 129 
34 1 31.4 87.00 149 93.8 46.0 3.00 3.8286 77.000 142 
60 1 22.2 104.67 221 105.4 60.0 3.68 5.6276 93.000 90 
64 1 21.0 92.33 227 146.8 65.0 3.49 4.3307 102.000 158 
39 2 21.2 90.00 182 110.4 60.0 3.00 4.0604 98.000 39 
71 2 26.5 105.00 281 173.6 55.0 5.00 5.5683 84.000 196 
48 2 29.2 110.00 218 151.6 39.0 6.00 4.9200 98.000 222 
79 2 27.0 103.00 169 110.8 37.0 5.00 4.6634 110.000 277 
40 1 30.7 99.00 177 85.4 50.0 4.00 5.3375 85.000 99 
49 2 28.8 92.00 207 140.0 44.0 5.00 4.7449 92.000 196 
51 1 30.6 103.00 198 106.6 57.0 3.00 5.1475 100.000 202 
57 1 30.1 117.00 202 139.6 42.0 5.00 4.6250 120.000 155 
59 2 24.7 114.00 152 104.8 29.0 5.00 4.5109 88.000 77 
51 1 27.7 99.00 229 145.6 69.0 3.00 4.2767 77.000 191 
74 1 29.8 101.00 171 104.8 50.0 3.00 4.3944 86.000 70 
67 1 26.7 105.00 225 135.4 69.0 3.00 4.6347 96.000 73 
49 1 19.8 88.00 188 114.8 57.0 3.00 4.3944 93.000 49 
57 1 23.3 88.00 155 63.6 78.0 2.00 4.2047 78.000 65 
56 2 35.1 123.00 164 95.0 38.0 4.00 5.0434 117.000 263 
52 2 29.7 109.00 228 162.8 31.0 8.00 5.1417 103.000 248 
69 1 29.3 124.00 223 139.0 54.0 4.00 5.0106 102.000 296 
37 1 20.3 83.00 185 124.6 38.0 5.00 4.7185 88.000 214 
24 1 22.5 89.00 141 68.0 52.0 3.00 4.6540 84.000 185 
55 2 22.7 93.00 154 94.2 53.0 3.00 3.5264 75.000 78 
36 1 22.8 87.00 178 116.0 41.0 4.00 4.6540 82.000 93 
42 2 24.0 107.00 150 85.0 44.0 3.00 4.6540 96.000 252 
21 1 24.2 76.00 147 77.0 53.0 3.00 4.4427 79.000 150 
41 1 20.2 62.00 153 89.0 50.0 3.00 4.2485 89.000 77 
57 2 29.4 109.00 160 87.6 31.0 5.00 5.3327 92.000 208 
20 2 22.1 87.00 171 99.6 58.0 3.00 4.2047 78.000 77 
67 2 23.6 111.33 189 105.4 70.0 2.70 4.2195 93.000 108 
34 1 25.2 77.00 189 120.6 53.0 4.00 4.3438 79.000 160 
41 2 24.9 86.00 192 115.0 61.0 3.00 4.3820 94.000 53 
38 2 33.0 78.00 301 215.0 50.0 6.02 5.1930 108.000 220 
51 1 23.5 101.00 195 121.0 51.0 4.00 4.7449 94.000 154 
52 2 26.4 91.33 218 152.0 39.0 5.59 4.9053 99.000 259 
67 1 29.8 80.00 172 93.4 63.0 3.00 4.3567 82.000 90 
61 1 30.0 108.00 194 100.0 52.0 3.73 5.3471 105.000 246 
67 2 25.0 111.67 146 93.4 33.0 4.42 4.5850 103.000 124 
56 1 27.0 105.00 247 160.6 54.0 5.00 5.0876 94.000 67 
64 1 20.0 74.67 189 114.8 62.0 3.05 4.1109 91.000 72 
58 2 25.5 112.00 163 110.6 29.0 6.00 4.7622 86.000 257 
55 1 28.2 91.00 250 140.2 67.0 4.00 5.3660 103.000 262 
62 2 33.3 114.00 182 114.0 38.0 5.00 5.0106 96.000 275 
57 2 25.6 96.00 200 133.0 52.0 3.85 4.3175 105.000 177 
20 2 24.2 88.00 126 72.2 45.0 3.00 3.7842 74.000 71 
53 2 22.1 98.00 165 105.2 47.0 4.00 4.1589 81.000 47 
32 2 31.4 89.00 153 84.2 56.0 3.00 4.1589 90.000 187 
41 1 23.1 86.00 148 78.0 58.0 3.00 4.0943 60.000 125 
60 1 23.4 76.67 247 148.0 65.0 3.80 5.1358 77.000 78 
26 1 18.8 83.00 191 103.6 69.0 3.00 4.5218 69.000 51 
37 1 30.8 112.00 282 197.2 43.0 7.00 5.3423 101.000 258 
45 1 32.0 110.00 224 134.2 45.0 5.00 5.4116 93.000 215 
67 1 31.6 116.00 179 90.4 41.0 4.00 5.4723 100.000 303 
34 2 35.5 120.00 233 146.6 34.0 7.00 5.5683 101.000 243 
50 1 31.9 78.33 207 149.2 38.0 5.45 4.5951 84.000 91 
71 1 29.5 97.00 227 151.6 45.0 5.00 5.0239 108.000 150 
57 2 31.6 117.00 225 107.6 40.0 6.00 5.9584 113.000 310 
49 1 20.3 93.00 184 103.0 61.0 3.00 4.6052 93.000 153 
35 1 41.3 81.00 168 102.8 37.0 5.00 4.9488 94.000 346 
41 2 21.2 102.00 184 100.4 64.0 3.00 4.5850 79.000 63 
70 2 24.1 82.33 194 149.2 31.0 6.26 4.2341 105.000 89 
52 1 23.0 107.00 179 123.7 42.5 4.21 4.1589 93.000 50 
60 1 25.6 78.00 195 95.4 91.0 2.00 3.7612 87.000 39 
62 1 22.5 125.00 215 99.0 98.0 2.00 4.4998 95.000 103 
44 2 38.2 123.00 201 126.6 44.0 5.00 5.0239 92.000 308 
28 2 19.2 81.00 155 94.6 51.0 3.00 3.8501 87.000 116 
58 2 29.0 85.00 156 109.2 36.0 4.00 3.9890 86.000 145 
39 2 24.0 89.67 190 113.6 52.0 3.65 4.8040 101.000 74 
34 2 20.6 98.00 183 92.0 83.0 2.00 3.6889 92.000 45 
65 1 26.3 70.00 244 166.2 51.0 5.00 4.8978 98.000 115 
66 2 34.6 115.00 204 139.4 36.0 6.00 4.9628 109.000 264 
51 1 23.4 87.00 220 108.8 93.0 2.00 4.5109 82.000 87 
50 2 29.2 119.00 162 85.2 54.0 3.00 4.7362 95.000 202 
59 2 27.2 107.00 158 102.0 39.0 4.00 4.4427 93.000 127 
52 1 27.0 78.33 134 73.0 44.0 3.05 4.4427 69.000 182 
69 2 24.5 108.00 243 136.4 40.0 6.00 5.8081 100.000 241 
53 1 24.1 105.00 184 113.4 46.0 4.00 4.8122 95.000 66 
47 2 25.3 98.00 173 105.6 44.0 4.00 4.7622 108.000 94 
52 1 28.8 113.00 280 174.0 67.0 4.00 5.2730 86.000 283 
39 1 20.9 95.00 150 65.6 68.0 2.00 4.4067 95.000 64 
67 2 23.0 70.00 184 128.0 35.0 5.00 4.6540 99.000 102 
59 2 24.1 96.00 170 98.6 54.0 3.00 4.4659 85.000 200 
51 2 28.1 106.00 202 122.2 55.0 4.00 4.8203 87.000 265 
23 2 18.0 78.00 171 96.0 48.0 4.00 4.9053 92.000 94 
68 1 25.9 93.00 253 181.2 53.0 5.00 4.5433 98.000 230 
44 1 21.5 85.00 157 92.2 55.0 3.00 3.8918 84.000 181 
60 2 24.3 103.00 141 86.6 33.0 4.00 4.6728 78.000 156 
52 1 24.5 90.00 198 129.0 29.0 7.00 5.2983 86.000 233 
38 1 21.3 72.00 165 60.2 88.0 2.00 4.4308 90.000 60 
61 1 25.8 90.00 280 195.4 55.0 5.00 4.9972 90.000 219 
68 2 24.8 101.00 221 151.4 60.0 4.00 3.8712 87.000 80 
28 2 31.5 83.00 228 149.4 38.0 6.00 5.3132 83.000 68 
65 2 33.5 102.00 190 126.2 35.0 5.00 4.9698 102.000 332 
69 1 28.1 113.00 234 142.8 52.0 4.00 5.2781 77.000 248 
51 1 24.3 85.33 153 71.6 71.0 2.15 3.9512 82.000 84 
29 1 35.0 98.33 204 142.6 50.0 4.08 4.0431 91.000 200 
55 2 23.5 93.00 177 126.8 41.0 4.00 3.8286 83.000 55 
34 2 30.0 83.00 185 107.2 53.0 3.00 4.8203 92.000 85 
67 1 20.7 83.00 170 99.8 59.0 3.00 4.0254 77.000 89 
49 1 25.6 76.00 161 99.8 51.0 3.00 3.9318 78.000 31 
55 2 22.9 81.00 123 67.2 41.0 3.00 4.3041 88.000 129 
59 2 25.1 90.00 163 101.4 46.0 4.00 4.3567 91.000 83 
53 1 33.2 82.67 186 106.8 46.0 4.04 5.1120 102.000 275 
48 2 24.1 110.00 209 134.6 58.0 4.00 4.4067 100.000 65 
52 1 29.5 104.33 211 132.8 49.0 4.31 4.9836 98.000 198 
69 1 29.6 122.00 231 128.4 56.0 4.00 5.4510 86.000 236 
60 2 22.8 110.00 245 189.8 39.0 6.00 4.3944 88.000 253 
46 2 22.7 83.00 183 125.8 32.0 6.00 4.8363 75.000 124 
51 2 26.2 101.00 161 99.6 48.0 3.00 4.2047 88.000 44 
67 2 23.5 96.00 207 138.2 42.0 5.00 4.8978 111.000 172 
49 1 22.1 85.00 136 63.4 62.0 2.19 3.9703 72.000 114 
46 2 26.5 94.00 247 160.2 59.0 4.00 4.9345 111.000 142 
47 1 32.4 105.00 188 125.0 46.0 4.09 4.4427 99.000 109 
75 1 30.1 78.00 222 154.2 44.0 5.05 4.7791 97.000 180 
28 1 24.2 93.00 174 106.4 54.0 3.00 4.2195 84.000 144 
65 2 31.3 110.00 213 128.0 47.0 5.00 5.2470 91.000 163 
42 1 30.1 91.00 182 114.8 49.0 4.00 4.5109 82.000 147 
51 1 24.5 79.00 212 128.6 65.0 3.00 4.5218 91.000 97 
53 2 27.7 95.00 190 101.8 41.0 5.00 5.4638 101.000 220 
54 1 23.2 110.67 238 162.8 48.0 4.96 4.9127 108.000 190 
73 1 27.0 102.00 211 121.0 67.0 3.00 4.7449 99.000 109 
54 1 26.8 108.00 176 80.6 67.0 3.00 4.9558 106.000 191 
42 1 29.2 93.00 249 174.2 45.0 6.00 5.0039 92.000 122 
75 1 31.2 117.67 229 138.8 29.0 7.90 5.7236 106.000 230 
55 2 32.1 112.67 207 92.4 25.0 8.28 6.1048 111.000 242 
68 2 25.7 109.00 233 112.6 35.0 7.00 6.0568 105.000 248 
57 1 26.9 98.00 246 165.2 38.0 7.00 5.3660 96.000 249 
48 1 31.4 75.33 242 151.6 38.0 6.37 5.5683 103.000 192 
61 2 25.6 85.00 184 116.2 39.0 5.00 4.9698 98.000 131 
69 1 37.0 103.00 207 131.4 55.0 4.00 4.6347 90.000 237 
38 1 32.6 77.00 168 100.6 47.0 4.00 4.6250 96.000 78 
45 2 21.2 94.00 169 96.8 55.0 3.00 4.4543 102.000 135 
51 2 29.2 107.00 187 139.0 32.0 6.00 4.3820 95.000 244 
71 2 24.0 84.00 138 85.8 39.0 4.00 4.1897 90.000 199 
57 1 36.1 117.00 181 108.2 34.0 5.00 5.2679 100.000 270 
56 2 25.8 103.00 177 114.4 34.0 5.00 4.9628 99.000 164 
32 2 22.0 88.00 137 78.6 48.0 3.00 3.9512 78.000 72 
50 1 21.9 91.00 190 111.2 67.0 3.00 4.0775 77.000 96 
43 1 34.3 84.00 256 172.6 33.0 8.00 5.5294 104.000 306 
54 2 25.2 115.00 181 120.0 39.0 5.00 4.7005 92.000 91 
31 1 23.3 85.00 190 130.8 43.0 4.00 4.3944 77.000 214 
56 1 25.7 80.00 244 151.6 59.0 4.00 5.1180 95.000 95 
44 1 25.1 133.00 182 113.0 55.0 3.00 4.2485 84.000 216 
57 2 31.9 111.00 173 116.2 41.0 4.00 4.3694 87.000 263 
64 2 28.4 111.00 184 127.0 41.0 4.00 4.3820 97.000 178 
43 1 28.1 121.00 192 121.0 60.0 3.00 4.0073 93.000 113 
19 1 25.3 83.00 225 156.6 46.0 5.00 4.7185 84.000 200 
71 2 26.1 85.00 220 152.4 47.0 5.00 4.6347 91.000 139 
50 2 28.0 104.00 282 196.8 44.0 6.00 5.3279 95.000 139 
59 2 23.6 73.00 180 107.4 51.0 4.00 4.6821 84.000 88 
57 1 24.5 93.00 186 96.6 71.0 3.00 4.5218 91.000 148 
49 2 21.0 82.00 119 85.4 23.0 5.00 3.9703 74.000 88 
41 2 32.0 126.00 198 104.2 49.0 4.00 5.4116 124.000 243 
25 2 22.6 85.00 130 71.0 48.0 3.00 4.0073 81.000 71 
52 2 19.7 81.00 152 53.4 82.0 2.00 4.4188 82.000 77 
34 1 21.2 84.00 254 113.4 52.0 5.00 6.0936 92.000 109 
42 2 30.6 101.00 269 172.2 50.0 5.00 5.4553 106.000 272 
28 2 25.5 99.00 162 101.6 46.0 4.00 4.2767 94.000 60 
47 2 23.3 90.00 195 125.8 54.0 4.00 4.3307 73.000 54 
32 2 31.0 100.00 177 96.2 45.0 4.00 5.1874 77.000 221 
43 1 18.5 87.00 163 93.6 61.0 2.67 3.7377 80.000 90 
59 2 26.9 104.00 194 126.6 43.0 5.00 4.8040 106.000 311 
53 1 28.3 101.00 179 107.0 48.0 4.00 4.7875 101.000 281 
60 1 25.7 103.00 158 84.6 64.0 2.00 3.8501 97.000 182 
54 2 36.1 115.00 163 98.4 43.0 4.00 4.6821 101.000 321 
35 2 24.1 94.67 155 97.4 32.0 4.84 4.8520 94.000 58 
49 2 25.8 89.00 182 118.6 39.0 5.00 4.8040 115.000 262 
58 1 22.8 91.00 196 118.8 48.0 4.00 4.9836 115.000 206 
36 2 39.1 90.00 219 135.8 38.0 6.00 5.4205 103.000 233 
46 2 42.2 99.00 211 137.0 44.0 5.00 5.0106 99.000 242 
44 2 26.6 99.00 205 109.0 43.0 5.00 5.5797 111.000 123 
46 1 29.9 83.00 171 113.0 38.0 4.50 4.5850 98.000 167 
54 1 21.0 78.00 188 107.4 70.0 3.00 3.9703 73.000 63 
63 2 25.5 109.00 226 103.2 46.0 5.00 5.9506 87.000 197 
41 2 24.2 90.00 199 123.6 57.0 4.00 4.5218 86.000 71 
28 1 25.4 93.00 141 79.0 49.0 3.00 4.1744 91.000 168 
19 1 23.2 75.00 143 70.4 52.0 3.00 4.6347 72.000 140 
61 2 26.1 126.00 215 129.8 57.0 4.00 4.9488 96.000 217 
48 1 32.7 93.00 276 198.6 43.0 6.42 5.1475 91.000 121 
54 2 27.3 100.00 200 144.0 33.0 6.00 4.7449 76.000 235 
53 2 26.6 93.00 185 122.4 36.0 5.00 4.8903 82.000 245 
48 1 22.8 101.00 110 41.6 56.0 2.00 4.1271 97.000 40 
53 1 28.8 111.67 145 87.2 46.0 3.15 4.0775 85.000 52 
29 2 18.1 73.00 158 99.0 41.0 4.00 4.4998 78.000 104 
62 1 32.0 88.00 172 69.0 38.0 4.00 5.7838 100.000 132 
50 2 23.7 92.00 166 97.0 52.0 3.00 4.4427 93.000 88 
58 2 23.6 96.00 257 171.0 59.0 4.00 4.9053 82.000 69 
55 2 24.6 109.00 143 76.4 51.0 3.00 4.3567 88.000 219 
54 1 22.6 90.00 183 104.2 64.0 3.00 4.3041 92.000 72 
36 1 27.8 73.00 153 104.4 42.0 4.00 3.4965 73.000 201 
63 2 24.1 111.00 184 112.2 44.0 4.00 4.9345 82.000 110 
47 2 26.5 70.00 181 104.8 63.0 3.00 4.1897 70.000 51 
51 2 32.8 112.00 202 100.6 37.0 5.00 5.7746 109.000 277 
42 1 19.9 76.00 146 83.2 55.0 3.00 3.6636 79.000 63 
37 2 23.6 94.00 205 138.8 53.0 4.00 4.1897 107.000 118 
28 1 22.1 82.00 168 100.6 54.0 3.00 4.2047 86.000 69 
58 1 28.1 111.00 198 80.6 31.0 6.00 6.0684 93.000 273 
32 1 26.5 86.00 184 101.6 53.0 4.00 4.9904 78.000 258 
25 2 23.5 88.00 143 80.8 55.0 3.00 3.5835 83.000 43 
63 1 26.0 85.67 155 78.2 46.0 3.37 5.0370 97.000 198 
52 1 27.8 85.00 219 136.0 49.0 4.00 5.1358 75.000 242 
65 2 28.5 109.00 201 123.0 46.0 4.00 5.0752 96.000 232 
42 1 30.6 121.00 176 92.8 69.0 3.00 4.2627 89.000 175 
53 1 22.2 78.00 164 81.0 70.0 2.00 4.1744 101.000 93 
79 2 23.3 88.00 186 128.4 33.0 6.00 4.8122 102.000 168 
43 1 35.4 93.00 185 100.2 44.0 4.00 5.3181 101.000 275 
44 1 31.4 115.00 165 97.6 52.0 3.00 4.3438 89.000 293 
62 2 37.8 119.00 113 51.0 31.0 4.00 5.0434 84.000 281 
33 1 18.9 70.00 162 91.8 59.0 3.00 4.0254 58.000 72 
56 1 35.0 79.33 195 140.8 42.0 4.64 4.1109 96.000 140 
66 1 21.7 126.00 212 127.8 45.0 4.71 5.2781 101.000 189 
34 2 25.3 111.00 230 162.0 39.0 6.00 4.9767 90.000 181 
46 2 23.8 97.00 224 139.2 42.0 5.00 5.3660 81.000 209 
50 1 31.8 82.00 136 69.2 55.0 2.00 4.0775 85.000 136 
69 1 34.3 113.00 200 123.8 54.0 4.00 4.7095 112.000 261 
34 1 26.3 87.00 197 120.0 63.0 3.00 4.2485 96.000 113 
71 2 27.0 93.33 269 190.2 41.0 6.56 5.2417 93.000 131 
47 1 27.2 80.00 208 145.6 38.0 6.00 4.8040 92.000 174 
41 1 33.8 123.33 187 127.0 45.0 4.16 4.3175 100.000 257 
34 1 33.0 73.00 178 114.6 51.0 3.49 4.1271 92.000 55 
51 1 24.1 87.00 261 175.6 69.0 4.00 4.4067 93.000 84 
43 1 21.3 79.00 141 78.8 53.0 3.00 3.8286 90.000 42 
55 1 23.0 94.67 190 137.6 38.0 5.00 4.2767 106.000 146 
59 2 27.9 101.00 218 144.2 38.0 6.00 5.1874 95.000 212 
27 2 33.6 110.00 246 156.6 57.0 4.00 5.0876 89.000 233 
51 2 22.7 103.00 217 162.4 30.0 7.00 4.8122 80.000 91 
49 2 27.4 89.00 177 113.0 37.0 5.00 4.9053 97.000 111 
27 1 22.6 71.00 116 43.4 56.0 2.00 4.4188 79.000 152 
57 2 23.2 107.33 231 159.4 41.0 5.63 5.0304 112.000 120 
39 2 26.9 93.00 136 75.4 48.0 3.00 4.1431 99.000 67 
62 2 34.6 120.00 215 129.2 43.0 5.00 5.3660 123.000 310 
37 1 23.3 88.00 223 142.0 65.0 3.40 4.3567 82.000 94 
46 1 21.1 80.00 205 144.4 42.0 5.00 4.5326 87.000 183 
68 2 23.5 101.00 162 85.4 59.0 3.00 4.4773 91.000 66 
51 1 31.5 93.00 231 144.0 49.0 4.70 5.2523 117.000 173 
41 1 20.8 86.00 223 128.2 83.0 3.00 4.0775 89.000 72 
53 1 26.5 97.00 193 122.4 58.0 3.00 4.1431 99.000 49 
45 1 24.2 83.00 177 118.4 45.0 4.00 4.2195 82.000 64 
33 1 19.5 80.00 171 85.4 75.0 2.00 3.9703 80.000 48 
60 2 28.2 112.00 185 113.8 42.0 4.00 4.9836 93.000 178 
47 2 24.9 75.00 225 166.0 42.0 5.00 4.4427 102.000 104 
60 2 24.9 99.67 162 106.6 43.0 3.77 4.1271 95.000 132 
36 1 30.0 95.00 201 125.2 42.0 4.79 5.1299 85.000 220 
36 1 19.6 71.00 250 133.2 97.0 3.00 4.5951 92.000 57 
;


proc stdize data=diabetes out=std_diabetes;
  var age bmi map tc ldl hdl tch ltg glu;
run;


ods graphics on;
ods output postintervals=intervals;
proc mcmc data=std_diabetes seed=45678 nmc=50000 propcov=quanew
          monitor=(b0 beta1-beta10 tau1-tau10 sigma2 lasso)
          outpost=posterior;

  array D[10,10];
  array beta[10] beta1-beta10;
  array mu0[10];
  array data[10] age sex bmi map tc ldl hdl tch ltg glu;

  begincnst;
   call identity(D);
   call zeromatrix(mu0);
  endcnst;

  beginnodata;
    lasso=sqrt(lambda);
    b=lambda/2;

    %macro loop;
      %do k = 1 %to 10;
         tau&k = exp(omega&k);
         D[&k,&k]=sigma2*tau&k;
         %end;
    %mend loop;
    %loop;
  endnodata;

  call mult(beta, data,xb);

  parms lambda ;
  prior lambda ~ gamma(1,scale=.1);
  parms omega1-omega10;
  prior omega: ~ expexpon(iscale=b);
  parms sigma2 1;
  prior sigma2 ~ igamma(shape = .1, iscale = .1);
  parms b0 0;
  prior b0 ~ general(0);
  parms beta;
  prior beta ~ mvn(mu0,D);

  model y ~ normal(b0 + xb,var=sigma2);

run;


proc means data=posterior mode;
  var b0 beta1-beta10;
  output out=parameters(drop=_TYPE_ _FREQ_) mode(b0 beta1-beta10)=b0 beta1-beta10;
run;

proc transpose data=parameters out=parameters;
run;


data parameters;
  length parameter $ 6;
  set parameters(rename=(col1=mode _NAME_=Parameter));
  label Parameter=;
  index=_N_;
run;


proc sort data=parameters out=parameters;
  by parameter;
run;

proc sort data=intervals out=intervals;
  by parameter;
run;


data parameters(where=(~missing(mode)));
  merge parameters intervals;
  by parameter;
  label parameter="Parameter" mode="Mode";
run;


proc sort data=parameters out=parameters;
  by index;
run;


proc print data=parameters noobs label;
  var parameter mode hpdlower hpdupper;
run;