Biostat 656: Homework 2

1. Binomial outcome -surgical data example in WinBUGs. This example concerns the mortality rates in 12 hospitals performing cardiac surgery in babies.

Hospital No of ops No of deaths

A 47 0 B 148 18 C 119 8 D 810 46 E 211 8 F 196 13 G 148 9 H 215 31 I 207 14 J 97 8 K 256 29 L 360 24

a. Run the fixed effects program in WinBUGS.

For fixed effect model, we assume that the true failure probabilities are independent for each hospital.

This is equivalent to assuming a standard non-informative prior distribution for Pi , Pi ~ beta(1.0,1.0) . The following is the programming in Winbugs.

**Fixed effect model for surgical data*** model { for( i in 1 : N ) { p[i] ~ dbeta(1.0, 1.0) r[i] ~ dbin(p[i], n[i]) } }

list(n = c(47, 148, 119, 810, 211, 196, 148, 215, 207, 97, 256, 360), r = c(0, 18, 8, 46, 8, 13, 9, 31, 14, 8, 29, 24), N = 12)

list(p = c(0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1))

Results from 5000 iterations with 200 burn in

node mean sd MC error2.5% median 97.5% start sample p[1] 0.02002 0.0191 3.061E-4 5.973E-4 0.01456 0.07009 200 4801 p[2] 0.1265 0.0274 3.973E-4 0.07774 0.1248 0.1839 200 4801 p[3] 0.07425 0.02375 3.597E-4 0.03508 0.07169 0.1281 200 4801 p[4] 0.05787 0.008141 1.167E-4 0.04272 0.05756 0.07487 200 4801 p[5] 0.0424 0.0139 1.818E-4 0.01939 0.04096 0.07355 200 4801 p[6] 0.07076 0.018 2.644E-4 0.04059 0.06912 0.1093 200 4801 p[7] 0.06664 0.0202 2.907E-4 0.03271 0.06469 0.11 200 4801 p[8] 0.1476 0.02426 3.662E-4 0.1032 0.1462 0.1991 200 4801 p[9] 0.07249 0.0177 2.005E-4 0.04181 0.07132 0.1094 200 4801 p[10] 0.09094 0.02915 4.353E-4 0.04364 0.08812 0.1547 200 4801 p[11] 0.116 0.02031 2.689E-4 0.07921 0.1151 0.1587 200 4801 p[12] 0.06902 0.01341 1.858E-4 0.04514 0.06811 0.09783 200 4801

1 b. Run the random effects program in WinBUGS.

For random effect model, we assume that the failure rates across hospitals are similar in some way which seems more appropriate for the surgical data. This is equivalent to specifying a random effects model for the true failure probabilities pi as follows:

Logit(pi)=bi,I=1,….12, bi~N(mu,tau)

Standard non-informative priors are specified for the population mean (logit) probability of failure(mu) , and precision(tau). The number of deaths ri for hospital I are modelled as a binary response variable with true failure probability Pi , ri~Bin(ni,pi).

**random effect model for surgical data*** model { for( i in 1 : N ) { b[i] ~ dnorm(mu,tau) r[i] ~ dbin(p[i],n[i]) logit(p[i]) <- b[i] } pop.mean <- exp(mu) / (1 + exp(mu)) mu ~ dnorm(0.0,1.0E-6) sigma <- 1 / sqrt(tau) tau ~ dgamma(0.001,0.001) }

list(n = c(47, 148, 119, 810, 211, 196, 148, 215, 207, 97, 256, 360), r = c(0, 18, 8, 46, 8, 13, 9, 31, 14, 8, 29, 24), N = 12) list(b = c( 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1,0.1, 0.1, 0.1),tau = 1, mu = 0)

Results from 5000 iterations with 200 burn in

node mean sd MC error2.5% median 97.5% start sample mu -2.563 0.1578 0.003854 -2.889 -2.556 -2.268 200 4801 pop.mean0.07226 0.0104 2.524E-4 0.05272 0.07203 0.09383 200 4801 sigma 0.4022 0.1631 0.005382 0.1526 0.3761 0.7991 200 4801 tau 10.78 15.6 0.7407 1.566 7.069 42.94 200 4801

b[1] -2.967 0.4539 0.01231 -4.026 -2.905 -2.267 200 4801 b[2] -2.187 0.2341 0.004292 -2.628 -2.192 -1.719 200 4801 b[3] -2.613 0.2685 0.004592 -3.189 -2.6 -2.125 200 4801 b[4] -2.774 0.1442 0.002749 -3.063 -2.771 -2.499 200 4801 b[5] -2.94 0.2811 0.006883 -3.543 -2.925 -2.443 200 4801 b[6] -2.623 0.2313 0.004186 -3.111 -2.611 -2.185 200 4801 b[7] -2.664 0.2618 0.004135 -3.215 -2.652 -2.191 200 4801 b[8] -1.985 0.2123 0.005785 -2.412 -1.979 -1.587 200 4801 b[9] -2.613 0.2249 0.003453 -3.072 -2.605 -2.191 200 4801 b[10] -2.501 0.2746 0.004172 -3.068 -2.496 -1.986 200 4801 b[11] -2.196 0.192 0.003624 -2.571 -2.194 -1.826 200 4801 b[12] -2.625 0.1853 0.002983 -3.01 -2.621 -2.277 200 4801

p[1] 0.05286 0.01965 5.226E-4 0.01753 0.05189 0.09393 200 4801 p[2] 0.1029 0.02185 4.005E-4 0.06738 0.1005 0.152 200 4801 p[3] 0.07026 0.01707 2.891E-4 0.0396 0.06914 0.1067 200 4801 p[4] 0.05927 0.008018 1.546E-4 0.04464 0.05892 0.07592 200 4801 p[5] 0.05187 0.01331 3.324E-4 0.02811 0.05095 0.07994 200 4801 p[6] 0.0691 0.0146 2.584E-4 0.04267 0.06845 0.1011 200 4801

2 p[7] 0.0669 0.01603 2.505E-4 0.03861 0.06587 0.1005 200 4801 p[8] 0.1225 0.02266 5.916E-4 0.08222 0.1215 0.1699 200 4801 p[9] 0.06966 0.01446 2.222E-4 0.04428 0.06879 0.1006 200 4801 p[10] 0.07801 0.0196 2.955E-4 0.04444 0.07614 0.1207 200 4801 p[11] 0.1014 0.01749 3.225E-4 0.071 0.1003 0.1387 200 4801 p[12] 0.06845 0.01168 1.86E-4 0.04698 0.06778 0.09306 200 4801 c. Run the fixed effects program in STATA Note that we have to use the asis command because nobody in hospital A died, so we have to force that covariate to stay in the model. Why is this a problem?

If we don’t force the covariate to stay in the model by using asis command, hospid_1 would be dropped and its 47 obs would not be used. Then no estimate could be obtained for hospital 1.

. xi, prefix(): logit death hospid_1 i.hospid, noconstant asis ------death | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------hospid_1 | -18.20289 . . . . . hospida2 | -1.977163 .2514913 -7.86 0.000 -2.470077 -1.484249 hospida3 | -2.630089 .3660724 -7.18 0.000 -3.347577 -1.9126 hospida4 | -2.809926 .1518158 -18.51 0.000 -3.10748 -2.512373 hospida5 | -3.233764 .3604526 -8.97 0.000 -3.940239 -2.52729 hospida6 | -2.644537 .2870323 -9.21 0.000 -3.20711 -2.081964 hospida7 | -2.737249 .3439555 -7.96 0.000 -3.41139 -2.063109 hospida8 | -1.780949 .1941465 -9.17 0.000 -2.161469 -1.400428 hospida9 | -2.623633 .276785 -9.48 0.000 -3.166121 -2.081144 hospida10 | -2.409195 .3691016 -6.53 0.000 -3.132621 -1.685769 hospida11 | -2.057654 .1972005 -10.43 0.000 -2.44416 -1.671148 hospida12 | -2.639057 .2112886 -12.49 0.000 -3.053175 -2.224939 ------d. Run the random effects program in STATA.

. xi: gllamm death, link(logit) fam(binom) i(hospid) adapt ------death | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------_cons | -2.54396 .1349977 -18.84 0.000 -2.808551 -2.279369 ------

Variances and covariances of random effects ------***level 2 (hospid)

var(1): .1279485 (.08651371) ------

. table hospid, c(mean phat)

------hospid | mean(phat) ------+------1 | .05548047 2 | .10271479 3 | .070913 4 | .0590393 5 | .05207246 6 | .0693002 7 | .06693963 8 | .12341625 9 | .07000539

3 10 | .07837136 11 | .10230842 12 | .06842877 ------e. Compare the fixed effects to random effects results. Compare the results from WinBUGS to those in STATA. Which model (fixed effects or random effects) do you think is the most appropriate for these data? Explain.

Fixed vs random:

From the table below, the coefficients are similar in fixed effect models and random effect model except for hospital A. But the interpretation for these two models is different. For fixed effect model, we assume that the true failure probabilities are independent for each hospital. We only make inference about these specific hospitals, and we can not draw any inference about other hospitals. Instead, for random effect, we consider that the failure rates across hospitals are similar in some way and assume the hospitals are sampled randomly from population. We can draw inference about the property of the population. The predicted probabilities are less spread out for the random effect model than for the fixed effect model, since we assume an underlying common distribution in the random effect model. From random effect model, the population mortality rate was estimated as 0.072 with variance 0.01^2.

WinBugs vs Stata:

Winbugs and stata got almost the same estimates for the surgical mortality rate in random effect model, and not quite the same in fixed effect model (but still similar) that is because in winbugs, we need to incorporate prior information. The random effect variance in winbugs is higher than what we got in stata(0.16 vs 0.12).

Although which approach is appropriate depends on the scientific question, in this particular example, random effect model is preferred. Since there is not enough information for hospital A, we cannot make inference about it unless we borrow strength from other hospitals, or from some prior information. Random effect model allows us to borrow information from other hospitals, which gives more reasonable inference. Also, it’s realistic to assume that there exists similarity in mortality across the hospitals and if we think the hospitals are samples from a larger population, random effect model is preferred. Please see Figure 1 for the estimated confidence intervals of morality rate for each hospital based on the random effect model fitted in winbugs. It seems that hospital H has higher failure rate than the other hospitals. And since there are lots of overlaps , it’s hard to tell which hospital has better performance.

Figure1 :95 % Cl for mortality rate across Hospitals

0.18 0.16 0.14 0.12 lower bound 0.1 mean 0.08 0.06 upper bound 0.04 0.02 0 A B C D E F G H I J K L

4 Fixed effects model (cluster: hospital) Winbug Output Stata Output

Hospital pˆ sd 2.50% 97.50%  pˆ A 0.020 0.019 0.001 0.070 -18.2029 0.000 B 0.127 0.027 0.078 0.184 -1.97716 0.122 C 0.074 0.024 0.035 0.128 -2.63009 0.067 D 0.058 0.008 0.043 0.075 -2.80993 0.057 E 0.042 0.014 0.019 0.074 -3.23376 0.038 F 0.071 0.018 0.041 0.109 -2.64454 0.066 G 0.067 0.020 0.033 0.110 -2.73725 0.061 H 0.148 0.024 0.103 0.199 -1.78095 0.144 I 0.072 0.018 0.042 0.109 -2.62363 0.068 J 0.091 0.029 0.044 0.155 -2.4092 0.082 K 0.116 0.020 0.079 0.159 -2.05765 0.113 L 0.069 0.013 0.045 0.098 -2.63906 0.067

Random effects model (cluster: hospital) Winbug Output Stata Output

Hospital pˆ sd 2.50% 97.50% pˆ A 0.053 0.020 0.018 0.094 0.055 B 0.103 0.022 0.067 0.152 0.103 C 0.070 0.017 0.040 0.107 0.071 D 0.059 0.008 0.045 0.076 0.059 E 0.052 0.013 0.028 0.080 0.052 F 0.069 0.015 0.043 0.101 0.069 G 0.067 0.016 0.039 0.101 0.067 H 0.123 0.023 0.082 0.170 0.123 I 0.070 0.014 0.044 0.101 0.070 J 0.078 0.020 0.044 0.121 0.078 K 0.101 0.017 0.071 0.139 0.102 L 0.068 0.012 0.047 0.093 0.068 population Mean 0.072 exp(-2.54)/(1+exp(-2.54))=0.073 variance 0.4022^2=0.161 0.1279485

5 2. Binomial outcome-blocker data example from WinBUGS. 22 clinical trials of beta-blockers are combined to determine if they are associated with lower mortality after myocardial infarction.

Study Mortality: deaths / total Treated Control ______1 3/38 3/39 2 7/114 14/116 3 5/69 11/93 4 102/1533 127/1520 ..... 20 32/209 40/218 21 27/391 43/364 22 22/680 39/674

a. Run the program in WinBUGS.

For this random effect model constructed, we assume that the failure rates for treatment group across study are similar in some way so as for the control group. This is equivalent to specifying a random effects model for the true failure probabilities pci(failure rate for control group) and pti(failure rate for treatment group) as follows:

Logit(pci)=mui,i=1,….22

Logit(pti)=mui+deltai,i=1,….22 mui~N(0,0.00001) deltai ~N(d,tau)

The number of deaths rci for control group in study i are modelled as a binary response variable with true failure probability Pci , rci~Bin(nci,pci) . Similarly, the number of deaths rti for treatment group in study i are modelled as a binary response variable with true failure probability Pti , rti~Bin(nti,pti). To determine if the treatment is associated with lower mortality after myocardial infarction in general, we can regard delta as the treatment effect and check the posterior distrubution of delta.

**Random effect model for outcome-blocker data (please see the appendix for output)*** model { for( i in 1 : Num ) { rc[i] ~ dbin(pc[i], nc[i]) rt[i] ~ dbin(pt[i], nt[i]) logit(pc[i]) <- mu[i] logit(pt[i]) <- mu[i] + delta[i] mu[i] ~ dnorm(0.0,1.0E-5) delta[i] ~ dnorm(d, tau) } d ~ dnorm(0.0,1.0E-6) tau ~ dgamma(0.001,0.001) delta.new ~ dnorm(d, tau) sigma <- 1 / sqrt(tau) }

6 list(rt = c(3, 7, 5, 102, 28, 4, 98, 60, 25, 138, 64, 45, 9, 57, 25, 33, 28, 8, 6, 32, 27, 22 ), nt = c(38, 114, 69, 1533, 355, 59, 945, 632, 278,1916, 873, 263, 291, 858, 154, 207, 251, 151, 174, 209, 391, 680), rc = c(3, 14, 11, 127, 27, 6, 152, 48, 37, 188, 52, 47, 16, 45, 31, 38, 12, 6, 3, 40, 43, 39), nc = c(39, 116, 93, 1520, 365, 52, 939, 471, 282, 1921, 583, 266, 293, 883, 147, 213, 122, 154, 134, 218, 364, 674),Num = 22)

list(d = 0, delta.new = 0, tau=1, mu = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), delta = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0))

b. Run the program in STATA.

. xi: gllamm death treated, link(logit) fam(binom) i(trial) adapt

Variances and covariances of random effects ------

***level 2 (trial)

var(1): .18772082 (.06666833) ------

. gllapred phat, mu

. table trial treated, c(mean phat)

------| treated trial | 0 1 ------+------1 | .08612787 .06950005 2 | .09209377 .0743559 3 | .09608922 .07766296 4 | .07760584 .06242816 5 | .07963423 .06409959 6 | .09210575 .07440005 7 | .12785818 .10396361 8 | .09963289 .08053195 9 | .10803029 .08748829 10 | .08666425 .06984821 11 | .08379102 .0674969 12 | .15691958 .12842081 13 | .05151127 .04121887 14 | .06265911 .0502478 15 | .16168221 .13248112 16 | .15179682 .12410057 17 | .10826446 .08769558 18 | .05794979 .04644393 19 | .04614189 .03688784 20 | .15142521 .12378612 21 | .09376663 .0757011 22 | .05049441 .04039004 ------

7 c. Interpret and compare the results from WinBUGS to those in STATA. In this exercise(Binomial outcome-blocker data) 22 clinical trials of beta-blockers are combined to determine if they are associated with lower mortality after myocardial infarction. To examine this we can check if the log odds ratio (delta) comparing treatment group to control group obtained from the random effect model is less than zero. From both output in winbugs and in stat, the estimated log odds ratio of failure when comparing treatment v,s control is about –0.2 and the 95% confidence intervals don’t cover zero which indicated that clinical trials of beta-blockers is associated with lower mortality after myocardial infarction. The standard deviation for delta in winbugs (0.05958) is similar to the standard error (0.0496209)for delta in stata. The estimated trial-specific death rates for both control & treatment groups are higher from Winbugs than from stata. In addition, the variance due to random effects is 0.013(=0.117^2) in winbugs, and in stata, it provides the variance between trials is 0.1877.

Random effects model(cluster:study) Winbug Output Stata Output Study pˆ (control) pˆ (treatment) pˆ (control) pˆ (treatment) 1 0.087 0.069 0.086 0.070 2 0.103 0.079 0.092 0.074 3 0.108 0.085 0.096 0.078 4 0.084 0.066 0.078 0.062 5 0.084 0.070 0.080 0.064 6 0.101 0.079 0.092 0.074 7 0.153 0.113 0.128 0.104 8 0.109 0.090 0.100 0.081 9 0.125 0.097 0.108 0.087 10 0.096 0.074 0.087 0.070 11 0.091 0.072 0.084 0.067 12 0.189 0.159 0.157 0.128 13 0.048 0.037 0.052 0.041 14 0.062 0.055 0.063 0.050 15 0.206 0.166 0.162 0.132 16 0.185 0.153 0.152 0.124 17 0.121 0.100 0.108 0.088 18 0.050 0.041 0.058 0.046 19 0.033 0.026 0.046 0.037 20 0.186 0.151 0.151 0.124 21 0.106 0.080 0.094 0.076 22 0.052 0.038 0.050 0.040 delta: Log odds ratio of failure for Treatment vs control -0.2525(95% CI : -0.3641,-0.1277) -0.234(95% CI -0.33,-0.137)

8 Appendix Output for 2.a Results from 5000 iterations with 200 burn in

node mean sd MC error2.5% median 97.5% start sample delta.new -0.2509 0.1505 0.004003 -0.564 -0.2581 0.08956 200 4801 d -0.2525 0.05958 0.0032 -0.3641 -0.2547 -0.1277 200 4801

delta[1] -0.2464 0.1425 0.00407 -0.519 -0.2542 0.07709 200 4801 delta[2] -0.2847 0.1379 0.003661 -0.6054 -0.2786 -0.0142 200 4801 delta[3] -0.2671 0.1429 0.004146 -0.5668 -0.2675 0.0297 200 4801 delta[4] -0.2507 0.09553 0.00274 -0.4432 -0.2527 -0.05452 200 4801 delta[5] -0.2028 0.132 0.005589 -0.4233 -0.2206 0.1114 200 4801 delta[6] -0.2642 0.1391 0.003583 -0.5647 -0.265 0.01371 200 4801 delta[7] -0.3537 0.107 0.005275 -0.5928 -0.342 -0.1738 200 4801 delta[8] -0.2106 0.1145 0.004717 -0.4112 -0.2213 0.04578 200 4801 delta[9] -0.2833 0.1258 0.003594 -0.5653 -0.2787 -0.04959 200 4801 delta[10] -0.2921 0.084 0.00239 -0.4724 -0.2891 -0.1305 200 4801 delta[11] -0.2467 0.1091 0.003925 -0.4548 -0.2541 -0.01649 200 4801 delta[12] -0.2084 0.1234 0.005032 -0.4268 -0.2208 0.06739 200 4801 delta[13] -0.2817 0.1321 0.003895 -0.5665 -0.2771 -0.01671 200 4801 delta[14] -0.1235 0.151 0.009372 -0.3476 -0.1492 0.2223 200 4801 delta[15] -0.2633 0.1226 0.00338 -0.518 -0.2647 -0.003809 200 4801 delta[16] -0.231 0.1266 0.004232 -0.4677 -0.2412 0.04716 200 4801 delta[17] -0.2117 0.138 0.005677 -0.4518 -0.2289 0.1148 200 4801 delta[18] -0.2219 0.1422 0.004582 -0.4806 -0.2349 0.1175 200 4801 delta[19] -0.2312 0.1467 0.004395 -0.5 -0.2446 0.1106 200 4801 delta[20] -0.2498 0.1181 0.0034 -0.4856 -0.2528 0.005685 200 4801 delta[21] -0.3169 0.1268 0.00441 -0.6104 -0.3023 -0.09962 200 4801 delta[22] -0.3131 0.1283 0.003763 -0.6164 -0.3018 -0.08961 200 4801

mu[1] -2.434 0.4472 0.007044 -3.402 -2.418 -1.622 200 4801 mu[2] -2.192 0.2459 0.004358 -2.688 -2.181 -1.737 200 4801 mu[3] -2.139 0.2722 0.004196 -2.701 -2.135 -1.628 200 4801 mu[4] -2.397 0.08075 0.001892 -2.556 -2.396 -2.241 200 4801 mu[5] -2.402 0.1513 0.003603 -2.712 -2.399 -2.109 200 4801 mu[6] -2.236 0.3468 0.004994 -2.957 -2.224 -1.595 200 4801 mu[7] -1.714 0.08225 0.002733 -1.874 -1.714 -1.551 200 4801 mu[8] -2.107 0.1171 0.003207 -2.344 -2.103 -1.886 200 4801 mu[9] -1.958 0.147 0.00286 -2.244 -1.96 -1.667 200 4801 mu[10] -2.241 0.06837 0.001367 -2.377 -2.241 -2.11 200 4801 mu[11] -2.308 0.1146 0.002855 -2.54 -2.307 -2.082 200 4801 mu[12] -1.462 0.1286 0.002929 -1.722 -1.459 -1.211 200 4801 mu[13] -2.999 0.2119 0.003782 -3.423 -2.993 -2.597 200 4801 mu[14] -2.725 0.1237 0.004902 -2.987 -2.718 -2.499 200 4801 mu[15] -1.356 0.1579 0.002477 -1.678 -1.352 -1.056 200 4801 mu[16] -1.488 0.1455 0.003119 -1.782 -1.484 -1.208 200 4801 mu[17] -1.994 0.1912 0.004857 -2.387 -1.986 -1.645 200 4801 mu[18] -2.974 0.2828 0.004517 -3.562 -2.964 -2.454 200 4801 mu[19] -3.434 0.3494 0.005109 -4.175 -3.41 -2.797 200 4801 mu[20] -1.486 0.139 0.002366 -1.77 -1.486 -1.221 200 4801 mu[21] -2.135 0.1363 0.002744 -2.409 -2.135 -1.868 200 4801 mu[22] -2.919 0.1407 0.002595 -3.198 -2.919 -2.651 200 4801

pc[1] 0.08664 0.0338 5.485E-4 0.03224 0.08178 0.1649 200 4801 pc[2] 0.1026 0.02235 3.917E-4 0.06369 0.1014 0.1497 200 4801 pc[3] 0.1081 0.02578 3.93E-4 0.06292 0.1057 0.1642 200 4801 pc[4] 0.0836 0.00618 1.446E-4 0.072 0.08346 0.09611 200 4801 pc[5] 0.08373 0.01158 2.689E-4 0.06225 0.08326 0.1082 200 4801 pc[6] 0.1007 0.03053 4.571E-4 0.04941 0.09761 0.1687 200 4801 pc[7] 0.1529 0.01068 3.566E-4 0.1331 0.1526 0.1749 200 4801 pc[8] 0.1089 0.01129 3.074E-4 0.08753 0.1088 0.1317 200 4801 pc[9] 0.1246 0.01603 3.136E-4 0.09587 0.1235 0.1589 200 4801 pc[10] 0.09625 0.005943 1.192E-4 0.08498 0.09617 0.1081 200 4801 pc[11] 0.0909 0.009435 2.31E-4 0.07309 0.0905 0.1108 200 4801 pc[12] 0.189 0.01962 4.429E-4 0.1516 0.1886 0.2295 200 4801 pc[13] 0.04837 0.009687 1.728E-4 0.03159 0.04776 0.06931 200 4801 pc[14] 0.0619 0.007099 2.794E-4 0.04804 0.06191 0.07589 200 4801

9 pc[15] 0.206 0.02565 3.975E-4 0.1574 0.2056 0.2581 200 4801 pc[16] 0.1851 0.02183 4.674E-4 0.1441 0.1848 0.2301 200 4801 pc[17] 0.1213 0.02013 4.935E-4 0.08415 0.1207 0.1617 200 4801 pc[18] 0.05025 0.01324 2.054E-4 0.0276 0.04907 0.07917 200 4801 pc[19] 0.03296 0.01076 1.598E-4 0.01514 0.03199 0.05751 200 4801 pc[20] 0.1855 0.0209 3.573E-4 0.1456 0.1845 0.2277 200 4801 pc[21] 0.1064 0.01295 2.661E-4 0.08252 0.1057 0.1337 200 4801 pc[22] 0.05167 0.006874 1.284E-4 0.03924 0.05124 0.06591 200 4801 pt[1] 0.0693 0.02797 4.77E-4 0.02571 0.06537 0.1341 200 4801 pt[2] 0.07938 0.01803 2.554E-4 0.04844 0.07773 0.1189 200 4801 pt[3] 0.08518 0.02165 3.334E-4 0.04824 0.08346 0.1328 200 4801 pt[4] 0.06633 0.005362 8.838E-5 0.05635 0.06629 0.07722 200 4801 pt[5] 0.0695 0.01018 2.221E-4 0.05125 0.06885 0.09065 200 4801 pt[6] 0.07938 0.02474 3.004E-4 0.0382 0.07658 0.1341 200 4801 pt[7] 0.1126 0.009114 2.773E-4 0.09456 0.1126 0.1307 200 4801 pt[8] 0.09012 0.009436 1.832E-4 0.07298 0.08951 0.1097 200 4801 pt[9] 0.09693 0.0133 1.898E-4 0.07226 0.09644 0.1242 200 4801 pt[10] 0.0737 0.005105 9.764E-5 0.0638 0.07356 0.08398 200 4801 pt[11] 0.07245 0.00731 1.22E-4 0.05875 0.07227 0.08737 200 4801 pt[12] 0.1592 0.01776 3.916E-4 0.1275 0.1582 0.1974 200 4801 pt[13] 0.03697 0.007584 1.253E-4 0.0238 0.03637 0.05297 200 4801 pt[14] 0.05515 0.006788 2.55E-4 0.04345 0.05456 0.06987 200 4801 pt[15] 0.1664 0.02218 3.544E-4 0.1256 0.1654 0.212 200 4801 pt[16] 0.1529 0.01928 3.325E-4 0.1176 0.152 0.1938 200 4801 pt[17] 0.1004 0.01593 2.765E-4 0.07155 0.09943 0.1335 200 4801 pt[18] 0.04071 0.01102 1.725E-4 0.02253 0.0396 0.06538 200 4801 pt[19] 0.02637 0.008764 1.281E-4 0.01186 0.02537 0.04622 200 4801 pt[20] 0.1508 0.01837 3.43E-4 0.1165 0.1504 0.1891 200 4801 pt[21] 0.07992 0.01057 2.067E-4 0.05986 0.07955 0.1019 200 4801 pt[22] 0.03837 0.005569 9.653E-5 0.02795 0.03814 0.04996 200 4801 sigma 0.117 0.06852 0.005002 0.02381 0.1071 0.2747 200 4801 tau 263.4 474.7 33.31 13.26 87.26 1764.0 200 4801