Metric data: regression and relatives
Peter Ralph
12 February 2018 – Advanced Biological Statistics
Overview
Summary
Last Thursday, we looked at:
Using non-Normal noise distributions (e.g., Cauchy) to make models robust to outliers and more generally model overdispersed data.
Interrogating hyperparameters to learn what we want (e.g., percent variation explained).
Today
Finish up robust ANOVA.
Problem #1: “too much” noise (i.e., non-Normal noise).
Problem #2: too many variables.
Robust regression
Some data
Relative axon growth for neurons after \(x\) hours:
Standard linear regression
## user system elapsed
## 0.005 0.000 0.004##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -127.829 -0.307 0.234 0.691 46.293
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.8164 0.7202 1.133 0.258
## x 2.9323 0.2582 11.356 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.18 on 198 degrees of freedom
## Multiple R-squared: 0.3944, Adjusted R-squared: 0.3913
## F-statistic: 129 on 1 and 198 DF, p-value: < 2.2e-16with Stan
srr_block <- "
data {
int N;
vector[N] x;
vector[N] y;
}
parameters {
real b0;
real b1;
real<lower=0> sigma;
}
model {
y ~ cauchy(b0 + b1*x, sigma);
}"
system.time( stanrr <- stan(model_code=srr_block,
data=list(N=length(x), x=x, y=y), iter=1e3) )## user system elapsed
## 36.170 0.947 42.192## $summary
## mean se_mean sd 2.5% 25%
## b0 1.0492653 0.0011858160 0.04901458 0.9556078 1.0153481
## b1 3.0175214 0.0003716323 0.01579547 2.9876462 3.0068720
## sigma 0.4887785 0.0011772173 0.05008859 0.3999234 0.4549012
## lp__ -143.9498680 0.0440483247 1.29259423 -147.3354709 -144.5436828
## 50% 75% 97.5% n_eff Rhat
## b0 1.0491830 1.081127 1.144147 1708.5037 0.9996744
## b1 3.0172806 3.027901 3.048448 1806.5022 0.9985911
## sigma 0.4863373 0.519511 0.598097 1810.3575 1.0001622
## lp__ -143.6147451 -143.008105 -142.518558 861.1239 1.0039008
##
## $c_summary
## , , chains = chain:1
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 1.0516070 0.04808494 0.9561784 1.0172640 1.0535676
## b1 3.0168313 0.01636800 2.9872796 3.0052616 3.0159255
## sigma 0.4872553 0.04932650 0.3966417 0.4543649 0.4835046
## lp__ -143.9923917 1.28438229 -147.4284683 -144.6003711 -143.6344760
## stats
## parameter 75% 97.5%
## b0 1.0846034 1.142480
## b1 3.0277263 3.050457
## sigma 0.5192919 0.591551
## lp__ -143.0595455 -142.523462
##
## , , chains = chain:2
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 1.0467663 0.04860953 0.9543881 1.0131220 1.0468940
## b1 3.0178001 0.01599980 2.9886282 3.0064272 3.0173143
## sigma 0.4910389 0.05262914 0.4006050 0.4540697 0.4899721
## lp__ -143.9991601 1.36601484 -147.4543730 -144.5897508 -143.6740284
## stats
## parameter 75% 97.5%
## b0 1.0773398 1.1438271
## b1 3.0283606 3.0478673
## sigma 0.5229438 0.6013051
## lp__ -143.0507556 -142.5151027
##
## , , chains = chain:3
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 1.0497458 0.05014097 0.9571373 1.0153348 1.0480027
## b1 3.0181037 0.01486087 2.9896974 3.0076207 3.0184036
## sigma 0.4879835 0.04999519 0.3955632 0.4555168 0.4857678
## lp__ -143.9280711 1.22779789 -146.8053849 -144.4929896 -143.6277472
## stats
## parameter 75% 97.5%
## b0 1.0803476 1.1503617
## b1 3.0276672 3.0465796
## sigma 0.5195939 0.5972677
## lp__ -142.9810258 -142.5383894
##
## , , chains = chain:4
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 1.0489423 0.04922303 0.9552911 1.0157737 1.048428
## b1 3.0173506 0.01593153 2.9864034 3.0071933 3.017419
## sigma 0.4888364 0.04837375 0.4060966 0.4562453 0.484854
## lp__ -143.8798493 1.28859894 -147.1772748 -144.5032436 -143.479043
## stats
## parameter 75% 97.5%
## b0 1.0795552 1.1438732
## b1 3.0276787 3.0476952
## sigma 0.5157548 0.5962235
## lp__ -142.9418798 -142.5069584Compare the results.
Make a table and/or a graph of estimates and confidence intervals obtained by (a) ordinary linear regression and (b) robust regression as we have done here.
Problem #2: too many variables
Example data
diabetes package:lars R Documentation
Blood and other measurements in diabetics
Description:
The ‘diabetes’ data frame has 442 rows and 3 columns. These are
the data used in the Efron et al "Least Angle Regression" paper.
Format:
This data frame contains the following columns:
x a matrix with 10 columns
y a numeric vector
x2 a matrix with 64 columnsThe dataset has
- 442 diabetes patients
- 10 main variables: age, gender, body mass index, average blood pressure (map), and six blood serum measurements (tc, ldl, hdl, tch, ltg, glu)
- 45 interactions, e.g.
age:ldl - 9 quadratic effects, e.g.
age^2 - measure of disease progression taken one year later,
y
Crossvalidation plan
Put aside 20% of the data for testing.
Refit the model.
Predict the test data; compute \[\begin{aligned} S = \sqrt{\frac{1}{M} \sum_{k=1}^M (\hat y_i - y_i)^2} \end{aligned}\]
Repeat for the other four 20%s.
Compare.
Crossvalidation
First let’s split the data into testing and training just once:
Ordinary linear regression
##
## Call:
## lm(formula = y ~ ., data = training_d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -116.452 -31.222 -2.942 30.048 115.597
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 150.404 2.925 51.421 < 2e-16 ***
## age 131.031 77.503 1.691 0.092046 .
## sex -286.088 77.795 -3.677 0.000284 ***
## bmi 587.855 99.149 5.929 9.22e-09 ***
## map 323.091 84.736 3.813 0.000170 ***
## tc 16796.166 63143.128 0.266 0.790440
## ldl -15061.254 55494.787 -0.271 0.786290
## hdl -6254.463 23596.136 -0.265 0.791162
## tch 330.715 316.398 1.045 0.296835
## ltg -4856.914 20761.268 -0.234 0.815207
## glu 19.128 79.886 0.239 0.810945
## `age^2` 123.345 78.097 1.579 0.115410
## `bmi^2` -57.665 91.584 -0.630 0.529454
## `map^2` -161.860 102.020 -1.587 0.113777
## `tc^2` 11301.512 7685.114 1.471 0.142563
## `ldl^2` 7536.397 5812.171 1.297 0.195848
## `hdl^2` 2298.167 1803.074 1.275 0.203545
## `tch^2` 756.037 717.721 1.053 0.293100
## `ltg^2` 1549.779 1936.210 0.800 0.424167
## `glu^2` 49.718 107.963 0.461 0.645518
## `age:sex` 149.665 89.058 1.681 0.093999 .
## `age:bmi` -51.026 89.830 -0.568 0.570480
## `age:map` 73.426 88.382 0.831 0.406825
## `age:tc` -147.148 718.847 -0.205 0.837960
## `age:ldl` -215.127 576.917 -0.373 0.709521
## `age:hdl` 349.106 337.121 1.036 0.301331
## `age:tch` 450.274 258.548 1.742 0.082718 .
## `age:ltg` 31.703 276.056 0.115 0.908655
## `age:glu` 65.267 92.163 0.708 0.479443
## `sex:bmi` -14.043 91.191 -0.154 0.877726
## `sex:map` 165.153 89.773 1.840 0.066907 .
## `sex:tc` -15.990 685.587 -0.023 0.981410
## `sex:ldl` 125.005 550.648 0.227 0.820583
## `sex:hdl` -108.670 328.604 -0.331 0.741123
## `sex:tch` -314.485 253.384 -1.241 0.215623
## `sex:ltg` 97.473 272.504 0.358 0.720849
## `sex:glu` 24.914 83.625 0.298 0.765983
## `bmi:map` 326.318 101.658 3.210 0.001487 **
## `bmi:tc` 21.507 738.126 0.029 0.976776
## `bmi:ldl` 92.412 616.938 0.150 0.881040
## `bmi:hdl` -101.632 364.919 -0.279 0.780836
## `bmi:tch` -162.323 250.337 -0.648 0.517262
## `bmi:ltg` 70.844 283.351 0.250 0.802760
## `bmi:glu` 11.216 102.970 0.109 0.913341
## `map:tc` 1003.122 781.272 1.284 0.200248
## `map:ldl` -916.970 661.618 -1.386 0.166897
## `map:hdl` -303.164 358.515 -0.846 0.398514
## `map:tch` 83.430 240.818 0.346 0.729277
## `map:ltg` -455.173 322.944 -1.409 0.159845
## `map:glu` -65.997 112.875 -0.585 0.559242
## `tc:ldl` -17718.758 12810.745 -1.383 0.167763
## `tc:hdl` -6298.383 4188.098 -1.504 0.133773
## `tc:tch` -2574.374 1995.618 -1.290 0.198142
## `tc:ltg` -10009.299 13661.300 -0.733 0.464387
## `tc:glu` -497.859 705.502 -0.706 0.480991
## `ldl:hdl` 5004.483 3442.436 1.454 0.147165
## `ldl:tch` 1731.892 1615.223 1.072 0.284567
## `ldl:ltg` 8124.594 11369.412 0.715 0.475468
## `ldl:glu` 265.993 594.756 0.447 0.655064
## `hdl:tch` 1076.355 1217.662 0.884 0.377503
## `hdl:ltg` 3597.256 4794.265 0.750 0.453708
## `hdl:glu` 560.737 366.606 1.530 0.127294
## `tch:ltg` 212.466 727.798 0.292 0.770563
## `tch:glu` 596.191 278.394 2.142 0.033119 *
## `ltg:glu` 172.802 328.491 0.526 0.599283
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 51.82 on 272 degrees of freedom
## Multiple R-squared: 0.6421, Adjusted R-squared: 0.5579
## F-statistic: 7.625 on 64 and 272 DF, p-value: < 2.2e-16With ordinary linear regression, we got a root-mean-square-prediction-error of 64.602275 (on the test data), compared to a root-mean-square-error of 46.5588313 for the training data.
This suggests there’s some overfitting going on.
layout(t(1:2))
plot(training_d$y, predict(ols), xlab="true values", ylab="OLS predicted", main="training data")
abline(0,1)
plot(test_d$y, ols_pred, xlab="true values", ylab="OLS predicted", main="test data")
abline(0,1)
A sparsifying prior
We have a lot of predictors: 64 of them. A good guess is that only a few are really useful. So, we can put a sparsifying prior on the coefficients, i.e., the \(\beta\)s in \[\begin{aligned} y = \beta_0 + \beta_1 x_1 + \cdots \beta_n x_n + \epsilon \end{aligned}\]
Sparseness and scale mixtures
Encouraging sparseness
Suppose we do regression with a large number of predictor variables.
The resulting coefficients are sparse if most are zero.
The idea is to “encourage” all the coefficients to be zero, unless they really want to be nonzero, in which case we let them be whatever they want.
This tends to discourage overfitting.
The idea is to “encourage” all the coefficients to be zero, unless they really want to be nonzero, in which case we let them be whatever they want.
To do this, we want a prior which is very peak-ey at zero but flat away from zero (“spike-and-slab”).
Compare the Normal
\[\begin{aligned} X \sim \Normal(0,1) \end{aligned}\]
to the “exponential scale mixture of Normals”,
\[\begin{aligned} X &\sim \Normal(0,\sigma) \\ \sigma &\sim \Exp(1) . \end{aligned}\]
Why use a scale mixture?
Lets the data choose the appropriate scale of variation.
Weakly encourages \(\sigma\) to be small: so, as much variation as possible is explained by signal instead of noise.
Gets you a prior that is more peaked at zero and flatter otherwise.
Implementation
Note that
\[\begin{aligned} \beta &\sim \Normal(0,\sigma) \\ \sigma &\sim \Exp(1) . \end{aligned}\]
is equivalent to
\[\begin{aligned} \beta &= \sigma \gamma \\ \gamma &\sim \Normal(0,1) \\ \sigma &\sim \Exp(1) . \end{aligned}\]
parameters {
real beta;
real<lower=0> sigma;
}
model {
beta ~ normal(0, sigma);
}is equivalent to
parameters {
real gamma;
real<lower=0> sigma;
}
transformed parameters {
real beta;
beta = gamma * sigma;
}
model {
beta ~ normal(0, sigma);
}The second version is better for Stan.
Why is it better?
parameters {
real beta;
real<lower=0> sigma;
}
model {
beta ~ normal(0, sigma);
}In the first, the optimal step size depends on sigma.
A strongly sparsifying prior
The “horseshoe”:
\[\begin{aligned} \beta_j &\sim \Normal(0, \lambda_j) \\ \lambda_j &\sim \Cauchy(0, \tau) \\ \tau &\sim \Unif(0, 1) \end{aligned}\]
parameters {
vector[p] d_beta;
vector[p] d_lambda;
real<lower=0, upper=1> tau;
}
transformed parameters {
vector[p] beta;
beta = d_beta .* d_lambda * tau;
}
model {
d_beta ~ normal(0, 1);
d_lambda ~ cauchy(0, 1);
// tau ~ uniform(0, 1); // uniform
}The Cauchy as a scale mixture
Inverse square-root gamma scaling
It turns out that if
\[\begin{aligned} \beta &\sim \Normal(0, 1/\sqrt{\lambda}) \\ \lambda &\sim \Gam(1/2, 1/2) \end{aligned}\]
then
\[\begin{aligned} \beta &\sim \Cauchy(0, 1). \end{aligned}\]
What black magic is this??
It says so here.
If you like to do integrals, you can check mathematically.
Or, you can check with simulation.
Using the horseshoe
What’s an appropriate noise distribution?
Aside: quantile-quantile plots
The idea is to plot the quantiles of each distribution against each other.
If these are datasets, this means just plotting their sorted values against each other.
x <- rnorm(1e4)
y <- rbeta(1e4, 2, 2)
layout(t(1:3))
plot(sort(x), sort(y))
qqplot(x, y, main="qqplot")
qqnorm(y, main="qnorm")
Regression with a horseshoe prior
Uses a reparameterization of the Cauchy as a scale mixture of normals.
horseshoe_block <- "
data {
int N;
int p;
vector[N] y;
matrix[N,p] x;
}
parameters {
real b0;
vector[p] d_beta;
vector[p] d_a;
vector<lower=0>[p] d_b;
real<lower=0, upper=1> tau;
real<lower=0> sigma;
}
transformed parameters {
vector[p] beta;
vector[N] f;
beta = d_beta .* d_a .* sqrt(d_b) * tau;
f = b0 + x * beta;
}
model {
y ~ normal(f, sigma);
// HORSESHOE PRIOR:
d_beta ~ normal(0, 1);
d_a ~ normal(0, 1);
d_b ~ inv_gamma(0.5, 0.5);
// tau ~ uniform(0, 1); // uniform
// priors on noise distribution:
sigma ~ normal(0, 10);
}"Note the data have already been normalized, with the exception of \(y\):
## y age sex
## Min. : 25.0 Min. :-0.1072256 Min. :-0.044642
## 1st Qu.: 88.0 1st Qu.:-0.0345749 1st Qu.:-0.044642
## Median :142.0 Median : 0.0053831 Median :-0.044642
## Mean :154.3 Mean : 0.0000151 Mean : 0.000615
## 3rd Qu.:215.0 3rd Qu.: 0.0380759 3rd Qu.: 0.050680
## Max. :346.0 Max. : 0.1107267 Max. : 0.050680
## bmi map tc
## Min. :-0.090275 Min. :-0.1123996 Min. :-0.126781
## 1st Qu.:-0.032073 1st Qu.:-0.0366565 1st Qu.:-0.033216
## Median :-0.005128 Median :-0.0056706 Median :-0.000193
## Mean : 0.002986 Mean :-0.0002015 Mean : 0.001795
## 3rd Qu.: 0.035829 3rd Qu.: 0.0333486 3rd Qu.: 0.032830
## Max. : 0.170555 Max. : 0.1079441 Max. : 0.153914
## ldl hdl tch
## Min. :-0.115613 Min. :-0.102307 Min. :-0.076395
## 1st Qu.:-0.028558 1st Qu.:-0.032356 1st Qu.:-0.039493
## Median :-0.002880 Median :-0.006584 Median :-0.002592
## Mean : 0.002058 Mean :-0.002635 Mean : 0.002334
## 3rd Qu.: 0.034698 3rd Qu.: 0.026550 3rd Qu.: 0.034309
## Max. : 0.155887 Max. : 0.177497 Max. : 0.155345
## ltg glu age^2
## Min. :-0.1043648 Min. :-0.1377672 Min. :-0.04130
## 1st Qu.:-0.0307512 1st Qu.:-0.0300725 1st Qu.:-0.03651
## Median : 0.0002715 Median :-0.0010777 Median :-0.01950
## Mean : 0.0029586 Mean : 0.0004587 Mean :-0.00153
## 3rd Qu.: 0.0336568 3rd Qu.: 0.0279170 3rd Qu.: 0.01646
## Max. : 0.1335990 Max. : 0.1356118 Max. : 0.18276
## bmi^2 map^2 tc^2
## Min. :-0.032976 Min. :-0.039369 Min. :-0.0319463
## 1st Qu.:-0.029542 1st Qu.:-0.034253 1st Qu.:-0.0291980
## Median :-0.016958 Median :-0.018662 Median :-0.0163678
## Mean : 0.001593 Mean :-0.001993 Mean : 0.0005684
## 3rd Qu.: 0.015956 3rd Qu.: 0.015789 3rd Qu.: 0.0090080
## Max. : 0.391017 Max. : 0.180473 Max. : 0.3025598
## ldl^2 hdl^2 tch^2
## Min. :-0.0296059 Min. :-0.027654 Min. :-0.0305374
## 1st Qu.:-0.0261629 1st Qu.:-0.025283 1st Qu.:-0.0301802
## Median :-0.0178284 Median :-0.014861 Median :-0.0094855
## Mean :-0.0003397 Mean :-0.002452 Mean :-0.0006151
## 3rd Qu.: 0.0073940 3rd Qu.: 0.005809 3rd Qu.:-0.0094855
## Max. : 0.2883956 Max. : 0.357531 Max. : 0.2952011
## ltg^2 glu^2 age:sex
## Min. :-0.0349354 Min. :-0.0319022 Min. :-0.120695
## 1st Qu.:-0.0304543 1st Qu.:-0.0293459 1st Qu.:-0.030888
## Median :-0.0191324 Median :-0.0191600 Median : 0.001365
## Mean :-0.0007457 Mean :-0.0001022 Mean : 0.001096
## 3rd Qu.: 0.0113077 3rd Qu.: 0.0106420 3rd Qu.: 0.036802
## Max. : 0.2406822 Max. : 0.2358489 Max. : 0.111614
## age:bmi age:map age:tc
## Min. :-0.1748118 Min. :-0.166429 Min. :-0.136218
## 1st Qu.:-0.0192374 1st Qu.:-0.021593 1st Qu.:-0.020113
## Median :-0.0065030 Median :-0.007572 Median :-0.008896
## Mean : 0.0005207 Mean :-0.001270 Mean :-0.001146
## 3rd Qu.: 0.0184220 3rd Qu.: 0.022763 3rd Qu.: 0.017210
## Max. : 0.1702584 Max. : 0.145881 Max. : 0.199426
## age:ldl age:hdl age:tch
## Min. :-0.155477 Min. :-0.130730 Min. :-0.2012887
## 1st Qu.:-0.019358 1st Qu.:-0.022842 1st Qu.:-0.0153943
## Median :-0.006878 Median : 0.001632 Median :-0.0080380
## Mean :-0.001112 Mean :-0.000360 Mean :-0.0008048
## 3rd Qu.: 0.014055 3rd Qu.: 0.018063 3rd Qu.: 0.0196049
## Max. : 0.206119 Max. : 0.206328 Max. : 0.1672338
## age:ltg age:glu sex:bmi
## Min. :-0.1600509 Min. :-0.119778 Min. :-0.1565731
## 1st Qu.:-0.0214797 1st Qu.:-0.022366 1st Qu.:-0.0317632
## Median :-0.0083190 Median :-0.010909 Median :-0.0016333
## Mean : 0.0001587 Mean :-0.001671 Mean : 0.0004026
## 3rd Qu.: 0.0172821 3rd Qu.: 0.012698 3rd Qu.: 0.0343105
## Max. : 0.1795324 Max. : 0.184369 Max. : 0.1791461
## sex:map sex:tc sex:ldl
## Min. :-0.113623 Min. :-0.136130 Min. :-1.297e-01
## 1st Qu.:-0.030841 1st Qu.:-0.029601 1st Qu.:-3.037e-02
## Median : 0.007036 Median :-0.001490 Median : 2.988e-03
## Mean : 0.001291 Mean : 0.000314 Mean : 4.536e-05
## 3rd Qu.: 0.037201 3rd Qu.: 0.033216 3rd Qu.: 3.224e-02
## Max. : 0.107407 Max. : 0.161566 Max. : 1.602e-01
## sex:hdl sex:tch sex:ltg
## Min. :-1.622e-01 Min. :-0.159990 Min. :-0.133887
## 1st Qu.:-3.080e-02 1st Qu.:-0.019548 1st Qu.:-0.027870
## Median : 2.594e-05 Median : 0.013582 Median : 0.008752
## Mean :-7.790e-04 Mean : 0.001308 Mean : 0.002488
## 3rd Qu.: 3.021e-02 3rd Qu.: 0.022402 3rd Qu.: 0.032245
## Max. : 1.748e-01 Max. : 0.157698 Max. : 0.136614
## sex:glu bmi:map bmi:tc
## Min. :-0.136467 Min. :-0.1672668 Min. :-0.2932071
## 1st Qu.:-0.029386 1st Qu.:-0.0219508 1st Qu.:-0.0193874
## Median :-0.002274 Median :-0.0104349 Median :-0.0085277
## Mean : 0.001033 Mean : 0.0003255 Mean : 0.0005746
## 3rd Qu.: 0.033874 3rd Qu.: 0.0168374 3rd Qu.: 0.0179394
## Max. : 0.137802 Max. : 0.2284830 Max. : 0.2269828
## bmi:ldl bmi:hdl bmi:tch
## Min. :-0.286577 Min. :-0.268316 Min. :-0.1318179
## 1st Qu.:-0.022307 1st Qu.:-0.015669 1st Qu.:-0.0237279
## Median :-0.008503 Median : 0.012965 Median :-0.0170881
## Mean :-0.000038 Mean : 0.001334 Mean : 0.0003022
## 3rd Qu.: 0.013517 3rd Qu.: 0.025325 3rd Qu.: 0.0203124
## Max. : 0.232069 Max. : 0.146477 Max. : 0.2764597
## bmi:ltg bmi:glu map:tc
## Min. :-1.851e-01 Min. :-0.1543918 Min. :-0.203608
## 1st Qu.:-2.464e-02 1st Qu.:-0.0240990 1st Qu.:-0.021147
## Median :-1.414e-02 Median :-0.0144630 Median :-0.008035
## Mean : 8.025e-05 Mean : 0.0001888 Mean :-0.001411
## 3rd Qu.: 2.215e-02 3rd Qu.: 0.0170453 3rd Qu.: 0.019085
## Max. : 2.189e-01 Max. : 0.2246097 Max. : 0.189747
## map:ldl map:hdl map:tch
## Min. :-0.203862 Min. :-0.217314 Min. :-0.185154
## 1st Qu.:-0.020688 1st Qu.:-0.018023 1st Qu.:-0.017297
## Median :-0.006742 Median : 0.005007 Median :-0.010037
## Mean :-0.002099 Mean : 0.001038 Mean :-0.000926
## 3rd Qu.: 0.017189 3rd Qu.: 0.018608 3rd Qu.: 0.021630
## Max. : 0.191608 Max. : 0.389867 Max. : 0.195708
## map:ltg map:glu tc:ldl
## Min. :-1.459e-01 Min. :-0.143393 Min. :-0.0476094
## 1st Qu.:-2.391e-02 1st Qu.:-0.023048 1st Qu.:-0.0273934
## Median :-1.015e-02 Median :-0.011865 Median :-0.0167630
## Mean :-1.827e-05 Mean :-0.001108 Mean : 0.0000648
## 3rd Qu.: 2.148e-02 3rd Qu.: 0.009924 3rd Qu.: 0.0086523
## Max. : 1.655e-01 Max. : 0.279359 Max. : 0.3086895
## tc:hdl tc:tch tc:ltg
## Min. :-0.1281815 Min. :-0.1223554 Min. :-0.1048969
## 1st Qu.:-0.0177190 1st Qu.:-0.0219653 1st Qu.:-0.0258567
## Median :-0.0028607 Median :-0.0164289 Median :-0.0144205
## Mean : 0.0006946 Mean :-0.0004093 Mean : 0.0009462
## 3rd Qu.: 0.0134764 3rd Qu.: 0.0109094 3rd Qu.: 0.0169747
## Max. : 0.2076833 Max. : 0.2288045 Max. : 0.2231252
## tc:glu ldl:hdl ldl:tch
## Min. :-0.100555 Min. :-0.253886 Min. :-0.1115079
## 1st Qu.:-0.021634 1st Qu.:-0.013653 1st Qu.:-0.0236124
## Median :-0.012260 Median : 0.006305 Median :-0.0140355
## Mean :-0.001407 Mean : 0.001842 Mean :-0.0008025
## 3rd Qu.: 0.011721 3rd Qu.: 0.019763 3rd Qu.: 0.0102002
## Max. : 0.237370 Max. : 0.160773 Max. : 0.1998614
## ldl:ltg ldl:glu hdl:tch
## Min. :-1.826e-01 Min. :-0.151734 Min. :-0.234890
## 1st Qu.:-2.081e-02 1st Qu.:-0.022529 1st Qu.:-0.010455
## Median :-8.185e-03 Median :-0.009924 Median : 0.017148
## Mean : 4.830e-06 Mean :-0.001870 Mean : 0.001743
## 3rd Qu.: 1.902e-02 3rd Qu.: 0.011544 3rd Qu.: 0.031343
## Max. : 2.034e-01 Max. : 0.227999 Max. : 0.080445
## hdl:ltg hdl:glu tch:ltg
## Min. :-0.254685 Min. :-0.2232546 Min. :-0.160745
## 1st Qu.:-0.015445 1st Qu.:-0.0138933 1st Qu.:-0.027375
## Median : 0.012004 Median : 0.0086822 Median :-0.014089
## Mean : 0.002202 Mean : 0.0003819 Mean :-0.001075
## 3rd Qu.: 0.022550 3rd Qu.: 0.0215139 3rd Qu.: 0.012681
## Max. : 0.163067 Max. : 0.1725039 Max. : 0.375845
## tch:glu ltg:glu
## Min. :-0.1176911 Min. :-0.0887652
## 1st Qu.:-0.0204024 1st Qu.:-0.0234429
## Median :-0.0155472 Median :-0.0130175
## Mean :-0.0006656 Mean : 0.0002682
## 3rd Qu.: 0.0176564 3rd Qu.: 0.0138425
## Max. : 0.3181041 Max. : 0.3381838horseshoe_fit <- stan(model_code=horseshoe_block,
data=list(N=nrow(training_d),
p=ncol(training_d)-1,
y=(training_d$y
- median(training_d$y))
/mad(training_d$y),
x=as.matrix(training_d[,-1])),
iter=1000,
control=list(adapt_delta=0.999,
max_treedepth=15))## Warning: There were 9 divergent transitions after warmup. Increasing adapt_delta above 0.999 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Warning: Examine the pairs() plot to diagnose sampling problems## $summary
## mean se_mean sd 2.5% 25%
## b0 0.096800745 0.0007377281 0.03299221 0.031390174 0.0752586406
## sigma 0.604787108 0.0005570903 0.02491383 0.558250917 0.5872577691
## beta[1] 0.148057745 0.0109305497 0.41805937 -0.495635089 -0.0471093191
## beta[2] -1.204385646 0.0345979818 0.98862695 -3.222027862 -1.9429444764
## beta[3] 7.350051339 0.0278128675 0.93474095 5.584185129 6.7157763171
## beta[4] 2.277303479 0.0423373558 1.15843418 -0.006683978 1.5352704427
## beta[5] -0.513431566 0.0269555581 0.90868693 -2.901725648 -0.8055820020
## beta[6] -0.121777929 0.0175443428 0.64054242 -1.596394583 -0.2652538749
## beta[7] -1.103833945 0.0437622103 1.18226429 -3.820279964 -1.9136968720
## beta[8] 0.403766066 0.0264164146 0.91425363 -0.695986725 -0.0388721449
## beta[9] 6.233609645 0.0258510753 1.06610839 4.135042677 5.4998642175
## beta[10] 0.103711901 0.0113641703 0.39802961 -0.588260927 -0.0658665380
## beta[11] 0.367509644 0.0156857828 0.59610529 -0.378129452 -0.0017942749
## beta[12] 0.001984490 0.0121072660 0.39711273 -0.887628567 -0.1350136052
## beta[13] -0.181940927 0.0131277693 0.49545925 -1.608175689 -0.2860604088
## beta[14] -0.040084678 0.0142996674 0.49783406 -1.293891620 -0.1691176191
## beta[15] -0.122228457 0.0154341999 0.53788348 -1.443505820 -0.2602545503
## beta[16] -0.052335111 0.0169673578 0.53698638 -1.322121648 -0.1718361918
## beta[17] 0.411706077 0.0251864935 0.85767494 -0.601311783 -0.0305926867
## beta[18] -0.191205994 0.0134165129 0.50235797 -1.515284030 -0.3322063927
## beta[19] 0.253340032 0.0140910943 0.51135941 -0.458717777 -0.0241955844
## beta[20] 1.501949075 0.0273019943 0.98412280 -0.050165848 0.7390433187
## beta[21] 0.146875761 0.0114853253 0.43243965 -0.568196688 -0.0477025255
## beta[22] 0.413263977 0.0155014383 0.61056288 -0.320046382 0.0008245203
## beta[23] -0.065179497 0.0162340684 0.50594530 -1.213694542 -0.1936016029
## beta[24] -0.220367603 0.0149009118 0.55336005 -1.693395563 -0.3474773073
## beta[25] 0.134296902 0.0158579709 0.49794682 -0.670275192 -0.0717375598
## beta[26] 0.040186360 0.0151931244 0.52433815 -0.915157458 -0.1415451888
## beta[27] 0.434596874 0.0183564106 0.70255957 -0.353124960 -0.0074949738
## beta[28] 0.263892746 0.0138721854 0.53338117 -0.448253336 -0.0204744858
## beta[29] 0.067378889 0.0105923209 0.39503427 -0.721663625 -0.0801207739
## beta[30] 0.444734542 0.0165891506 0.66647791 -0.340316319 -0.0002432803
## beta[31] -0.099177806 0.0136434371 0.46901285 -1.243031042 -0.2564471319
## beta[32] -0.182373498 0.0136708855 0.51892805 -1.559542747 -0.3128892214
## beta[33] 0.131832611 0.0121984753 0.43653538 -0.616826932 -0.0567908880
## beta[34] -0.138428391 0.0137140764 0.50427947 -1.456616191 -0.2733217779
## beta[35] 0.010278834 0.0117435899 0.40540415 -0.813716792 -0.1387939885
## beta[36] 0.067932904 0.0106321280 0.37893229 -0.666240995 -0.0867538553
## beta[37] 1.685579427 0.0297282807 1.07141740 -0.064677669 0.8395771235
## beta[38] -0.021351244 0.0116995739 0.42473452 -0.959904519 -0.1462526653
## beta[39] -0.037146326 0.0129950160 0.41757569 -1.013351236 -0.1514916025
## beta[40] -0.019595910 0.0131165807 0.46542959 -1.070985581 -0.1713730456
## beta[41] 0.106274175 0.0139569995 0.49472332 -0.765726254 -0.0945270510
## beta[42] 0.115106253 0.0128018687 0.45648034 -0.726731733 -0.0682009663
## beta[43] 0.087326880 0.0113271341 0.39662659 -0.691810592 -0.0734540022
## beta[44] -0.024818917 0.0108920146 0.39290099 -0.989385713 -0.1387048587
## beta[45] -0.053951606 0.0116731363 0.41097421 -1.034393415 -0.1807392216
## beta[46] 0.181412548 0.0132146618 0.50308674 -0.507646511 -0.0572035075
## beta[47] -0.168847411 0.0139043202 0.50524901 -1.584479382 -0.3082805076
## beta[48] 0.030921049 0.0124374895 0.44037088 -0.932989982 -0.1302554820
## beta[49] -0.085249214 0.0125876062 0.43595250 -1.159881317 -0.2069066339
## beta[50] 0.012144693 0.0146928229 0.50207431 -1.057582894 -0.1475289241
## beta[51] 0.161262994 0.0151984694 0.54397108 -0.646935081 -0.0627006090
## beta[52] -0.283173868 0.0266882100 0.85466081 -2.719617102 -0.3888031142
## beta[53] -0.096545864 0.0150783510 0.52487838 -1.449141128 -0.2178617656
## beta[54] 0.192424623 0.0164115603 0.56626549 -0.733332957 -0.0452525746
## beta[55] 0.004703693 0.0139243482 0.49267419 -1.105409972 -0.1335667006
## beta[56] 0.160798800 0.0221464484 0.71542572 -0.989116068 -0.0646225923
## beta[57] 0.240024963 0.0171541011 0.59712377 -0.585365509 -0.0437433551
## beta[58] 0.296298625 0.0177218461 0.61143468 -0.474826327 -0.0265672831
## beta[59] -0.174561340 0.0182424229 0.60911994 -1.820805684 -0.2902358278
## beta[60] 0.358482879 0.0160736686 0.61774534 -0.399340993 -0.0081089410
## beta[61] 0.137998478 0.0166204573 0.56127480 -0.788739862 -0.0749583847
## beta[62] -0.353522241 0.0211709625 0.74083703 -2.437839055 -0.4824600899
## beta[63] 0.457375386 0.0243986751 0.81962720 -0.473226102 -0.0108862796
## beta[64] 0.053008482 0.0137594696 0.45296975 -0.874008732 -0.1047452558
## 50% 75% 97.5% n_eff Rhat
## b0 9.650898e-02 0.119195144 0.1602656 2000.0000 0.9989130
## sigma 6.042950e-01 0.621087362 0.6534551 2000.0000 1.0015059
## beta[1] 3.134983e-02 0.273859073 1.2800457 1462.8235 0.9996546
## beta[2] -1.163392e+00 -0.287647608 0.1479235 816.5135 1.0053306
## beta[3] 7.352664e+00 7.983749393 9.1787197 1129.5124 1.0040270
## beta[4] 2.340058e+00 3.104664136 4.4102073 748.6783 1.0069451
## beta[5] -1.725640e-01 0.006100276 0.5269065 1136.4018 1.0008742
## beta[6] -3.204547e-02 0.072503731 0.9395877 1332.9737 0.9993283
## beta[7] -8.105861e-01 -0.069179772 0.2841098 729.8450 1.0038576
## beta[8] 7.447353e-02 0.534730711 3.0826922 1197.8037 0.9993356
## beta[9] 6.219964e+00 6.960795008 8.2320850 1700.7696 1.0014602
## beta[10] 2.629784e-02 0.224387227 1.0765856 1226.7479 1.0011508
## beta[11] 1.452309e-01 0.616118365 1.9422449 1444.2207 0.9987717
## beta[12] 1.087336e-03 0.133924232 0.8733495 1075.8097 1.0023349
## beta[13] -3.779729e-02 0.051601975 0.5445642 1424.4067 1.0034937
## beta[14] -6.476186e-03 0.120986507 0.9596036 1212.0411 1.0004960
## beta[15] -2.843583e-02 0.083924146 0.8281380 1214.5298 1.0000143
## beta[16] -3.406762e-03 0.123044577 0.9741382 1001.6087 1.0017051
## beta[17] 9.171008e-02 0.612252362 2.8556339 1159.6048 1.0025822
## beta[18] -4.614337e-02 0.042787442 0.5230953 1401.9977 1.0040739
## beta[19] 9.219333e-02 0.445871101 1.6099054 1316.9311 1.0000307
## beta[20] 1.485492e+00 2.234409378 3.4358149 1299.3012 0.9995558
## beta[21] 4.087306e-02 0.289595077 1.2545295 1417.6349 1.0006187
## beta[22] 1.887150e-01 0.714947719 2.0820662 1551.3750 0.9989734
## beta[23] -8.366392e-03 0.097026483 0.8334770 971.2978 1.0020796
## beta[24] -5.514247e-02 0.036864042 0.5088934 1379.0815 1.0028113
## beta[25] 2.872789e-02 0.239378652 1.4446361 985.9859 1.0067042
## beta[26] -8.631173e-05 0.158585436 1.2165624 1191.0465 1.0011136
## beta[27] 1.630007e-01 0.687886233 2.3323283 1464.8420 1.0024640
## beta[28] 9.647147e-02 0.427911911 1.7129741 1478.3783 0.9988235
## beta[29] 1.072545e-02 0.168685272 1.0628734 1390.8724 0.9991848
## beta[30] 1.862698e-01 0.760926082 2.2079361 1614.0732 1.0004663
## beta[31] -2.286213e-02 0.081146382 0.7771172 1181.7399 0.9999779
## beta[32] -4.283305e-02 0.049705924 0.5286878 1440.8574 0.9998730
## beta[33] 3.265539e-02 0.260274667 1.3284102 1280.6422 1.0008032
## beta[34] -2.718756e-02 0.075372541 0.7106186 1352.1020 0.9998384
## beta[35] 2.742249e-03 0.142047344 1.0014172 1191.7211 1.0005057
## beta[36] 1.299972e-02 0.187374643 1.0668694 1270.2314 1.0000822
## beta[37] 1.726574e+00 2.409029716 3.8307750 1298.9063 1.0005679
## beta[38] -7.607924e-03 0.115931255 0.9259206 1317.9382 1.0032408
## beta[39] -4.191319e-03 0.118217399 0.7783973 1032.5635 0.9998153
## beta[40] -2.638830e-03 0.137968722 0.9092323 1259.1188 0.9996931
## beta[41] 1.031600e-02 0.209939740 1.4856652 1256.4368 0.9993350
## beta[42] 2.583178e-02 0.243750560 1.2698513 1271.4446 0.9997268
## beta[43] 1.877351e-02 0.216946070 1.1038814 1226.0934 1.0005459
## beta[44] -5.555789e-04 0.122203549 0.7687124 1301.2173 0.9992789
## beta[45] -7.937788e-03 0.093216662 0.7493574 1239.5214 0.9997529
## beta[46] 4.220337e-02 0.297640899 1.6459243 1449.3512 1.0006258
## beta[47] -4.822399e-02 0.058969288 0.6748588 1320.4179 1.0011824
## beta[48] 5.947299e-03 0.183195867 1.0929164 1253.6368 1.0000397
## beta[49] -1.507508e-02 0.078328658 0.7439193 1199.4774 1.0020309
## beta[50] 3.813026e-04 0.151179088 1.1751295 1167.6845 0.9990124
## beta[51] 3.418521e-02 0.286090071 1.5798943 1281.0082 1.0031372
## beta[52] -3.638296e-02 0.070640687 0.7710908 1025.5307 1.0051598
## beta[53] -1.667603e-02 0.084334764 0.8696586 1211.7406 1.0003921
## beta[54] 4.381272e-02 0.339458843 1.7786208 1190.5303 1.0032486
## beta[55] 1.787067e-03 0.158021814 0.9713906 1251.9007 1.0019401
## beta[56] 3.041852e-02 0.249548457 1.9181104 1043.5684 1.0017113
## beta[57] 5.797979e-02 0.379843638 1.9615755 1211.6936 1.0014872
## beta[58] 8.023542e-02 0.459054373 2.0266783 1190.3707 1.0000956
## beta[59] -2.763325e-02 0.061804646 0.6862684 1114.9120 1.0003219
## beta[60] 1.280235e-01 0.572198633 2.0368150 1477.0287 1.0005896
## beta[61] 2.532298e-02 0.231892281 1.5782635 1140.4208 1.0001674
## beta[62] -7.529499e-02 0.021590673 0.4673873 1224.5152 1.0006860
## beta[63] 1.456097e-01 0.709721936 2.6199230 1128.4966 0.9993906
## beta[64] 9.284657e-03 0.180310526 1.1213950 1083.7647 1.0037735
##
## $c_summary
## , , chains = chain:1
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 0.096043255 0.03370987 0.02704341 0.071600437 0.094657779
## sigma 0.604995610 0.02431772 0.56137593 0.587244673 0.604577602
## beta[1] 0.145789362 0.38379211 -0.44378226 -0.044905826 0.025365573
## beta[2] -1.213246860 1.02416541 -3.20583848 -1.972631618 -1.144522454
## beta[3] 7.383114557 0.89067475 5.68530972 6.837834951 7.385963400
## beta[4] 2.299721401 1.11963428 0.03363081 1.579095144 2.271788133
## beta[5] -0.580647928 1.00231965 -3.23420358 -0.841057690 -0.215168288
## beta[6] -0.098255415 0.68015749 -1.48381404 -0.279858548 -0.023253103
## beta[7] -1.092493810 1.20086939 -4.03484086 -1.959633416 -0.762059622
## beta[8] 0.407550528 0.94386794 -0.72007422 -0.048295192 0.068958820
## beta[9] 6.250616833 1.08760811 4.06526228 5.497082285 6.318552138
## beta[10] 0.087649196 0.43497841 -0.75270211 -0.093529399 0.020623200
## beta[11] 0.374764001 0.60392956 -0.35286356 -0.008540096 0.145230932
## beta[12] 0.015217682 0.40311001 -0.78426696 -0.116007899 0.001208401
## beta[13] -0.162735515 0.52366082 -1.58524437 -0.235723737 -0.029633699
## beta[14] -0.057855841 0.50637843 -1.25387506 -0.189734678 -0.011722606
## beta[15] -0.092815115 0.48983139 -1.33523878 -0.222322009 -0.033642443
## beta[16] -0.047676432 0.56945858 -1.44617051 -0.195068933 -0.004425763
## beta[17] 0.421119148 0.91397336 -0.60818250 -0.036557733 0.087885563
## beta[18] -0.209400462 0.56412756 -1.72435289 -0.375092616 -0.056872354
## beta[19] 0.291009023 0.54540990 -0.49316429 -0.008072179 0.145946500
## beta[20] 1.480571579 0.93695015 -0.03785704 0.761562059 1.475263995
## beta[21] 0.164947444 0.45569576 -0.48509914 -0.063726104 0.032609242
## beta[22] 0.401285227 0.58238643 -0.29648939 0.003210501 0.180548201
## beta[23] -0.096383578 0.55104037 -1.38108413 -0.224872718 -0.012540336
## beta[24] -0.176140186 0.50208690 -1.52635509 -0.268228336 -0.035128796
## beta[25] 0.158423045 0.54024027 -0.59045018 -0.073992155 0.030904220
## beta[26] 0.048991185 0.51585360 -0.82856444 -0.148188722 -0.005089784
## beta[27] 0.467735996 0.70880500 -0.36294259 0.003366895 0.200763451
## beta[28] 0.271277297 0.53185116 -0.39314028 -0.026739412 0.083399555
## beta[29] 0.079982196 0.37760203 -0.52230749 -0.087615414 0.011931552
## beta[30] 0.453388249 0.67475414 -0.30199073 -0.003246616 0.184021906
## beta[31] -0.074141651 0.45556556 -1.12642425 -0.225858835 -0.016261385
## beta[32] -0.185007548 0.48715362 -1.53759170 -0.302864944 -0.049595576
## beta[33] 0.126140815 0.41856396 -0.57710370 -0.055925581 0.037962213
## beta[34] -0.159575725 0.51732249 -1.52744405 -0.305762970 -0.037583731
## beta[35] -0.008822837 0.42014013 -0.83275032 -0.170599779 0.003853561
## beta[36] 0.069236429 0.40893326 -0.71764897 -0.100443649 0.013684134
## beta[37] 1.679011830 1.09299819 -0.08963171 0.860737739 1.739556611
## beta[38] -0.010093177 0.41372296 -0.89468786 -0.136736795 -0.008382645
## beta[39] -0.037644441 0.40900583 -0.94788634 -0.144069908 -0.006006124
## beta[40] -0.047426340 0.44121591 -1.03219435 -0.218090876 -0.012570691
## beta[41] 0.101708612 0.48688025 -0.81965501 -0.084556619 0.024019229
## beta[42] 0.082137517 0.44098361 -0.77701065 -0.122295220 0.011803594
## beta[43] 0.070230622 0.38517888 -0.75336667 -0.080456197 0.011742135
## beta[44] -0.021158858 0.37969429 -0.86271383 -0.150766507 -0.013124197
## beta[45] -0.054081436 0.41310332 -0.99458620 -0.177353020 -0.005557781
## beta[46] 0.209001589 0.53084922 -0.49510306 -0.028420177 0.067402154
## beta[47] -0.170899308 0.49680324 -1.71947343 -0.274307573 -0.053944957
## beta[48] 0.007022844 0.46820479 -1.07489551 -0.155484150 -0.001134159
## beta[49] -0.096618793 0.42614588 -1.17799906 -0.214623730 -0.021212001
## beta[50] 0.002787401 0.43129729 -0.87667587 -0.142250734 0.002122639
## beta[51] 0.237378907 0.61209647 -0.53907690 -0.039252469 0.051416806
## beta[52] -0.340299396 0.90638824 -3.08998249 -0.374818478 -0.037001423
## beta[53] -0.115331416 0.53946233 -1.37617555 -0.205443530 -0.014712446
## beta[54] 0.257991186 0.61432989 -0.59498359 -0.030995874 0.065448389
## beta[55] 0.001265655 0.49389986 -1.23850895 -0.138063714 0.005329492
## beta[56] 0.194952775 0.80126225 -0.89886795 -0.076858506 0.034767077
## beta[57] 0.228892684 0.53157477 -0.52533825 -0.019034366 0.071198762
## beta[58] 0.294074519 0.60949856 -0.47411970 -0.031089560 0.097103431
## beta[59] -0.172214433 0.59225720 -1.85522252 -0.297799394 -0.027192721
## beta[60] 0.348923422 0.64578679 -0.38416557 -0.014724533 0.102109376
## beta[61] 0.095391621 0.53852654 -0.83790129 -0.081176267 0.021695397
## beta[62] -0.337433537 0.74185922 -2.42750151 -0.413622941 -0.080953328
## beta[63] 0.428202539 0.78937195 -0.51685731 -0.006522981 0.156366412
## beta[64] 0.043233199 0.43081561 -0.82878595 -0.108388983 0.001814935
## stats
## parameter 75% 97.5%
## b0 0.120946240 0.1592675
## sigma 0.620797781 0.6531221
## beta[1] 0.258304054 1.2295684
## beta[2] -0.230859629 0.1511064
## beta[3] 8.027439434 8.9647040
## beta[4] 3.099262159 4.3408438
## beta[5] -0.005857782 0.4713624
## beta[6] 0.074341678 0.9720115
## beta[7] -0.059292749 0.3068976
## beta[8] 0.511876508 3.1604980
## beta[9] 6.975133962 8.1675404
## beta[10] 0.209583240 1.1526731
## beta[11] 0.576222208 1.9393828
## beta[12] 0.140293985 0.9718253
## beta[13] 0.070944984 0.6201343
## beta[14] 0.126220593 0.9075197
## beta[15] 0.086361562 0.8826077
## beta[16] 0.140554040 1.0864658
## beta[17] 0.615549228 3.0699077
## beta[18] 0.057648022 0.6195850
## beta[19] 0.469665128 1.7438884
## beta[20] 2.211829189 3.2404234
## beta[21] 0.332877799 1.3337647
## beta[22] 0.703329562 1.9064928
## beta[23] 0.090081614 0.8335657
## beta[24] 0.055339697 0.5309573
## beta[25] 0.279325441 1.5847465
## beta[26] 0.137548937 1.1966511
## beta[27] 0.727220908 2.1907324
## beta[28] 0.404205501 1.7721020
## beta[29] 0.177450605 1.0011747
## beta[30] 0.763871848 2.1859661
## beta[31] 0.079431073 0.8256627
## beta[32] 0.029789087 0.5157569
## beta[33] 0.221507905 1.4723614
## beta[34] 0.063839229 0.7699274
## beta[35] 0.127799864 1.0294184
## beta[36] 0.189143111 1.1665577
## beta[37] 2.363913452 3.9202462
## beta[38] 0.131798986 0.8573691
## beta[39] 0.120162082 0.6819696
## beta[40] 0.108664434 0.8023568
## beta[41] 0.225805437 1.3118542
## beta[42] 0.247151717 1.1920543
## beta[43] 0.211060764 1.0205250
## beta[44] 0.116910924 0.8034665
## beta[45] 0.099004045 0.7200431
## beta[46] 0.320150859 1.7889741
## beta[47] 0.046675151 0.5365188
## beta[48] 0.179542137 1.0146447
## beta[49] 0.070369599 0.7618872
## beta[50] 0.139873025 0.9305689
## beta[51] 0.406381707 1.8634864
## beta[52] 0.046064250 0.5536887
## beta[53] 0.076721234 0.8135630
## beta[54] 0.394356783 2.0493617
## beta[55] 0.178487059 0.9308485
## beta[56] 0.291127179 1.9597541
## beta[57] 0.363313450 1.7437266
## beta[58] 0.468983842 2.0547741
## beta[59] 0.052215441 0.6566924
## beta[60] 0.532104748 2.0636872
## beta[61] 0.164378094 1.4186008
## beta[62] 0.037565978 0.4129595
## beta[63] 0.687495839 2.5811196
## beta[64] 0.145885256 1.1007628
##
## , , chains = chain:2
##
## stats
## parameter mean sd 2.5% 25%
## b0 0.097863082 0.03312878 0.030301076 0.0766758056
## sigma 0.604446666 0.02514548 0.555941084 0.5867280327
## beta[1] 0.145224768 0.44810068 -0.521348357 -0.0489199128
## beta[2] -1.228857390 0.96624664 -3.255502882 -1.9437359819
## beta[3] 7.326688099 0.97618114 5.560967321 6.6956470919
## beta[4] 2.369088616 1.16371700 -0.007685079 1.6599659349
## beta[5] -0.500126293 0.85001519 -2.674698386 -0.7315312826
## beta[6] -0.108593749 0.54146286 -1.504603244 -0.2264235566
## beta[7] -1.098690314 1.22010311 -3.839107976 -1.9692545066
## beta[8] 0.397948759 0.82973689 -0.587011982 -0.0258467559
## beta[9] 6.173098755 1.09628725 3.963181456 5.4049318572
## beta[10] 0.091542782 0.32790878 -0.467914546 -0.0676025806
## beta[11] 0.375615292 0.59970145 -0.348440360 -0.0008684619
## beta[12] 0.021002674 0.42582100 -0.906318896 -0.1221650491
## beta[13] -0.141306698 0.42409526 -1.339290160 -0.2179355619
## beta[14] -0.024069266 0.50307215 -1.373880738 -0.1688233868
## beta[15] -0.146818166 0.51928873 -1.470019817 -0.2801568056
## beta[16] -0.082473932 0.57541816 -1.384300214 -0.1608664583
## beta[17] 0.439368247 0.87305397 -0.398334553 -0.0243020254
## beta[18] -0.182210563 0.48076384 -1.472295702 -0.3047635530
## beta[19] 0.270567484 0.49331017 -0.447393996 -0.0129180856
## beta[20] 1.511949745 1.01060906 -0.080138814 0.7260842849
## beta[21] 0.123455588 0.39579640 -0.657494413 -0.0327905413
## beta[22] 0.411182059 0.60222592 -0.268690533 0.0008245203
## beta[23] -0.034793639 0.47429448 -1.082294691 -0.1779979705
## beta[24] -0.230155309 0.52318147 -1.704197543 -0.3786497098
## beta[25] 0.139733717 0.51479790 -0.749578330 -0.0793635169
## beta[26] 0.054962756 0.54697231 -0.886755415 -0.1276546732
## beta[27] 0.421200515 0.66881961 -0.325690494 -0.0084643828
## beta[28] 0.275000112 0.54234936 -0.431237753 -0.0228721005
## beta[29] 0.052195761 0.41684319 -0.910779797 -0.0797980837
## beta[30] 0.456438284 0.66810710 -0.334186020 -0.0001619620
## beta[31] -0.134671086 0.43320674 -1.344577051 -0.2858425489
## beta[32] -0.147004244 0.50457162 -1.302384066 -0.2684600916
## beta[33] 0.125895336 0.45489764 -0.542556546 -0.0813635225
## beta[34] -0.141423201 0.47790001 -1.460700477 -0.2537832701
## beta[35] 0.026045310 0.44211248 -0.797592834 -0.1316845733
## beta[36] 0.081913647 0.38579107 -0.554386825 -0.0894192342
## beta[37] 1.648748077 0.99616916 -0.007438141 0.8170893082
## beta[38] -0.008723044 0.44328298 -0.878743447 -0.1606750589
## beta[39] -0.067986086 0.44864984 -1.311085556 -0.1815960524
## beta[40] -0.024761878 0.47054117 -1.172615998 -0.1643637791
## beta[41] 0.099636470 0.50789584 -0.865429147 -0.1034534098
## beta[42] 0.139389103 0.46764841 -0.703923959 -0.0412191249
## beta[43] 0.116948511 0.39736112 -0.486803180 -0.0564683595
## beta[44] -0.047376088 0.42221231 -1.135213997 -0.1479052555
## beta[45] -0.071254300 0.41215730 -1.004786522 -0.2079147483
## beta[46] 0.152430892 0.45274040 -0.462527041 -0.0613096289
## beta[47] -0.164580246 0.47516385 -1.438511929 -0.3185415674
## beta[48] 0.012933709 0.41032691 -0.874436677 -0.1423919027
## beta[49] -0.122738844 0.46462516 -1.277434484 -0.2599500682
## beta[50] 0.009368875 0.48999158 -1.194052113 -0.1501792650
## beta[51] 0.138115357 0.59665861 -0.706862560 -0.0786821302
## beta[52] -0.273143198 0.76698653 -2.555352280 -0.4045591640
## beta[53] -0.072259477 0.45727339 -1.059576330 -0.2145653947
## beta[54] 0.193174239 0.57622013 -0.719424697 -0.0568049111
## beta[55] -0.039015096 0.52522080 -1.108259691 -0.1804298711
## beta[56] 0.164443130 0.60274342 -0.918509895 -0.0592996035
## beta[57] 0.229808622 0.55051989 -0.521391319 -0.0490464572
## beta[58] 0.283122213 0.60125244 -0.496050966 -0.0173523194
## beta[59] -0.158270233 0.62993344 -1.801824142 -0.2870117835
## beta[60] 0.380413184 0.62292087 -0.363521498 0.0005595968
## beta[61] 0.184097507 0.58841029 -0.862488620 -0.0517607249
## beta[62] -0.358028832 0.75950023 -2.314449591 -0.4930923867
## beta[63] 0.477105484 0.84568947 -0.432172033 -0.0086127927
## beta[64] 0.060067502 0.49340066 -0.862049773 -0.1111779175
## stats
## parameter 50% 75% 97.5%
## b0 0.0983291598 0.12057281 0.1638375
## sigma 0.6058074455 0.62023563 0.6528083
## beta[1] 0.0330896764 0.28679687 1.2271710
## beta[2] -1.2649638751 -0.31378045 0.1844630
## beta[3] 7.3222455999 7.96840764 9.3989267
## beta[4] 2.4716333694 3.14891789 4.4737615
## beta[5] -0.1759553447 0.01219901 0.4387424
## beta[6] -0.0246219837 0.08599405 0.6392261
## beta[7] -0.6766445144 -0.04518592 0.2878568
## beta[8] 0.0780424066 0.56178593 2.8452628
## beta[9] 6.1565684096 6.95043715 8.1709283
## beta[10] 0.0283568765 0.22361960 0.8869880
## beta[11] 0.1394076255 0.64224521 1.9797657
## beta[12] 0.0046606300 0.18568125 1.0442417
## beta[13] -0.0324995527 0.04559562 0.5509822
## beta[14] -0.0027478321 0.14663774 1.0369085
## beta[15] -0.0301099791 0.06861713 0.7360260
## beta[16] -0.0036485069 0.09511069 0.7283029
## beta[17] 0.1097650338 0.61552687 2.8564969
## beta[18] -0.0450689752 0.03627951 0.4503278
## beta[19] 0.1283815248 0.51050874 1.4486375
## beta[20] 1.4863667267 2.27621887 3.4864057
## beta[21] 0.0404310887 0.27833208 1.0357377
## beta[22] 0.1808239018 0.72054760 2.1219887
## beta[23] -0.0058255261 0.11215290 0.8817749
## beta[24] -0.0608787538 0.02731006 0.4768778
## beta[25] 0.0299145891 0.22256099 1.4304001
## beta[26] 0.0060528258 0.19335475 1.3266900
## beta[27] 0.1471225824 0.70106992 2.2864997
## beta[28] 0.0894365245 0.43255922 1.7486068
## beta[29] 0.0121204010 0.17537740 0.9790295
## beta[30] 0.2304610628 0.74516051 2.1836862
## beta[31] -0.0524931437 0.05289075 0.5132697
## beta[32] -0.0260860390 0.06558419 0.5738092
## beta[33] 0.0128475786 0.26782374 1.2546919
## beta[34] -0.0268133556 0.08628742 0.6120409
## beta[35] 0.0018990015 0.17882371 1.1060702
## beta[36] 0.0051763384 0.18523929 1.0931118
## beta[37] 1.6676555210 2.38875474 3.5749428
## beta[38] -0.0069549244 0.11412554 1.0281651
## beta[39] -0.0067119544 0.11660333 0.6635317
## beta[40] 0.0016308233 0.13386008 0.9181422
## beta[41] 0.0038553444 0.23418117 1.3052699
## beta[42] 0.0357398440 0.23368250 1.3704825
## beta[43] 0.0254481199 0.24535671 1.0750118
## beta[44] 0.0001171708 0.13638823 0.7131980
## beta[45] -0.0111523704 0.07625630 0.7463435
## beta[46] 0.0195149453 0.26144475 1.3161496
## beta[47] -0.0534358814 0.03792003 0.6694155
## beta[48] -0.0029900703 0.15103644 0.8956869
## beta[49] -0.0242921450 0.07183460 0.6575819
## beta[50] 0.0003891385 0.16091299 1.1106848
## beta[51] 0.0155896151 0.24794619 1.4668260
## beta[52] -0.0393924906 0.08120025 0.7612110
## beta[53] -0.0138703041 0.07971401 0.8105731
## beta[54] 0.0270620290 0.31515369 1.8282145
## beta[55] -0.0008713000 0.11475097 0.8804458
## beta[56] 0.0570717737 0.24639028 1.7748681
## beta[57] 0.0680547977 0.38831859 1.7256219
## beta[58] 0.0760302331 0.45456092 2.0058215
## beta[59] -0.0342535752 0.08533285 0.7676128
## beta[60] 0.1582884043 0.58068543 2.0439788
## beta[61] 0.0437676762 0.29437119 1.7798463
## beta[62] -0.0671238878 0.02446569 0.4579827
## beta[63] 0.1732226319 0.70368220 2.7428420
## beta[64] 0.0094120070 0.19128402 1.2757906
##
## , , chains = chain:3
##
## stats
## parameter mean sd 2.5% 25%
## b0 0.096750534 0.03393401 0.032734310 0.0750657059
## sigma 0.603340785 0.02411071 0.560005071 0.5871170313
## beta[1] 0.166424556 0.44436377 -0.497341038 -0.0428261233
## beta[2] -1.307380419 0.98384448 -3.234348192 -2.0753758828
## beta[3] 7.276798792 0.95436819 5.402051700 6.6069217864
## beta[4] 2.317862239 1.16544213 0.002362751 1.5475810719
## beta[5] -0.555434965 0.90810727 -2.779299728 -0.9294002335
## beta[6] -0.123978102 0.64817610 -1.601150717 -0.2551240681
## beta[7] -1.153597681 1.12990145 -3.732901945 -1.8338402637
## beta[8] 0.405613452 0.95167082 -0.707048962 -0.0577375905
## beta[9] 6.316772749 1.02025437 4.389750560 5.5491065995
## beta[10] 0.089587622 0.42505001 -0.696855570 -0.0750111675
## beta[11] 0.368951755 0.61289867 -0.411411559 -0.0036770500
## beta[12] -0.004166531 0.41410205 -0.986809073 -0.1476432059
## beta[13] -0.252375685 0.52841382 -1.767235934 -0.4165123005
## beta[14] -0.063122787 0.51862142 -1.350489072 -0.1884960507
## beta[15] -0.145588384 0.60368517 -1.793984252 -0.2847139528
## beta[16] -0.061834063 0.54995945 -1.511261383 -0.1819369550
## beta[17] 0.477039553 0.91827533 -0.687477937 -0.0140982411
## beta[18] -0.192275205 0.46704027 -1.461210724 -0.3425033710
## beta[19] 0.223313102 0.51801059 -0.461913064 -0.0489740158
## beta[20] 1.539444759 0.95225076 -0.015418008 0.8187438363
## beta[21] 0.150815612 0.43534160 -0.511677280 -0.0406610756
## beta[22] 0.417337861 0.61438570 -0.386328652 -0.0002418128
## beta[23] -0.073070636 0.56280431 -1.380663279 -0.2198837696
## beta[24] -0.270848069 0.66106431 -1.961499805 -0.4598077856
## beta[25] 0.146050195 0.51237448 -0.705110620 -0.0747546041
## beta[26] 0.051830884 0.56742778 -0.867117737 -0.1456944564
## beta[27] 0.441320025 0.74038137 -0.364561032 -0.0107363626
## beta[28] 0.260132945 0.52610070 -0.454144392 -0.0096880845
## beta[29] 0.078251509 0.40364275 -0.707829894 -0.0817613783
## beta[30] 0.434301858 0.66192337 -0.364293179 -0.0027183069
## beta[31] -0.101254942 0.50178960 -1.242418108 -0.2790069605
## beta[32] -0.199264540 0.53859729 -1.596819282 -0.3172858933
## beta[33] 0.123931629 0.42672182 -0.670324991 -0.0543525801
## beta[34] -0.143359266 0.55224435 -1.504804216 -0.2630534246
## beta[35] 0.018000943 0.37647353 -0.765695043 -0.1305583148
## beta[36] 0.058538779 0.37291533 -0.699947351 -0.0791027699
## beta[37] 1.718524618 1.10304516 -0.094070684 0.8732578204
## beta[38] -0.038073623 0.45322946 -1.118445977 -0.1401499835
## beta[39] -0.038549186 0.43775311 -0.894615272 -0.1677839305
## beta[40] 0.012205662 0.50945393 -1.118533194 -0.1423813028
## beta[41] 0.082499607 0.48266142 -0.735477423 -0.1013413103
## beta[42] 0.116672381 0.45178608 -0.723743946 -0.0538085353
## beta[43] 0.082220233 0.39838762 -0.677579095 -0.0833229395
## beta[44] -0.031745801 0.38735955 -1.013312637 -0.1486430887
## beta[45] -0.026869508 0.39439592 -1.009089140 -0.1322340255
## beta[46] 0.206927337 0.51713982 -0.499194519 -0.0575688024
## beta[47] -0.170979233 0.53696919 -1.563814405 -0.3216526402
## beta[48] 0.053620778 0.43478535 -0.863117607 -0.0955547399
## beta[49] -0.093603162 0.44553412 -1.177801634 -0.2266249441
## beta[50] 0.040535593 0.58921209 -1.032330854 -0.1497235926
## beta[51] 0.140524161 0.47893979 -0.560259255 -0.0673875104
## beta[52] -0.333009301 0.94415157 -2.813643845 -0.4627338203
## beta[53] -0.122781410 0.54969896 -1.608262264 -0.2444734195
## beta[54] 0.169270656 0.52910992 -0.599203773 -0.0510934259
## beta[55] 0.003981441 0.49774158 -0.949051605 -0.1259188240
## beta[56] 0.197579840 0.80044163 -0.899788420 -0.0618682011
## beta[57] 0.288552711 0.69319319 -0.636844246 -0.0426753144
## beta[58] 0.340650278 0.67350705 -0.495150503 -0.0272980993
## beta[59] -0.150048751 0.54313799 -1.378883844 -0.2619190614
## beta[60] 0.382176436 0.61040963 -0.376088028 -0.0017401266
## beta[61] 0.149377983 0.56810294 -0.726145460 -0.0953938835
## beta[62] -0.354454218 0.70511360 -2.399070399 -0.5171962304
## beta[63] 0.497274135 0.86921490 -0.449846244 -0.0146039333
## beta[64] 0.098332817 0.43578217 -0.752845136 -0.0787644129
## stats
## parameter 50% 75% 97.5%
## b0 0.0970338646 0.119486820 0.16006756
## sigma 0.6023666549 0.617888780 0.65203278
## beta[1] 0.0376975690 0.284696529 1.33986904
## beta[2] -1.3140532259 -0.416726824 0.06024406
## beta[3] 7.2737082172 7.926729470 9.19209667
## beta[4] 2.3717962681 3.215698788 4.43507938
## beta[5] -0.1947770047 -0.000215417 0.46168192
## beta[6] -0.0448951482 0.080193804 1.08342694
## beta[7] -0.9903898449 -0.137769462 0.17994152
## beta[8] 0.0638767802 0.540289125 3.25314491
## beta[9] 6.3067155948 7.034400439 8.32270825
## beta[10] 0.0125431218 0.215503829 1.07346854
## beta[11] 0.1735123442 0.626857475 1.94590295
## beta[12] -0.0019824543 0.117262092 0.88909872
## beta[13] -0.0858388598 0.034166885 0.47015861
## beta[14] -0.0064761860 0.105199959 1.03103622
## beta[15] -0.0190764657 0.085362834 0.70044552
## beta[16] -0.0049008157 0.116871743 1.02739354
## beta[17] 0.1146910150 0.739773011 3.18470762
## beta[18] -0.0461477346 0.035886435 0.46509369
## beta[19] 0.0397056785 0.391724026 1.62956348
## beta[20] 1.5780864812 2.236841086 3.37919389
## beta[21] 0.0524984106 0.277994818 1.31421071
## beta[22] 0.1926241905 0.716556606 1.97437199
## beta[23] -0.0077132298 0.101970458 0.78065344
## beta[24] -0.0606261314 0.039093031 0.46203758
## beta[25] 0.0261918747 0.245007705 1.51659310
## beta[26] 0.0029116906 0.191492952 1.19011905
## beta[27] 0.1423930498 0.713920229 2.51126968
## beta[28] 0.1246797861 0.441369551 1.62926102
## beta[29] 0.0050639156 0.159883913 1.17957731
## beta[30] 0.1809641497 0.778032535 2.19543034
## beta[31] -0.0210915686 0.085595633 0.81814290
## beta[32] -0.0284671888 0.057978243 0.50108207
## beta[33] 0.0254593958 0.266646710 1.13764383
## beta[34] -0.0233136775 0.070738072 0.68730491
## beta[35] 0.0020386478 0.151708098 0.95860096
## beta[36] 0.0187721747 0.186990901 1.04379734
## beta[37] 1.7428021418 2.424314784 4.00030275
## beta[38] -0.0065737173 0.103778210 0.87843187
## beta[39] -0.0032032428 0.108137646 0.82834883
## beta[40] 0.0008359924 0.169449635 0.98221100
## beta[41] 0.0051336273 0.171871414 1.45986260
## beta[42] 0.0298321850 0.235036228 1.23931087
## beta[43] 0.0187767589 0.199318120 1.05959201
## beta[44] 0.0003183083 0.110557379 0.77894323
## beta[45] -0.0049235595 0.133791544 0.70479219
## beta[46] 0.0625556049 0.338126696 1.61054760
## beta[47] -0.0399062083 0.087039514 0.75092893
## beta[48] 0.0204702134 0.208265300 1.09603805
## beta[49] -0.0169452582 0.082673635 0.72320316
## beta[50] 0.0017584081 0.168141510 1.33736026
## beta[51] 0.0312291180 0.239131917 1.55429295
## beta[52] -0.0534667171 0.065356581 0.72449662
## beta[53] -0.0233950831 0.078165615 0.85146683
## beta[54] 0.0561823834 0.348329987 1.32936274
## beta[55] 0.0014247424 0.157731894 0.99454269
## beta[56] 0.0259180805 0.296598251 1.95406242
## beta[57] 0.0697551982 0.465413747 2.19454240
## beta[58] 0.1014221756 0.484742828 2.20963405
## beta[59] -0.0280347403 0.056140704 0.64872808
## beta[60] 0.1482180438 0.592202421 2.06558317
## beta[61] 0.0202579758 0.236361216 1.90136110
## beta[62] -0.0887302682 0.018240668 0.48304053
## beta[63] 0.1445586561 0.828770180 2.73270332
## beta[64] 0.0229665494 0.242535694 1.13856099
##
## , , chains = chain:4
##
## stats
## parameter mean sd 2.5% 25% 50%
## b0 0.096546109 0.03119814 0.03395806 0.077447745 0.0965095621
## sigma 0.606365370 0.02601545 0.55888247 0.588350196 0.6051786083
## beta[1] 0.134792293 0.39251495 -0.45534817 -0.050204199 0.0316981567
## beta[2] -1.068057917 0.96694432 -3.06514838 -1.796898313 -0.9238968042
## beta[3] 7.413603908 0.91219501 5.61078760 6.756476080 7.4423340418
## beta[4] 2.122541661 1.17272372 -0.03214448 1.301502627 2.2044262070
## beta[5] -0.417517079 0.86040240 -2.74420720 -0.686475160 -0.1053228717
## beta[6] -0.156284450 0.68248812 -1.78088896 -0.288960938 -0.0388863655
## beta[7] -1.070553973 1.17823428 -3.75289738 -1.892006598 -0.7661059973
## beta[8] 0.403951524 0.92918512 -0.85355128 -0.025696427 0.1019137322
## beta[9] 6.193950243 1.05596438 4.13845647 5.536986032 6.0964358343
## beta[10] 0.146068004 0.39352310 -0.38785335 -0.037186990 0.0347360379
## beta[11] 0.350707529 0.56840243 -0.33621216 0.001517974 0.1288057578
## beta[12] -0.024115866 0.33938481 -0.88596422 -0.146188716 -0.0034642372
## beta[13] -0.171345812 0.49293933 -1.68908165 -0.276011152 -0.0230133392
## beta[14] -0.015290820 0.46114522 -1.04555045 -0.137679703 -0.0048112318
## beta[15] -0.103692162 0.53161903 -1.39657129 -0.249554098 -0.0310001471
## beta[16] -0.017356015 0.44165397 -1.02940990 -0.159484394 -0.0003919004
## beta[17] 0.309297358 0.70029027 -0.62504006 -0.053299569 0.0659460915
## beta[18] -0.180937746 0.49290100 -1.41614007 -0.283951594 -0.0371937834
## beta[19] 0.228470518 0.48489347 -0.41688172 -0.032366918 0.0695854104
## beta[20] 1.475830216 1.03497593 -0.05190627 0.660631041 1.4488697016
## beta[21] 0.148284401 0.44091990 -0.58074430 -0.055827953 0.0411691746
## beta[22] 0.423250762 0.64327663 -0.35024286 -0.001763475 0.1995558016
## beta[23] -0.056470135 0.42198689 -1.08751179 -0.176357958 -0.0102009410
## beta[24] -0.204326846 0.50890765 -1.56174360 -0.337839685 -0.0664825137
## beta[25] 0.092980653 0.41403994 -0.62819505 -0.062420306 0.0252204292
## beta[26] 0.004960615 0.46091351 -0.97960300 -0.134688447 -0.0044246808
## beta[27] 0.408130959 0.69094943 -0.39912827 -0.010513680 0.1206067636
## beta[28] 0.249160630 0.53431447 -0.48870831 -0.018321634 0.0794747011
## beta[29] 0.059086089 0.38118232 -0.68374425 -0.069460437 0.0164495672
## beta[30] 0.434809778 0.66273590 -0.30493028 0.001435845 0.1645989511
## beta[31] -0.086643546 0.48183164 -1.22932722 -0.209075325 -0.0107216054
## beta[32] -0.198217659 0.54310971 -1.66071143 -0.352966969 -0.0685537658
## beta[33] 0.151362663 0.44572024 -0.56298294 -0.044021376 0.0501036917
## beta[34] -0.109355374 0.46519544 -1.29350785 -0.233817836 -0.0201705784
## beta[35] 0.005891920 0.37946143 -0.80753683 -0.108664324 0.0028116083
## beta[36] 0.062042759 0.34608416 -0.64403134 -0.086753855 0.0157197062
## beta[37] 1.696033184 1.09194149 -0.05466979 0.817392231 1.7656743500
## beta[38] -0.028515133 0.38597976 -0.88543400 -0.145004447 -0.0096725940
## beta[39] -0.004405590 0.36918367 -0.82951958 -0.126668548 -0.0019244500
## beta[40] -0.018401085 0.43630236 -1.04895133 -0.171113159 -0.0042289876
## beta[41] 0.141252010 0.50064578 -0.58262850 -0.083481042 0.0121680857
## beta[42] 0.122226011 0.46448308 -0.72043695 -0.060445905 0.0330657894
## beta[43] 0.079908154 0.40493419 -0.63175059 -0.088765612 0.0210775567
## beta[44] 0.001005078 0.38038779 -0.92394620 -0.107286841 0.0006368025
## beta[45] -0.063601182 0.42356851 -1.15345668 -0.197176905 -0.0158965326
## beta[46] 0.157290374 0.50681227 -0.57848394 -0.085276009 0.0200104162
## beta[47] -0.168930859 0.51155047 -1.49594730 -0.322138103 -0.0481591655
## beta[48] 0.050106865 0.44549763 -0.77501573 -0.105762944 0.0100948284
## beta[49] -0.028036058 0.40061927 -0.99261573 -0.144641557 -0.0041327906
## beta[50] -0.004113098 0.48516183 -1.07529346 -0.141861334 -0.0040897514
## beta[51] 0.129033550 0.46640788 -0.70388789 -0.074359914 0.0312289306
## beta[52] -0.186243579 0.78081247 -2.10210458 -0.326639629 -0.0211603294
## beta[53] -0.075811151 0.54707675 -1.50713381 -0.206610231 -0.0131417243
## beta[54] 0.149262412 0.53707912 -0.78685277 -0.049896044 0.0371059986
## beta[55] 0.052582773 0.44789010 -0.84649523 -0.092785314 0.0035208611
## beta[56] 0.086219455 0.62950334 -1.17749690 -0.065058696 0.0154939968
## beta[57] 0.212845834 0.59906723 -0.59113470 -0.056482944 0.0376803732
## beta[58] 0.267347491 0.55503383 -0.39874925 -0.024715881 0.0671812083
## beta[59] -0.217711945 0.66409051 -1.97110764 -0.337879094 -0.0206872799
## beta[60] 0.322418474 0.59044164 -0.47294447 -0.015716642 0.1174859852
## beta[61] 0.123126799 0.54656507 -0.72153136 -0.075345447 0.0191426785
## beta[62] -0.364172375 0.75755037 -2.62379808 -0.462862511 -0.0734422675
## beta[63] 0.426919386 0.77044999 -0.49978769 -0.009193722 0.1232542847
## beta[64] 0.010400408 0.44607663 -1.00581239 -0.124430837 0.0026485640
## stats
## parameter 75% 97.5%
## b0 0.11532454 0.1588412
## sigma 0.62389415 0.6575069
## beta[1] 0.25869810 1.2543498
## beta[2] -0.21673321 0.1826845
## beta[3] 8.00662542 9.1519377
## beta[4] 2.95206989 4.2689175
## beta[5] 0.02177994 0.7387188
## beta[6] 0.04271197 1.0711325
## beta[7] -0.04496668 0.3521680
## beta[8] 0.52942373 3.0966434
## beta[9] 6.88928881 8.2305780
## beta[10] 0.23593774 1.1671557
## beta[11] 0.58719401 1.7988301
## beta[12] 0.11731477 0.6330601
## beta[13] 0.05410160 0.4392717
## beta[14] 0.12154803 0.8854676
## beta[15] 0.09184914 0.9041607
## beta[16] 0.11411800 0.9584540
## beta[17] 0.54107493 2.2854979
## beta[18] 0.05210473 0.5497280
## beta[19] 0.39956793 1.4887159
## beta[20] 2.22781676 3.5487697
## beta[21] 0.28843009 1.2147244
## beta[22] 0.71977556 2.1977597
## beta[23] 0.07704070 0.6800647
## beta[24] 0.02649505 0.5950662
## beta[25] 0.21609099 1.1731736
## beta[26] 0.10335974 1.0320278
## beta[27] 0.57313594 2.2421853
## beta[28] 0.42014957 1.5453318
## beta[29] 0.15370075 1.0275464
## beta[30] 0.72749763 2.2662753
## beta[31] 0.10491340 0.8203729
## beta[32] 0.03127546 0.5296128
## beta[33] 0.30977704 1.3396136
## beta[34] 0.08152119 0.7557737
## beta[35] 0.12989660 0.8501700
## beta[36] 0.18022421 0.9007207
## beta[37] 2.46585702 3.8206777
## beta[38] 0.11057769 0.7658582
## beta[39] 0.12440654 0.8293644
## beta[40] 0.13673095 0.8512379
## beta[41] 0.22061268 1.5640561
## beta[42] 0.25720122 1.2235647
## beta[43] 0.20427252 1.2053780
## beta[44] 0.12814593 0.8949197
## beta[45] 0.06978069 0.8546703
## beta[46] 0.25262860 1.5940772
## beta[47] 0.06957541 0.7047576
## beta[48] 0.16732765 1.3135822
## beta[49] 0.09213477 0.8457497
## beta[50] 0.13966205 1.0919625
## beta[51] 0.29034475 1.3679455
## beta[52] 0.08463103 1.0607000
## beta[53] 0.11819276 0.9025530
## beta[54] 0.30770524 1.4930241
## beta[55] 0.18967722 1.0539308
## beta[56] 0.18212825 1.3860326
## beta[57] 0.34173677 1.9991118
## beta[58] 0.41579681 1.8002621
## beta[59] 0.06126964 0.5687689
## beta[60] 0.56726528 1.8606980
## beta[61] 0.21117149 1.4984123
## beta[62] 0.01255208 0.5103501
## beta[63] 0.64219280 2.5380032
## beta[64] 0.16556018 0.9009418First compare the resulting regression parameters to OLS values.
hs_samples <- extract(horseshoe_fit, pars=c("b0", "sigma", "beta"))
# rescale to real units
post_med_intercept <- median(hs_samples$b0) * mad(training_d$y) + median(training_d$y)
post_med_sigma <- median(hs_samples$sigma) * mad(training_d$y)
post_med_slopes <- colMedians(hs_samples$beta) * mad(training_d$y)
The coefficient estimates from OLS are wierd.
## ols stan
## (Intercept) 150.40351 150.441968698
## `tc:ldl` -17718.75773 0.033353836
## tc 16796.16643 -15.094758817
## ldl -15061.25410 -2.803126530
## `tc^2` 11301.51242 -0.566494009
## `tc:ltg` -10009.29895 -1.458708736
## `ldl:ltg` 8124.59404 5.071689135
## `ldl^2` 7536.39662 -2.487378411
## `tc:hdl` -6298.38254 2.990296176
## hdl -6254.46319 -70.904720898
## `ldl:hdl` 5004.48290 0.156320821
## ltg -4856.91364 544.081396337
## `hdl:ltg` 3597.25613 11.198654641
## `tc:tch` -2574.37368 -3.182541198
## `hdl^2` 2298.16669 -0.298001012
## `ldl:tch` 1731.89234 2.660811733
## `ltg^2` 1549.77906 -4.036317748
## `hdl:tch` 1076.35481 -2.417174388
## `map:tc` 1003.12227 -0.048598374
## `map:ldl` -916.96996 -0.694345338
## `tch^2` 756.03682 8.022192132
## `tch:glu` 596.19136 12.736974731
## bmi 587.85483 643.162520011
## `hdl:glu` 560.73667 2.215087063
## `tc:glu` -497.85900 3.832447892
## `map:ltg` -455.17300 0.520230473
## `age:tch` 450.27383 -0.007549981
## `age:hdl` 349.10626 2.512926502
## tch 330.71500 6.514453051
## `bmi:map` 326.31803 151.029322377
## map 323.09057 204.692791124
## `sex:tch` -314.48509 -2.378188538
## `map:hdl` -303.16364 3.691671900
## sex -286.08810 -101.765851787
## `ldl:glu` 265.99329 7.018465292
## `age:ldl` -215.12650 -4.823499031
## `tch:ltg` 212.46622 -6.586308746
## `ltg:glu` 172.80190 0.812160488
## `sex:map` 165.15332 16.293653062
## `bmi:tch` -162.32294 0.902375176
## `map^2` -161.86028 -3.306257826
## `age:sex` 149.66531 129.941006661
## `age:tc` -147.14797 -0.731836716
## age 131.03118 2.742276426
## `sex:ldl` 125.00529 -3.746752079
## `age^2` 123.34475 12.703843426
## `sex:hdl` -108.67040 2.856477713
## `bmi:hdl` -101.63216 -0.230827419
## `sex:ltg` 97.47348 0.239873862
## `bmi:ldl` 92.41196 -0.366628908
## `map:tch` 83.43006 -4.218316446
## `age:map` 73.42593 16.507539257
## `bmi:ltg` 70.84372 2.259593599
## `map:glu` -65.99667 -1.318668568
## `age:glu` 65.26740 8.438687365
## `bmi^2` -57.66550 0.095112982
## `age:bmi` -51.02631 3.575305840
## `glu^2` 49.71817 8.064464238
## `age:ltg` 31.70304 14.258226696
## `sex:glu` 24.91435 1.137130088
## `bmi:tc` 21.50722 -0.665490953
## glu 19.12792 2.300361215
## `sex:tc` -15.98984 -1.999827883
## `sex:bmi` -14.04317 0.938191184
## `bmi:glu` 11.21618 1.642182837And, quite different than what Stan gets.
## ols stan
## (Intercept) 150.40351 150.441968698
## bmi 587.85483 643.162520011
## ltg -4856.91364 544.081396337
## map 323.09057 204.692791124
## `bmi:map` 326.31803 151.029322377
## `age:sex` 149.66531 129.941006661
## sex -286.08810 -101.765851787
## hdl -6254.46319 -70.904720898
## `age:map` 73.42593 16.507539257
## `sex:map` 165.15332 16.293653062
## tc 16796.16643 -15.094758817
## `age:ltg` 31.70304 14.258226696
## `tch:glu` 596.19136 12.736974731
## `age^2` 123.34475 12.703843426
## `hdl:ltg` 3597.25613 11.198654641
## `age:glu` 65.26740 8.438687365
## `glu^2` 49.71817 8.064464238
## `tch^2` 756.03682 8.022192132
## `ldl:glu` 265.99329 7.018465292
## `tch:ltg` 212.46622 -6.586308746
## tch 330.71500 6.514453051
## `ldl:ltg` 8124.59404 5.071689135
## `age:ldl` -215.12650 -4.823499031
## `map:tch` 83.43006 -4.218316446
## `ltg^2` 1549.77906 -4.036317748
## `tc:glu` -497.85900 3.832447892
## `sex:ldl` 125.00529 -3.746752079
## `map:hdl` -303.16364 3.691671900
## `age:bmi` -51.02631 3.575305840
## `map^2` -161.86028 -3.306257826
## `tc:tch` -2574.37368 -3.182541198
## `tc:hdl` -6298.38254 2.990296176
## `sex:hdl` -108.67040 2.856477713
## ldl -15061.25410 -2.803126530
## age 131.03118 2.742276426
## `ldl:tch` 1731.89234 2.660811733
## `age:hdl` 349.10626 2.512926502
## `ldl^2` 7536.39662 -2.487378411
## `hdl:tch` 1076.35481 -2.417174388
## `sex:tch` -314.48509 -2.378188538
## glu 19.12792 2.300361215
## `bmi:ltg` 70.84372 2.259593599
## `hdl:glu` 560.73667 2.215087063
## `sex:tc` -15.98984 -1.999827883
## `bmi:glu` 11.21618 1.642182837
## `tc:ltg` -10009.29895 -1.458708736
## `map:glu` -65.99667 -1.318668568
## `sex:glu` 24.91435 1.137130088
## `sex:bmi` -14.04317 0.938191184
## `bmi:tch` -162.32294 0.902375176
## `ltg:glu` 172.80190 0.812160488
## `age:tc` -147.14797 -0.731836716
## `map:ldl` -916.96996 -0.694345338
## `bmi:tc` 21.50722 -0.665490953
## `tc^2` 11301.51242 -0.566494009
## `map:ltg` -455.17300 0.520230473
## `bmi:ldl` 92.41196 -0.366628908
## `hdl^2` 2298.16669 -0.298001012
## `sex:ltg` 97.47348 0.239873862
## `bmi:hdl` -101.63216 -0.230827419
## `ldl:hdl` 5004.48290 0.156320821
## `bmi^2` -57.66550 0.095112982
## `map:tc` 1003.12227 -0.048598374
## `tc:ldl` -17718.75773 0.033353836
## `age:tch` 450.27383 -0.007549981Now let’s look at out-of-sample prediction error, using the posterior median coefficient estimates:
pred_stan <- function (x) {
post_med_intercept + as.matrix(x) %*% post_med_slopes
}
pred_y <- pred_stan(test_d[,-1])
stan_pred_error <- sqrt(mean((test_d$y - pred_y)^2))
stan_mse_resid <- sqrt(mean((training_d$y - pred_stan(training_d[,-1]))^2))
plot(test_d$y, pred_y, xlab="true values", ylab="predicted values", main="test data")
abline(0,1)
Conclusions?
Our “sparse” model is certainly more sparse, and arguably more interpretable.
It has a root-mean-square prediction error of 55.7274879 on the test data, and 52.083831 on the training data.
This is substantially better than ordinary linear regression, which had a root-mean-square prediction error of 64.602275 on the test data, and a root-mean-square-error of 46.5588313 on the training data.
The sparse model is more interpretable, and more generalizable.