We use the api
dataset from package survey to illustrate estimation of a population mean from a sample using a linear regression model. First let’s estimate the population mean of the academic performance indicator 2000 from a simple random sample, apisrs
. Using package survey’s GREG estimator [@Sarndal1992], we find
library(survey)
data(api)
# define the regression model
model <- api00 ~ ell + meals + stype + hsg + col.grad + grad.sch
# compute corresponding population totals
XpopT <- colSums(model.matrix(model, apipop))
N <- XpopT[["(Intercept)"]] # population size
# create the survey design object
des <- svydesign(ids=~1, data=apisrs, weights=~pw, fpc=~fpc)
# compute the calibration or GREG estimator
cal <- calibrate(des, formula=model, population=XpopT)
svymean(~ api00, des) # equally weighted estimate
## mean SE
## api00 656.58 9.2497
## mean SE
## api00 663.86 4.1942
The true population mean in this case can be obtained from the apipop
dataset:
## [1] 664.7126
Note that the GREG estimate is more accurate than the simple equally weighted estimate, which is also reflected by the smaller estimated standard error of the former.
We can run the same linear model using package mcmcsae. In the next code chunk, function create_sampler
sets up a sampler function that is used as input to function MCMCsim
, which runs a simulation to obtain draws from the posterior distribution of the model parameters. By default three chains with independently generated starting values are run over 1250 iterations with the first 250 discarded as burnin. As the starting values for the MCMC simulation are randomly generated, we set a random seed for reproducibility.
The results of the simulation are subsequently summarized, and the DIC model criterion is computed. The simulation summary shows several statistics for the model parameters, including the posterior mean, standard error, quantiles, as well as the Rhat convergence diagnostic.
library(mcmcsae)
set.seed(21)
sampler <- create_sampler(model, data=apisrs)
sim <- MCMCsim(sampler, verbose=FALSE)
(summ <- summary(sim))
## llh_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## -1.10e+03 2.20e+00 -5.02e+02 4.07e-02 -1.11e+03 -1.10e+03 -1.10e+03 2.91e+03 1.00e+00
##
## sigma_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 6.06e+01 3.14e+00 1.93e+01 6.07e-02 5.57e+01 6.04e+01 6.61e+01 2.67e+03 1.00e+00
##
## reg1 :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## (Intercept) 777.821 24.804 31.36 0.46695 737.793 778.032 818.0877 2822 0.999
## ell -1.716 0.309 -5.55 0.00575 -2.224 -1.716 -1.2074 2890 1.000
## meals -1.744 0.276 -6.32 0.00504 -2.198 -1.746 -1.2904 3000 1.000
## stypeH -108.765 14.060 -7.74 0.25669 -131.772 -109.147 -85.6129 3000 0.999
## stypeM -59.444 12.316 -4.83 0.22486 -79.715 -59.410 -39.4082 3000 0.999
## hsg -0.695 0.435 -1.60 0.00831 -1.412 -0.703 0.0248 2743 1.001
## col.grad 1.225 0.499 2.46 0.01072 0.418 1.224 2.0599 2164 1.000
## grad.sch 2.214 0.499 4.43 0.00912 1.374 2.216 3.0150 3000 1.002
## DIC p_DIC
## 2217.807439 9.104826
The output of function MCMCsim
is an object of class draws
. Methods for the generic functions summary
, plot
, predict
, residuals
and fitted
are currently implemented for this class.
To compute a model-based estimate of the population mean, we need to predict the values of the target variable for the unobserved units. Let \(U\) denote the set of population units, \(s \subset U\) the set of sample (observed) units, and let \(y_i\) denote the value of the variable of interest taken by the \(i\)th unit. The population mean of the variable of interest is \[ \bar{Y} = \frac{1}{N}\sum_{i=1}^N y_i = \frac{1}{N}\left(\sum_{i\in s} y_i + \sum_{i\in U\setminus s} y_i \right)\,. \]
Posterior draws for \(\bar{Y}\) can be obtained by generating draws for the non-sampled population units, summing them and adding the sample sum. This is done in the next code chunk, where method predict
is used to generate draws from the posterior predictive distribution for the new, unobserved, units.
m <- match(apisrs$cds, apipop$cds) # population units in the sample
# use only a sample of 250 draws from each chain
predictions <- predict(sim, newdata=apipop[-m, ], iters=sample(1:1000, 250), show.progress=FALSE)
str(predictions)
## List of 3
## $ : num [1:250, 1:5994] 746 670 672 706 748 ...
## $ : num [1:250, 1:5994] 662 759 694 731 633 ...
## $ : num [1:250, 1:5994] 784 752 711 620 630 ...
## - attr(*, "class")= chr "dc"
samplesum <- sum(apisrs$api00)
summary(transform_dc(predictions, fun = function(x) (samplesum + sum(x))/N))
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 664.119 4.206 157.900 0.155 656.896 664.527 670.574 732.308 1.000
The result for the population mean can also be obtained directly (which is more efficient memory wise) by supplying the appropriate aggregation function to the prediction method:
summary(predict(sim, newdata=apipop[-m, ], fun=function(x, p) (samplesum + sum(x))/N,
show.progress=FALSE)
)
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 6.64e+02 4.20e+00 1.58e+02 7.78e-02 6.57e+02 6.64e+02 6.71e+02 2.91e+03 1.00e+00
For any linear model one can obtain the same result more efficiently by precomputing covariate population totals. Posterior draws for \(\bar{Y}\) are then computed as
\[ \bar{Y}_r = \frac{1}{N} \left( n\bar{y} + \beta_r'(X - n\bar{x}) + e_r\right)\,, \]
where \(e_r \sim {\cal N}(0, (N-n)\sigma_r^2)\), representing the sum of \(N-n\) independent normal draws. The code to do this is
n <- nrow(apisrs)
XsamT <- colSums(model.matrix(model, apisrs))
XpopR <- matrix(XpopT - XsamT, nrow=1)
predictions <- predict(sim, X=list(reg1=XpopR), var=N-n, fun=function(x, p) (samplesum + x)/N,
show.progress=FALSE)
summary(predictions)
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 6.64e+02 4.17e+00 1.59e+02 7.79e-02 6.57e+02 6.64e+02 6.71e+02 2.87e+03 1.00e+00
To compute weights corresponding to the population total:
sampler <- create_sampler(model, data=apisrs,
linpred=list(reg1=matrix(XpopT/N, nrow=1)),
compute.weights=TRUE)
sim <- MCMCsim(sampler, verbose=FALSE)
plot(weights(cal)/N, weights(sim)); abline(0, 1)
## [1] 663.8594
## linpred_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 663.850 4.217 157.407 0.077 656.883 663.964 670.629 3000.000 1.001
Note the small difference between the weighted sample sum of the target variable and the posterior mean of the linear predictor. This is due to Monte Carlo error; the weighted sum is exact for the simple linear regression case.
One possible way to deal with outliers is to use a Student-t sampling distribution, which has fatter tails than the normal distribution. In the next example, the formula.V argument is used to add local variance parameters with inverse chi-squared distributions. The marginal sampling distribution then becomes Student-t. Here the degrees of freedom parameter is modeled, i.e. assigned a prior distribution and inferred from the data.
sampler <- create_sampler(model, formula.V=~vfac(prior=pr_invchisq(df="modeled")),
linpred=list(reg1=matrix(XpopR/N, nrow=1)),
data=apisrs, compute.weights=TRUE)
sim <- MCMCsim(sampler, n.iter=5000, burnin=1000, verbose=FALSE)
(summ <- summary(sim))
## llh_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## -1081.809 8.610 -125.640 0.865 -1095.666 -1081.948 -1067.340 99.097 1.019
##
## sigma_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 50.159 4.642 10.806 0.422 42.689 50.096 57.899 120.752 1.017
##
## linpred_ :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 6.43e+02 3.92e+00 1.64e+02 3.93e-02 6.36e+02 6.43e+02 6.49e+02 9.91e+03 1.00e+00
##
## reg1 :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## (Intercept) 792.406 26.230 30.21 0.61985 748.817 792.670 834.5690 1791 1
## ell -1.496 0.359 -4.17 0.01079 -2.073 -1.503 -0.8972 1105 1
## meals -2.060 0.357 -5.77 0.01476 -2.669 -2.051 -1.4888 586 1
## stypeH -105.292 12.916 -8.15 0.13531 -126.366 -105.212 -83.8361 9111 1
## stypeM -56.992 11.096 -5.14 0.11264 -75.283 -56.985 -38.8345 9703 1
## hsg -0.676 0.455 -1.49 0.00526 -1.421 -0.674 0.0698 7459 1
## col.grad 1.001 0.486 2.06 0.00684 0.209 0.996 1.7966 5058 1
## grad.sch 2.114 0.467 4.52 0.00489 1.358 2.112 2.8990 9145 1
##
## vfac1_df :
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 8.817 5.705 1.546 0.783 3.686 7.077 20.569 53.136 1.042
## [[1]]
## [[1]][[1]]
## [1] 0.2696
##
## [[1]][[2]]
## [1] 0.2702
##
## [[1]][[3]]
## [1] 0.2854
## DIC p_DIC
## 2195.4577 31.8395
predictions <- predict(sim, newdata=apipop[-m, ], show.progress=FALSE,
fun=function(x, p) (samplesum + sum(x))/N)
summary(predictions)
## Mean SD t-value MCSE q0.05 q0.5 q0.95 n_eff R_hat
## 663.99 4.00 166.09 0.04 657.48 663.98 670.52 9981.56 1.00
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2342 0.9359 1.0695 1.0002 1.1181 1.1509