This example illustrates how PROC MCMC treats missing at random (MAR) data. For a short overview of missing data problems, see the section Handling of Missing Data.
Researchers studied the effects of air pollution on respiratory disease in children. The response variable (y
) represented whether a child exhibited wheezing symptoms; it was recorded as 1 for symptoms exhibited and 0 for no symptoms
exhibited. City of residency (x1
) and maternal smoking status (x2
) were the explanatory variables. The variable x1
was coded as 1 if the child lived in the more polluted city, Steel City, and 0 if the child lived in Green Hills. The variable
x2
was the number of cigarettes the mother reported that she smoked per day. Both the covariates contain missing values: 17
for x1
and 30 for x2
, respectively. The total number of observations in the data set is 390. The following statements generate the data set air
:
title 'Missing at Random Analysis'; data air; input y x1 x2 @@; datalines; 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 1 7 0 0 8 0 1 10 0 1 9 0 0 0 1 1 6 0 1 10 0 1 12 0 0 . 0 0 0 0 1 0 0 1 7 1 1 15 0 0 8 0 0 0 1 1 0 1 0 6 0 0 0 1 1 11 0 0 0 1 0 0 1 0 5 0 0 8 0 0 0 0 1 9 ... more lines ... 0 0 11 0 0 0 0 0 6 0 0 12 0 0 10 0 1 10 0 1 11 0 0 9 1 0 11 0 1 7 0 0 7 0 0 0 0 . 11 1 1 6 0 0 8 0 0 0 0 1 12 0 0 0 0 1 0 1 1 8 0 0 0 0 1 11 0 1 0 0 1 8 0 . 0 1 0 0 1 1 10 0 . 4 1 1 16 0 . 13 ;
Suppose you want to fit a logistic regression model for whether the subject develops wheezing symptoms with density for the subjects as follows:
Suppose you specify a joint distribution of x1
and x2
in terms of the product of a conditional and marginal distribution; that is,
where could be a logistic model and could be a Poisson distribution that models the following counts:
The researchers are interested in interpreting how the odds of developing a wheeze changes for a child living in the more polluted city. The odds ratio can be written as the follows:
Similarly, the odds ratio for the maternal smoking effect can be written as follows:
The following statements fit a Bayesian logistic regression with missing covariates:
proc mcmc data=air seed=1181 nmc=10000 monitor=(_parms_ orx1 orx2) diag=none plots=none; parms beta0 -1 beta1 0.1 beta2 .01; parms alpha10 0 alpha11 0 alpha20 0; prior beta: alpha1: ~ normal(0,var=10); prior alpha20 ~ normal(0,var=2); beginnodata; pm = exp(alpha20); orx1 = exp(beta1); orx2 = exp(beta2); endnodata; model x2 ~ poisson(pm) monitor=(1 3 10); p1 = logistic(alpha10 + alpha11 * x2); model x1 ~ binary(p1) monitor=(random(3)); p = logistic(beta0 + beta1*x1 + beta2*x2); model y ~ binary(p); run;
The PARMS
statements specify the parameters in the model and assign initial values to each of them. The PRIOR
statements specify priors for all the model parameters. The notations beta:
and alpha:
in the PRIOR
statements are shorthand for all variables that start with "beta
," and "alpha
," respectively. The shorthand notation is not necessary, but it keeps your code succinct.
The BEGINNODATA
and ENDNODATA
statements enclose three programming statements that calculate the Poisson mean (pm
) and the two odds ratios (ORX1
and ORX2
). These enclosed statements are independent of any data set variables, and they are run only once per iteration to reduce
unnecessary observation-level computations.
The first MODEL
statement assigns a Poisson likelihood with mean pm
to x2
. The statement models missing values in x2
automatically, creating one variable for each of the missing values, and augments them accordingly. By default, PROC MCMC
does not output analyses of the posterior samples of the missing values. You can use the MONITOR=
option to choose the missing values that you want to monitor. In the example, the first, third, and tenth missing values
are monitored.
The P1 assignment statement calculates . The second MODEL
statement assigns a binary likelihood with probability p1
and requests a random choice of three missing data variables of x1
to monitor.
The P assignment statement calculates in the logistic model. The third MODEL statement specifies the complete data likelihood function for Y.
Output 61.10.1 displays the number of observations read from the DATA= data set, the number of observations used in the analysis, and the "Missing Data Information" table. No observations were omitted from the data set in the analysis.
The "Missing Data Information" table lists the variables that contain missing values, which are x1
and x2
, the number of missing observations in each variable, the observation indices of these missing values, and the sampling algorithms
used. By default, the first 20 observation indices of each variable are printed in the table.
There are 30 missing values in the variable x2
, and 17 in x1
. Internally, PROC MCMC creates 30 and 17 variables for the missing values in x2
and x1
, respectively. The default naming convention for these missing values is to concatenate the response variable and the observation
number. For example, the first missing value in x2
is the fourteenth observation, and the corresponding variable is x2_14
.
Output 61.10.2 displays the summary and interval statistics for each parameter, the odds ratios, and the monitored missing values.
Output 61.10.2: Posterior Summary and Interval Statistics
Missing at Random Analysis |
Posterior Summaries and Intervals | |||||
---|---|---|---|---|---|
Parameter | N | Mean | Standard Deviation |
95% HPD Interval | |
beta0 | 10000 | -1.3732 | 0.2078 | -1.7909 | -0.9676 |
beta1 | 10000 | 0.4797 | 0.2387 | 0.0268 | 0.9491 |
beta2 | 10000 | 0.0156 | 0.0227 | -0.0265 | 0.0642 |
alpha10 | 10000 | -0.2166 | 0.1422 | -0.4662 | 0.0874 |
alpha11 | 10000 | 0.0126 | 0.0201 | -0.0267 | 0.0521 |
alpha20 | 10000 | 1.5635 | 0.0235 | 1.5199 | 1.6105 |
orx1 | 10000 | 1.6627 | 0.4094 | 0.9205 | 2.4493 |
orx2 | 10000 | 1.0160 | 0.0231 | 0.9739 | 1.0663 |
x2_14 | 10000 | 4.9022 | 2.2083 | 1.0000 | 9.0000 |
x2_50 | 10000 | 4.8924 | 2.1626 | 1.0000 | 9.0000 |
x2_90 | 10000 | 4.8263 | 2.0816 | 1.0000 | 8.0000 |
x1_296 | 10000 | 0.4160 | 0.4929 | 0 | 1.0000 |
x1_304 | 10000 | 0.4460 | 0.4971 | 0 | 1.0000 |
x1_373 | 10000 | 0.4443 | 0.4969 | 0 | 1.0000 |
The odds ratio for x1
is the multiplicative change in the odds of a child wheezing in Steel City compared to the odds of the child wheezing in
Green Hills. The estimated odds ratio (ORX1
) value is 1.6736 with a corresponding 95% equal-tail credible interval of . City of residency is a significant factor in a childâ€™s wheezing status. The estimated odds ratio for x2
is the multiplicative change in the odds of developing a wheeze for each additional reported cigarette smoked per day. The
odds ratio of ORX2
indicates that the odds of a child developing a wheeze is 1.0150 times higher for each reported cigarette a mother smokes.
The corresponding 95% equal-tail credible interval is . Since this interval contains the value 1, maternal smoking is not considered to be an influential effect.