# SAS Macro Programs: `missrc`

$Version: 1.4 (6 Nov 2001)

**
Michael Friendly**

York University

### MLEs for incomplete n-way contingency tables

The `missrc` macro estimates cell probabilities in an n-way table with
ignorable missing data (missing completely at random [MCAR], or missing at
random [MAR]) on the table variables. For example, in a longitudinal
survey, some respondents may have missing responses on one or more
occasions, and it is desired to estimate the cell probabilities using
the available information. The results are equivalent to the
use of the EM algorithm.
### Method

Lipsitz etal (1998) show
that in this case the cell probabilities may be estimated as a
specially constructed Poisson generalized linear model with a structured
design matrix and an offset containing various marginal totals.
The equivalent Poisson GLM is
*
E(f) = X p + g
*

where **f** is a vector containing the cell frequencies, **X** is the design matrix, **p** is the vector of cell
probabilities, and **g** is a vector of marginal totals used
as an offset variable.
Friendly (1999)
generalizes their method to n-way tables.
The MISSRC macro constructs this design matrix and offset variable.
It then estimates
the cell probabilities using PROC GENMOD, and returns a table with
the estimates, their standard errors, and fitted cell frequencies.

## Usage

The `missrc` macro is called with keyword parameters. Only the VAR=
parameters is required. The arguments may be listed within
parentheses in any order, separated by commas. For example:
%missc(var=R C, count=count);

### Parameters

Default values are shown after the name of each parameter.
- DATA=_LAST_
- Specifies the name of the input data set to be analyzed.
If omitted, the most recently created data set is used.
- VAR=
- Specifies the names of the table variables. In this version,
all VAR= variables must be numeric, with non-negative
integer levels. The missing level must have the SAS
missing value,
`.`
- COUNT=COUNT
- Specifies the name of the variable holding the cell frequencies.
If not specified, COUNT=COUNT is assumed.
- OUT=CELLS
- Specifies the name of the output data set containing estimated
cell probabilities, standard errors, etc.
- DESIGN=DESIGN
- Specifies the name of the output design matrix data set

### Example

Little & Rubin (1987, p. 183) gave the following table:
| C1 C2 Missing
--------+---------------------
R1 | 100 50 30
R2 | 75 75 60
Missing | 28 60

Create a SAS data set, as follows. The frequency variable is named COUNT,
the default for MISSRC. Note that missing values for R and C are specified
as `.`
data little;
input R C count @@;
cards;
1 1 100 1 2 50 1 . 30
2 1 75 2 2 75 2 . 60
. 1 28 . 2 60
;
%include macros(missrc); *-- or include in an autocall library;
%missrc(data=little, var=R C);

The following output data set is produced. The variable P is the observed
cell probability, ESTIMATE is the MLE, and FITTED is the estimated cell
frequency.
R C COUNT P PARM ESTIMATE STDERR FITTED
. 1 28 . . . .
. 2 60 . . . .
1 . 30 . . . .
1 1 100 0.33333 P11 0.27947 0.022310 133.589
1 2 50 0.16667 P12 0.17402 0.020978 83.184
2 . 60 . . . .
2 1 75 0.25000 P21 0.23872 0.022660 114.108
2 2 75 0.25000 P22* 0.30778 0.025298 147.120

### See also

powerrxc Power analysis for Chi-sqare tests of independence

### References

Friendly, M. (1999). Note on ``Obtaining the maximum
likelihood estimates in incomplete R x C Contingency tables...'',
J. Computational. and Graphical Statistics, 8,
in press.
Lipsitz, S. R., Parzen, M. and Molenberghs, G. (1998). ``Obtaining the maximum
likelihood estimates in incomplete R x C Contingency tables using a
Poisson generalized linear model''. J. Computational. and Graphical Statistics,
7, 356--376.