MATH-MATIC(ID:435/mat008)

Mathematically oriented autocode 

• UNIVAC Remington Rand
• UNIVAC II Sperry Univac

Illustrative Problem
The purpose of the example below is to show how automatic programming can simplify the process of coding a statistical application for a computer. The MATH-MATIC compiler system is used to develop a program which will compute the Snedecor F statistic of the one-way Analysis of Variance.17 Chart 749 gives the pseudo-code program for this computation.
This program is based on formula (1) which is fairly well-known.

The pseudo-code is entered on the appropriate tape reel. The process of compilation results in a running program which will process the input data, compute F} and display it on another tape reel. The operator types in the values of N and K at the control console before the processing begins.
The sequence of operations during the processing of the data is a successive series of accumulations to sums of values and sums of squares. These accumulations as defined in the program are:

Using these definitions the formula for F reduces to (2)

The pseudo-code program of Chart 749 is developed in such a manner that as the input data are processed, their values and the squares of their values are added to the S sums in the desired sequence. When all accumulations are complete, F is computed by formula (2). The pseudo-code program of Chart 749 is relatively easy to follow with a small amount of additional explanation which is given below.

The data to be reduced to an F statistic (the Xt/s) are entered onto the input data tape reel by group. The end of the data is indicated by twelve Z's (ZZZZZZZZZZZZ) which is called the sentinel18
The first seven statements or "sentences" in the program set up the condition for processing each unit of input data. Sentence (1) enables the operator to type in the values of N and K. Sentences (2), (3), and (4) insure that the accumulations of Sij &, and S$ all start at zero at the outset.
Sentence (5) sets up a "loop" or major subroutine which is repeated for each of the K groups starting with the first group and advancing in stages of one group at a time. Sentence (6) clears So to zero at the beginning of each of these K loops. Sentence (7) sets up a loop for reading each of the N observations within each group.
Sentence (8) is the beginning of the actual data reduction. It enables the reading (from the input data tape reel) of the next value and testing to determine whether it is the sentinel. If it is not the sentinel, accumulations are made to So and Si by sentences (9) and (10). When all of the N values for a group have been entered and reduced, accumulations are made for #2 and $3 by Sentences (11) and (13). This process is repeated for each of the K groups.
If the 12 Z sentinel is the item, this indicates the end of the data. The next step is to compute F. This is all performed under the control of sentence (13). This is a remarkable instruction because it is simply a statement of formula (2).19 It is worth noting that on the basis of this single instruction three multiplications, four subtractions, and three divisions are all performed. In addition, all significant digits are properly positioned and all decimal points evaluated with no additional coding.
Sentence (14) instructs the control typewriter to print out the values of the statistic F, and the constants K and N. Sentence (15) is the instruction that stops all calculations and rewinds the magnetic tapes.
This problem required about four working days of preparation. This included learning to use the compiler, planning input data layout, and flow charting. Only two hours were required to prepare the pseudo-code. The 15 sentences of pseudo-code correspond to a running program of approximately 1,000 computer code instructions.20
About four minutes of compiling time was required (i.e., for the computer to prepare the running program). The running time for the computation itself was negligible. The program ran without errors the first time it was tried. Thus, no debugging time was required.
For a set of test data with 30 observations and K = 3 and N = 10, the running time was about three seconds. Extrapolation indicates that it would take a very large number of observations to require a substantial running time, such as a minute..
  

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Footnotes
17 This method is frequently used in analysis of the results of experiments for which N random values are observed in each of K different groups. By reducing the data to an F statistic and computing its value to use in an BF" probability distribution, a judgement is made as to whether some cause or pure chance alone might have accounted for the variations between grouped values. See Snedecor's Statistical Methods for a number of illustrations of the uses of this statistic  
18 Sentinels are needed in computer operation to indicate end of data, end of file and of tape reel, or any other point of differentiation between one type of data and another. Twelve Z'a are often employed for this purpose because it is unlikely that anyone would use them for data or any other purpose in programming.
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