How to Use this Program:

First enter your Maxiumim Possible raw score in the Maximum Possible Score box below. If you had a 100 word list you would enter 100, if you had a 25 word list you would enter 25, if you had a 10 word list enter 10, etc.

Second, in the Raw Scores box below enter your individual raw scores, pressing the enter key after each score entered. i.e. one score per line. You should end up with a single column of numbers (one score per line) in the Raw Scores box. DO NOT ENTER PERCENTAGES, enter the raw scores. If your max score was 10 and the first score was 8 correct, you would enter 8 then press enter. If the next score was 3 correct you would enter 3 on the second line and then press enter, etc.

Third, press the Submit button, a new page will come up. The top table will show your raw scores, max scores, percentage scores, and rationalized arcsine unit (rau) scores. The rau scores are repeated in a column below the table in the same order to facilitate copying and pasting this data into other programs/documents.

Maximum Possible Score:

Raw Scores (Not Percentages):

Background:
One of the requirements of inferential statistics such as ANOVA is that the dependent or repsonse variables are quantitative (interval scale) (Agresti & Finlay, 1997). Specifically, interval data assumes that intervals between different pairs of values are the same, i.e. the difference between 15 and 20 is the same as the difference between 90 and 100. Thornton & Raffin (1978) demonstrated that the variability among word recognition scores can be described using the binomial model. The authors demonstrated that critical differences (CD) between two scores vary, so that a single CD interval could not be used to find statistically significant differences for a range of scores. Tables were provided so that readers could look up a given percentage score for either a 25 or 50 word list and then find the critical difference interval for that score. The heterogeneity of CD ranges were visually demonstrated by Thibodeau (1999) in a useful chart that not only facilitates the identification of CD intervals for a given score, but also allows the user to determine if a PBMax score is appropriate for a given PTA score.

Figure 1. Critical Differences for lists containing 50 words.
Note: Adapted from Thornton & Raffin (1978) and Thibodeau (2000).

To illustrate the Thornton & Raffin (1978) data for a list of 50 words and the visual methodology after Thobodeau (2000) please see Figure 1. Statistically significant differences between two word recognition scores are determined by finding your first score along the X axis. Draw an imaginary line directly up from that point. If your second score plotted from the Y axis falls outside of the orange area it is significantly different. Readers are encouranged to review the original chart in Thibodeau (2000) for use in clinical practice as the chart is more detailed and user friendly.

The reader should note from Figure 1 that the CD range changes as a function of the initial score. The function is fairly symmetrical, in that, the CD range is wider for the middle scores and narrower for the more extreme scores. This aspect of the data violates the concept of interval data, in that, differences between scores are not the same. As shown in Figure 1 the CD range for 92% is 78-98% a 20% range, whereas the CD range for 50% is 32-68% a 36% range.

One way to think of the function of the arcsine transform is to modify the scores to help account for these differences. Studebaker's (1985) "rationalized" arcsine transform converts the scores into rationalized arcsine units (rau). Compared to other arcsine transforms the advantage of the rau conversion is that the rau are closer to the original percentage scores to help make the rau more easily interpreted when comparing to the original data. To give a visual example of what the rau transform does, visualize Figure 1 and imagine that the tips of each end of the CD area were expanded so that the width of the figure was constant. The CD range would look less like an oviod and more like a rectangle. The function of the rau conversion is to try and minimize the changes along the CD range. Numerically it makes the largest changes above 85% and below 15% (see Figure 1 in Studebaker (1985)). As a numeric example when the CD ranges for the two scores given previously (92% CD range 78-98% and 50% CD range 32-68%) are transformed into rau, the new ranges in rau are now 77.02-107.21 rau and 33.23-66.77 rau. The CD ranges are now 30.2 and 33.54 rau respectively, which reduces the differences between the two CD ranges from 16 to 3.

If you would like further information on this an other arcsine transforms, please see articles in Works Cited list below.

Program Information:
RATARC Online was written in Perl to perform the Studebaker (1985) "Rationalized" Arcsine Transform. We have a "RATARC" program in our lab which runs on DOS, but with Microsoft having moved away from DOS, this version was created to run in a non Microsoft environment. Additionally, we wanted to make it accessible to others via the internet. The program uses Equation 3 and 7 from the Studebaker (1985) article which takes into account the total number of presentations given to a subject. You may verify this program by entering values to compare to the results in Table 1 from Studebaker (1985).

I would appreciate any and all comments/suggestions/criticisms regarding information on this page.

Works Cited:
Agresti, A., & Finlay, B. (1997). Statistical Methods for the Social Sciences (3rd ed.). Upper Saddle River: Prentice Hall. Textbook @ Amazon
Thibodeau, L. M. (2000). Speech Audiometry. In R. Roeser, Valente, M., Hosford-Dunn, H. (Ed.), Audiology: Diagnosis. New York: Thieme. Textbook @ Amazon
Thornton, A. R., & Raffin, M. J. (1978). Speech-discrimination scores modeled as a binomial variable. J Speech Hear Res, 21(3), 507-518. Medline Abstract.
Studebaker, G.A. (1985) A "Rationalized" Arcsine Transform. Journal of Speech and Hearing Research, 28, 455-462. Medline Abstract.