A BASIC program to determine regions of significance using the Johnson-Neyman technique. CHARLES T. SCIALFA. The Pennsylvania State University, ...
Behavior Research Methods, Instruments, & Computers 1987, 19 (3), 349-352
A BASIC program to determine regions of significance using the Johnson-Neyman technique
tween groups. If one or both regions of significance are in the range of theoretical or applied interest, then the researcher can make more explicit statements concerning the person-in-situation (Cronbach, 1957)relationship CHARLES T. SCIALFA The Pennsylvania State University, University Park, Pennsylvania manifest in his/her data. Although Borich, Godbout, and Wunderlich (1976) In many experimental designs, the researcher's intent have developed mainframe programs for conducting the is to determine the relationship between some criterion Johnson-Neyman tests, to date noneof thecommonly used and two independent factors, one of which is continuous packages (e.g., SPSS-X, SAS, BMDP) include algorithms and the second categorical (e.g., representing indepen- for establishing regionsof significance. The BASIC prodent groupsor treatment conditions). The analysis of data gram described here (see Appendix) allows the user to from such designs involves the comparisonof regression quickly determine the point of intersection, as well as slopesacross levelsof the categorical variable, where the regions of significance, from ATI data. The user is refull model being tested allows for group differences in quired only to provide (for each of two groups or condiboth slopes and intercepts (Huitema, 1980; Pedhazur, tions) the predictor mean, standard deviation, sample size, 1982; Rogosa, 1980). This test is referred to as a test of residual sums of squares, and least squares estimates for homogeneity of regression, or the aptitude X treatment slope and intercept parameters. These data can be most interaction (ATI). As with the more common factorial easily obtainedby conductingsimple regressionanalyses model involving two discrete independent factors, after for each group, using either large-scale software or any obtaininga significant ATI, one would typically conduct of a variety of microcomputer-based statistical packages follow-up tests, analogous to tests of simple effects, in (e.g., HSD Resources Regress Il, MINITAB). The only order to specify the natureof the interaction. One applica- additional information necessary at inputis the tabled critble test of significance involves establishing regions of ical F with 1 and N -4 df, where N is the combined samsignificance using the Johnson-Neyman technique (John- ple size. son & Neyman, 1936). At output, the user is provided with three values. The Assume, for example, that an investigator has examined first is the point of intersection, or the value along the the relationship between classroom performance and need covariate continuum at which the two groups exhibit for achievement (nAch), measured as continuous varia- equivalent performance. Also included at output are the bles, under both traditional and self-directed teaching two regions of significance ("pick-a-point" or simultanestyles. Data from this hypothetical study are shown in ous, depending on user preference), at which there exist Figure I, where performance is plotted as a function of significant group differences for that covariate value. nAch for both treatment groups. It can be seen that there Again, if one or both of these regions are within a range exists a disordinal ATI, wherein slopes relating nAch that is of theoretical or applied interest, the user is better to performance differ for the two conditions. Assuming able to determine person-in-situation relationships. that the interaction is significant, one might wish to The regionsof significance establishedusing this techdetermine points along the nAch continuum where there niqueare nonsimultaneous regions. That is, these regions are significantperformance differences between the two are appropriate when one wishesto make inferences congroups. The Johnson-Neyman technique accomplishes this goal Region I Region 2 Region 3 by establishing regions of significance beyond which the groups display criterion differences. In Figure 1, Self-Directed Region 1 indicates the nAch level at which performance in the traditional group is superior, whereas in Region 3 the self-directed group is superior. Obviously then, in Region 2 performance does not differ significantly beThis work was supportedin part by a grant from the Gerontological Education, Research, and ServicesProgram(GERAS), The University of Notre Dame, Notre Dame, IN 46556, and in part by National Institute on Aging Grant AGOOllO--Q2. The authorwishes to express many thanks to Scott MaxweU for providing the motivation for this project as well as insightful comments concerning the manuscript. Reprint requests should be addressed to the author, The Pennsylvania StateUniversity, Collegeof HumanDevelopment, Amy Gardner House, University Park, PA 16802.
Traditional
nAch
Figure 1. Performanc:eas a function of nAch for traditional and self-directed teaching conditions, with regions of signific:ance.
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Copyright 1987 Psychonomic Society, Inc.
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ceming group differences at a specific point along the predictor continuum. However, it does not follow that one can state with 95 % confidence that there exist group criterion differences at all points within the regions. Potthoff (1964) described a modification of the JohnsonNeyman technique for establishing simultaneous regions of significance. His modification can be easily implemented with the present program by inputting a critical F of 2F(2, N-4). Finally, the program is restricted to the analysis of ATI data where there are only two groups and one continuous predictor. This is not a limitation of the technique, because extensions to more complex independent-groups designs are available (Abelson, 1953). The author is presently working to develop BASIC programs that will treat data from more complex ATI designs.
REFERENCES ABELSON, R. P. (1953). A noteon the Neyman-Johnson technique. Psychometrika, 18, 213-218. BORlCH, G. D., GODBOUT, R. C., & WUNDERLICH, K. W. (1976). The analysisofaptitude-treatment interactions: Computer programsand calculations. Chicago: International Educational Services. CRONBACH, L. J. (1957). The two disciplinesin scientificpsychology. American Psychologist, 12, 671-684. HUITEMA, B. E. (1980). Analysisofcovariance and alternatives. New York: Wiley. JOHNSON, P.O., & NEYMAN, J. (1936). Testsof certain1inear hypotheses and theirapplication to someeducational problems. Statistical Research Memoirs, 1, 57-93. PEDHAZUR, E. J. (1982). Multiple regression in behavioral research (2nd ed.). New York: Holt, Rinehart & Winston. POTTHOFF, R. F. (1964). On the Johnson-Neyman techniqueand some extensions thereof. Psychometrika, 29, 241-256. ROGOSA, D. (1980). Comparing non-parallel regression lines. Psychological Review, 88, 307-321.
APPENDIX Listing for Program to Determine Regions of Significance Using the Johnson-Neyman Technique 100 110 120 130 140 150
REM
*••••••••••••••••••••••••••••••••
REM JOHNSON-NEYMAN TECHNIQUE PROGRAM REM •••••••••••••••••••••••••••••••••
REM REM TIllS PROGRAM TAKESUSERINPUT REM ANDCALCULATES POINTOF INTERSECI10N 160 REM FOR 2 REGRESSION LINES. 170 REM TIffiN DETERMINES REGIONS OF SIGNIFICANCE 180 REM USINGJOHNSON-NEYMAN TECHNIQUE 190 REM ffiM AND APPLEUSERS SHOULD 200 REM ADJUSTCODEAS INDICATED IN 210 REM SUBROUTINES 2000, AND 2200
220 230 240 250 260 270 280 290 300
310 320 330 340 350 360 370 380 390 400 410 420 430
440 450
460 470 480 490 500 510 520 530 540 550
560 570 580 590
600 610
REM ••••••••••••••••••••••••••••••••
GOSUB 2000 REM SUBROUTINE AT 2000 ALLOWS ffiM OR APPLESCREENCLEAR PRINT" "; PRINT "•••••••••"'• ••• *•••*••••••••••••••••
PRINT" PRINT'" PRINT" PRINT'" PRINT" PRINT". PRINT" PRINT'" PRINT"
"; ";
." JOHNSON-NEYMAN
"; TECHNIQUE "; ";
."
."
11
."
PRINT "***********************************" FOR 1= 1 TO 3000: NEXTI: REM PAUSETO READFRAME GOSUB 2000
PRINT"TIllS PROGRAM ASKSYOUFOR" PRINT"INFORMATION CONTAINED ON THE" PRINT"OUTPUTFROMREGRESSION ANALYSES" PRINT"IN MOSTSTATISTICAL PACKAGES" PRINT"(E.G., SPSSX,SAS,BMDP), AND" PRINT"DETERMINES THE POINTOF INTERSECI10N" PRINT"AND REGIONS OF SIGNIFJCANCE USING" PRINT"THEJOHNSON-NEYMAN TECHNIQUE." PRINT PRINT"THE PROGRAM IS RESTRICTED TO THE" PRINT"TWO-GROUP, SIMPLEREGRESSION CASE." PRINT"IN ORDERTO USE THE PROGRAM" PRINT"YOUNEEDTO PROVIDE INFORMATION" PRINT"THATCAN BE MOSTEASILY OBTAINED" PRINT"BY CONDUCI1NG SEPARATE SIMPLE" PRINT"REGRESSION ANALYSES FOR BOTHGROUPS." PRINT PRINT"PRESS ANY KEYTO CONTINUE." GOSUB 2200: REM ffiM OR APPLEKEYSTROKE CHECK GOSUB 620: REM INPUTDATAROUTINE GOSUB 1420: REM CALCULATIONS ROUTINE GOSUB 1620: REM OUTPUTROUTINE END
JOHNSON-NEYMAN TECHNIQUE APPENDIX (Continued) 620 REM ***************************** 630 REM INPUTDATA ROUTINE 640 REM ***************************** 650 GOSUB 2000 u. 660 PRINT" 670 PRINT
"*********************"
1440 REM
*****************************
680 PRINT" ; 690 PRINT"* FORGROUP I *" 700 PRINT" "; 710 PRINT u*****.***.*••••••••••" 720 PRINT 730 INPUT"WHATIS THE SAMPLE SIZE?";NI 740 IFNI < =OTHEN PRINT"IIl..EGALVALUE!": GOT0730 750 INPUT"WHATARETHERESIDUAL SUMSOF SQUARES? ";SI 760 IF SI < =OTHEN PRINT"ILLEGAL VALUE!": GOT0750 770 INPUT"WHATIS THEPREDICfORMEAN?";MI 780 INPUT"WHATIS THEPREDIcroR STANDARD DEVIATION? ";VI 790 IFVI < = o THEN PRINT"ILLEGAL VALUE!": GOTO780 800 INPUT"WHATIS THEREGRESSION WEIGlIT(UNSTANDARDIZED)? ";BI 810 INPUT"WHATIS THE INTERCEPT? ";11 820 GOSUB2000 830 PRINT"INPUTDATAFORGROUP I" 840 PRINT 850 PRINT"SAMPLE SIZE = ";NI 860 PRINT"RESID. SUMSOF SQUARES = ";SI 870 PRINT"PREDICfORMEAN= ";MI 880 PRINT"PREDICfOR S.D. = ";VI 890 PRINT"REGRESSION WEIGlIT= ";BI 900 PRINT"INTERCEPT = ";11 910 PRINT: PRINT 920 PRINT"IS TIllS INFORMATION CORRECT?" 930 PRINT"ENTER'Y'ESOR 'N'O" 940 PRINT 950 GOSUB2200 960 IF A$ = "N"OR A$ = "n" THEN PRINT"TRYAGAIN": FOR I = I TO 3000: NEXTI: GOTO650 970 IF A$ < > "Y"OR A$ < > "y"THEN GOTO940 1000 GOSUB2000 1010 PRINT" «, 1020 PRINT"***************************" 1030 PRINT" u, 1040 PRINT "* FORGROUP2 *" 1050 PRINT" "; 1060 PRINT "••••••••••• *••• *.**••••••••" 1070 PRINT 1080 INPUT"WHATIS THE SAMPLE SIZE?";N2 1090 IFN2 < = o THEN PRINT"ILLEGAL VALUE!": GOTO1080 1100 INPUT"WHATARETHERESIDUAL SUMSOF SQUARES? ";S2 1110 IF S2 < = 0 THEN PRINT"ILLEGAL VALUE!": GOTO1100 1120 INPUT"WHATIS THEPREDIcroR MEAN?";M2 1130 INPUT"WHATIS THEPREDlcroR STANDARD DEVIATION? ";V2 1140 IF V2 < = 0 THEN PRINT"ILLEGAL VALUE!": GOTO1130 1150 INPUT"WHATIS THEREGRESSION WEIGHT? ";B2 1160 INPUT"WHATIS THE INTERCEPT? ";12 1170 GOSUB2000 1180 PRINT"INPUTDATAFORGROUP2" 1190 PRINT 1200 PRINT"SAMPLE SIZE = ";N2 1210 PRINT"RESID. SUMSOF SQUARES = ";S2 1220 PRINT"PREDIcroR MEAN= ";M2 1230 PRINT"PREDICfOR S.D. = ";V2 1240 PRINT"REGRESSION WEIGHT= ";B2 1250 PRINT"INTERCEPT = ";12 1260 PRINT 1270 PRINT 1280 PRINT"IS TIllS INFORMATION CORRECfT 1290 PRINT"TYPE'Y'ESOR 'N'O" 1300 PRINT 1310 GOSUB2200 1320 IF A$ = "N"OR A$ = "n" THEN PRINT"TRYAGAIN": FOR I = I TO 3000: NEXTI: GOTO1000 1340 IF A$ < > "Y" OR A$ < > "y"THEN GOTO1300 1360 GOSUB2000 1370 PRINT"FINALLY, WHATIS THE CRITICAL F' 1380 PRINT"FOR ALPHA=.05 ANDD.F.=" 1390 INPUT"I ANDTOTALN-4?";F 1400 IF F < = 0 THEN PRINT"ILLEGAL VALVE!": GOTO1390 1410 RETIJRN 1420 REM ***************************** 1430 REM CALCULATIONS ROUTINE 1450 CROSS= (11 - 12)I (B2 - BI): REM POINTOF INTERSECTION 1460 SS = SI + S2: REM POOLEDSUMSOF SQUARES RESIDUAL 1470 N=(NI +N2):REM TOTALN 1480 VI = «VI A 2) * (NI - 1»:V2 = «V2 A 2) * (N2 - I»: REM PREDICTOR SUMS OF SQUARES
351
352
SCIALFA APPENDIX (Continued) 1490 A=(I/VI)+(1/V2) 1500 A = A * SS * (- F I (N - 4» 1510 A=A+«BI-B2)A2) 1520 B = «MIl VI) + (M21 V2» 1530 B = B * SS * (F / (N - 4» 1540 B = B + «11- 12) * (Bl- B2» 1550 C = «N / (NI * N2» + «MI A2) / VI) + «M2 A2) / V2» 1560 C = C * SS * (- F / (N - 4» 1570 C = C + «11 - 12) A2) 1580 Z = ( SQR «B A2) - (A * C») 1590 ZIANS = ( - B + Z) / A 1600 Z2ANS= (- B -Z)/ A 1610 RETIJRN 1620 ~ ***************************** 1630 REM OUTPUT ROUTINE 1640 REM ***************************** 1650 GOSUB2000 1660 PRINT"THEPOINTOF INTERSEcnON FOR" 1670 PRINT"THElWO FUNcnONS = ";CROSS 1680 PRINT: PRINT 1690 PRINT"THEGROUPS ARE SIGNIFICANTI..Y DIFFERENT' 1700 PRINT"ABOVE ";ZIANS;" AND BELOW ";Z2ANS 1710 RETIJRN 2000 REM APPLEUSERSDELETE LINE2030 2010 REM ffiM USERSDELETE LINE2020 2020 HOME 2030 CLS 2040 RETIJRN 2200 REM APPLEUSERSDELETE LINE2230 2210 REM ffiM USERSDELETE LINE2220 2220 GET A$:IF A$ = "" THEN GOTO2220 2230 INKEY$: IF A$=""THEN GOTO2230 2240 RETIJRN
(Revision accepted for publication October 2, 1986.)
