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360 References

Index

A

Abdi, H., 324

Abrahamowicz, M., 244

Abramowitz, M., 129, 140, 163, 305

Achenbach, T.M., 74

ACT, 48, 149

Aitchison, J., 226

Aitkin, M., 268

Akaike criterion, 4

Akaike, H., 323

Al-Karni, A., 271

Allen, N.L., 48, 57

Alternative kernels, 13, 159–173

Anchor test, 21, 24–25, 27–31, 34–41, 71, 139,

145, 152, 160, 175, 181, 183, 186,

193–195, 199, 218–220, 225–229, 231,

233, 234, 281–284, 288, 289, 291–296,

319, 326, 333

Andersen, E.B., 312

Angle bisector method, 89, 94–98

Angoff, W.H., 22, 24, 26, 29, 33, 46, 89, 94,

181, 186, 225

Armstrong, R.D., 329

Asymptotic distribution, 170, 321

Asymptotic expansion, 261–280

Asymptotic standard deviation, 125, 132, 133,

138, 139

B

Baker, F.B., 254, 271

Balakrishnan, N., 162

Bandwidth, 13, 132, 139, 148, 156, 160, 161,

165–166, 168–171, 173, 185, 193–198,

200, 245, 250, 254, 319, 323, 324

Ban, J.-C., 47, 58

Barron, S., viii

Bayesian nonparametrics, 176

Bayes’ theorem, 176, 179

Bayley, N., 72, 75

Beguin, A.A., 8, 297–316

Bennett, R.E., 44

Berkeley Growth Study, 72, 75

Bernstein polynomial prior, 178–180

Bertenthal, M.W., 22

Best practices, 21–42

Beta distribution, 176, 177, 179, 181

Bifactor model, 2, 8, 9

Billeaud, K., 51

BILOG-MG, 254, 301

Birnbaum, A., 300

Bishop, Y.M.M., 4

Bivariate Bernstein-Dirichlet process, 177

Bock, R.D., 56, 230, 254, 261, 262, 268, 301, 316

Bootstrap, 247, 256, 261, 310–313

Bowles, R., 74

Box and Jenkins models, 330

Box, G.E.P., 3, 330, 332, 339

Bozdogan, H., 4

Bradlow, E.T., 8

Bradway, K.P., 77

Bradway–McArdle Longitudinal Study, 74–77

Braiding plan, 328

Braun, H.I., 3–7, 11, 34, 186, 194, 212, 283

Braun–Holland equating, 194, 196

Brennan, R.L., 2, 16, 22–24, 26, 27, 30, 32, 33,

35, 37, 41–58, 60, 62, 65–67, 110, 111,

118, 143, 149, 152, 173, 175, 183, 186,

187, 190, 194, 196, 199, 221, 222, 261,

281, 282, 286, 296, 317, 328, 330, 343

Brockwell, P.J., 330, 331, 333, 345

B-splines, 244–247, 252

C

Calibration, 2, 10, 23, 64, 65, 68, 69, 226,

228–232, 237–241, 243, 247, 254, 299

California First-Year Mental Scale, 72

361

Campbell, N.R., 214

Carlson, J.E., 48, 59–70

Carmrer, S.G., 326

Casabianca, J., 282, 289

Cattell, R.B., 86

CDF. See Cumulative distribution function

CE. See Chained equating

Chained equating, 14, 15, 183–185, 187,

193–196, 200, 294, 295, 331

Chain equipercentile equating, 145

Chatfield, C., 330–332, 339, 345

Cheng, P.E., 213, 261

Chen, H.H., 8, 132, 185–200

Chia, M., 241

Circle-arc equating, 111–113, 121–122

Classical test theory, 31, 35, 190, 222

Cochran, D., 333

Cognitive abilities, 72, 73, 81, 82

Cohen, A.S., 299

Collateral information, 109, 116–119, 121, 222

Common-item design, 60–61, 64–69

Conditional standard error of equating (CSEE),

113, 114, 117, 119

Continuization, 4, 13, 15, 16, 37, 141–157,

159–161, 163, 164, 172, 193–195,

286, 323

Continuization with alternative kernels,

164–167

Continuized log-linear method (CLL),

141–157

Cook, L.L., 22, 328

Cornish-Fisher expansion, 268

Cross-grade scaling. See Vertical linkingCSEE. See Conditional standard error of

equating

Cudeck, R., 71

Cumulant-generating function, 162

Cumulants, 161–164, 166, 167, 169, 170,

266–277, 280

Cumulative distribution function (CDF),

14–17, 32, 33, 37, 146, 156, 160–165,

169, 173, 175–177, 179, 203, 247–249,

283, 285–287, 319, 320

D

Dalal, S., 179

Damien, P., 176

Data collection designs, 3, 4, 6, 9–12, 15, 21,

26–29, 31, 32, 37, 39, 40, 71, 141, 142,

159, 160, 225, 298, 326, 334

Data processing, 21, 35–37

Davis, R.A., 330, 333, 345

De Ayala, R.J., 250

Del Pino, G., 220

Design function (DF), 12, 14, 141–144, 147,

152–155, 160, 173, 319, 320, 322

Diaconis, P., 176, 179

Difference that matters, 38, 326

Direct equating methods, 328

Dirichlet process prior, 177–178

Divgi, D.R., 111

Dorans, N.J., 1, 2, 4, 6, 11, 21–42, 45, 48, 60,

62, 71, 110, 213, 282, 326

Douglas, J.A., 243–258

Draper, N.R., 3

Drasgow, F., 44

DTM. See Difference that matters

Durbin-Watson test, 332, 339, 341, 342

E

Ebel, R.L., 47

Educational measurement, 1, 2, 5, 8, 329

Edwards, M.C., 74, 86, 262

Efron, B., 116, 310

Eignor, D.R., 1, 21–42

Embretson, S.E., 71

Empirical Bayes, 116–118

Equating, 1–17, 21–42, 45, 48, 59, 70, 71, 74,

87–107, 109–122, 125–141, 143–147,

152–157, 159–173, 175–223, 225, 227,

241, 261, 281–331, 333–337, 340–343,

345

Equating assumptions, 193, 330

Equating in a circle, 328

Equating requirements, 11, 23

Equipercentile equating, 13, 17, 110, 113–115,

125, 126, 143, 145, 152, 159, 175, 176,

181, 184, 185, 193–197, 207, 208, 211,

213, 217–219, 222, 302, 308, 323

Equipercentile equating function, 13–17,

32–33, 37, 89, 113, 141, 159, 161, 176,

198, 296, 319, 325, 326

Equity, 23, 24, 41, 57, 58, 201, 204, 205,

207–209

Equivalent groups design, 4, 12, 26–29, 31–33,

35, 60–63, 68–69, 126, 139, 142–147,

152–154, 159, 161, 167, 175–177,

181–184, 318, 323, 326

Exponential families, 125–140

F

Fairbank, B.A., 37

Feigenbaum, M.D., 326

Ferguson, T., 179

362 Index

Ferrer-Caja, E., 81

Feuer, M.J., 22

Fienberg, S.E., 4

Fisher, R.A., 324

Fitzpatrick, A.R., 2, 49, 53, 62–66

Fitzpatrick, S.J., 328, 329

Five requirements, 11, 23, 24

Flanagan, J.C., 22, 46

Forsyth, R.A., 261

Fournier-Zajac, S., 16, 89, 196

Freeman, M.F., 47

Freeman-Tukey residuals, 4, 167, 287, 293

Frequency estimation equipercentile equating,

185, 196

G

Gao, X., 30

Gaussian kernel, 13, 141, 143, 156, 159, 168,

171, 172, 245, 319, 323

Generalized equating function, 185–200

Generalized least square estimates, 332

Generic kernel function, 161

Gibbons, R.D., 8, 316

Gibbs algorithm, 180, 182

Gilula, Z., 127, 128

Glas, C.A.W., 8, 226, 241, 297, 301, 303,

312, 313, 316, 329

Goodness-of-fit measures, 287, 292–294

Grant, M.C., 119, 120

Green, B.F., 22

Grimm, K.J., 71, 74

Guo, H., 25

Guttman items, 206

H

Haberman, S.J., 13, 17, 25, 31, 41, 42, 89,

119, 125, 127, 128

Haebara characteristic-curves approach, 226

Hall, P., 179

Hall, W., 262, 267

Hamagami, F., 74, 77, 81

Hambleton, R.K., 48, 49, 225, 227,

229, 334

Hammond, S., 213

Han, N., 132, 282, 287

Hanson, B.A., 47, 56, 57, 261, 299

Harris, D.J., 23, 30, 57

Hatrak, N., 328

Hedeker, D., 8

Hemphill, F.C., 22

He, X., 245, 246, 250

Holland, P.W., 1–8, 11–13, 16, 17, 22–26, 28,

29, 32–34, 36–38, 41, 42, 45, 48, 60, 62,

71, 89, 92, 94, 98, 112, 125, 141, 142,

146, 147, 159, 160, 175, 176, 185, 186,

194–196, 199, 212, 218, 225, 243, 261,

281–284, 287, 288, 291, 296, 317, 321

Hoover, H.D., 22, 232, 243, 261, 308

Horn, J.L., 86

Hybrid equating functions, 89

I

Indirect equating methods, 328, 329

Information theory, 350

Inverse operator, 92

IRT. See Item response theory

Item characteristic curves (ICC), 67, 244–248,

250, 253, 254, 257

Item response theory (IRT), 2, 43, 59, 160, 201,

225, 243, 261, 282, 297, 327

J

Jacobian, 14, 173, 235, 237, 321

Jaffa, A.S., 72

Janssen, R., 220

Jarjoura, D., 37

Jenkins, G.M., 330, 331

Johnson, E.G., 261

K

Kane, M.T., 187

Kao, C.W., 328

Karabatsos, G., 8, 116, 175, 176, 180

Kaskowitz, G.S., 250

Kendall, M.G., 14

Kernel equating, 11, 13, 90, 125, 132, 138–140,

159–173, 185, 193–199, 261, 326

Kernel equating framework, 164, 165, 317–326

Kernel equating method, 16, 318–320, 323,

325

Kernel smoothing, 37, 139, 160, 164, 244–247,

250, 319

Kim, S., 40, 42, 89, 109, 112, 114, 115, 118,

119, 328

King, B.F., 146

Koenker, R., 246

Kolen, M.J., 2, 6, 16, 22–24, 26, 27, 32, 33, 35,

37, 41, 43–48, 51, 54, 57, 58, 60, 62,

65–68, 110, 111, 118, 143, 149, 152,

175, 183, 186, 187, 190, 199, 217, 221,

222, 225, 226, 229, 232, 233, 236, 243,

261, 281, 282, 286, 296, 297, 299, 304,

308, 317, 328, 330, 343

Kong, N., 187

Koretz, D., 22

Index 363

Kronecker delta, 272

Kuehl, R.O., 326

Kupperman, M., 163

Kurtosis, 148, 149, 151, 161, 168, 197, 262,

270, 271, 277, 323

L

Lagrange multiplier test, 226, 241

Laud, P., 176

Leary, L.F., 30

Lee, W., 194, 199

Lee, W.-C., 47, 58

Lee, Y.-H., 159

Lee, Y.-S., 244

Lehmann, E.L., 321

Levine equating, 14, 185, 187–192, 194, 195,

197–199, 221

Levine observed-score equipercentile equating,

195, 196

Levine, R., 185

Lewis, C., 117–119

Lewis, D., 241

Liang, L., 35, 36

Li, D., 327

Lieberman, M., 262, 268

Lijoi, A., 179

Li, M.-Y., 213

Lindquist, E.F.,

Linear equating, 11, 13, 15, 16, 22, 31, 33, 37,

89, 91, 93–98, 101, 103–104, 111, 156,

176, 185–196, 221, 222, 324–326

Linear interpolation, 13, 16, 17, 37, 161, 165,

168, 286

Linking, 1, 21, 59, 71, 89, 139, 185, 225, 243,

261, 283, 297, 333

Linking procedures, 21, 60, 66, 225, 226, 256

Liou, M., 213, 244, 261

Li, S., 4, 42, 327

Liu, J., 25, 41, 213

Livingston, S.A., 8, 22, 40, 109, 110, 112, 114,

115, 117–119, 185, 282, 328

Ljung-Box test, 332, 339–344

Ljung, G., 332, 339

Local equating, 10, 201–223

Local observed-score equating, 201–223

Logistic distribution, 162

Logistic kernel (LK), 161–163, 166, 168–173

Log-linear models, 4, 12, 15, 37, 132, 138,

146–149, 151, 160, 193, 227, 284, 319,

321–323

Log-linear smoothing, 125, 141–143, 148,

152, 153

Longitudinal data, 74, 82, 85, 87

Longitudinal models, 74, 81–82

Loomis, S.C., 47

Lord, F.M., 2, 6, 7, 24, 28, 51–53, 55, 56, 58,

63, 65, 66, 71, 87, 201–209, 212, 214,

216, 223, 225, 228–230, 234, 235, 243,

253, 261, 300, 305, 334

Lord’s theorem, 2, 201, 205–206, 208–210, 215

Lorentz, G., 178

Loyd, B.H., 232, 243, 261

Lp-circles, 96–98, 104

Luecht, R.M., 44

M

Magis, D., 220

Marco, G.L., 41, 226, 233, 243, 261

Marginal maximum likelihood (MML),

226, 230, 235, 264, 269, 299–302,

313, 316

Martin, K.,

Mason, R., 86

Masters, G.N., 81

McArdle, J.J., 71, 77, 86

McCall, W.A., 46

McHale, F., 119, 120

mdltm software, 230, 237, 238

Mean-preserving linear transformation

(MPLT), 194–196, 199, 200

Measurement model, 2–10, 17, 79–81

Meijer, R.R., 329

Meredith, W., 77

Merrill, M.A., 72, 76, 77

Mislevy, R.J., 254, 261, 301

MML. See Marginal maximum likelihood

Model fit, 69–70, 85

Modified post-stratification equating, 186, 193,

194, 196, 281–296

Molenaar, I.W., 312

Moment-generating function, 162

Moments, 13, 14, 16, 17, 34, 125, 137, 138,

142, 143, 148–151, 161, 164, 166–168,

221, 261, 266, 284, 288, 293–294

Montgomery, D.C., 332, 339

Morris, C.N., 6

Morrison, C.A., 329

Moses, T.P., 21–42, 120

Mosteller, F., 287

Moustaki, I., 261

MPLT. See Mean-preserving linear

transformation

Muliere, P., 179

M€uller, P., 176Multiple-group, 68, 229, 238, 300

Muraki, E., 53, 63, 64

364 Index

N

Nadaraya, E.A., 245

NAEP. See National Assessment of

Educational Progress

National Assessment of Educational Progress

(NAEP)

NEAT design. See Nonequivalent groups withanchor test design

Nelson, L., 51, 52

Ng, P., 246

Nicewander, W.A., 56

Nonequivalent groups design, 183–184, 338,

342, 344

Nonequivalent groups with anchor test design

(NEAT), 5, 12, 14, 16, 27–28, 34–35,

39, 40, 89, 143, 145, 152, 154–156,

185–200, 218–221, 225–229, 243,

281–284, 291, 296, 319, 326

Nonparametric IRT models, 243–258

Normal distribution, 10, 13, 46, 160, 162, 176,

193, 262, 301, 305, 316, 334

Novick, M.R., 6

O

Observed-score equating (OSE), 2, 5, 21, 30,

32, 34, 35, 159, 176, 177, 191, 196–199,

201–223, 281, 283, 286, 297–316

Observed-score equating (OSE) framework,

10–17

Ogasawara, H., 261–280

One-parameter logistic (1PL) model, 64, 79,

226, 297, 300, 304–310, 312, 316

Orcutt, G.H., 333

Ord, K., 273, 276

Orlando, M., 53

OSE. See Observed-score equating

P

2-Parameter logistic (2PL) model, 49, 64, 226,

229, 231, 237, 238, 248, 250, 262, 268,

271, 334

Patz, R., 23, 62, 241, 242

PDF. See Probability density function

Peck, E.A., 332

Penalty function, 148, 149, 156, 165, 166, 247,

323

Petersen, N.S., 22, 24, 26, 37, 41, 226, 308,

328, 329

Petrone, S., 179, 180

Pitoniak, M.J., 48

1PL model. See One-parameter logistic (1PL)

model

2PL model. See Two-parameter logistic (2PL)

model

3PL model. See Three-parameter logistic (3PL)

model

Point-wise average, 92–95, 97, 98, 101, 104

Pommerich, M., 22, 23, 51, 56, 71

Population invariance, 4, 24, 33–35, 41, 188,

189, 191, 203–205, 209, 281, 309

Portnoy, S., 246

Posterior distribution, 51, 176–182, 184, 216

Poststratification equating (PSE), 14, 34, 35,

145, 193–200, 220, 281–296

Presmoothing, 12, 15, 37, 141, 142, 146, 160,

166, 173, 193, 194, 199, 200, 221, 284,

285, 321

Prior distribution, 176–182, 184, 241

Probability density function (PDF), 142–145,

149, 150, 153, 161–165, 169, 179

Pr€unster, I., 179PSE. See Poststratification equating

Q

Quadrature, 142, 143, 230, 235, 264, 265,

269, 305

Quintana, F., 176

Qu, Y., 38, 317

R

Ramsay, J.O., 244, 245, 258

Randomly equivalent groups, 61, 68, 69,

126–133

Rao, C.R., 15

Rasch, G., 79, 300

Rasch model, 49, 79, 229, 300

Reinsel, G.C., 330

Reise, S.P., 22

Reliability, 21, 39

Resampling studies, 114–116, 118–120

Rescorla, L.A., 74

Revised Stanford-Binet, 77

Ricker, K., 282

Rijmen, F., 8, 9, 38, 317

Rock, D.A., 10

Rodriguez, M.C., 56

Rogers, H.J., 225

Rosa, K., 51, 52

Rosenbaum, P.R., 28

Rossi, N., 244

S

Same construct, 17, 21, 23, 24, 27, 41,

72, 159, 176, 183, 225

Samejima, F., 53, 64

San Martin, E., 220

SAT, 25, 31, 35, 42

Index 365

Scale aligning, 22, 23, 25

Scale drift, 4, 70, 327–346

Scaling, 21, 23, 25, 28, 42–64, 68–70, 73,

84, 228, 297, 327, 328

Scaling test, 60

Schwarz, G., 239

Score equity assessment, 41

Score probability, 11, 12, 141, 146, 161,

166, 167, 285–287, 319, 320

Second order model, 8, 9

SEE. See Standard error of equating

SEED. See Standard error of equating

difference

Senturk, D., 47

Sethuraman, J., 177

Shealy, R., 254

Shi, M., 329

Shi, P., 245, 246, 250

Shumway, R.H., 330, 345

SiGNET design, 120, 121

Silvey, S.D., 226

Single-group design, 4, 12, 26–29, 137,

139, 143, 144, 175, 181, 221

Sinharay, S., 5, 35, 41, 281

Skaggs, G., 118

Skewness, 37, 40, 111, 139, 148–151, 161,

167, 168, 197, 262, 269–271, 275

Small samples, 24, 26, 40, 89, 109–122,

220, 221, 300, 315

Smith, A., 176

Smoothing, 14, 35, 37, 40, 125, 139,

141–143, 148, 152, 159, 160, 164, 221,

223, 244–247, 249, 250, 254, 308,

319, 320

Smoothness penalty, 166

Standard error of equating (SEE), 4, 14, 15, 28,

38, 113, 141, 146–147, 152, 157, 160,

161, 171, 172, 199, 281, 320–322

Standard error of equating difference (SEED),

4, 14, 38, 160, 161, 172

Stanford-Binet, 72, 73, 77, 78, 83

Stecher, B.M.,

Stegun, I.A., 129, 163, 305

Steinberg, L., 62, 64

Stewart, E.E., 41, 226

Stocking and Lord method, 66–68, 226, 228,

230, 235, 236, 243 See also Stocking

and Lord scale linkage; Stocking–Lord

test characteristic curve method

Stocking and Lord scale linkage, 234–236

Stocking-Lord test characteristic curves (TCC)

method, 66–68, 235, 253

Stocking, M.L., 66, 229, 234–236, 243, 253,

261, 328, 334

Stoffer, D.S., 330

Stout, W.F., 254

Strawderman, W.E., 89

Stuart, A., 14, 273, 276

Swaminathan, H., 225, 334

Swanson, M.R., 326

Swave, 99–104

Swygert, K.A., 51, 52

Symmetric average, 98, 100–103

Symmetry property, 92–95, 101, 102, 209

Synthetic population. See Target population

T

Target population, 11, 14, 32–35, 40, 110, 111,

114, 141, 144, 145, 155, 161, 218, 219,

249, 283, 319

Taylor expansion, 266

Terman, L.M., 72, 76, 77

Test equating, 1–17, 21–42, 45, 87, 109, 116,

141, 143, 159, 160, 164, 172, 175–184,

186, 190, 223, 317–319

Testlet model, 2, 8–10

Testlets, 2, 8–10, 119, 120

Thayer, D.T., 4, 12, 13, 17, 22, 37, 71, 89, 112,

125, 141, 142, 146, 147, 159, 160, 175,

176, 185, 218, 225, 243, 261, 281, 284,

287, 288, 317, 321

Thissen, D., 48, 51–53, 56, 57, 62, 64, 225,

230, 312

Thisted, R.A., 143

Three-parameter logistic (3PL) model, 49, 52,

64, 217, 222, 226, 228, 229, 231, 236,

237, 243, 254, 263, 267, 268, 297, 300,

304–308, 312–316

Tibshirani, R.J., 310

Time series, 4, 14, 327–346

Tong, Y., 6, 43

True-score equating, 10, 192, 197, 201, 206,

207, 212, 222, 331, 337, 341

Tsai, T.-H., 261

Tucker equating, 185, 187–190, 192, 194–199

Tukey, J.W., 38, 47

Two-parameter logistic (2PL) model, 49, 64,

226, 229, 231, 237, 238, 248, 250, 262,

268, 271, 334

U

Uniform distribution, 128, 134, 161, 163, 165,

176, 334

Uniform kernel, 13, 141, 142, 160–171

366 Index

V

van der Linden, W.J., 8, 10, 49, 201

Van Krimpen-Stoop, E.M.L.A., 329

Veerkamp, W.J.J., 329

Verhelst, N.D., 303, 312, 313

Vertical linking, 59–70, 241, 242

Vertical scaling. See Vertical linkingVining, G.G., 332

von Davier, A.A., 1, 22, 26, 27, 29, 33, 34, 37,

38, 41, 42, 71, 89, 90, 112, 119, 125,

132, 137–141, 143, 146, 148, 152, 156,

159, 175, 176, 181, 182, 185, 187, 196,

197, 218, 221, 225, 243, 261, 281, 299,

317, 327

von Davier, M., 2, 225, 299

W

Wainer, H., 8, 36, 48, 56, 225

WAIS. See Wechsler Adult Intelligence Scale

WAIS-R. See Wechsler Adult Intelligence

Scale-Revised

Wald, A., 313, 317

Wald test, 312–315, 317, 318, 322–323,

325–326

Walker, S.G., 8, 175

Wang, J., 47

Wang, T., 57, 58, 127, 132, 137, 141, 194, 196,

199, 282

Wang, X., 8, 244

Wasserman, L., 160

Watson, G.S., 245

Wechsler Adult Intelligence Scale (WAIS), 72,

73, 80

Wechsler Adult Intelligence Scale-Revised

(WAIS-R), 72, 73, 80

Wechsler-Bellevue Intelligence Scale, 72, 77

Wechsler, D., 72, 77

Weights, 45, 66, 87, 89, 90, 95, 97, 101, 117,

119, 144, 188, 189, 191, 192, 195–198,

212, 213, 265

Wiberg, M., 211, 213, 219, 221, 222

Williams, V.S.L., 51

Wingersky, M.S., 51, 216, 305

Wirth, R.J., 74, 86, 262

Woodcock, R.W., 81

Wright, N.K., 213, 282

X

Xu, X., 227, 243

Y

Yamamoto, K., 227

Yan, D., 132

Yao, L., 23, 62, 241

Yen, W.M., 2, 7, 23, 49, 53, 62–66

Ylvisaker, D., 176, 179

Youtz, C., 287

Z

Zelenak, C.A., 48

Zeng, L., 57, 217, 297

Zimowski, M.F., 56, 254, 301

Zwick, R., 47

Index 367