equate (version 1.2-0)

equate: Statistical Equating

Description

This function equates test scores under the equivalent groups and nonequivalent groups with anchor test designs.

Usage

equate(x, y, type, method = NA, name = NULL, ident = 0,
	bootse = FALSE, ...)

Arguments

x, y

score distributions of class freqtab for forms X and Y. These may be univariate or bivariate distributions (see below for details). Alternatively, if y is an object of class

type

the type of equating, as the string "mean", "linear", "equipercentile", "circle-arc", or abbreviations of these.

method

string specifying the equating method, where NA (default) indicates equating under the equivalent groups design, and "nominal weights", "tucker", "levine", "frequency estimation", "bra

name

An optional name, used to label the output.

ident

value between 0 (default) and 1 specifying the proportional representation of the identity function in the equating function.

bootse

logical indicating whether or not bootstrap standard errors of equating should be returned (default is FALSE).

...

further arguments passed to or from other methods (see below for details).

Value

• When y contains equating output, a vector of equated scores is returned. Otherwise, an object of class equate is returned, listing the following components, some of which are conditional on the equating type, method, and smoothing:
• name
  name or NULL if not specified
• type
  equating type
• method
  equating method
• design
  equating design, inferred from the method
• ident
  weight given to identity function
• concordance
  conversion table containing scores on X with their form Y equivalents.
• bootsee
  vector of bootstrap standard errors of equating
• yx
  Form Y equivalents for each score on X
• stats
  descriptive statistics for X, Y, and YX (equated scores) distributions
• x, y
  original frequency tables for X and Y
• coefficients
  conversion coefficients intercept and slope. Center points of circle and radius are also included for circle-arc equating
• points
  X and Y coordinates for low, middle, and high points used in circle-arc equating
• anchorstats
  descriptive statistics for the anchor distributions
• anchortab
  frequency table for the anchor distributions
• synthstats
  means and standard deviations for the synthetic distributions
• w
  weight applied to X in synthetic estimation
• xsynthtab, ysynthtab
  frequency tables for the synthetic distributions, used in frequency estimation equipercentile equating
• smoothmethod
  smoothing method
• smoothout
  smoothed frequencies for X and Y

Details

For the equivalent groups design x and y are separate frequency tables where each row contains a score scale point and the total number of examinees obtaining that score. For the nonequivalent groups design each row contains a score scale point on x or y, a score scale point on the anchor test, and the total number of examinees obtaining that score combination (see freqtab for details). When equating output is supplied for y, no other arguments are required and scores from x are converted directly to the scale indicated in y. In this case x must be a vector of scores. Four equating types are currently supported: mean, linear, and equipercentile, as described by Kolen and Brennan (2004), and circle-arc equating, as described by Livingston and Kim (2009). The equating design is implied by the method argument, where NA (default) indicates that no method is needed (because examinees taking forms X and Y are assumed to be equivalent). The nominal weights, Tucker, Levine observed score, Levine true score, frequency estimation, Braun/Holland, and chained equating methods are supported for the nonequivalent groups with anchor test design. All but the Levine true score and chained method rely on a synthetic distribution of scores (Braun and Holland, 1982), a weighted combination of x and y. Depending on the equating method, the following additional arguments may be required: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object] For equipercentile equating under the random groups design, three smoothing options are available: "bump" and "average" (see freqbump and freqavg) require the additional argument jmin, and loglinear smoothing ("loglin") requires the score functions xscorefun and yscorefun (see loglinear). For frequency estimation and chained methods, only smoothing methods "freqbump" and "loglin" are supported. The parameter Ky is used to obtain form Y equivalents of form X scores with percentile ranks of 100. Typically Ky is set to be the number of score points in the form Y scale, which assumes that scores are integers ranging from 1 (or 0) to the total number of items, and that each item is scored correct/incorrect. Scores on other scales (such as scales which include negative values, or which exclude zero) may also be used. In such cases Ky can be set to the highest possible score on form Y, or alternatively the highest observed score on Y. Specifying Ky = "maxobs" sets Ky to the maximum observed score on Y. This may be useful for calculating bootstrap standard errors, where Ky would depend on the particular bootstrap sample. Standard errors are returned for the equivalent groups design only, using equations derived by Lord (1982, p. 168) for equipercentile equating and Braun and Holland (1982) for linear. Bootstrap standard errors are estimated using a default of reps=100 replications, sampling the maximum amount from each score distribution (controlled by the arguments xn and yn; see se.boot).

References

Braun, H. I., and Holland, P. W. (1982). Observed-score test equating: A mathematical analysis of some ETS equating procedures. In P. W. Holland and D. B. Rubin (Eds.), Test Equating (pp. 9--49). New York: Academic. Kolen, M. J., and Brennan, R. L. (2004). Test Equating, Scaling, and Linking. (2nd ed.), New York: Springer. Livingston, S. A., and Kim, S. (2009). The circle-arc method for equating in small samples, Journal of Educational Measurement, 46, 330--343. Lord, F. (1982). The standard error of equipercentile equating. Journal of Educational Statistics, 7, 165--174.

See Also

freqbump, freqavg, loglinear, se.boot

Examples

# see vignette("equatevignette") for a description of methods
# and additional examples

# random groups equating for (1) mean, (2) linear, 
# (3) equipercentile, and (4) equipercentile with 
# loglinear smoothing:
rscale <- ACTmath[, 1]
rx <- as.freqtab(cbind(rscale, ACTmath[, 2]))
ry <- as.freqtab(cbind(rscale, ACTmath[, 3]))
set.seed(2007)

req1 <- equate(rx, ry, type = "m", bootse = TRUE, reps = 50)
req2 <- equate(rx, ry, type = "l", bootse = TRUE, reps = 50)
req3 <- equate(rx, ry, type = "e", bootse = TRUE, reps = 50)
req4 <- equate(rx, ry, type = "e", bootse = TRUE, reps = 50,
	smooth = "loglin", degree = 3)

# compare equated scores:
cbind(rscale, mean = req1$conc[, 2], linear = req2$conc[, 2],
	equip = req3$conc[, 2], equipS = req4$conc[, 2])

# compare boostrap standard errors:
cbind(rscale, linear = req2$bootsee, equip = req3$bootsee,
	equipS = req4$bootsee)


# nonequivalent groups design for (1) Tucker linear,
# (2) frequency estimation , and (3) Braun/Holland linear
nx <- freqtab(KBneat$x[, 1], xscale = 0:36,
	v = KBneat$x[, 2], vscale = 0:12)
ny <- freqtab(KBneat$y[, 1], xscale = 0:36,
	v = KBneat$y[, 2], vscale = 0:12)

neq1 <- equate(nx, ny, type = "Linear", method = "Tuck", w = 1)
neq2 <- equate(nx, ny, type = "equip", method = "freq", w = 1)
neq3 <- equate(nx, ny, type = "linear", method = "braun", w = 1)

# compare equated scores:
cbind(0:36, Tucker = neq1$conc[, 2], FE = neq2$conc[, 2],
	Braun = neq3$conc[, 2])