Package equate. February 15, 2013

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1 Package equate February 15, 2013 Version Date Title Statistical Methods for Test Score Equating Author Anthony Albano Maintainer Anthony Albano Depends R (>= ) The equate package contains functions for non-irt equating under both random groups and nonequivalent groups with anchor test designs. Mean, linear, equipercentile and circle-arc equating are supported, as are methods for univariate and bivariate presmoothing of score distributions. Specific equating methods currently supported include synthetic, nominal weights, Tucker, Levine observed score, Levine true score,braun/holland, frequency estimation, and chained equating. LazyLoad yes License GPL (>= 2) URL Repository CRAN Date/Publication :11:10 NeedsCompilation no R topics documented: equate-package ACTmath descriptives equate freqadjust

2 2 equate-package freqtab KBneat loglinear percentiles plot.freqtab se.boot Index 19 equate-package Statistical Methods for Test Score Equating Details The equate package contains functions for non-irt equating under both random groups and nonequivalent groups with anchor test designs. Mean, linear, equipercentile and circle-arc equating are supported, as are methods for univariate and bivariate presmoothing of score distributions. Specific equating methods currently supported include synthetic, nominal weights, Tucker, Levine observed score, Levine true score, Braun/Holland, frequency estimation, and chained equating. Package: equate Version: Date: Depends: R (>= ) License: GPL (>= 2) Index: ACTmath KBneat descript equate freqavg freqtab loglinear plot.freqtab px se.boot ACT Mathematics Test Scores Test Scores under a NEAT design Descriptive Statistics for a Frequency Table Statistical Equating Frequency Adjusting and Averaging Frequency Tabulation Loglinear Smoothing Plotting of Frequency Distributions Percentile Rank and Cumulative Distribution Functions Bootstrap Standard Errors of Equating Further information is available in the following vignettes: equatevignette equate vignette (source, pdf)

3 ACTmath 3 The package vignette provides an introduction to statistical equating and includes descriptions of the supported equating methods and examples. The help page for the main function of the package, equate, contains additional examples. Author(s) Anthony Albano <tony.d.albano@gmail.com> ACTmath ACT Mathematics Test Scores This dataset contains score distributions for two forms of the ACT mathematics test, as presented in table 2.5 of Test Equating, Scaling, and Linking (Kolen and Brennan, 2004; p. 50). ACTmath Format A 41 by 3 matrix containing the score scale, frequencies for form X, and frequencies for form Y Source Kolen, M. J., and Brennan, R. L. (2004) Test Equating, Scaling, and Linking. (2nd ed.), New York: Springer. The dataset is also provided with the equating software RAGE, available at the following link: descriptives Descriptive Statistics for a Frequency Table The functions mean.freqtab, sd.freqtab, skew.freqtab, and kurt.freqtab return the mean, standard deviation, skewness, and kurtosis for a frequency table of class freqtab. descript returns all four of these values along with the number of observations. min.freqtab and max.freqtab return the minimum and maximum. cov.freqtab returns the variance of a univariate frequency table and the covariance of a bivariate.

4 4 descriptives mean.freqtab(x,...) sd.freqtab(x) skew.freqtab(x) kurt.freqtab(x) descript(x) min.freqtab(x,..., na.rm = FALSE) max.freqtab(x,..., na.rm = FALSE) cov.freqtab(x) Arguments x two-column matrix of class freqtab, where column 1 specifies the score scale and column 2 the frequencies for each score point. If x is a three-column bivariate frequency table, only the first distribution is used, except with cov.freqtab (see examples)... further arguments passed to or from other methods na.rm logical indicating whether or not missing values should be removed Value descript returns a vector which includes the mean, standard deviation, skewness, kurtosis, and number of observations. Otherwise a vector of length one is returned. Author(s) Anthony Albano <tony.d.albano@gmail.com> See Also freqtab Examples xtab <- freqtab(actmath[, 1], ACTmath[, 2]) descript(xtab) mean(xtab) set.seed(2008) y <- round(rnorm(100, 100, 10)) ytab <- freqtab(70:130, y) min(y); min(ytab) max(y); max(ytab) cov.freqtab(ytab); var(y) yscale <- 0:36 vscale <- 0:12 ny <- freqtab(yscale, KBneat$y[, 1], vscale, KBneat$y[, 2]) cov.freqtab(ny); cov(kbneat$y[, 1], KBneat$y[, 2])

5 equate 5 equate Statistical Equating This function equates test scores under the random groups and nonequivalent groups with anchor test designs. 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 equate (the output of a previous equating) and x a vector of scores, the equating function from y is used to convert scores in x to the Y scale. type method name Details ident bootse the type of equating, as the string "mean", "linear", "equipercentile", "circle-arc", or abbreviations of these string specifying the equating method, where NA (default) indicates equating under the random groups design, and "nominal weights", "tucker", "levine", "frequency estimation", "braun/holland", and "chained" (or abbreviations of these) indicate the respective methods under the nonequivalent groups design An optional name, used to label the output value between 0 (default) and 1 specifying the proportional representation of the identity function in the equating function. 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) For the random 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,

6 6 equate Value 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: internal logical indicating whether or not the anchor item scores are included in the total scores. This applies only to the Levine method, as all other methods assume an internal anchor test. Default is TRUE Ky integer, typically indicating the number of items on form Y or the maximum possible score on Y (default). This value is used in equipercentile equating for the extremes of the scale (see below) smoothmeth string indicating one of four smoothing methods to be used in equipercentile equating: "none" (default), "bump", "average", and "loglin" (see below) w value between 0 and 1 specifying the weight applied to form X scores (and implicitly specifying the form Y weight as 1 - w) when estimating the synthetic population. When set to -1, proportional weights are calculated for X and Y based on sample size lts logical indicating whether or not to use levine true score (hence lts ) equating. Default is FALSE lowp, highp two vectors, each of length 2, recycled if necessary, specifying the coordinates for the low and high points used in circle-arc equating. lowp defaults to the minimums of the X and Y scales. highp defaults to the number of items, i.e., the maximum possible scores midp string used to specify the type of chained linear equating used to obtain the midpoint in circlearc equating, whether "mean" (default) or "linear" 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 random 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). An object of class equate is returned, listing the following components, some of which are conditional on the equating type, method, and smoothing:

7 equate 7 name type method design ident concordance bootsee yx stats freqtab coefficients points w anchorstats anchortab synthstats synthtab smoothmethod smoothout name or NULL if not specified equating type equating method equating design, inferred from the method weight given to identity function conversion table containing scores on X with their form Y equivalents. vector of bootstrap standard errors of equating Form Y equivalents for each score on X descriptive statistics for X, Y, and YX (equated scores) distributions table containing the score scale, cumulative distribution functions, frequencies, and smoothed frequencies for forms X and Y conversion coefficients intercept and slope. Center points of circle and radius are also included for circle-arc equating X and Y coordinates for low, middle, and high points used in circle-arc equating weight applied to X in synthetic estimation descriptive statistics for the anchor distributions frequency table for the anchor distributions means and standard deviations for the synthetic distributions frequency table for the synthetic distribution, used in frequency estimation equipercentile equating smoothing method Smoothed frequencies for X and Y Author(s) Anthony Albano <tony.d.albano@gmail.com> 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, Lord, F. (1982). The standard error of equipercentile equating. Journal of Educational Statistics, 7, See Also freqbump, freqavg, loglinear, se.boot

8 8 freqadjust Examples # see vignette("equatevignette") for 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(rscale, ACTmath[, 2]) ry <- as.freqtab(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$bootsee) # nonequivalent groups design for (1) Tucker linear, # (2) frequency estimation, and (3) Braun/Holland linear nscale <- 0:36 vscale <- 0:12 nx <- freqtab(nscale, KBneat$x[, 1], vscale, KBneat$x[, 2]) ny <- freqtab(nscale, KBneat$y[, 1], vscale, KBneat$y[, 2]) neq1 <- equate(nx, ny, type = "Linear", method = "Tuck", w = 1) neq2 <- equate(nx, ny, type = "equip", method = "freq", w = 1) neq3 <- equate(nx, ny, type = "linear", method = "braun", w = 1) # compare equated scores: cbind(nscale, Tucker = neq1$conc[, 2], FE = neq2$conc[, 2], Braun = neq3$conc[, 2]) freqadjust Frequency Adjusting and Averaging These functions provide simple methods for smoothing score distributions, typically ones with unobserved score points, prior to performing equipercentile equating. freqbump smooths by adding a small relative frequency to each score while adjusting the probabilities to sum to one (see Kolen

9 freqadjust 9 and Brennan, 2004, p. 48). freqavg replaces frequencies falling below a specified threshold with an average of adjacent values (see Moses and Holland, 2008). freqbump(x, jmin = 10^-6, Kx = max(x[,1])) freqavg(x, jmin = 1) Arguments x jmin Kx two-column matrix of class freqtab, where column 1 specifies the score scale and column 2 the frequencies for each score point. For freqbump, x can also be a bivariate frequency table. for freqbump, the frequency to be added to each score point (default is 10^-6) for freqavg, the minimum frequency, as an integer, below which frequencies will be replaced (default is 1) integer indicating the number of items in the scale Value Returns a vector of adjusted frequencies. For freqavg, if no values fall below jmin these will simply be the original frequencies. For freqbump, frequencies are in the form of probabilities, summing to 1. Author(s) Anthony Albano <tony.d.albano@gmail.com> References Kolen, M. J., and Brennan, R. L. (2004) Test Equating, Scaling, and Linking. (2nd ed.), New York: Springer. Moses, T., and Holland, P. W. (2008). Notes on a general framework for observed score equating (ETS Research Rep. No. RR-08-59). Princeton, NJ: ETS. See Also loglinear Examples set.seed(2005) x <- round(rnorm(1000, 100, 10)) xscale <- 70:130 xtab <- freqtab(xscale, x) xavg <- freqavg(xtab) xbump <- freqbump(xtab) cbind(xtab, avg = xavg, bump = round(xbump * sum(xtab[, 2]), 6))

10 10 freqtab freqtab Frequency Tabulation freqtab creates a frequency table for a vector, or vectors, of scores. as.freqtab converts to class freqtab freqtab(xscale, x, vscale, v) as.freqtab(xscale, x) Arguments xscale x vscale v the score scale for form X, as a vector vector of scores (one per examinee) on items in form X (see details for other options) the score scale for form V, as a vector vector of scores (one per examinee) for form X examinees on items in form V (i.e., the anchor test) Details If x is a vector of scores, one per examinee, values are factored with levels = xscale. This ensures that unobtained scores (i.e., scores not in x) will still be represented in the tabulation. If x is a two-column matrix, as.freqtab serves to simply change its class to freqtab, as required by equate. In this case the first column should contain the score scale and the second column the number of examinees obtaining each score. Alternatively, both xscale and x can be included in as.freqtab, with the same result. Value In the univariate case (i.e., form V is not specified) a two-column matrix is returned, where column 1 contains the score scale and column 2 the frequency for each scale value. In the bivariate case a three-column matrix is returned where columns 1 and 2 contain all combinations of form X score scale points (in column 1) and score scale points on the anchor test (in column 2), and column 3 contains the total number of examinees obtaining the corresponding score combination. Author(s) Anthony Albano <tony.d.albano@gmail.com> See Also table

11 KBneat 11 Examples set.seed(2005) x1 <- round(rnorm(1000, 100, 10)) x1scale <- 70:130 freqtab(x1scale, x1) x2 <- ACTmath[, 2] x2scale <- ACTmath[, 1] as.freqtab(x2scale, x2) xscale <- 0:36 vscale <- 0:12 freqtab(xscale, KBneat$x[, 1], vscale, KBneat$x[, 2]) KBneat Test Scores under a NEAT design This dataset contains scores for two forms of a 36-item test administered under a nonequivalent groups with anchor test design. The anchor test is internal and consists of twelve items taken by both groups of examinees. The data is distributed as part of the equating software CIPE (see link below), and additional examples using this dataset are presented in Test Equating, Scaling, and Linking (Kolen and Brennan, 2004). KBneat Format A list of length two, where each element ($x and $y) contains two columns of data, one for the total score and another for the anchor test score for each examinee. Source Kolen, M. J., & Brennan, R. L. (2004) Test Equating, Scaling, and Linking. (2nd ed.), New York: Springer. The dataset can be downloaded as part of the CIPE software, available at the following link: http: //

12 12 loglinear loglinear Loglinear Smoothing This function smooths univariate and bivariate score distributions via polynomial loglinear modeling (Holland & Thayer, 2000; Moses & von Davier, 2006). loglinear(x, scorefun, degree, raw = TRUE, convergecrit =.0001, verbose = TRUE,...) Arguments x scorefun degree raw Details score distribution of class freqtab. Under the random groups design (i.e., no anchor test) x will contain the score scale in column 1, and the number of examinees obtaining each score in column 2. For the nonequivalent groups design a bivariate frequency table is used, where columns 1 and 2 include all score combinations for the total and anchor test score scales, and column 3 contains the number of examinees obtaining each combination (see freqtab for details) matrix of score functions, where each column represents a transformation of the score scale (or the crossed score scales, in the bivariate case) integer indicating a maximum polynomial transformation to be computed (passed to poly; ignored if scorefun is provided) logical. If TRUE (default), raw polynomials will be used, if FALSE, orthogonal (passed to poly) convergecrit convergence criteria used in maximum likelihood estimation (default is.0001) verbose logical, with default TRUE, indicating whether or not output in addition to the fitted values should be returned... further arguments passed to or from other methods Loglinear smoothing is a flexible procedure for reducing irregularities in a raw score distribution. The loglinear function fits a polynomial loglinear model to a distribution of scores, where the degree of each polynomial term determines the specific moment of the raw distribution that is preserved in the fitted distribution (see below for examples). scorefun must contain at least one score function of the scale score values. While there is no explicit limit on the number of columns in scorefun, models with more than ten may not converge on a solution. Specifying degree is an alternative to scorefun. It takes a maximum polynomial degree and constructs the score functions accordingly. For example, degree=3 will result in a model with three terms: the score scale raised to the first, second, and third powers. In the bivariate case powers 1 through 3 would be included for each variable. Maximum likelihood estimates are obtained using a Newton-Raphson algorithm, with slightly smoothed frequencies (all nonzero) as the basis for starting values. Calculating standard errors

13 loglinear 13 for these estimates requires matrix inversion, which for complex models may not be possible. In this case the standard errors will be omitted. The tolerance level for detecting singularity may be modified with the argument tol, which is passed to solve(). For a detailed description of the estimation procedures, including examples, see Holland and Thayer, 1987 and For a more recent discussion, including the macro after which the loglinear function is modeled, see Moses and von Davier, Value Returns a list including the following components: modelfit rawbetas alpha iterations fitted.values residuals cmatrix scorefun table of model fit statistics: likelihood ratio chi-square, Pearson chi-square, Freeman-Tukey chi-square, AIC, and CAIC two-column matrix of raw maximum likelihood estimates for the beta coefficients and corresponding standard errors normalizing constant number of iterations reached before convergence vector of estimated frequencies vector of residuals the C matrix, a factorization of the covariance matrix of the fitted values matrix of score functions Author(s) Anthony Albano <tony.d.albano@gmail.com> References Holland, P. W., and Thayer, D. T. (1987). Notes on the use of log-linear models for fitting discrete probability distributions (PSR Technical Rep. No ; ETS RR-87-31). Princeton, NJ: ETS. Holland, P. W., and Thayer, D. T. (2000). Univariate and bivariate loglinear models for discrete test score distributions. Journal of Educational and Behavioral Statistics, 25, Moses, T. P., and von Davier, A. A. (2006). A SAS macro for loglinear smoothing: Applications and implications (ETS Research Rep. No. RR-06-05). Princeton, NJ: ETS. Wang, T. (2009). Standard errors of equating for the percentile rank-based equipercentile equating with log-linear presmoothing. Journal of Educational and Behavioral Statistics, 34, See Also glm, loglin

14 14 loglinear Examples set.seed(2010) x <- round(rnorm(1000, 100, 15)) xscale <- 50:150 # smooth x preserving first 3 moments: xtab <- freqtab(xscale, x) xlog1 <- loglinear(xtab, degree = 3) plot(xtab, cbind(xscale, xlog1$fit), col1 = 2, col2 = 4) # add "teeth" and "gaps" to x: teeth <- c(.5, rep(c(1, 1, 1, 1,.5), 20)) xt <- xtab[, 2] * teeth cbind(xtab, xt) xttab <- as.freqtab(xscale, xt) xlog2 <- loglinear(xttab, degree = 3) cbind(xscale, xt, xlog2$fit) # smooth xt using score functions that preserve # the teeth structure (also 3 moments): teeth2 <- c(1, rep(c(0, 0, 0, 0, 1), 20)) xt.fun <- cbind(xscale, xscale^2, xscale^3) xt.fun <- cbind(xt.fun, xt.fun * teeth2) xlog3 <- loglinear(xttab, xt.fun) cbind(xscale, xt, xlog3$fit) par(mfrow = c(2, 1)) plot(xscale, xt, type = "h", ylab = "count", main = "X teeth raw") plot(xscale, xlog3$fit, type = "h", ylab = "count", main = "X teeth smooth") # bivariate example, preserving first 3 moments of total # and v (anchor) each of x and y, and the covariance # between anchor and total # see equated scores in Wang (2009), Table 4 xscale <- 0:36 vscale <- 0:12 xvtab <- freqtab(xscale, KBneat$x[, 1], vscale, KBneat$x[, 2]) yvtab <- freqtab(xscale, KBneat$y[, 1], vscale, KBneat$y[, 2]) Y <- yvtab[, 1] V <- yvtab[, 2] scorefun <- cbind(y, Y^2, Y^3, V, V^2, V^3, V * Y) wang09 <- equate(xvtab, yvtab, type = "equip", method = "chained", smooth = "loglin", xscorefun = scorefun, yscorefun = scorefun) wang09$concordance # replicate Moses and von Davier, 2006, univariate example: uv <- c(0, 4, 11, 16, 18, 34, 63, 89, 87, 129, 124,

15 percentiles , 125, 131, 109, 98, 89, 66, 54, 37, 17) loglinear(as.freqtab(0:20, uv), degree = 3) percentiles Percentile Rank and Cumulative Distribution Functions px returns the percentile ranks and fx the cumulative frequencies for each score point in a frequency distribution. px(x,y) fx(x) Arguments x y two-column matrix of class freqtab, where column 1 specifies the score scale and column 2 the frequencies for each score point. For px, x can also be a vector of score points (see details) optional frequency distribution of class freqtab (see details) Details px can be used to find the percentile ranks for a distribution of scores. When x is a frequency table and y is missing, the X distribution is used. If a frequency table is supplied for y, the percentile ranks are found for each score from x in the Y distribution. In this latter case x can either be a frequency table or simply a vector of scores. Value px returns a vector of percentile ranks, and fx a vector of cumulative frequencies. Author(s) Anthony Albano <tony.d.albano@gmail.com> Examples x <- freqtab(0:40, ACTmath[, 2]) px(x) fx(x) px(c(8.2, 40, 41), x)

16 16 plot.freqtab plot.freqtab Plotting of Frequency Distributions This function plots univariate and bivariate frequency tables of class freqtab plot.freqtab(x, y, col1 = "gray27", col2 = "gray74", pch=16, yoffset = 0.3,...) Arguments x y col1 col2 pch Details Value yoffset univariate or bivariate score distribution of class freqtab univariate score distribution, also of class freqtab, to be added to the plot of univariate x (see details) color of lines used in plotting univarite x, and in plotting both variables in bivariate x color of lines used in plotting y plotting point (see par) used in bivariate scatterplot integer indicating amount of offset, in terms of x scale points, when plotting lines for y... further arguments passed to or from other methods, such as graphical parameters besides col When x is bivariate y is ignored. Otherwise x and y are (currently) assumed to be on the same scale, which determines the x-axis. Colors must be specified using col1 and col2 The univariate option produces a single, simple, line plot of type="h". The bivariate option produces a scatterplot with a marginal barplot for each distribution. Author(s) Anthony Albano <tony.d.albano@gmail.com> See Also plot.table, lines

17 se.boot 17 Examples xscale <- 0:36 vscale <- 0:12 neat.xv <- freqtab(xscale, KBneat$x[, 1], vscale, KBneat$x[, 2]) plot(neat.xv) neat.x <- freqtab(xscale, KBneat$x[, 1]) neat.v <- freqtab(vscale, KBneat$x[, 2]) plot(neat.x, neat.v, lwd = 2) se.boot Bootstrap Standard Errors of Equating This function returns the standard deviation of equated scores over multiple replications of a specified equating. se.boot(x, y, xn = sum(x[,ncol(x)]), yn = sum(y[,ncol(y)]), reps = 100, eqfun, returnboots = FALSE,...) Arguments x, y score distributions of class freqtab for forms X and Y. These may be univariate or bivariate distributions, as required by equate xn, yn reps Details Value eqfun returnboots integers specifying the number of scores to sample from each distribution at each replication (default is the total number of scores in each) number of replications string indicating the equating function logical (default is FALSE) indicating whether or not the matrix of equated scores should be returned... further arguments passed to or from other methods Samples are drawn of size xn and yn, with replacement, from each score distribution. Form Y equivalents of each form X score are then obtained using the specified equating function eqfun. This process is repeated reps times. This function is most easily called from equate. Returns either a vector of standard deviations, one per score scale point, or a matrix of equated scores with number of columns equal to reps

18 18 se.boot Author(s) Anthony Albano Examples xtab <- freqtab(0:36, KBneat$x[, 1]) ytab <- freqtab(0:36, KBneat$y[, 1]) se.boot(xtab, ytab, reps = 100, eqfun = "equate.ln")

19 Index Topic datasets ACTmath, 3 KBneat, 11 Topic distribution percentiles, 15 Topic methods equate, 5 se.boot, 17 Topic misc plot.freqtab, 16 Topic package equate-package, 2 Topic smooth freqadjust, 8 loglinear, 12 Topic univar descriptives, 3 Topic utilities freqtab, 10 ACTmath, 3 as.freqtab (freqtab), 10 cov.freqtab (descriptives), 3 descript (descriptives), 3 descriptives, 3 KBneat, 11 kurt.freqtab (descriptives), 3 lines, 16 loglin, 13 loglinear, 6, 7, 9, 12 max.freqtab (descriptives), 3 mean.freqtab (descriptives), 3 min.freqtab (descriptives), 3 par, 16 percentiles, 15 plot.freqtab, 16 plot.table, 16 poly, 12 print.equate (equate), 5 px (percentiles), 15 sd.freqtab (descriptives), 3 se.boot, 6, 7, 17 se.eq (equate), 5 se.ln (equate), 5 skew.freqtab (descriptives), 3 synthetic (equate), 5 table, 10 equate, 3, 5, 5, 6, 10, 17 equate-package, 2 equipercentile (equate), 5 freqadjust, 8 freqavg, 6, 7 freqavg (freqadjust), 8 freqbump, 6, 7 freqbump (freqadjust), 8 freqtab, 3 5, 9, 10, 12, fx (percentiles), 15 glm, 13 19