Package 'emreliability'

Cronbach's Coefficient Alpha

Description

Compute Cronbach's coefficient alpha and the associated standard error of measurement (SEM) for a set of items.

Usage

alpha(x)

Arguments

x	A data frame or matrix containing item responses, with rows as respondents (subjects) and columns as items.

Details

Cronbach's alpha is an estimate of the internal consistency reliability of a test. This implementation:

• removes rows with any missing values using stats::na.exclude(),

• computes the sample covariance matrix of the items,

• uses the classical formula

  $\alpha = \frac{k}{k-1} \left(1 - \frac{\sum \sigma_i^2}{\sigma_X^2}\right),$

  where $k$ is the number of items, $\sigma_i^2$ are item variances, and $\sigma_X^2$ is the variance of the total score,

• computes SEM as $\text{SD}(X) \sqrt{1 - \alpha}$.

Value

A named list with the following elements:

alpha

Cronbach's coefficient alpha.

sem

Standard error of measurement (SEM) based on alpha.

Examples

data(data.u)
alpha(data.u)

---

CSEM and CSSEM with Binomial Model

Description

Compute the conditional standard error of measurement (CSEM) and conditional standard error of scaled scores (CSSEM) under the binomial model.

Usage

csem_binomial(ni, ct = NULL)

Arguments

ni	A single numeric value indicating the number of items.
ct	An optional data frame or matrix containing a conversion table with two columns: the first column as raw scores (0 to ni) and the second column as scale scores.

Details

Under the binomial model, for a test with $n_i$ items and a true-score proportion $\pi$, the distribution of raw scores is assumed to be $\mathrm{Binomial}(n_i, \pi)$. This function treats each possible raw score $k = 0, 1, \ldots, n_i$ as the true-score value (i.e., $\pi_k = k / n_i$) and computes:

• the CSEM of the raw scores; and

• if ct is provided, the CSSEM of the scale scores defined in the conversion table.

Value

A list with:

x

A vector of raw scores from 0 to ni.

csem

A vector of CSEM values (on the raw-score metric) for each raw score.

cssem

If ct is provided, a vector of CSSEM values for the scale scores corresponding to each raw score.

Examples

csem_binomial(40)
csem_binomial(40, ct.u)

---

CSEM, CSSEM, and Reliability under the Compound Binomial Model

Description

Compute the CSEM, CSSEM, and reliability coefficients for raw scores and scaled scores using the full compound binomial error model.

Usage

csem_compound_binomial(x, s, ct = NULL, w = NULL)

Arguments

x	Examinee-by-item matrix/data frame of item responses, ordered by stratum.
s	Numeric vector of number of items in each stratum. Sum(s) must equal ncol(x).
ct	Optional conversion table with maxZ + 1 rows. The second column is the scale score corresponding to composite score Z = 0, 1, ..., maxZ.
w	Optional numeric vector of weights for each stratum. Defaults to 1 per stratum.

Value

A list containing:

x

Raw total scores (row sums of x).

total_scale

If ct is provided, the composite scale score for each examinee.

csem

CSEM on the raw-score metric for each examinee.

cssem

If ct is provided, CSSEM on the scale-score metric.

reliability_raw

Reliability coefficient for raw scores.

reliability_scale

If ct is provided, reliability coefficient for scale scores.

Examples

data(data.m)
data(ct.m)
csem_compound_binomial(data.m, c(13, 12, 6))

csem_compound_binomial(data.m, c(13, 12, 6), ct.m)

---

CSEM of IRT Model via Information

Description

Compute the CSEM for a unidimensional IRT model using either MLE- or EAP-based test information.

Usage

csem_info(theta, ip, est = c("MLE", "EAP"))

Arguments

theta	A numeric vector (or object coercible to a numeric vector) containing the ability values at which to compute CSEM.
ip	A data frame or matrix of item parameters. Columns are interpreted in the same way as in info(): 3 columns: b, a, c (3PL; a on the D = 1.702 metric), 2 columns: b, a (2PL; c internally set to 0), 1 column: b (1PL/Rasch; a = 1, c = 0).
est	A character string specifying the estimation method: "MLE" for maximum likelihood or "EAP" for empirical Bayes.

Value

A list containing:

• theta — vector of ability values.

• csemMLE — CSEM values for MLE (if est = "MLE").

• csemEAP — CSEM values for EAP (if est = "EAP").

---

CSEM Lord Method

Description

Compute Lord's CSEM in classical test theory under the binomial model.

Usage

csem_lord(ni)

Arguments

ni	A numeric value indicating the number of items (must be at least 2).

Value

A list with:

x

Vector of raw scores from 0 to ni.

csem

Vector of Lord CSEM values corresponding to each raw score.

Examples

csem_lord(40)

---

CSEM: Lord Keats Method

Description

Compute CSEM using the Lord Keats approach, which rescales Lord's binomial-model CSEM using empirical KR-20 and KR-21 reliability estimates.

Usage

csem_lord_keats(x)

Arguments

x	A data frame or matrix of item responses, with rows as persons and columns as items. Items are assumed to be dichotomous (0/1).

Details

This function first computes Lord's CSEM under the binomial model via csem_lord(ni), where ni = ncol(x). It then rescales the resulting CSEM curve using the ratio

$\sqrt{\frac{1 - \text{KR-20}}{1 - \text{KR-21}}},$

where KR-20 and KR-21 are computed from the observed data via kr20(x) and kr21(x), respectively.

Value

A list with:

x

Vector of raw scores from 0 to ni.

csem

Vector of CSEM values under the Lord Keats method.

Examples

data(data.u)
csem_lord_keats(data.u)

---

Polynomial Method for CSSEM

Description

Implement the polynomial method for computing conditional standard errors of measurement for scale scores (CSSEM). A polynomial regression of scale scores on raw scores is fit for degrees 1 through K; for each degree k, the transformation derivative is used to map raw-score CSEM values to scale-score CSSEM values.

Usage

cssem_polynomial(csemx, ct, K = 10, gra = TRUE)

Arguments

csemx	A data frame or matrix containing raw scores and their CSEM on the raw-score metric. It must have at least the following numeric columns: x: raw scores, csem: conditional standard errors of measurement on the raw-score metric.
ct	A data frame or matrix containing the score conversion table. It must have at least the following numeric columns: x: raw scores (matching those in csemx), ss: scale scores corresponding to each raw score.
K	Integer. Highest polynomial degree to fit. Defaults to 10.
gra	Logical. If TRUE, a plot of the fitted polynomial curve and the observed conversion points is produced for each degree k.

Details

At the beginning of the function, csemx and ct are merged by the x column (inner join) to create an internal data frame . Only rows with x values present in both inputs are used. The polynomial model is then fit to ss ~ poly(x, k, raw = TRUE) for k = 1, ..., K.

Value

A list with two components:

rsquared

A matrix with one column containing the R-squared values from polynomial fits of degree k = 1, ..., K, where K is the largest successfully fitted degree.

cssempoly

A data frame containing the merged data (x, csem, ss) and, for each degree k, the additional columns:

• fx_k1, fx_k2, ...: transformation derivatives $f'_k(x)$ for each raw score,

• ss_k1, ss_k2, ...: fitted (rounded) scale scores from the polynomial of degree k,

• cssem_k1, cssem_k2, ...: CSSEM values on the scale-score metric, computed as $f'_k(x)\,\mathrm{CSEM}_x$.

Examples

data(ct.u)
cssem_polynomial(as.data.frame(csem_lord(40)), ct.u, K = 4, gra = TRUE)

---

Conversion table for multidimensional data.

Description

A dataset containing the conversion table for the multidimensional data, with first column as raw scores and second column as scale scores

Usage

ct.m

Format

A data frame with 32 rows and 2 variables:

x

raw score

ss

scale score

---

Conversion table for unidimensional data.

Description

A dataset containing the conversion table for the unidimensional data, with first column as raw scores and second column as scale scores

Usage

ct.u

Format

A data frame with 41 rows and 2 variables:

x

raw score

ss

scale score

---

Multidimensional data

Description

A dataset containing the responses of 3000 subjects to 31 items on three subscales (13, 12, and 6 items respectively).

Usage

data.m

Format

A data frame with 3000 rows and 31 numeric variables named V1–V31, each representing the response to one item.

---

Unidimensional data

Description

A dataset containing the responses of 3000 subjects to 40 items.

Usage

data.u

Format

A data frame with 3000 rows and 40 numeric variables named V1–V40, each representing the response to one item.

---

Feldt's Coefficient

Description

Compute Feldt's coefficient as an estimate of internal consistency reliability.

Usage

feldt(x)

Arguments

x	A data frame or matrix containing item responses, with rows as subjects and columns as items.

Value

A named list with:

feldt

Feldt's coefficient.

Examples

data(data.u)
feldt(data.u)

---

Information for IRT Model

Description

Compute test information for a unidimensional IRT model (1PL/2PL/3PL) across a vector of ability values.

Usage

info(theta, ip, est = c("MLE", "EAP"), D = 1.702)

Arguments

theta	Numeric vector of ability values at which to compute test information.
ip	A data frame or matrix of item parameters. Columns are interpreted in order as: 3 columns: b, a, c (3PL, with a on the 1.702 metric), 2 columns: b, a (2PL, c internally set to 0), 1 column: b (1PL/Rasch, a = 1, c = 0).
est	Character string indicating the estimation method: "MLE" for maximum likelihood or "EAP" for empirical Bayes.
D	A numeric constant representing the scaling factor of the IRT model. Defaults to 1.702.

Details

Test information at each $\theta$ is the sum of item information. For est = "EAP", this function returns

$I_{\mathrm{EAP}}(\theta) = I_{\mathrm{MLE}}(\theta) + 1,$

where the additional 1 reflects the prior (population) contribution under a standard normal prior.

Value

A list with:

theta

Vector of ability values.

infoMLE

If est = "MLE", vector of test information at each theta.

infoEAP

If est = "EAP", vector of test information at each theta.

---

Item parameters for unidimensional data.

Description

A dataset containing the item parameters for the unidimensional data, with first column as b parameters and second column as a parameters

Usage

ip.u

Format

A data frame with 40 rows and 2 variables:

b

b parameter

a

a parameter

---

KR-20

Description

Compute the KR-20 reliability coefficient for dichotomously scored items (e.g., 0/1).

Usage

kr20(x)

Arguments

x	A data frame or matrix of item responses, with rows as persons and columns as items. Items are assumed to be dichotomous (0/1).

Details

KR-20 is an internal consistency reliability estimate for tests with dichotomously scored items. Rows containing missing values are removed using stats::na.exclude().

Value

A single numeric value: the KR-20 reliability coefficient.

Examples

data(data.u)
kr20(data.u)

---

KR-21

Description

Compute the KR-21 reliability coefficient for dichotomously scored items (0/1), assuming equal item difficulty.

Usage

kr21(x)

Arguments

x	A data frame or matrix of item responses, with rows as persons and columns as items. Items are assumed to be dichotomous (0/1).

Details

KR-21 is a simplified alternative to KR-20, assuming equal item difficulty. Rows containing missing values are removed using stats::na.exclude().

Value

A single numeric value: the KR-21 reliability coefficient.

Examples

data(data.u)
kr21(data.u)

---

Lord-Wingersky Recursive Formula

Description

Compute the raw score distribution for a given theta value using the Lord-Wingersky recursive formula, given item-level probabilities of a correct response.

Usage

lord_wingersky(probs)

Arguments

probs

A numeric vector (or matrix) of probabilities that a given theta value will correctly answer each item. If a matrix is provided, it will be coerced to a numeric vector.

Value

A list with:

x

Vector of possible raw scores, from 0 to ni.

probability

Vector of probabilities for each raw score.

---

Gaussian Quadrature Points and Weights

Description

Generate Gaussian quadrature points and corresponding normalized weights based on the standard normal density over a symmetric interval.

Usage

normal_quadra(n, mm)

Arguments

n	Integer. Number of quadrature points (must be >= 2).
mm	Numeric. Positive value giving the maximum absolute value of the quadrature nodes (range will be from -mm to +mm).

Value

A list with:

nodes

Quadrature nodes from -mm to +mm.

weights

Normalized weights proportional to the standard normal density at each node.

Examples

normal_quadra(41, 5)

---

Marginal Reliability of a Unidimensional IRT Model

Description

Compute marginal reliability for a unidimensional IRT model using either MLE-based or EAP-based information, via Gaussian quadrature over a standard normal ability distribution.

Usage

rel_info(ip, est)

Arguments

ip	A data frame or matrix of item parameters with columns in the order b, a, c, where a is on the D = 1.702 metric. If only 1 or 2 columns are supplied, the info() function is expected to treat them as 1PL/2PL accordingly.
est	A character string specifying the ability estimation method: "MLE" for maximum likelihood or "EAP" for empirical Bayes.

Details

Gaussian quadrature with 41 nodes on [-5, 5] is used to approximate the integrals.

Value

A single numeric value: the marginal reliability (MLE or EAP, depending on est).

Examples

data(ip.u)
rel_info(ip.u, "MLE")

---

Test Reliability and CSEMs for IRT Scores

Description

Compute test reliability for raw scores (and optionally scale scores), along with associated conditional standard errors of measurement (CSEMs), for a unidimensional IRT model.

Usage

rel_test(ip, ct = NULL, nq = 11, D = 1.702)

Arguments

ip	A data frame or matrix of item parameters. Columns are interpreted in order as: 3 columns: b, a, c (3PL; a on the D metric), 2 columns: b, a (2PL; c internally set to 0), 1 column: b (1PL/Rasch; a = 1, c = 0).
ct	Optional. A data frame or matrix containing the score conversion table. If supplied, it must have ni + 1 rows (for raw scores 0:ni) and a column named ss giving the corresponding scale scores. If ct = NULL (default), only raw-score reliability and CSEMs are computed.
nq	Integer. Number of quadrature points used to approximate the standard normal ability distribution. Defaults to 11.
D	Numeric. Scaling constant for the logistic IRT model. Defaults to 1.702.

Value

A list with three components:

fx

A data frame containing the estimated marginal score distribution for raw scores (and scale scores if ct is provided).

rel

A data frame with overall error variance, true score variance, observed score variance, and reliability for raw scores, and additionally for scale scores if ct is provided.

csem

A data frame with theta, weights, expected raw scores and corresponding CSEMs. If ct is provided, expected scale scores and scale-score CSEMs are also included.

Examples

data(ip.u)
data(ct.u)
rel_test(ip.u)
rel_test(ip.u, ct.u)

---

Spearman–Brown Prophecy Formula

Description

Compute the predicted test reliability after changing test length, or compute the required test-length ratio to achieve a desired reliability, using the Spearman–Brown prophecy formula.

Usage

spearman_brown(rxx, input, type = c("r", "l"))

Arguments

rxx	A numeric value indicating the original reliability (must be between 0 and 1, exclusive).
input	A numeric value indicating either: the ratio of new test length to original test length (if type = "r"), or the desired reliability of the new test (if type = "l").
type	Character string specifying the calculation type: "r" Compute new reliability given the length ratio. "l" Compute the length ratio required to achieve a desired reliability.

Details

The Spearman–Brown prophecy formula is:

$r_{yy} = \frac{k r_{xx}}{1 + (k - 1) r_{xx}},$

where $r_{xx}$ is the original reliability and $k$ is the ratio of the new test length to the original test length.

Solving for $k$ gives:

$k = \frac{r_{yy}(1 - r_{xx})}{r_{xx}(1 - r_{yy})}.$

Value

A named list depending on type:

reliability

Predicted reliability of the new test (if type = "r").

ratio

Required ratio of new test length to original test length (if type = "l").

Examples

spearman_brown(0.7, 3.86, "r")
spearman_brown(0.7, 0.90, "l")

---

Stratified Cronbach's Coefficient Alpha

Description

Compute the stratified Cronbach's coefficient alpha for a test composed of several item strata (e.g., subtests or subscales).

Usage

stratified_alpha(x, s)

Arguments

x	A data frame or matrix containing item responses, with rows as subjects and columns as items. Items are assumed to be ordered by stratum.
s	A numeric vector giving the number of items in each stratum. The sum of s must equal ncol(x).

Details

Stratified alpha is an estimate of the internal consistency reliability of a composite test formed by multiple item strata (e.g., subtests). Each stratum reliability is computed using alpha(), and combined using the classical stratified-alpha formula.

Value

A named list with:

stratified_alpha

Stratified Cronbach's coefficient alpha.

Examples

data(data.m)
stratified_alpha(data.m, c(13, 12, 6))

---

Stratified Feldt's Coefficient

Description

Compute the stratified Feldt's coefficient for a test composed of several item strata (e.g., subtests or subscales).

Usage

stratified_feldt(x, s)

Arguments

x	A data frame or matrix containing item responses, with rows as subjects and columns as items. Items are assumed to be ordered by stratum.
s	A numeric vector giving the number of items in each stratum. The sum of s must equal ncol(x).

Details

Stratified Feldt's coefficient is an estimate of internal consistency reliability for a composite test formed by multiple strata.

Value

A named list with:

stratified.feldt

Stratified Feldt's coefficient.

Examples

data(data.m)
stratified_feldt(data.m, c(13, 12, 6))

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