Author: Hume Winzar

# likertMakeR::reliability()

#### Hume Winzar

#### December 2025

    knitr::opts_chunk$set(
      collapse = TRUE,
      comment = "#>"
    )

## Reliability estimation with `LikertMakeR::reliability()`

The `reliability()` function estimates a range of internal consistency reliability coefficients for single-factor Likert and rating-scale measures. It is designed to work naturally with synthetic data generated by LikertMakeR, but applies equally to real survey data.

Unlike many reliability functions, `reliability()`:

- presents multiple coefficients in a tidy table,
- provides bootstrap confidence intervals when requested,
- supports ordinal (polychoric-based) reliability, and
- includes explicit diagnostics explaining when ordinal estimates are not feasible.

### When should you use `reliability()`?

Use `reliability()` when:

- your scale is intended to measure one underlying construct,
- items are Likert-type or bounded rating scales, and
- you want transparent, reproducible reliability estimates, especially for teaching, simulation, or methods work.

The function is not intended for multidimensional scales or SEM models; excellent alternatives already exist for those purposes (e.g. `lavaan`, `semTools`).

## Function usage

    reliability(
      data,
      include = "none",
      ci = FALSE,
      ci_level = 0.95,
      n_boot = 1000,
      na_method = c("pairwise", "listwise"),
      min_count = 2,
      digits = 3,
      verbose = TRUE
    )

## Arguments

### `data`

An `n × k` data frame or matrix containing item responses, where rows correspond to respondents and columns correspond to items.

### `include`

A character vector specifying which additional reliability coefficients to compute.

Possible values are:

- `"none"` (default)
  Computes Cronbach’s alpha and McDonald’s omega (total) using Pearson correlations.

- `"lambda6"`
  Adds Guttman’s lambda-6, computed via `psych::alpha()`
  (requires the optional package psych).

- `"omega_h"`
  Adds McDonald’s omega hierarchical (\(\omega_h\)), also known as Coefficient H.
  This coefficient estimates the maximum reliability of the general factor under a single-factor model, assuming optimal weighting of items.

- `"polychoric"`
  Adds ordinal reliability estimates, computed from polychoric correlations:

  - ordinal alpha (Zumbo’s alpha),
  - ordinal omega (total).

Multiple options may be supplied, for example:

    include <- c("lambda6", "polychoric")

### `ci`

Logical.
If `TRUE`, confidence intervals are computed using a nonparametric bootstrap.

Default is `FALSE`.

### `ci_level`

Confidence level for bootstrap intervals.
Default is `0.95`.

### `n_boot`

Number of bootstrap resamples used when `ci = TRUE`.
Default is `1000`.

Larger values reduce Monte Carlo error but increase computation time, especially for ordinal (polychoric-based) reliability estimates.

### `na_method`

How missing values are handled:

- `"pairwise"` (default): correlations use all available pairs,
- `"listwise"`: rows with any missing values are removed before analysis.

### `min_count`

Minimum observed frequency per response category required to attempt polychoric correlations.
Default is `2`.

Ordinal reliability estimates are skipped if any item contains categories with fewer than `min_count` observations. When this occurs, diagnostics are stored in the returned object and may be inspected using `ordinal_diagnostics()`.

### `digits`

Number of decimal places used when printing estimates.
Default is `3`.

### `verbose`

Logical.
If `TRUE`, warnings and progress indicators are displayed.

Default is `TRUE`.

## Reliability coefficients returned

### Pearson-based coefficients (always available)

- Cronbach’s alpha
  Computed from the Pearson correlation matrix.

- McDonald’s omega (total)
  Computed from the leading eigenvalue of the correlation matrix, assuming a single common factor.

These estimates are appropriate when Likert-scale responses are treated as approximately interval-scaled.

### Ordinal (polychoric-based) coefficients

When `include = "polychoric"`:

- Ordinal alpha (Zumbo’s alpha)
  Cronbach’s alpha computed from the polychoric correlation matrix.

- Ordinal omega (total)
  McDonald’s omega computed from the polychoric correlation matrix.

These estimates are often preferred when items are clearly ordinal, response distributions are skewed, or floor/ceiling effects are present.

## Ordinal diagnostics and safeguards

Ordinal reliability estimation can fail when response categories are sparse (e.g., very few observations in extreme categories).

When this occurs:

- ordinal estimates are skipped rather than forced,
- a warning is issued,
- diagnostics are stored in the returned object.

Diagnostics may be inspected using:

    ordinal_diagnostics(result)

### Hierarchical reliability: \(\omega_h\) (Coefficient H)

When `include = "omega_h"`, `reliability()` reports McDonald’s omega hierarchical (\(\omega_h\)), also known as Coefficient H.

\(\omega_h\) answers a different question from \(\alpha\) or \(\omega\) (total):

> How well would the underlying latent factor be measured if the best possible linear combination of items were used?

Key characteristics of \(\omega_h\):

- It is a model-based upper bound on reliability
- It reflects factor determinacy, not observed-score reliability
- It assumes a single dominant common factor
- It is insensitive to scale length but sensitive to factor structure

\(\omega_h\) is therefore best interpreted as a diagnostic index, rather than as a direct estimate of the reliability of observed summed scores.

#### Why no confidence intervals for \(\omega_h\)?

Confidence intervals are not reported for \(\omega_h\).

This is intentional:

- \(\omega_h\) is a maximal reliability bound, not a descriptive statistic
- Its sampling distribution is highly non-normal
- Bootstrap confidence intervals are often unstable or misleading
- There is no agreed inferential framework for \(\omega_h\) in the literature

Accordingly, \(\omega_h\) is reported as a point estimate only, with explanatory notes in the output table.

## Examples

### Create a synthetic dataset

The example below generates a four-item single-factor scale with a target Cronbach’s alpha of 0.80, using functions from LikertMakeR.

    # example correlation matrix
    my_cor <- LikertMakeR::makeCorrAlpha(
      items = 4,
      alpha = 0.80
    )

    # example correlated dataframe
    my_data <- LikertMakeR::makeScales(
      n = 64,
      means = c(2.75, 3.00, 3.25, 3.50),
      sds = c(1.25, 1.50, 1.30, 1.25),
      lowerbound = rep(1, 4),
      upperbound = rep(5, 4),
      cormatrix = my_cor
    )
    #> Variable  1 :  item01  -
    #> reached maximum of 4096 iterations
    #> Variable  2 :  item02  -
    #> best solution in 545 iterations
    #> Variable  3 :  item03  -
    #> reached maximum of 4096 iterations
    #> Variable  4 :  item04  -
    #> reached maximum of 4096 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

### Basic reliability estimates

By default, `reliability()` returns Pearson-based Cronbach’s alpha and McDonald’s omega (total), assuming a single common factor.

    # $\alpha$ and $\omega$

    reliability(my_data)
    #>    coef_name estimate n_items n_obs                notes
    #>        alpha    0.799       4    64 Pearson correlations
    #>  omega_total    0.869       4    64 1-factor eigen omega

### Including additional coefficients

Additional reliability coefficients may be requested using the `include` argument.

    # $\alpha$, $\omega$ (total), $\lambda 6$, $\omega_h$, and ordinal variants

    reliability(
      my_data,
      include = c("lambda6", "omega_h", "polychoric")
    )
    #>            coef_name estimate n_items n_obs
    #>                alpha    0.799       4    64
    #>          omega_total    0.869       4    64
    #>              lambda6    0.756       4    64
    #>              omega_h    0.803       4    64
    #>        ordinal_alpha    0.759       4    64
    #>  ordinal_omega_total    0.848       4    64
    #>                                                notes
    #>                                 Pearson correlations
    #>                                 1-factor eigen omega
    #>                                       psych::alpha()
    #>     Coefficient H (1-factor FA, maximal reliability)
    #>                              Polychoric correlations
    #>  Polychoric correlations | Ordinal CIs not requested

The available options are:

- `"lambda6"`
  Adds Guttman’s lambda-6, computed using `psych::alpha()`.
  This option requires the suggested package psych.

- `"omega_h"`
  Adds omega hierarchical (Coefficient H), a model-based upper bound on reliability that reflects how well the general factor is measured. \(\omega_h\) is reported as a point estimate only and is best used as a diagnostic indicator of factor strength rather than as observed-score reliability.

- `"polychoric"`
  Adds ordinal (polychoric-based) reliability estimates, including ordinal alpha (Zumbo’s alpha) and ordinal omega (total).

Multiple options may be supplied simultaneously. If `"none"` is included alongside other options, it is ignored.

If ordinal reliability estimates cannot be computed — most commonly due to sparse response categories — they are skipped automatically. In such cases, the returned object contains diagnostic information explaining why the estimates were omitted.

### When should I use each option?

By default, `reliability()` reports Cronbach’s alpha and McDonald’s omega computed from Pearson correlations. This is appropriate for most teaching, exploratory, and applied settings, especially when Likert items have five or more categories and reasonably symmetric distributions.

Use `include = "lambda6"` when you want an additional lower-bound reliability estimate that is less sensitive to tau-equivalence assumptions. Guttman’s lambda-6 is often reported alongside alpha and omega in methodological comparisons and requires the psych package.

Use `include = "omega_h"` when you want to assess the strength and clarity of the general factor underlying a scale. \(\omega_h\) is particularly useful when evaluating whether a set of items meaningfully reflects a single latent construct, but it should not be interpreted as the reliability of summed or averaged scores.

Use `include = "polychoric"` when item responses are clearly ordinal and category distributions are well populated. In this case, the function computes ordinal alpha (Zumbo’s alpha) and ordinal omega based on polychoric correlations. Ordinal methods are most appropriate when response categories are few (e.g., 4–5 points) and when treating items as continuous may be questionable. If response categories are sparse, ordinal estimates are skipped and diagnostics are provided to explain why.

### Notes on computation

All reliability coefficients in `reliability()` are computed under the assumption of a single common factor. The function is intended for unidimensional scales and does not perform factor extraction or dimensionality testing.

Cronbach’s alpha and McDonald’s omega are computed from Pearson correlations by default. When `include = "polychoric"` is specified, ordinal reliability estimates are computed using polychoric correlations, corresponding to Zumbo’s ordinal alpha and ordinal omega total.

Ordinal reliability estimates may be skipped automatically when:

- an item has fewer than two observed response categories, or
- one or more response categories occur fewer than `min_count` times.

In these cases, the function returns `NA` for ordinal estimates and stores diagnostic information explaining the decision. These diagnostics can be inspected using `ordinal_diagnostics()`.

When `ci = TRUE`, confidence intervals are obtained using a nonparametric bootstrap. For ordinal reliability estimates, bootstrap resamples may fail if polychoric correlations cannot be estimated in some resampled datasets. Such failures are tracked internally and reported in the output notes. Increasing `n_boot` can improve the stability of ordinal confidence intervals when the proportion of successful bootstrap draws is high but not complete.

For transparency, methodological details about estimation methods and bootstrap performance are reported alongside point estimates in the returned table.

## Choosing a Reliability Coefficient: A Practical Decision Guide

Researchers and students are often faced with multiple reliability coefficients and little guidance on when each should be used. This section provides a practical, defensible guide for choosing among Cronbach’s alpha, McDonald’s omega, and their ordinal counterparts when working with Likert-type and rating-scale data.

This guidance assumes a single-factor scale, which is the design focus of LikertMakeR.

### Step 1: What kind of data do you have?

#### Continuous or approximately continuous items

Examples:

- Scale scores with many response options

- Visual analogue scales

- Aggregated or averaged ratings

→ Pearson correlations are usually appropriate.

#### Ordinal (Likert-type) items

Examples:

- Single 5-point or 7-point agreement scales

- Frequency scales with clear category boundaries

→ Ordinal (polychoric-based) methods are often more appropriate, especially when responses are skewed or unevenly distributed.

### Step 2: Choosing between \(\alpha\) and \(\omega\)

#### Cronbach’s alpha (\(\alpha\))

Cronbach’s alpha is the most widely reported reliability coefficient and is based on average inter-item correlations.

Use alpha when:

- You need comparability with legacy literature

- Items are roughly tau-equivalent (all items make equal contributions to the underlying factor)

- You want a simple baseline estimate

Limitations:

- Assumes equal factor loadings

- Can underestimate reliability when loadings differ

- Sensitive to the number of items

Alpha should be viewed as a descriptive lower bound, not a definitive measure of internal consistency.

#### McDonald’s omega (\(\omega\))

McDonald’s omega estimates the proportion of variance attributable to a single common factor, allowing items to have different loadings.

Use omega when:

- Items vary in strength or discrimination

- You want a model-based reliability estimate

- A single factor is theoretically justified

Advantages:

- Fewer restrictive assumptions than alpha

- Better behaved in simulations

- Increasingly recommended in methodological literature

As a general rule, omega is preferred to alpha for single-factor scales when factor loadings are unequal.

#### Where does Guttman’s \(\lambda_6\) fit?

Guttman’s lambda-6 (\(\lambda_6\)) is a lower-bound estimate of reliability that relaxes Cronbach’s assumption of equal error variances across items.

Use \(\lambda_6\) when:

- You want a reliability estimate that is:
  - more defensible than \(\alpha\),
  - but does not rely on a factor model
- You are comparing multiple lower-bound estimates
- You want a conservative benchmark alongside \(\omega\)

Key points:

- \(\lambda_6\) is always \(\geqslant\) \(\alpha\) for the same data
- Like \(\alpha\), it is a lower bound — not an estimate of true reliability
- Unlike \(\omega\), it does not assume a latent factor structure

In practice, \(\lambda_6\) is most useful when reported alongside \(\alpha\) and \(\omega\) to show how sensitive conclusions are to different reliability assumptions.

### Step 3: When should I use ordinal reliability?

Ordinal reliability coefficients are computed from polychoric correlations, which estimate associations between latent continuous variables underlying ordinal responses.

In reliability(), these correspond to:

- Ordinal alpha (often called Zumbo’s alpha)

- Ordinal omega

Use ordinal reliability when:

- Items are ordinal (e.g., 5- or 7-point Likert scales)

- Response distributions are skewed or uneven

- You wish to respect the ordinal measurement scale

Important caveats:

- Polychoric correlations require sufficient observations per category

- Sparse categories can cause estimation failure

- Diagnostics should always be inspected

If ordinal estimation is not feasible, reliability() reports this transparently and falls back to Pearson-based estimates.

### Step 4: \(\alpha\) vs \(\omega\) vs ordinal \(\omega\) — a practical summary

Situation | Recommended.coefficient
--- | ---
Legacy comparison, simple reporting | \(\alpha\), Cronbach’s alpha
Single-factor scale, unequal loadings | \(\omega\), McDonalds omega
Strength of general factor | \(\omega_h\), Coefficient H
Likert items with skew or ceiling effects | Ordinal \(\omega\)
Teaching or demonstration | \(\alpha\) and \(\omega\)
Ordinal data, small samples or sparse categories | \(\omega\) (Pearson-based)

When in doubt:

    Report omega, and optionally alpha for comparison.

If your data are clearly ordinal and diagnostics permit:

    Ordinal omega is the most defensible choice.

    “ordinal $\omega$” refers to omega total computed from the
    polychoric correlation matrix.

### Step 5: Confidence intervals

When `ci = TRUE`, LikertMakeR computes nonparametric bootstrap confidence intervals.

Why bootstrap?

- No closed-form CI exists for omega

- Ordinal reliability has no reliable analytic CI

- Bootstrap intervals are flexible and robust

Practical advice:

- Use at least 1,000 resamples for stable intervals

- Expect longer runtimes for ordinal bootstraps

- Always report the method used to compute CIs

Confidence intervals are intentionally not provided for \(\omega_h\), as it represents a model-based upper bound on reliability rather than an inferential estimate.

## Recommended reading

- For readers who want to go a little deeper

If you use reliability() for teaching or applied research, the following sources provide accessible explanations of the ideas behind the coefficients reported here.

### Understanding Cronbach’s alpha and its limitations

- Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. (Cronbach (1951))
  The original source for alpha; still worth reading to understand what alpha does—and does not—measure.

- Revelle, W., & Zinbarg, R. E. (2009). Coefficients alpha, beta, omega, and the glb. (Revelle and Zinbarg (2009))
  A clear discussion of why alpha can be misleading and when omega is preferable.

### Omega and factor-based reliability

- McDonald, R. P. (1999). Test theory: A unified treatment. (McDonald (2013))
  The definitive reference for omega; recommended for readers comfortable with factor analysis concepts.

- Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. (Hancock and Mueller (2001))
  Introduces Coefficient H and discusses its interpretation as factor determinacy rather than observed-score reliability.

### Comparative studies

- Xiao L, Hau KT. (2022). Performance of Coefficient Alpha and Its Alternatives: Effects of Different Types of Non-Normality. (Xiao and Hau (2023))

### Ordinal reliability for Likert-type data

- Zumbo, B. D., Gadermann, A. M., & Zeisser, C. (2007). Ordinal versions of coefficients alpha and theta for Likert rating scales. (Zumbo et al. (2007))
  Introduces ordinal (polychoric-based) alpha—often called Zumbo’s alpha.

- Gadermann, A. M., Guhn, M., & Zumbo, B. D. (2012). Estimating ordinal reliability for Likert-type and ordinal item response data. (Gadermann et al. (2012))
  A practical, non-technical guide that is especially suitable for teaching.

### Polychoric correlations in practice

- Holgado–Tello, F. P., et al. (2010). Polychoric versus Pearson correlations in factor analysis of ordinal variables. (Holgado–Tello et al. (2010))
  A helpful applied comparison explaining why Pearson correlations can distort analyses of Likert-type data.

## Teaching tip

For most classroom examples, start with Pearson-based alpha and omega. Introduce ordinal reliability only after students understand:

1. factor models, and
2. why Likert responses are not truly continuous.

This mirrors the progressive structure used in reliability() and helps students see why additional assumptions are required for ordinal methods.

## Citations

Cronbach, Lee J. 1951. “Coefficient Alpha and the Internal Structure of Tests.” Psychometrika 16 (3): 297–334.
Gadermann, Anne M, Martin Guhn, and Bruno D Zumbo. 2012. “Estimating Ordinal Reliability for Likert-Type and Ordinal Item Response Data: A Conceptual, Empirical, and Practical Guide.” Practical Assessment, Research & Evaluation 17 (3): n3.
Hancock, Gregory R, and Ralph O Mueller. 2001. “Rethinking Construct Reliability Within Latent Variable Systems.” Structural Equation Modeling: Present and Future 195 (216): 60–70.
Holgado–Tello, Francisco Pablo, Salvador Chacón–Moscoso, Isabel Barbero–Garcı́a, and Enrique Vila–Abad. 2010. “Polychoric Versus Pearson Correlations in Exploratory and Confirmatory Factor Analysis of Ordinal Variables.” Quality & Quantity 44 (1): 153–66.
McDonald, Roderick P. 2013. Test Theory: A Unified Treatment. Psychology press.
Revelle, William, and Richard E Zinbarg. 2009. “Coefficients Alpha, Beta, Omega, and the Glb: Comments on Sijtsma.” Psychometrika 74 (1): 145–54.
Xiao, Leifeng, and Kit-Tai Hau. 2023. “Performance of Coefficient Alpha and Its Alternatives: Effects of Different Types of Non-Normality.” Educational and Psychological Measurement 83 (1): 5–27.
Zumbo, Bruno D, Anne M Gadermann, and Cornelia Zeisser. 2007. “Ordinal Versions of Coefficients Alpha and Theta for Likert Rating Scales.” Journal of Modern Applied Statistical Methods 6 (1): 4.