My Reliability Is Lower Than Expected

01

Symptoms

Your published scale had alpha around .85, but your dissertation sample produces .61. A committee member asks whether the scale is unusable or whether you can remove a few items to reach .70. You may also see negative item correlations, inconsistent subscale results, or a higher coefficient after software automatically reverses an item.

The coefficient deserves investigation. It does not identify the cause by itself, and a value from an earlier study is not a guarantee for different participants, translations, administrations, or scoring rules.

02

What This Usually Means

Reliability concerns the consistency of scores under a specified measurement model or replication condition. It is a property of the scores and their use in a population and setting, rather than a permanent property of an instrument. Internal-consistency coefficients do not directly assess stability over time or agreement between raters.

Raw Cronbach alpha is k / (k − 1) × [1 − sum of item variances / variance of the item sum]. It responds to item covariance, scale length, and item variances. For a standardized alpha, the calculation uses the correlation matrix instead. Specify which score and coefficient you report; standardizing items changes their weights in the composite.

Interpreting alpha as score reliability requires assumptions, including appropriate relationships among the true item scores and uncorrelated errors. Alpha is not a test of unidimensionality, and a high value is not evidence that the score measures the intended construct. A low estimate can reflect scoring errors, weak relationships, heterogeneous content, a short scale, or uncertainty in the estimated covariance structure.

03

Common Causes

• A reverse-keyed item was not recoded, an item was recoded twice, or missing-value codes were treated as responses.
• The item set combines distinct constructs or subscales that should not automatically form one total score.
• A short scale has fewer items, or the sample has restricted response variation or floor/ceiling effects.
• Item wording, translation, administration, or interpretation differs from the validation setting.
• Missing-data handling, inappropriate numeric coding, or small-sample uncertainty changes the estimate.
• Correlated errors or redundant items can inflate a coefficient, so a reassuringly high number may also conceal a measurement problem.

04

Run These Checks

1. Reproduce the intended scoring. Check the instrument manual, item list, allowable responses, reverse keys, subscale membership, and missing-item rules. For an item ranging from L to U, a direction reversal is L + U − response. Do not select item direction by whichever coding maximizes alpha.
2. Audit the item data before calculating the score. Inspect response counts, impossible values, missingness, floor/ceiling patterns, duplicate items, and variance. Treat special codes such as 99 as missing when the codebook requires it. Document who contributes to the estimate.
3. Examine relationships and content together. Review the item correlation matrix and corrected item–total correlations, with each item excluded from the rest-score used for its correlation. Unexpected negative relationships can suggest a coding error or incompatible content; they do not justify automatic reversal or deletion.
4. Evaluate the proposed score structure. Use theory, existing validation, and an appropriate exploratory or confirmatory measurement analysis. Consider response categories and distributions when choosing an estimator. Alpha does not decide the number of dimensions.
5. Match the reliability estimate to the score and assumptions. Consider model-based omega when a defensible factor model fits; identify the exact omega and its target. Omega total and omega hierarchical answer different questions. Neither is a repair for a poor measurement model, and a larger estimate is not automatically the correct one.
6. Report uncertainty and consequences. Use an interval method appropriate to the coefficient, sampling design, item type, and missingness. A bootstrap should resample independent sampling units, not treat clustered rows as independent. Explain how measurement limitations affect the intended comparisons, rather than declaring a universal cutoff decisive.

05

What Not to Do

• Do not delete items solely to push alpha above .70 or another conventional threshold.
• Do not accept automatic item reversal without checking the instrument and item meaning.
• Do not treat alpha if item deleted as validation of a revised instrument.
• Do not cite the original validation coefficient as though it described your current scores.
• Do not interpret alpha as a percentage of participants answering correctly, a validity coefficient, or evidence of a single factor.
• Do not present omega as automatically superior without specifying its model, target score, and assumptions.
• Do not claim that increasing sample size necessarily raises reliability; it primarily helps estimate the relevant quantities more precisely.

06

Treatment Options

Correct a verified scoring or data error. Retain an audit trail, recompute affected scores and estimates, and rerun analyses that used the incorrect score.

Report the intended subscales when theory and measurement evidence support them. Splitting a score changes the questions you can answer; do not invent subscales merely because their coefficients are higher.

Retain the established score with qualified interpretation when the scoring is correct and the construct remains defensible. Report the estimate, its uncertainty, population, and limits for the intended use. Requirements for individual decisions may differ from those for group-level research.

Revise item content or the instrument when substantive and measurement evidence supports a revision. Label data-driven changes, preserve content coverage, and seek validation in new data rather than claiming the same sample confirms the revision.

Use an appropriate measurement model when latent-variable analysis addresses the question and the data support it. It can represent measurement error under assumptions, but cannot manufacture construct validity or rescue an unidentified model.

Limit claims when the score is too poorly supported for its intended interpretation. A change of instrument, additional validation, or a narrower question may be more defensible than chasing a coefficient.

07

Worked Example

The code enumerates all 256 equally weighted combinations of two independent centered factors and six independent centered error terms, each taking −1 and +1. These are constructed covariance examples, not randomly sampled participants or Likert responses. We use raw alpha for an equally weighted item sum; every item has the same variance.

In the one-factor world, each of six items equals f1 plus its own error. Each pair has correlation .50. Alpha is .857143 for all six items and .75 for three of them. This isolates scale length while holding pairwise relationships constant; it is not a recommendation to add redundant items.

In the separate two-factor world, items 1–3 measure f1 and items 4–6 measure independent f2. Within each group the correlation is .50; between groups it is zero. The combined six-item alpha is .60, while each intended three-item subscale has alpha .75. The known construction explains the difference. In real data, alpha values alone cannot establish this structure or justify splitting a scale.

We then intentionally reverse the direction of item 1 in that two-factor example. The combined alpha becomes .30. Correcting this known coding error restores .60, but does not make the item set unidimensional. Because these items are centered, multiplying by −1 is the correct demonstration reversal; actual questionnaire scoring must follow its response bounds and manual.

The teaching example demonstrates scoring, length, and covariance structure. It does not estimate omega, test a factor model, handle missing data, or provide sampling intervals. Those steps require methods matched to the real score, design, and responses.

See it in R and Python

Base R; Python requires NumPy: python -m pip install numpy. Both scripts construct the same item patterns; no CSV is needed.

# Case 008: deterministic item-scoring and structure example. Base R.
# 256 equally weighted combinations; no random sampling or ordinal data.
names <- c("f1","f2",paste0("e",1:6))
args <- setNames(rep(list(c(-1,1)),8),names)
dat <- do.call(expand.grid,args)
errors <- as.matrix(dat[paste0("e",1:6)])
one_factor <- errors + dat$f1
two_factor <- errors + cbind(dat$f1,dat$f1,dat$f1,dat$f2,dat$f2,dat$f2)
miskeyed <- two_factor
miskeyed[,1] <- -miskeyed[,1]  # deliberately wrong direction
alpha <- function(items) {
  S <- cov(items)
  k <- ncol(items)
  stopifnot(k>1, sum(S)>0)
  k/(k-1)*(1-sum(diag(S))/sum(S))
}
results <- c(one_factor_6=alpha(one_factor),one_factor_3=alpha(one_factor[,1:3]),
 two_factor_6=alpha(two_factor),wrong_key_6=alpha(miskeyed),
 subscale_1=alpha(two_factor[,1:3]),subscale_2=alpha(two_factor[,4:6]))
print(round(results,6))
print(round(cor(two_factor),3))
corrected <- miskeyed
corrected[,1] <- -corrected[,1]  # known coding error, not data-driven key selection
print(c(after_known_key_correction=alpha(corrected)))
stopifnot(nrow(dat)==256L, max(abs(results-c(6/7,.75,.6,.3,.75,.75)))<1e-10,
 abs(alpha(corrected)-.6)<1e-10)
sessionInfo()

"""Case 008: deterministic scoring and structure example. NumPy only.
256 equally weighted combinations, not sampled participants or ordinal items.
"""
import itertools
import numpy as np
dat=np.array(list(itertools.product([-1.,1.],repeat=8)))
f1,f2=dat[:,0],dat[:,1]
errors=dat[:,2:]
one_factor=errors+f1[:,None]
two_factor=errors+np.column_stack([f1,f1,f1,f2,f2,f2])
miskeyed=two_factor.copy()
miskeyed[:,0]*=-1  # deliberately wrong direction
def alpha(items):
    S=np.cov(items,rowvar=False,ddof=1)
    k=items.shape[1]
    assert k>1 and S.sum()>0
    return k/(k-1)*(1-np.trace(S)/S.sum())
results=np.array([alpha(one_factor),alpha(one_factor[:,:3]),alpha(two_factor),
                  alpha(miskeyed),alpha(two_factor[:,:3]),alpha(two_factor[:,3:])])
print("one_factor_6 one_factor_3 two_factor_6 wrong_key_6 subscale_1 subscale_2")
print(np.round(results,6))
print("Two-factor item correlations:\n",np.round(np.corrcoef(two_factor,rowvar=False),3))
corrected=miskeyed.copy()
corrected[:,0]*=-1  # correct only the known coding error
print("after_known_key_correction:",round(alpha(corrected),6))
assert len(dat)==256
assert np.allclose(results,[6/7,.75,.6,.3,.75,.75],atol=1e-10,rtol=0)
assert np.isclose(alpha(corrected),.6,atol=1e-10,rtol=0)
print("All assertions passed; NumPy",np.__version__)

Download verified R script · Download verified Python script

08

What to Tell Your Committee

“For [intended score and use], we estimated [coefficient and interval] in [sample and setting]. We verified [item keys, ranges, missingness, and scoring rules] and examined [item behavior and measurement structure]. We [retained/revised] the score because [substantive and measurement rationale]. Any data-driven revisions are labeled and require further validation. The estimate limits [specific interpretation], rather than proving [unsupported claim].”

Bring the scoring manual, an item summary, the correlation pattern, the proposed measurement structure, and the estimate with uncertainty. Explain the decision for the score you actually use, rather than defending a number against a universal threshold.

09

When You Need More Help

Bring the instrument and scoring instructions, permitted response values, intended total and subscale definitions, sample description, missingness summary, reliability output, and any factor-model results. A consultation can help distinguish scoring issues, structural questions, and what the scores can support.

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Related case: My Sample Is Smaller Than Planned

Technical references:

• Cronbach: Coefficient alpha and the internal structure of tests
• Sijtsma: On the use, misuse, and limited usefulness of alpha
• Dunn, Baguley & Brunsden: From alpha to omega