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Tests for comparing elements of a correlation matrix
Abstract
In a variety of situations in psychological research, it is desirable to be able to make statistical comparisons between correlation coefficients measured on the same individuals. For example, an experimenter may wish to assess whether two predictors correlate equally with a criterion variable. In another situation, the experimenter may wish to test the hypothesis that an entire matrix of correlations has remained stable over time. The present article reviews the literature on such tests, points out some statistics that should be avoided, and presents a variety of techniques that can be used safely with medium to large samples. Several illustrative numerical examples are provided.
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“As seen in Table 4, all between-subscale differences in correlation coefficients were statistically significant (all ps < 0.001).”
From Spanish adaptation of the experiential approach scale (Eas) (2021) · cites this paper as [33]
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“The formula given by Steiger (1980) was used to test the statistical difference in predictive utility between antenatal anxiety as assessed by diagnostic interview and antenatal anxiety as assessed using the STAI trait anxiety scale.”
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“scriptions of the partner, following a procedure proposed by Steiger (1980). For four different partners, six paired comparisons can be made for each social value orientation group.”
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“However, it is important to note that the version of Williams’s t that May and Hittner simulated, and which was originally published in Hendrickson, Stanley, and Hills (1970), is mathematically different from the Williams’s t that has been reported by others (e.g., Neill & Dunn, 1975; Steiger, 1980).”
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“…Higgins, & Rothstein, 2009 or Lipsey & Wilson, 2001), we used the Fisher z (hyperbolic arctangent) transformation (z ¼ tanh 1(r)) of the Pearson correlation coefficients for the analysis, and used formulas given in Stieger (1980) to find the variance and covariance of z -transformed correlations.”
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“…and the relations between the RDI-P total externalizing symptoms and informant characteristics, as recommended by previous investigators (see Steiger, 1980). examining informant discrepancies in clinical child research must pay careful attention to the measure of informant discrepancies…”
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Adaptation canadienne-française de la forme révisée du State-Trait Anxiety Inventory de Spielberger
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“In both cases, the correlations for the two outcome measures differed significantly when Steiger’s (1980) method for comparing dependent correlations was used: t(110) 2.”
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“Steiger’s (1980) tests for differences between correlations in the same sample showed that extraversion, conscientiousness, and agreeableness were more stable than both emotional stability and culture (in each case, z42.”
BackgroundWhere to recover from attentional fatigue: An expectancy-value analysis of environmental preference
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“We then used the Hotelling–William test (Steiger 1980) to assess differences in the relative strength of correlations between the measured constructs and the two states of customer inspiration.”
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“Values between 0 and .05 indicate a close fit of the model in relation to its degrees of freedom, .05 to .08 indicates a satisfactory fit of the model, and models with values greater than .10 should be rejected (Steiger 1980; Steiger and Lind 1980).”
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“We tested our hypothesis with a test given by Steiger (1980).”
MethodPersonality and citizenship behavior: The mediating role of job satisfaction.
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“T2 statistic −3 80 −15 18 −18 13 Df 5,849 5,849 5,849 which generates a t-statistic (T2 for comparing elements of a correlation matrix (Steiger 1980).”
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“25), as indicated by Williams’s t test (Steiger, 1980).”
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“Based on Steiger's Z1 Ã statistic for comparing correlations from the same sample (Steiger, 1980), age-related changes in recognition memory for faces and for names significantly differed over this interval (see Table 1).”
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“To detect differences among the correlation coefficients for Blame, paranoia, and hostility, we used Steiger’s Z -test for dependent correlations (based on the r to z transformation; Steiger, 1980).”
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“Correlations coefficients were converted into Fisherʼs z scores (Steiger, 1980) for the purposes of visual comparisons of correlations between conditions.”
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EMPLOYEE ATTITUDES AND CUSTOMER SATISFACTION: MAKING THEORETICAL AND EMPIRICAL CONNECTIONS
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Visceral adipose tissue: relations between single-slice areas and total volume
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“Differences between correlation coefficients from correlated samples were tested by using the method of Steiger (34).”
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Dynamic interactions between musical, cardiovascular, and cerebral rhythms in humans.
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A comparison of Likert scale and traditional measures of self-efficacy.
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Assessment of Cognitive Decline in the Elderly by Informant Interview
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“To see whether these pairs of correlations differed significantly, they were compared using the @‘¿ 2 statistic (Steiger, 1980).”
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Basal forebrain degeneration precedes and predicts the cortical spread of Alzheimer’s pathology
Schmitz, T. W., Spreng, R. N., Initiative, T. A. D. N., Weiner, M. W., Aisen, P. S. S., Petersen, R. C., Jack, C. R., Jagust, W. J., Trojanowki, J. Q., Toga, A. W., et al. (2016) · Nature Communications · doi:10.1038/ncomms13249 · Cited by 345
“N (male, female) 150 (77, 73) 103 (69, 34) 84 (52, 32) 97 (49, 48)”
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“The significance of the differences between WD and PC correlations was tested using the z statistic described by Steiger ( 1980 ) to compare dependent correlations (two-tailed test).”
Estimating Facets of Psychopathy From Normal Personality Traits
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“NOTE: For each variable, a t test for the difference between dependent correlations (Steiger, 1980) was used to evaluate whether correlations differed significantly for PPI-I versus PPI-II.”
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A Health-Related Quality of Life Instrument for Patients Evaluated for Epilepsy Surgery
Vickrey, B. G., Hays, R. D., Graber, J., Rausch, R., Engel, J., & Brook, R. H. (1992) · Medical Care · doi:10.1097/00005650-199204000-00002 · Cited by 332
An Expanded Anxiety Sensitivity Index
Taylor, S., & Cox, B. J. (1998) · Journal of Anxiety Disorders · doi:10.1016/s0887-6185(98)00028-0 · Cited by 399
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“Differences in intra-individual and sex correlation coefficients were examined by using the methods of Steiger and Fisher’s Z transformation, respectively (33, 34).”
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“Specifically, Yuan [86] adapt Steiger’s Z test for a difference in correlations [87] to assess which of the two traits in an analysis has the stronger correlation with the SNP instrument.”
MethodComparing correlations based on individual-level and aggregated data.
Ostroff, C. L. (1993) · Journal of Applied Psychology · doi:10.1037/0021-9010.78.4.569 · Cited by 323
“Although significance tests for comparing correlations from the same sample have been developed ( Steiger, 1980 ), these tests assume one”
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“Indeed, Williams’ test (Steiger 1980) confirmed that the NHI explained significantly more of the variance in average masculinity preference than did average SOI (t19 1⁄4 2.”
BackgroundInfluentialCan inattention/overactivity be an institutional deprivation syndrome?
Kreppner, J. M., O'Connor, T. G., Rutter, M., & English and Romanian Adoptees Study Team (2001) · Journal of Abnormal Child Psychology · doi:10.1023/a:1012229209190 · Cited by 317
Individual Differences Reveal Correlates of Hidden Hearing Deficits
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“To compare the correlations, we used the T2 test, which has been shown to perform well when comparing dependent correlations (Steiger, 1980).”
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Hypertension and the Brain: Vulnerability of the Prefrontal Regions and Executive Functions.
Raz, N., Rodrigue, K. M., & Acker, J. D. (2003) · Behavioral Neuroscience · doi:10.1037/0735-7044.117.6.1169 · Cited by 312
“deepwhite matter) with Steiger’s Z* statistic (Steiger, 1980).”
Gaidatzis, D., Burger, L., Florescu, M., & Stadler, M. B. (2015) · Nature Biotechnology · doi:10.1038/nbt.3269 · Cited by 313
“Notably, the correlation is reduced significantly ( P = 3.3 × 10 −20 , test of difference between two correlated correlations 34 ) when comparing half-lives to ∆ exon ( R = 0.12) and absent when comparing them to ∆ intron ( R = −0.002, P = 1.2 × 10 −50 ).”
BackgroundInfluentialPsychometric properties of the Fatigue Severity Scale and the Modified Fatigue Impact Scale
Learmonth, Y. C., Dlugonski, D., Pilutti, L. A., Sandroff, B. M., Klaren, R., & Motl, R. W. (2013) · Journal of the Neurological Sciences · doi:10.1016/j.jns.2013.05.023 · Cited by 349
“Correlations were compared between FSS and MFIS scores with other outcomes based on the approach provided by Dunn and Clark as referred to by Steiger [50] using the Z1* values; values greater than 1.”
BackgroundMethodDörnyei, Z., & Chan, L. (2013) · Language Learning · doi:10.1111/lang.12005 · Cited by 484
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“To examine whether reframing was associated more strongly with better adjustment than other coping styles, we compared related correlations with Williams’s (1959) modification of T (Steiger, 1980).”
MethodProtected Values and Omission Bias
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Seeing is believing: Trustworthiness as a dynamic belief
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“To evaluate whether the ratings produced by the model were better than the initial normed trustworthiness ratings at predicting the actual post-task participant ratings, we used a Williams's T2 statistic (Steiger, 1980) to compare Fig.”
MethodMentoring provided: Relation to mentor’s career success, personality, and mentoring received
Bozionelos, N. (2003) · Journal of Vocational Behavior · doi:10.1016/s0001-8791(03)00033-2 · Cited by 315
“in the two regression models were compared with the use of Williams formula for non-independent variables (Steiger, 1980; Williams, 1959).”
Measurement of perceived control over anxiety-related events
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Evaluating Holland's and Gati's vocational-interest models: A structural meta-analysis.
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“In addition, the differences in the Gati and Holland correlations with sample size were examined by means of a t test for differences between two correlation coefficients from correlated samples (Steiger, 1980), and they were not found to differ from each other, ;(101) = 1.”
MethodInput normalization by global feedforward inhibition expands cortical dynamic range
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“02), assessed using one-tail test of correlation for dependent variable”
Elliott, M. L., Knodt, A. R., Cooke, M. E., Kim, M. J., Melzer, T. R., Keenan, R. J., Ireland, D. R., Ramrakha, S., Poulton, R., Caspi, A., et al. (2019) · NeuroImage · doi:10.1016/j.neuroimage.2019.01.068 · Cited by 308
“Differences in predictive utility between models were compared using Steiger's z (Steiger, 1980).”
Posttraumatic growth and posttraumatic distress: a longitudinal study.
Dekel, S., Ein-Dor, T., & Solomon, Z. (2012) · Psychological Trauma: Theory, Research, Practice, and Policy · doi:10.1037/a0021865 · Cited by 304
“Of importance, Steiger’s (1980) test for equality of dependent correlations revealed that the magnitude of the cross-sectional association between PTSD and PTG was not significantly different between 2003 and 2008, z(168) .”
BackgroundStallings, L. M., MacDonald, M. C., & O’Séaghdha, P. G. (1998) · Journal of Memory and Language · doi:10.1006/jmla.1998.2586 · Cited by 301
“The Z 1* statistic (Steiger, 1980) is appropriate for this case, in which the two correlations share one factor (shifting frequency), and independence between the correlations cannot be assumed.”
BackgroundCameron, C. D., Brown-Iannuzzi, J. L., & Payne, B. K. (2012) · Personality and Social Psychology Review · doi:10.1177/1088868312440047 · Cited by 352
“In addition, 28 samples reported all three relationships—priming behavior, priming attitudes, and attitudes behavior—allowing us to examine whether implicit attitudes predicted behavior significantly better than explicit attitudes (Steiger, 1980).”
BackgroundWells, A. (2005) · Cognitive Therapy and Research · doi:10.1007/s10608-005-1652-0 · Cited by 300
“In order to test for the significance of differences between correlations as predicted above, Steiger’s (1980) formula for dependent correlations was used.”
MethodExpectation in Melody: The Influence of Context and Learning
Pearce, M. T., & Wiggins, G. A. (2006) · Music Perception An Interdisciplinary Journal · doi:10.1525/mp.2006.23.5.377 · Cited by 297
“…of the two models, since the individual participant data were not available for any of the experiments and the two models are not nested, Williams’ t statistic for comparing dependent correlations (Hittner, May, & Silver, 2003; Steiger, 1980) was used to compare the two models in each experiment.”
MethodNgo, L. V., & O’Cass, A. (2012) · Journal of Product Innovation Management · doi:10.1111/j.1540-5885.2012.00939.x · Cited by 328
“In particular, a Hotelling–Williams test as recommended by Steiger (1980) was computed.”
MethodSkeem, J. L., & Mulvey, E. P. (2001) · Journal of Consulting and Clinical Psychology · doi:10.1037/0022-006x.69.3.358 · Cited by 303
“…and violence was moderately strong (eta = 0.36), but violence was significantly better predicted by scores on the Antisocial Behavior factor (Factor 2, eta = 0.38) than scores on the Emotional Detachment factor (Factor 1, eta = 0.28), T 2 (860) = 3.42,/> < .001 (see Steiger, 1980; Williams, 1959).”
BackgroundInfluentialMarković, S., Iglesias, O., Singh, J. J., & Sierra, V. (2015) · Journal of Business Ethics · doi:10.1007/s10551-015-2985-6 · Cited by 310
Kollias, A., Lagou, S., Zeniodi, M. E., Boubouchairopoulou, N., & Stergiou, G. S. (2016) · Hypertension · doi:10.1161/hypertensionaha.115.06066 · Cited by 287
“In comparing correlation coefficients (central versus brachial), Steiger z test was used.(6) Because several studies did not report correlations between central and brachial BP, the respective correlation coefficient derived from a large French cohort with cardiovascular risk factors was used in the Steiger z test.”
MethodDiener, E., Scollon, C. N., Oishi, S., Dzokoto, V. A. A., & Suh, E. M. (2000) · Journal of Happiness Studies · doi:10.1023/a:1010031813405 · Cited by 293
A meta-analytic examination of work and general locus of control.
Wang, Q., Bowling, N. A., & Eschleman, K. J. (2010) · Journal of Applied Psychology · doi:10.1037/a0017707 · Cited by 292
“k # number of samples; N # total sample size; mean r # average weighted correlation coefficient; CI # confidence interval; z # Steiger’s (1980) test, which was used to examine whether two correlations were significantly different from each other.”
MethodInfluentialT1 Mapping for Myocardial Extracellular Volume Measurement by CMR
White, S. K., Sado, D. M., Fontana, M., Banypersad, S. M., Maestrini, V., Flett, A. S., Piechnik, S. K., Robson, M. D., Hausenloy, D. J., Sheikh, A. M., et al. (2013) · JACC. Cardiovascular imaging · doi:10.1016/j.jcmg.2013.01.011 · Cited by 294
Social-Cognitive and Behavioral Correlates of Aggression and Victimization in Boys' Play Groups
Schwartz, D., Dodge, K. A., Coie, J. D., Hubbard, J. A., Cillessen, A. H. N., Lemerise, E. A., & Bateman, H. V. (1998) · Research on Child and Adolescent Psychopathology · doi:10.1023/a:1022695601088 · Cited by 300
“In addition, t-tests for dependent rs (Steiger, 1980) were conducted, in order to determine whether the correlation coefficients found for reactive and proactive aggression were significantly different.”
Stress at work and psychosomatic complaints: A causal interpretation.
Fresé, M. (1985) · Journal of Applied Psychology · doi:10.1037/0021-9010.70.2.314 · Cited by 302
Waist Circumference Correlates with Metabolic Syndrome Indicators Better Than Percentage Fat
Shen, W., Punyanitya, M., Chen, J., Gallagher, D., Albu, J., Pi-Sunyer, X., Lewis, C. E., Grünfeld, C., Heshka, S., & Heymsfield, S. B. (2006) · Obesity · doi:10.1038/oby.2006.83 · Cited by 288
“For each health risk factor, we tested whether the correlation coefficients were significantly different from each other (33).”
MethodZhu, S., Heymsfield, S. B., Toyoshima, H., Wang, Z., Pietrobelli, A., & Heshka, S. (2005) · American Journal of Clinical Nutrition · doi:10.1093/ajcn.81.2.409 · Cited by 288
“Statistical comparisons of 2 dependent correlation coefficients between risk factor correlating with WC or BMI were made by using z tests (17).”
MethodReflexivity in Teams: A Measure and Correlates
Schippers, M. C., Den Hartog, D. N., & Koopman, P. L. (2007) · Applied Psychology · doi:10.1111/j.1464-0597.2006.00250.x · Cited by 287
Hays, R. D., & Hayashi, T. (1990) · Behavior Research Methods, Instruments, & Computers · doi:10.3758/bf03203140 · Cited by 287
Reciprocity of social exchange and social satisfaction among older women.
Rook, K. S. (1987) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.52.1.145 · Cited by 281
SPSS and SAS programs for comparing Pearson correlations and OLS regression coefficients
Weaver, B., & Wuensch, K. L. (2013) · Behavior Research Methods · doi:10.3758/s13428-012-0289-7 · Cited by 279
“Pearson and Filon (1898) devised a method for comparing two nonindependent correlations with no variables in common, but a revised version of it by Steiger (1980) yields a “ theoretically better test statistic ” (Raghunathan et al., 1996, p. 179).”
MethodContinuity and Change of Psychopathology From Childhood Into Adulthood: A 14-Year Follow-up Study
HOFSTRA, M. B., VAN der ENDE, J., & VERHULST, F. C. (2000) · Journal of the American Academy of Child & Adolescent Psychiatry · doi:10.1097/00004583-200007000-00013 · Cited by 279
Differential aging of the human striatum: longitudinal evidence.
Raz, N., Rodrigue, K. M., Kennedy, K. M., Head, D. M., Gunning‐Dixon, F. M., & Acker, J. D. (2003) · PubMed Central · Cited by 278
Metadata
- Authors
- Steiger, J. H.
- Affiliations
- University of British Columbia (CA)
- Corresponding author
- Steiger, James H.
- Venue
- Psychological Bulletin
- Published
- 1980-03
- Volume
- 87
- Issue
- 2
- Pages
- 245-251
- ISSN
- 0033-2909 (ISSN-L)
- 1939-1455
- OpenAlex
- W2097137621
- Cited by
- 4,621
- References
- 17
- Citations by year
- 2026: 227 · 2025: 157 · 2024: 152 · 2023: 168 · 2022: 212 · 2021: 214 · 2020: 242 · 2019: 259 · 2018: 302 · 2017: 265 · 2016: 261 · 2015: 260 · 2014: 189 · 2013: 197 · 2012: 148
- FWCI
- 16.67
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Psychological Bulletin (publishedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Multidisciplinary Science and Engineering Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- correlation coefficient, hypothesis testing
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