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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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“Further, in order to determine whether the magnitude of the estimate between IA and T2 ER was significantly different from the magnitude of the estimate between HI and T2 ER, we conducted a Steiger’s Z (Steiger 1980; Uitenbroek 1997) test to examine which ADHD variable (i.”
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“…of the difference between two dependent correlations is a function of the correlations between the three variables Y, A, and B and sample size N (Steiger, 1980a; Williams, 1959), TðN 3Þ ¼ rYA rYBð Þ…”
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“Significant sex differences in all correlations were assessed using the Fisher r-to-z transformation formula (Steiger, 1980).”
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“H 0 denotes the best-performing condition and all differences between other conditions and H 0 are significant ( P < 0.05, Steiger’s test 53 ). c , The effect of including different feature groups (individually and cumulatively) on the correlation of predictions aggregated from the heldout target…”
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Self-Construals Across Cultures
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“Dependent r tests (Steiger, 1980) showed that these predicted correlations are significantly higher than the highest nonpredicted correlations presented in Table 3.”
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“whether there was a significant difference among the strengths of these associations, Steiger’s Z test was chosen as a more conservative estimate than Hotelling’s t statistic to account for conducting multiple comparisons (Steiger, 1980).”
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“It was tested following a procedure designed to test for the significance of the difference between two correlated correlations (Tabachnick & Fidell, 2001, p. 146) introduced by Steiger (1980).”
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“Signal changes for IA and EA showed a significant differential association to BHS, as revealed by using a paired t-test for r-values according to Williams (Steiger, 1980; Williams, 1959) (T = −2.”
InfluentialHäusler, M., Strecker, C., Huber, A., Brenner, M., Höge, T., & Höfer, S. (2017) · Frontiers in Psychology · doi:10.3389/fpsyg.2017.01159 · Cited by 144
“We conducted Steiger tests (one-tailed) for comparing dependent correlations (Steiger, 1980).”
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“If so, statistical procedures for testing multivariate independence (e.g., Bartlett, 1950, 1954; Box, 1949; Steiger, 1980a, 1980b) may be used to help determine dimensionality.”
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“The difference of the size of partial correlations was determined by Steiger’s Z test [28].”
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“This was confirmed by tests for the equality of two dependent correlations (Steiger, 1980).”
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“Differences between related correlations were evaluated using the procedure described by Steiger (1980) .”
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“09 (Steiger, 1980).”
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“Rosenthal, & Rubin, 1992; Steiger, 1980 ).2 These tests demon-”
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“A test comparing the magnitude of the correlations between fluency and liking in the in-group and out-group conditions using Steiger's (1980) method revealed that, not surprisingly, there was no significant difference between them, Z2⁎=−0.”
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“The formulae developed by Steiger (1980) are used and show that: (1) the correlations between CSP (T+1) and CFP (T+1) are significantly higher than that of CSP (T) and CFP (T+1) for T=2010 and T=2012 at the p<0.1 level; and (2) the correlation between CFP (T) and CSP (T) is not significantly…”
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“Steiger’s test (which analyzes the difference in the strength of dependent correlations; Steiger, 1980) revealed a significant difference between these relationships t(169) = 6.01 (p < .001).”
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“A special statistical test is required to test if, within each group, the correlation between measure j (gain) and k (motion threshold) is significantly different from the correlation between j and h (measures of attention) (Steiger, 1980).”
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“Given the limitations of using Hotelling’s t test to compare related correlation coefficients (Steiger, 1980), we used Steiger’s z to compare the correlation coefficients between the RTSHIA scales and relevant standardized measures.”
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“We compared the correlation coefficients between each voxel and its mirror image in the other hemisphere using Steiger’s test (89).”
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“Although a t test revealed no significant differences between the two span–cognition correlations [t(145) � 1; Steiger, 1980 ], we conducted another set of regressions similar to those described above, except that here, individual tasks, rather than composites, were used.”
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“Calculations for the test of the difference between two independent correlation coefficients [54] were conducted with computer software available at http://quantpsy.”
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“Differences in correlations between cognitive tests and speech perception tests were assessed by computing z-values for differences between correlations following Steiger (1980).”
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“The subsequent model, removing direct paths from baseline symptoms to follow-up functioning, was an adequate fit and all paths were significant (x2 = 27.32, df = 31, p = .66, CFI = 1.00, NFI = 0.90, RMSEA = 0.00, 90% CI = 0.00–0.07).”
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“Regarding the third condition, we conducted a t-test of dependent correlations (Steiger, 1980) and found that the observed relationship (r = – 0.”
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Cognitive function and brain structure correlations in healthy elderly East Asians
Chee, M. W. L., Chen, K. H. M., Zheng, H., Chan, K. P. L., Isaac, V., Sim, S. K. Y., Chuah, L. Y. M., Schuchinsky, M., Fischl, B. R., & Ng, T. P. (2009) · NeuroImage · doi:10.1016/j.neuroimage.2009.01.036 · Cited by 122
“We applied Steiger's Z⁎ statistic (Stieger, 1980) when determining whether slopes of cognitive decline (or volume decline) vs.”
Evaluation of the Psychometric Properties of the Revised Short-Form McGill Pain Questionnaire
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“Model fit was assessed using the following indices: w2 divided by degrees of freedom (w2/df), the Non-Normed Fit Index (NNFI; Bentler & Bonett, 1980), the Comparative Fit Index (CFI; Bentler, 1990), and the Root Mean Square of Approximation (RMSEA; Steiger, 1980).”
MethodWolf, S. T., Cohen, T. R., Panter, A. T., & Insko, C. A. (2009) · Self and Identity · doi:10.1080/15298860903106843 · Cited by 122
“We used Steiger's ( 1980 ) method for evaluating dependent correlations as implemented in the MULTICORR v2.”
MethodAkimoto, T., Yoshizawa, H., Watanabe, Y., Numata, A., Yamazaki, T., Takeshima, E., Iwazu, K., Komada, T., Otani, N., Morishita, Y., et al. (2012) · BMC Nephrology · doi:10.1186/1471-2369-13-155 · Cited by 120
“The differences between the independent correlation coefficients were compared statistically by performing a Fisher’s r-to-z transformation [13-15].”
MethodOjo, E. R., Bullock, P., L’Heureux, J., Powers, J., McNairn, H. E., & Pacheco, A. (2015) · Vadose Zone Journal · doi:10.2136/vzj2014.08.0114 · Cited by 120
“his was determined by conducting a test of the equality of two dependent correlation coeicients with one variable in common (Steiger, 1980).”
MethodMemory-Guided Attention: Independent Contributions of the Hippocampus and Striatum
Goldfarb, E. V., Chun, M. M., & Phelps, E. A. (2016) · Neuron · doi:10.1016/j.neuron.2015.12.014 · Cited by 137
“We then correlated these responses with attention benefits from each mnemonic cue and compared the dependent correlations (Steiger, 1980, Equations 3, 10, and 14).”
Nature and measurement of attention control.
Burgoyne, A. P., Tsukahara, J. S., Mashburn, C. A., Pak, R., & Engle, R. W. (2023) · Journal of Experimental Psychology General · doi:10.1037/xge0001408 · Cited by 140
Anxiety sensitivity and pain: Generalisability across noxious stimuli
Thompson, T., Keogh, E., French, C. C., & Davis, R. (2007) · Pain · doi:10.1016/j.pain.2007.04.018 · Cited by 120
Wenzel, M. (2000) · Personality and Social Psychology Bulletin · doi:10.1177/0146167200264004 · Cited by 123
Oakhill, J. V., & Yuill, N. (1986) · Language and Speech · doi:10.1177/002383098602900104 · Cited by 120
Neural mechanisms underlying heterogeneity in the presentation of anxious temperament
Shackman, A. J., Fox, A. S., Oler, J. A., Shelton, S. E., Davidson, R. J., & Kalin, N. H. (2013) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1214364110 · Cited by 113
“Between one-third and two-thirds of anxiety patients are treatment-resistant or refractory (33), underscoring the need to develop more efficacious interventions.”
Judging a Man by the Width of His Face
Valentine, K. A., Li, N. P., Penke, L., & Perrett, D. I. (2014) · Psychological Science · doi:10.1177/0956797613511823 · Cited by 119
Taubner, S., White, L. O., Zimmermann, J., Fonagy, P., & Nolte, T. (2013) · Research on Child and Adolescent Psychopathology · doi:10.1007/s10802-013-9736-x · Cited by 118
“Psychopathy and proactive aggression were correlated with a higher effect size than psychopathy and reactive aggression, z=2.66, p 0.01 (Steiger 1980).”
InfluentialMotivations to volunteer: The role of altruism
Burns, D. J., Reid, J. S., Toncar, M., Fawcett, J., & Anderson, C. E. (2006) · International Review on Public and Nonprofit Marketing · doi:10.1007/bf02893621 · Cited by 119
Case-sensitive letter and bigram frequency counts from large-scale English corpora
Jones, M. N., & Mewhort, D. J. K. (2004) · Behavior Research Methods, Instruments, & Computers · doi:10.3758/bf03195586 · Cited by 119
“The pattern supports our presumption that uppercase decision latency is predicted better by upper- than by lowercase counts.3 We tested the difference between these correlations with Williams’ (1959) ratio for nonindependent correlations (see also Steiger, 1980 ).”
BackgroundArab-Moghaddam, N., & Sénéchal, M. (2001) · International Journal of Behavioral Development · doi:10.1080/01650250042000320 · Cited by 119
“, adjusted for differences in the reliability of the measures), was statistically signi cant (z ˆ 2.19, p < .05; Steiger, 1980).”
BackgroundInfluentialAnthony, J. L., Lonigan, C. J., Hooe, E. S., & Phillips, B. M. (2002) · Journal of Clinical Child & Adolescent Psychology · doi:10.1207/s15374424jccp3104_7 · Cited by 119
Chio, F. H. N., Mak, W. W. S., & Yu, B. C. L. (2021) · Clinical Psychology Review · doi:10.1016/j.cpr.2021.101986 · Cited by 135
“To further examine if the predictability of positive and negative components on mental well-being and psychological distress were different, method recommended by Steiger (1980) in comparing dependent correlations was used.”
MethodHyde, L. W., Shaw, D. S., Gardner, F. E. M., Cheong, J., Dishion, T. J., & Wilson, M. N. (2013) · Development and Psychopathology · doi:10.1017/s0954579412001101 · Cited by 132
“…cross-sectionally or longitudinally predicting problem behavior scores (e.g., age 3 deceitfulness and age 4 problem behavior r¼ .37; age 3 callousness and age 4 problem behavior r¼ .40) using Steiger’s Z test (Steiger, 1980) from the FZT computator (http://psych.unl.edu/psycrs/statpage/regression).”
Age-related functional reorganization, structural changes, and preserved cognition
David Meunier, A., Stamatakis, E. A., & TYLER, L. K. (2013) · Neurobiology of Aging · doi:10.1016/j.neurobiolaging.2013.07.003 · Cited by 121
Brain-age prediction: A systematic comparison of machine learning workflows
More, S., Antonopoulos, G., Hoffstaedter, F., Caspers, J., Eickhoff, S. B., Patil, K. R., & Initiative, T. A. D. N. (2023) · NeuroImage · doi:10.1016/j.neuroimage.2023.119947 · Cited by 118
Post-traumatic stress disorder and medication adherence: Results from the Mind Your Heart Study
Kronish, I. M., Edmondson, D. E., Li, Y., & Cohen, B. E. (2012) · Journal of Psychiatric Research · doi:10.1016/j.jpsychires.2012.06.011 · Cited by 118
“To determine if there were differences in the strength of the association between PTSD and type of non-adherence (forgetting to take and deciding to skip medications), we transformed the odds ratios for the association between PTSD and non-adherence into rscores and then derived a Z-statistic according to the method r…”
MethodValidity of the Five-Factor Model for the Implicit Self-Concept of Personality
Schmukle, S. C., Back, M. D., & Egloff, B. (2008) · European Journal of Psychological Assessment · doi:10.1027/1015-5759.24.4.263 · Cited by 117
Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators
Couillet, R., & McKay, M. R. (2014) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2014.06.018 · Cited by 117
“a group of tested individuals (Steiger, 1980), or electrical engineering at large where signal samples extracted from a possibly short time window are used to retrieve parameters of the signal (Scharf, 1991).”
Background
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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