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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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Statistical power analyses using G∗Power 3.1: tests for correlation and regression analyses
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“45) than to the BIS (T = 5.38, p< .01; Steiger, 1980).”
InfluentialOptimized sgRNA design to maximize activity and minimize off-target effects of CRISPR-Cas9
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“Thus, to determine whether the estimated mean validities for each leadership criterion were significantly different, we conducted the Steiger (1980) test, which takes dependency into account and allowed us to calculate a test statistic with a Student's t distribution .”
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“Although effort attributions for success were correlated with both mastery (r = .37) and performance (r = .14) goal orientation, a T 2 test (Steiger, 1980) for testing differences between nonindependent correlations showed that the perceived covariation between effort and success was more related…”
BackgroundCrawford, J. R., & Henry, J. D. (2004) · British Journal of Clinical Psychology · doi:10.1348/0144665031752934 · Cited by 2,957
“Steiger’s (1980) test revealed that the proportion explained by PA was significantly greater than that explained by NA (z = 2.”
Orienting the causal relationship between imprecisely measured traits using GWAS summary data.
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“This can be achieved by assessing which of the two variables has the biggest absolute correlation with g (S2 Text), formalised by testing for a difference in the correlations ρgx and ρgy using Steiger’s Z-test for correlated correlations within a population [55].”
MethodComparing correlated correlation coefficients
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First Impressions: Making Up Your Mind After a 100-Ms Exposure to a Face
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Thinning of the Cerebral Cortex in Aging
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“We applied Steiger’s Z* statistic which determines the significance of slope differences while accounting for the correlation between the nonoverlapping variables as described in Steiger (1980) and employed in Raz et al. (1997) to determine whether slopes differed within and across lobes.”
Methodcocor: A Comprehensive Solution for the Statistical Comparison of Correlations
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“For the case of dependent nonoverlapping correlations, the pros and cons of various tests are discussed in Raghunathan, Rosenthal, and Rubin [14], Silver, Hittner, and May [15], and Steiger [13].”
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“In order to account for ependencies between these correlations the covariance between he values was calculated (Steiger, 1980).”
BackgroundDiamantopoulos, A., Sarstedt, M., Fuchs, C., Wilczynski, P., & Kaiser, S. (2012) · Journal of the Academy of Marketing Science · doi:10.1007/s11747-011-0300-3 · Cited by 1,812
“However, this test is only appropriate when correlations from two independent (as opposed to paired) samples are to be compared (e.g., Steiger 1980); for related correlation coefficients, Ferguson’s (1971) or Meng et al.’s (1992) procedures should be employed.”
MethodShen, W., Punyanitya, M., Wang, Z., Gallagher, D., St.-Onge, M.-P., Albu, J., Heymsfield, S. B., & Heshka, S. (2004) · Journal of Applied Physiology · doi:10.1152/japplphysiol.00744.2004 · Cited by 1,692
“Differences between correlated correlation coefficients were tested using the method of Steiger (26).”
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Dimensions of achievement motivation in schoolwork and sport.
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An Empirical Assessment of Organizational Commitment and Organizational Effectiveness
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Comparing network structures on three aspects: A permutation test.
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“Steiger’s test is about testing dependent correlations (i.e., in the same sample) but between different variables (Steiger, 1980).”
BackgroundDevelopment and validation of a revised short version of the working alliance inventory
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“WAI-SR /Working Alliance Inventory /Short Revised developed in this study; WAIS /Working Alliance Inventory /Short ([Tracey & Kokotovic, 1989]); WAI /Working Alliance Inventory (Horvath & Greenburg, 1989); z /Steiger (1980).”
MethodWang, G., Oh, I.-S., Courtright, S. H., & Colbert, A. E. (2011) · Group & Organization Management · doi:10.1177/1059601111401017 · Cited by 1,315
“However, because the two corrected mean correlations for task and contextual performance were not based on completely independent samples, we conducted the Hotelling-Williams Test, which compares two dependent correlations (Judge & Piccolo, 2004, p. 759; Steiger, 1980).”
MethodSubtlex-UK: A New and Improved Word Frequency Database for British English
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“For instance, the t -value of the Hotelling – Williams test (Steiger, 1980) 8 of the difference in correlation with SUBTLEX-UK (Zipf) and BNC (Zipf) equals 16.8 ( df = 28,282, p , .001).”
MethodInfluentialThe Forgotten Ones? The Validity of Consideration and Initiating Structure in Leadership Research.
Judge, T. A., Piccolo, R. F., & Ilies, R. (2004) · Journal of Applied Psychology · doi:10.1037/0021-9010.89.1.36 · Cited by 1,265
“k number of correlations; N combined sample size; r mean observed correlation; ̂ estimated true score correlation; Z Steiger (1980) statistic for differences in correlations of Consideration and Initiating Structure with the criteria.”
MethodInfluentialDeterminants of Entrepreneurial Intent: A Meta–Analytic Test and Integration of Competing Models
Schlaegel, C., & Koenig, M. (2013) · Entrepreneurship Theory and Practice · doi:10.1111/etap.12087 · Cited by 1,409
“We followed the recommendations in the literature for comparing nonindependent correlations (Meng, Rosenthal, & Rubin, 1992) and applied Steiger’s z-test (Steiger, 1980) as well as the procedure suggested by Zou (2007), which takes into account the confidence limits around overlapping effect sizes.”
MethodChu, C., Buchman-Schmitt, J. M., Stanley, I. H., Hom, M. A., Tucker, R. P., Hagan, C. R., Rogers, M. L., Podlogar, M. C., Chiurliza, B., Ringer, F. B., et al. (2017) · Psychological Bulletin · doi:10.1037/bul0000123 · Cited by 1,231
“To account for the dependency between these two correlations, z values were corrected by incorporating the association between perceived burdensomeness and thwarted belongingness, according to procedures recommended by Steiger (1980).”
MethodRaz, N. (1997) · Cerebral Cortex · doi:10.1093/cercor/7.3.268 · Cited by 1,292
“between the variables correlated with age (Steiger, 1980).”
MethodLeader-member exchange and citizenship behaviors: A meta-analysis
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“A Hotelling–Williams test, recommended when comparing nonindependent correlations that share a variable (see Steiger, 1980), showed that the two correlations were significantly different from each other.”
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Microcomputer Playfulness: Development of a Measure With Workplace Implications
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Pie-Expansion Efforts: Collaboration Processes in Buyer-Supplier Relationships
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Large-scale physical activity data reveal worldwide activity inequality
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“For example, results of the first-order oblique model initially developed by Sullivan et al. (1995) and confirmed by Osman et al.”
ResultBringing order out of chaos: Psychometric characteristics of the confusion, hubbub, and order scale
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A longitudinal study of abnormalities on MRI and disability from multiple sclerosis
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Robust prediction of individual creative ability from brain functional connectivity
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Enhancement and denial in socially desirable responding.
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“19) is significantly lower than the corresponding intercorrelations for the SE Scale (.74), Z = 6.02, p .001, and for the IMS (.47), Z = 2.63, p .01 (Steiger, 1980).”
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“Furthermore, we hypothesized specificity in the relationship between OBN and depression changes by comparing the strength of this correlation with that between the perceptual factors from the ASC, namely VRS and AUA, and depression changes (Steiger, 1980; Lee and Preacher, 2013).”
BackgroundInfluentialZheng, J., Haberland, V., Baird, D., Walker, V., Haycock, P., Hurle, M., Gutteridge, A., Erola, P., Liu, Y., Luo, S., et al. (2019) · doi:10.1101/627398 · Cited by 812
Social Cognition, Prosocial Behavior, and Emotion in Preschoolers: Contextual Validation
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Automatic personality assessment through social media language.
Park, G., Schwartz, H. A., Eichstaedt, J. C., Kern, M. L., Kosinski, M., Stillwell, D. J., Ungar, L. H., & Seligman, M. E. P. (2015) · Journal of Personality and Social Psychology · doi:10.1037/pspp0000020 · Cited by 802
“To test whether aggregate ratings were significantly more accurate than informants alone, we used a significance test for dependent correlations as described by Steiger (1980; as implemented by Revelle, 2014).”
MethodDeveloping and investigating the use of single-item measures in organizational research.
Fisher, G. G., Matthews, R. A., & Gibbons, A. M. (2015) · Journal of Occupational Health Psychology · doi:10.1037/a0039139 · Cited by 794
Reliability and validity of the Beck anxiety inventory
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Learmonth, Y. C., Motl, R. W., Sandroff, B. M., Pula, J. H., & Cadavid, D. (2013) · BMC Neurology · doi:10.1186/1471-2377-13-37 · Cited by 790
“We examined the significance of differences in the magnitude of dependent correlations between PDDS and EDSS scores with other variables [31]; the significance of differences was based on an alpha value of .”
MethodTurnley, W. H., Bolino, M. C., Lester, S. W., & Bloodgood, J. M. (2003) · Journal of Management · doi:10.1177/014920630302900204 · Cited by 776
McVay, J. C., & Kane, M. J. (2009) · Journal of Experimental Psychology Learning Memory and Cognition · doi:10.1037/a0014104 · Cited by 849
“39 for Blocks 1–4, respectively), as indicated by Williams’s t test (Steiger, 1980).”
Toward using confidence intervals to compare correlations.
Zou, G. (2007) · Psychological Methods · doi:10.1037/1082-989x.12.4.399 · Cited by 744
Evaluating Faces on Trustworthiness After Minimal Time Exposure
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Raz, N., Gunning‐Dixon, F. M., Head, D. M., Rodrigue, K. M., Williamson, A. L., & Acker, J. D. (2003) · Neurobiology of Aging · doi:10.1016/s0197-4580(03)00118-0 · Cited by 748
“The magnitude of correlations was compared among the regional volumes adjusted for height using Steiger’s Z∗ test that takes into account the strength of association between the members of all relevant pairs of variables [ 71 ].”
MethodA large-scale meta-analysis relating the Hare measures of psychopathy to antisocial conduct.
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“We used the Z statistic derived from Steiger (1980) to compare the relative abilities of ?x and F2 to explain future antisocial conduct.”
MethodThe proxy problem: child report versus parent report in health-related quality of life research
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“Agreement between the children and parents was further quantified using the means of the absolute differences, the means of the differences (mean bias) and the associated standard deviations,29 Pearson correlation coefficients and the intraclass correlations (ICC).30,31 Differences between the HRQoL and health status…”
MethodSafeguarding Interorganizational Performance and Continuity Under Ex Post Opportunism
Jap, S. D., & Anderson, E. (2003) · Management Science · doi:10.1287/mnsc.49.12.1684.25112 · Cited by 694
“If the ceiling effect does not hold in the other group, a difference in the impact of governance mechanism across opportunism levels may appear.”
BackgroundKlonsky, E. D., May, A. M., & Glenn, C. R. (2013) · Journal of Abnormal Psychology · doi:10.1037/a0030278 · Cited by 692
“Next, following the procedures of Steiger (1980), we examined whether the association between NSSI and attempted suicide varied by gender (see Table 2).”
MethodConcepts and measures related to connection to nature: Similarities and differences
Tam, K. (2013) · Journal of Environmental Psychology · doi:10.1016/j.jenvp.2013.01.004 · Cited by 691
Asendorpf, J. B., Banse, R., & Mücke, D. (2002) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.83.2.380 · Cited by 721
A comparison of the discrete and dimensional models of emotion in music
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“…prediction rates of the three alternative dimensional models were evaluated using a comparison of the difference between two multiple correlations (Steiger, 1980), which involves transforming the multiple correlations of the predicted models into Z scores and adjusting for mutual correlation and…”
MethodArciniega, G. M., Anderson, T. C., Tovar‐Blank, Z. G., & Tracey, T. J. G. (2008) · Journal of Counseling Psychology · doi:10.1037/0022-0167.55.1.19 · Cited by 681
“In addition, differences between correlations on these two scales were evaluated with the Steiger (1980) modified z statistics, which are reported in the right column of Table 3.”
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Reasons for Studying: Motivational Orientations and Study Strategies
Nolen, S. B. (1988) · Cognition and Instruction · doi:10.1207/s1532690xci0504_2 · Cited by 670
“associations with a common index (Steiger, 1980).”
Negative Affects Deriving From the Behavioral Approach System.
Carver, C. S. (2004) · Emotion · doi:10.1037/1528-3542.4.1.3 · Cited by 691
“Comparisons of the final betas (Steiger, 1980) confirmed that BIS was a significantly stronger predictor of nervousness than any BAS scale, smallest T(463) 5.”
BackgroundMethodResultInfluentialThe validity of pain intensity measures: what do the NRS, VAS, VRS, and FPS-R measure?
Thong, I. S. K., Jensen, M. P., Miró, J., & Tan, G. (2018) · Scandinavian Journal of Pain · doi:10.1515/sjpain-2018-0012 · Cited by 652
Benning, S. D., Patrick, C. J., Hicks, B. M., Blonigen, D. M., & Krueger, R. F. (2003) · Psychological Assessment · doi:10.1037/1040-3590.15.3.340 · Cited by 691
“Therefore, we performed statistical tests of the difference between the correlations for the two PPI factors—that is, t tests for dependent correlations (Steiger, 1980)—to determine whether these divergences were reliable.”
MethodYoungstrom, E. A., Loeber, R., & Stouthamer‐Loeber, M. (2000) · Journal of Consulting and Clinical Psychology · doi:10.1037/0022-006x.68.6.1038 · Cited by 669
Extending the challenge-hindrance model of occupational stress: The role of appraisal
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Representing word meaning and order information in a composite holographic lexicon.
Jones, M. N., & Mewhort, D. J. K. (2007) · Psychological Review · doi:10.1037/0033-295x.114.1.1 · Cited by 635
“We tested the difference between these correlations with E. J. Williams’s (1959) ratio for nonindependent correlations (see also Steiger, 1980).”
MethodMar, R. A., Oatley, K., Hirsh, J., dela Paz, J., & Peterson, J. B. (2006) · Journal of Research in Personality · doi:10.1016/j.jrp.2005.08.002 · Cited by 614
“Fiction was positively correlated with MIE scores whereas Non-Wction was unrelated, and this diVerence in association was statistically signiWcant ( Steiger, 1980 ), t(91) D 3.38, p < .05.”
MethodInfluentialInvestigating an alternate form of the cognitive reflection test
Thomson, K. S., & Oppenheimer, D. M. (2016) · Judgment and Decision Making · doi:10.1017/s1930297500007622 · Cited by 612
An Algorithm for Predicting the Intelligibility of Speech Masked by Modulated Noise Maskers
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Mutuality and Reciprocity in the Psychological Contracts of Employees and Employers.
Dabos, G. E., & Rousseau, D. M. (2004) · Journal of Applied Psychology · doi:10.1037/0021-9010.89.1.52 · Cited by 605
Job satisfaction: Environmental and genetic components.
Arvey, R. D., Bouchard, T. J., Segal, N. L., & Abraham, L. M. (1989) · Journal of Applied Psychology · doi:10.1037/0021-9010.74.2.187 · Cited by 616
“To determine if the intraclass correlation for the intrinsic satisfaction subscale differed from that of the extrinsic satisfaction composite, we performed a one-tailed test for the equality of two dependent correlations (Snedecor & Cochran, 1967, p. 295; Steiger, 1980).”
MethodMaster, A., Cheryan, S., & Meltzoff, A. N. (2015) · Journal of Educational Psychology · doi:10.1037/edu0000061 · Cited by 662
“Using Fisher’s r -to- z transformation, we compared the size of the correlations (Steiger, 1980).”
BackgroundMethodEdwards, J. R., & Rothbard, N. P. (1999) · Organizational Behavior and Human Decision Processes · doi:10.1006/obhd.1998.2813 · Cited by 626
“We adapted procedures for testing dependent correlations (Steiger, 1980; Tabachnick & Fidell, 1989) to compare the variance explained by the five quadratic terms, controlling for age, gender, and income.”
MethodEnhanced Brain Correlations during Rest Are Related to Memory for Recent Experiences
Tambini, A., Ketz, N., & Davachi, L. (2010) · Neuron · doi:10.1016/j.neuron.2010.01.001 · Cited by 596
“, looking for differences in the extent to which Hippocampal-LO and PFC-LO correlations correlate with memory performance), a Hotelling-William test for comparing dependent correlations was used (Steiger, 1980).”
Adolescents' theories of education.
Nicholls, J. G., Patashnick, M., & Nolen, S. B. (1985) · Journal of Educational Psychology · doi:10.1037/0022-0663.77.6.683 · Cited by 574
Assessing the non-random sampling effects of subject attrition in longitudinal research
Goodman, J. S., & Blum, T. C. (1996) · Journal of Management · doi:10.1177/014920639602200405 · Cited by 766
Placebo effects on human μ-opioid activity during pain
Wager, T. D., Scott, D. J., & Zubieta, J.-K. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0702413104 · Cited by 562
Shore, L. M., & Martin, H. J. (1989) · Human Relations · doi:10.1177/001872678904200705 · Cited by 545
Quantifying construct validity: Two simple measures.
Westen, D., & Rosenthal, R. (2003) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.84.3.608 · Cited by 543
“More accurate results are obtained when the correlations ( r s) are first transformed into their Fisher Z r equivalents in order to improve normality (Meng et al., 1992; Steiger, 1980).”
MethodKring, A. M., Smith, D. A., & Neale, J. M. (1994) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.66.5.934 · Cited by 525
Implicit learning as an ability
Kaufman, S. B., DeYoung, C. G., Gray, J. R., Jiménez, L., Brown, J., & Mackintosh, N. (2010) · Cognition · doi:10.1016/j.cognition.2010.05.011 · Cited by 370
“Also in support of our hypotheses, g is significantly more strongly correlated with explicit associative learning than with implicit learning, according to a test of equality of correlations (Steiger, 1980), t(150) = 2.”
ResultInfluentialThe undesired self: A neglected variable in personality research.
Ogilvie, D. M. (1987) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.52.2.379 · Cited by 516
“The most conservative statistic to test this null hypothesis in a small sample is a t test proposed by Williams 0959) and recommended by Steiger (1980). This statistic shows the two corre-”
MethodRubenstein, A. L., Eberly, M. B., Lee, T., & Mitchell, T. R. (2017) · Personnel Psychology · doi:10.1111/peps.12226 · Cited by 641
“If samples are independent and confidence intervals are available, these approaches would be recommended to test for significant effect differences (Steiger, 1980; Zou, 2007).”
BackgroundTask-induced brain state manipulation improves prediction of individual traits
Greene, A. S., Gao, S., Scheinost, D., & Constable, R. T. (2018) · Nature Communications · doi:10.1038/s41467-018-04920-3 · Cited by 498
Feingold, A. (1990) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.59.5.981 · Cited by 508
Ho, A. K., Sidanius, J., Pratto, F., Levin, S., Thomsen, L., Kteily, N., & Sheehy-Skeffington, J. A. (2012) · Personality and Social Psychology Bulletin · doi:10.1177/0146167211432765 · Cited by 543
Butts, M. M., Casper, W. J., & Yang, T. S. (2012) · Journal of Applied Psychology · doi:10.1037/a0030389 · Cited by 494
Boswell, W. R., Olson‐Buchanan, J. B., & LePine, M. A. (2003) · Journal of Vocational Behavior · doi:10.1016/s0001-8791(03)00049-6 · Cited by 513
A New Dataset and Method for Automatically Grading ESOL Texts
Yannakoudakis, H., Briscoe, T., & Medlock, B. (2011) · Cited by 482
Schwarz, J. C., Barton-Henry, M. L., & Pruzinsky, T. (1985) · Child Development · doi:10.1111/j.1467-8624.1985.tb00120.x · Cited by 481
The SAGE Dictionary of Statistics
Cramer, D., & Howitt, D. (2004) · SAGE Publications Ltd eBooks · doi:10.4135/9780857020123 · Cited by 480
Assessment of life history of aggression: development and psychometric characteristics
Coccaro, E. F., Berman, M. E., & Kavoussi, R. J. (1997) · Psychiatry Research · doi:10.1016/s0165-1781(97)00119-4 · Cited by 478
“Significant differences between correlations were determined using the t statistic for independent (Fisher, 1921) and non-independent correlations (Williams, 1959; Steiger, 1980).”
Method
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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