{"title":"在对 z 变形相关性进行元分析时纠正测量误差。","authors":"Qian Zhang, Qi Wang","doi":"10.1111/bmsp.12328","DOIUrl":null,"url":null,"abstract":"<p>This study mainly concerns correction for measurement error using the meta-analysis of Fisher's z-transformed correlations. The disattenuation formula of Spearman (American Journal of Psychology, <b>15</b>, 1904, 72) is used to correct for individual raw correlations in primary studies. The corrected raw correlations are then used to obtain the corrected z-transformed correlations. What remains little studied, however, is how to best correct for within-study sampling error variances of corrected z-transformed correlations. We focused on three within-study sampling error variance estimators corrected for measurement error that were proposed in earlier studies and is proposed in the current study: (1) the formula given by Hedges (<i>Test validity</i>, Lawrence Erlbaum, 1988) assuming a linear relationship between corrected and uncorrected z-transformed correlations (linear correction), (2) one derived by the first-order delta method based on the average of corrected z-transformed correlations (stabilized first-order correction), and (3) one derived by the second-order delta method based on the average of corrected z-transformed correlations (stabilized second-order correction). Via a simulation study, we compared performance of these estimators and the sampling error variance estimator uncorrected for measurement error in terms of estimation and inference accuracy of the mean correlation as well as the homogeneity test of effect sizes. In obtaining the corrected z-transformed correlations and within-study sampling error variances, coefficient alpha was used as a common reliability coefficient estimate. The results showed that in terms of the estimated mean correlation, sampling error variances with linear correction, the stabilized first-order and second-order corrections, and no correction performed similarly in general. Furthermore, in terms of the homogeneity test, given a relatively large average sample size and normal true scores, the stabilized first-order and second-order corrections had type I error rates that were generally controlled as well as or better than the other estimators. Overall, stabilized first-order and second-order corrections are recommended when true scores are normal, reliabilities are acceptable, the number of items per psychological scale is relatively large, and the average sample size is relatively large.</p>","PeriodicalId":55322,"journal":{"name":"British Journal of Mathematical & Statistical Psychology","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Correcting for measurement error under meta-analysis of z-transformed correlations\",\"authors\":\"Qian Zhang, Qi Wang\",\"doi\":\"10.1111/bmsp.12328\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This study mainly concerns correction for measurement error using the meta-analysis of Fisher's z-transformed correlations. The disattenuation formula of Spearman (American Journal of Psychology, <b>15</b>, 1904, 72) is used to correct for individual raw correlations in primary studies. The corrected raw correlations are then used to obtain the corrected z-transformed correlations. What remains little studied, however, is how to best correct for within-study sampling error variances of corrected z-transformed correlations. We focused on three within-study sampling error variance estimators corrected for measurement error that were proposed in earlier studies and is proposed in the current study: (1) the formula given by Hedges (<i>Test validity</i>, Lawrence Erlbaum, 1988) assuming a linear relationship between corrected and uncorrected z-transformed correlations (linear correction), (2) one derived by the first-order delta method based on the average of corrected z-transformed correlations (stabilized first-order correction), and (3) one derived by the second-order delta method based on the average of corrected z-transformed correlations (stabilized second-order correction). Via a simulation study, we compared performance of these estimators and the sampling error variance estimator uncorrected for measurement error in terms of estimation and inference accuracy of the mean correlation as well as the homogeneity test of effect sizes. In obtaining the corrected z-transformed correlations and within-study sampling error variances, coefficient alpha was used as a common reliability coefficient estimate. The results showed that in terms of the estimated mean correlation, sampling error variances with linear correction, the stabilized first-order and second-order corrections, and no correction performed similarly in general. Furthermore, in terms of the homogeneity test, given a relatively large average sample size and normal true scores, the stabilized first-order and second-order corrections had type I error rates that were generally controlled as well as or better than the other estimators. 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引用次数: 0
摘要
本研究主要涉及利用费舍尔 z 变形相关系数的元分析来校正测量误差。斯皮尔曼(Spearman)的失调公式(《美国心理学杂志》,15,1904,72)用于校正原始研究中的个体原始相关性。然后使用校正后的原始相关性来获得校正后的 z 转换相关性。然而,对于如何最好地校正校正过的 z 转换相关性的研究内部抽样误差方差的研究仍然很少。我们重点研究了三种校正了测量误差的研究内部抽样误差方差估计方法,这三种方法在以前的研究中提出过,在本次研究中也提出了:(1) Hedges(《测试有效性》,Lawrence Erlbaum,1988 年)给出的公式,假设校正过的和未校正过的 z 转换相关系数之间存在线性关系(线性校正);(2) 根据校正过的 z 转换相关系数的平均值,通过一阶三角法得出的公式(稳定一阶校正);(3) 根据校正过的 z 转换相关系数的平均值,通过二阶三角法得出的公式(稳定二阶校正)。通过模拟研究,我们比较了这些估计器和未修正测量误差的抽样误差方差估计器在平均相关性的估计和推断准确性以及效应大小的同质性检验方面的性能。在获得经校正的 z 变形相关性和研究内部抽样误差方差时,使用了系数 α 作为通用的可靠性系数估计值。结果表明,在估计平均相关性方面,采用线性校正、稳定的一阶和二阶校正以及不采用校正的抽样误差方差总体表现相似。此外,就同质性检验而言,在平均样本量相对较大且真实分数正常的情况下,稳定化一阶和二阶校正的 I 类错误率通常控制得与其他估计器一样好,甚至更好。总体而言,当真实得分正常、信度可接受、每个心理量表的项目数相对较多、平均样本量相对较大时,建议使用稳定一阶和二阶修正法。
Correcting for measurement error under meta-analysis of z-transformed correlations
This study mainly concerns correction for measurement error using the meta-analysis of Fisher's z-transformed correlations. The disattenuation formula of Spearman (American Journal of Psychology, 15, 1904, 72) is used to correct for individual raw correlations in primary studies. The corrected raw correlations are then used to obtain the corrected z-transformed correlations. What remains little studied, however, is how to best correct for within-study sampling error variances of corrected z-transformed correlations. We focused on three within-study sampling error variance estimators corrected for measurement error that were proposed in earlier studies and is proposed in the current study: (1) the formula given by Hedges (Test validity, Lawrence Erlbaum, 1988) assuming a linear relationship between corrected and uncorrected z-transformed correlations (linear correction), (2) one derived by the first-order delta method based on the average of corrected z-transformed correlations (stabilized first-order correction), and (3) one derived by the second-order delta method based on the average of corrected z-transformed correlations (stabilized second-order correction). Via a simulation study, we compared performance of these estimators and the sampling error variance estimator uncorrected for measurement error in terms of estimation and inference accuracy of the mean correlation as well as the homogeneity test of effect sizes. In obtaining the corrected z-transformed correlations and within-study sampling error variances, coefficient alpha was used as a common reliability coefficient estimate. The results showed that in terms of the estimated mean correlation, sampling error variances with linear correction, the stabilized first-order and second-order corrections, and no correction performed similarly in general. Furthermore, in terms of the homogeneity test, given a relatively large average sample size and normal true scores, the stabilized first-order and second-order corrections had type I error rates that were generally controlled as well as or better than the other estimators. Overall, stabilized first-order and second-order corrections are recommended when true scores are normal, reliabilities are acceptable, the number of items per psychological scale is relatively large, and the average sample size is relatively large.
期刊介绍:
The British Journal of Mathematical and Statistical Psychology publishes articles relating to areas of psychology which have a greater mathematical or statistical aspect of their argument than is usually acceptable to other journals including:
• mathematical psychology
• statistics
• psychometrics
• decision making
• psychophysics
• classification
• relevant areas of mathematics, computing and computer software
These include articles that address substantitive psychological issues or that develop and extend techniques useful to psychologists. New models for psychological processes, new approaches to existing data, critiques of existing models and improved algorithms for estimating the parameters of a model are examples of articles which may be favoured.