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1. Shard Calculation

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Response:

Calculated Shard: 25 (from laksa007)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa025.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.ets.org/research/policy_research_reports/publications/report/2021/kcvs.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.ets.org/research/policy_research_reports/publications/report/2021/kcvs.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.ets.org\/research\/policy_research_reports\/publications\/report\/2021\/kcvs.html","crawl_time":1769691835,"first_indexed_time":1734558102,"http_code":200,"src_unparsed":"org,ets!www,\/research\/policy_research_reports\/publications\/report\/2021\/kcvs.html s443","src_root_hash":"1516433742396401625","history_drop_reason":null,"meta_title":"Symmetric Least Squares Estimates of Functional Relationships","meta_descriptions":["Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of |rxy|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y."],"attrs_boilerpipe_text":"Author(s):\nKane, Michael\nPublication Year:\n2021\nReport Number:\nRR-21-21\nSource:\nETS Research Report\nDocument Type:\nReport\nPage Count:\n14\nSubject\/Key Words:\nGeometric Mean Regression,\nOrdinary Least Squares Regression,\nLinear Relationships,\nLinear Equating,\nLinear Models,\nPrediction,\nDependent Variables,\nFunctional Relationship,\nIndependent Variables,\nSocial Sciences,\nTest Bias,\nSymmetry,\nTest Equating,\nValidity\nAbstract\nOrdinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of |rxy|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y.\nRead More\nRequest Copy\n (specify title and report number, if any)\nhttps:\/\/doi.org\/10.1002\/ets2.12331","attrs_markdown":"[skip to main content](https:\/\/www.ets.org\/research\/policy_research_reports\/publications\/report\/2021\/kcvs.html#content) [skip to footer](https:\/\/www.ets.org\/research\/policy_research_reports\/publications\/report\/2021\/kcvs.html#page-footer)\n\n1. [publications](https:\/\/www.ets.org\/content\/ets-org\/research\/policy_research_reports\/publications.html)\n\n1. [Go Back]()\n\n# Symmetric Least Squares Estimates of Functional Relationships\nAuthor(s):\n\n[Kane, Michael](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/author-bio\/Kane-Michael.html)\n\nPublication Year:\n\n2021\n\nReport Number:\n\nRR-21-21\n\nSource:\n\nETS Research Report\n\nDocument Type:\n\nReport\n\nPage Count:\n\n14\n\nSubject\/Key Words:\n\n[Geometric Mean Regression,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Geometric%20Mean%20Regression)   [Ordinary Least Squares Regression,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Ordinary%20Least%20Squares%20Regression)   [Linear Relationships,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Relationships)   [Linear Equating,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Equating)   [Linear Models,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Models)   [Prediction,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Prediction)   [Dependent Variables,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Dependent%20Variables)   [Functional Relationship,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Functional%20Relationship)   [Independent Variables,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Independent%20Variables)   [Social Sciences,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Social%20Sciences)   [Test Bias,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Test%20Bias)   [Symmetry,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Symmetry)   [Test Equating,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Test%20Equating)   [Validity](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Validity)\n## Abstract\nOrdinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \\|rxy\\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y.\n\n## Read More\n- [Request Copy](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/contact.html \"Request a copy of this report\") (specify title and report number, if any)\n- <https:\/\/doi.org\/10.1002\/ets2.12331>\n\n## Sign up for ETS updates\nStay up to date with the latest news, announcements and articles\n\nSign up for updates\n\nDialog box is opened\n\nETS Updates\n\nTo ensure we provide you with the most relevant content, please tell us a little more about yourself. Your choice helps us customize our communications to fit your needs.\n\nI AM A TEST TAKER\n\nor\n\nI REPRESENT AN ORGANIZATION OR INSTITUTION\n\nThank you for subscribing\\!\n\nClose","attrs_readable_markdown":"Author(s):\n\n[Kane, Michael](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/author-bio\/Kane-Michael.html)\n\nPublication Year:\n\n2021\n\nReport Number:\n\nRR-21-21\n\nSource:\n\nETS Research Report\n\nDocument Type:\n\nReport\n\nPage Count:\n\n14\n\nSubject\/Key Words:\n\n[Geometric Mean Regression,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Geometric%20Mean%20Regression)   [Ordinary Least Squares Regression,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Ordinary%20Least%20Squares%20Regression)   [Linear Relationships,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Relationships)   [Linear Equating,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Equating)   [Linear Models,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Linear%20Models)   [Prediction,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Prediction)   [Dependent Variables,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Dependent%20Variables)   [Functional Relationship,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Functional%20Relationship)   [Independent Variables,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Independent%20Variables)   [Social Sciences,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Social%20Sciences)   [Test Bias,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Test%20Bias)   [Symmetry,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Symmetry)   [Test Equating,](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Test%20Equating)   [Validity](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/research\/researcher.html?qt=Validity)\n## Abstract\nOrdinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \\|rxy\\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y.\n\n## Read More\n- [Request Copy](https:\/\/www.ets.org\/content\/ets-org\/language-master\/en\/home\/contact.html \"Request a copy of this report\") (specify title and report number, if any)\n- <https:\/\/doi.org\/10.1002\/ets2.12331>","meta_canonical":null}

3. Robots.txt Check

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Response:

4. Spam/Ban Check

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5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

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CRAWLED

2 months ago

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Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	2.4 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://www.ets.org/research/policy_research_reports/publications/report/2021/kcvs.html
Last Crawled	2026-01-29 13:03:55 (2 months ago)
First Indexed	2024-12-18 21:41:42 (1 year ago)
HTTP Status Code	200
Meta Title	Symmetric Least Squares Estimates of Functional Relationships
Meta Description	Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \|rxy\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y.
Meta Canonical	null
Boilerpipe Text	Author(s): Kane, Michael Publication Year: 2021 Report Number: RR-21-21 Source: ETS Research Report Document Type: Report Page Count: 14 Subject/Key Words: Geometric Mean Regression, Ordinary Least Squares Regression, Linear Relationships, Linear Equating, Linear Models, Prediction, Dependent Variables, Functional Relationship, Independent Variables, Social Sciences, Test Bias, Symmetry, Test Equating, Validity Abstract Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \|rxy\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y. Read More Request Copy (specify title and report number, if any) https://doi.org/10.1002/ets2.12331
Markdown	[skip to main content](https://www.ets.org/research/policy_research_reports/publications/report/2021/kcvs.html#content) [skip to footer](https://www.ets.org/research/policy_research_reports/publications/report/2021/kcvs.html#page-footer) 1. [publications](https://www.ets.org/content/ets-org/research/policy_research_reports/publications.html) 1. [Go Back]() # Symmetric Least Squares Estimates of Functional Relationships Author(s): [Kane, Michael](https://www.ets.org/content/ets-org/language-master/en/home/research/author-bio/Kane-Michael.html) Publication Year: 2021 Report Number: RR-21-21 Source: ETS Research Report Document Type: Report Page Count: 14 Subject/Key Words: [Geometric Mean Regression,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Geometric%20Mean%20Regression) [Ordinary Least Squares Regression,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Ordinary%20Least%20Squares%20Regression) [Linear Relationships,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Relationships) [Linear Equating,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Equating) [Linear Models,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Models) [Prediction,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Prediction) [Dependent Variables,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Dependent%20Variables) [Functional Relationship,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Functional%20Relationship) [Independent Variables,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Independent%20Variables) [Social Sciences,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Social%20Sciences) [Test Bias,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Test%20Bias) [Symmetry,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Symmetry) [Test Equating,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Test%20Equating) [Validity](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Validity) ## Abstract Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \\|rxy\\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y. ## Read More - [Request Copy](https://www.ets.org/content/ets-org/language-master/en/home/contact.html "Request a copy of this report") (specify title and report number, if any) - <https://doi.org/10.1002/ets2.12331> ## Sign up for ETS updates Stay up to date with the latest news, announcements and articles Sign up for updates Dialog box is opened ETS Updates To ensure we provide you with the most relevant content, please tell us a little more about yourself. Your choice helps us customize our communications to fit your needs. I AM A TEST TAKER or I REPRESENT AN ORGANIZATION OR INSTITUTION Thank you for subscribing\! Close
Readable Markdown	Author(s): [Kane, Michael](https://www.ets.org/content/ets-org/language-master/en/home/research/author-bio/Kane-Michael.html) Publication Year: 2021 Report Number: RR-21-21 Source: ETS Research Report Document Type: Report Page Count: 14 Subject/Key Words: [Geometric Mean Regression,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Geometric%20Mean%20Regression) [Ordinary Least Squares Regression,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Ordinary%20Least%20Squares%20Regression) [Linear Relationships,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Relationships) [Linear Equating,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Equating) [Linear Models,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Linear%20Models) [Prediction,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Prediction) [Dependent Variables,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Dependent%20Variables) [Functional Relationship,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Functional%20Relationship) [Independent Variables,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Independent%20Variables) [Social Sciences,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Social%20Sciences) [Test Bias,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Test%20Bias) [Symmetry,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Symmetry) [Test Equating,](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Test%20Equating) [Validity](https://www.ets.org/content/ets-org/language-master/en/home/research/researcher.html?qt=Validity) ## Abstract Ordinary least squares (OLS) regression provides optimal linear predictions of a dependent variable, y, given an independent variable, x, but OLS regressions are not symmetric or reversible. In order to get optimal linear predictions of x given y, a separate OLS regression in that direction would be needed. This report provides a least squares derivation of the geometric mean (GM) regression line, which is symmetric and reversible, as the line that minimizes a weighted sum of the mean squared errors for y, given x, and for x, given y. It is shown that the GM regression line is symmetric and predicts equally well (or poorly, depending on the absolute value of rxy) in both directions. The errors of prediction for the GM line are, naturally, larger for the predictions of both x and y than those for the two OLS equations, each of which is specifically optimized for prediction in one direction, but for high values of \\|rxy\\|, the difference is not large. The GM line has previously been derived as a special case of principal-components analysis and gets its name from the fact that its slope is equal to the geometric mean of the slopes of the OLS regressions of y on x and x on y. ## Read More - [Request Copy](https://www.ets.org/content/ets-org/language-master/en/home/contact.html "Request a copy of this report") (specify title and report number, if any) - <https://doi.org/10.1002/ets2.12331>
Shard	25 (laksa)
Root Hash	1516433742396401625
Unparsed URL	org,ets!www,/research/policy_research_reports/publications/report/2021/kcvs.html s443