Equidistant likert as weighted sum of response categories

Introduction: Addition of scores of Likert items may not be meaningful since equidistant property is not satisfied. This implies computation of mean, standard deviation, correlation, regression and Cronbach alpha using sum of item variances and test variance could be problematic. Objective: Avoiding...

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Autores:: Chakrabartty, Satyendra Nath

Tipo de recurso:: Article of journal

Fecha de publicación:: 2022

Institución:: Corporación Universidad de la Costa

Repositorio:: REDICUC - Repositorio CUC

Idioma:: eng

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dc.title.eng.fl_str_mv	Equidistant likert as weighted sum of response categories
dc.title.translated.none.fl_str_mv	Likert equidistante como suma ponderada de categorías de respuesta
title	Equidistant likert as weighted sum of response categories
spellingShingle	Equidistant likert as weighted sum of response categories Likert items Weighted sum Monotonic Equidistant Normal distribution Ítems tipo Likert Suma ponderada Monotónico Equidistante Distribución normal
title_short	Equidistant likert as weighted sum of response categories
title_full	Equidistant likert as weighted sum of response categories
title_fullStr	Equidistant likert as weighted sum of response categories
title_full_unstemmed	Equidistant likert as weighted sum of response categories
title_sort	Equidistant likert as weighted sum of response categories
dc.creator.fl_str_mv	Chakrabartty, Satyendra Nath
dc.contributor.author.none.fl_str_mv	Chakrabartty, Satyendra Nath
dc.subject.proposal.eng.fl_str_mv	Likert items Weighted sum Monotonic Equidistant Normal distribution
topic	Likert items Weighted sum Monotonic Equidistant Normal distribution Ítems tipo Likert Suma ponderada Monotónico Equidistante Distribución normal
dc.subject.proposal.spa.fl_str_mv	Ítems tipo Likert Suma ponderada Monotónico Equidistante Distribución normal
description	Introduction: Addition of scores of Likert items may not be meaningful since equidistant property is not satisfied. This implies computation of mean, standard deviation, correlation, regression and Cronbach alpha using sum of item variances and test variance could be problematic. Objective: Avoiding limitation of summative Likert scores by transforming raw item scores to continuous monotonic scores satisfying equidistant property and evaluate the methods with respect to desired properties and testing normality of transformed test scores. Methodology: The methodological paper gives three methods of transforming discrete, ordinal item scores to continuous scores by weighted sum where weights consider frequencies of different response-categories of different items and generate continuous data satisfying equidistant and monotonic properties. Results and discussions: All the proposed methods avoided major limitations of summative Likert scores, generates continuous data satisfying equidistant and monotonic properties. The method based on frequencies of response-categories for different items (Method 3) passed the normality test unlike the Method 1 and Method 2. Normally distributed transformed scores in Method 3 facilitate undertaking analysis under parametric set up. Conclusions: Proposed methods having high correlations with summative Likert scores, retained similar factor structure and provides reconciliation to the debate on ordinal vs. interval nature of data generated from a Likert questionnaire. Considering the theoretical advantages, the Method 3 is recommended for scoring Likert items primarily due to Normal distribution of individual scores facilitating meaningfulness of operations and to undertake parametric statistical analysis
publishDate	2022
dc.date.issued.none.fl_str_mv	2022-11-29
dc.date.accessioned.none.fl_str_mv	2023-01-28T03:14:29Z
dc.date.available.none.fl_str_mv	2023-01-28T03:14:29Z
dc.type.spa.fl_str_mv	Artículo de revista
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dc.identifier.citation.spa.fl_str_mv	Chakrabartty, S. (2022). Equidistant Likert as weighted sum of Response Categories. Cultura, Educación y Sociedad, 14(1), 75–92. DOI: http://dx.doi.org/10.17981/cultedusoc.14.1.2023.04
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dc.identifier.doi.none.fl_str_mv	10.17981/cultedusoc.14.1.2023.04
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identifier_str_mv	Chakrabartty, S. (2022). Equidistant Likert as weighted sum of Response Categories. Cultura, Educación y Sociedad, 14(1), 75–92. DOI: http://dx.doi.org/10.17981/cultedusoc.14.1.2023.04 2145-9258 10.17981/cultedusoc.14.1.2023.04 2389-7724 Corporación Universidad de la Costa REDICUC - Repositorio CUC
url	https://hdl.handle.net/11323/9835 https://repositorio.cuc.edu.co/
dc.language.iso.spa.fl_str_mv	eng
language	eng
dc.relation.ispartofjournal.spa.fl_str_mv	Cultura Educación y Sociedad
dc.relation.references.spa.fl_str_mv	Arvidsson, R. (2019). On the use of ordinal scoring scales in social life cycle assessment. The International Journal of Life Cycle Assessment, 24(3), 604–606. https://doi. org/10.1007/s11367-018-1557-2 Barua, A. (2013). Methods for Decision–making in Survey Questionnaires Based on Likert Scale. Journal of Asian Scientific Research, 3(1), 35–38. https://archive.aessweb.com/ index.php/5003/article/view/3446 Bürkner, P.C. & Vuorre, M. (2019). Ordinal Regression Models in Psychology: A Tutorial. Advances in Methods and Practices in Psychological Science, 2(1), 77–101. https://doi. org/10.1177/2515245918823199 Carifio, J. & Perla, R. (2007). Ten Common Misunderstandings, Misconceptions, Persistent Myths and Urban Legends about Likert Scales and Likert Response Formats and their Antidotes. Journal of Social Sciences, 3, 106–116. http://dx.doi.org/10.3844/ jssp.2007.106.116 Chakrabartty, S. N. (2021). Optimum number of Response Categories. Current Psychology, 104(1), 1–15. https://doi.org/10.1007/s12144-021-01866-6 Dawes, J. (2007). Do data characteristics change according to the number of scale points used? International Journal of Market Research, 50(1), 61–77. https://doi. org/10.1177/147078530805000106 cFlora, D. B. & Curran, P. J. (2004). An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis with Ordinal Data. Psychological Methods, 9(4), 466–491. https://doi.org/10.1037/1082-989X.9.4.466 Granberg-Rademacker, J. S. (2010). An Algorithm for Converting Ordinal Scale Measurement Data to Interval/Ratio Scale. Educational and Psychological Measurement, 70(1), 74–90. https://doi.org/10.1177/0013164409344532 Harwell, M. R. & Gatti, G. G. (2001). Rescaling ordinal data to interval data in educational research. Review of Educational Research, 71, 105–131. https://doi. org/10.3102/00346543071001105 Hinne, M. (2013). Additive conjoint measurement and the resistance toward falsifiability in psychology. Frontiers in Psychology, 4(1), 1–4. https://doi.org/10.3389/fpsyg.2013.00246 Huiping, W. & Leung, S-O. (2017). Can Likert Scales be Treated as Interval Scales?—A Simulation Study. Journal of Social Service Research, 43(4), 527–532. https://doi.org/ 10.1080/01488376.2017.1329775 Jamieson, S. (2005, Aug. 11). Likert scale. Encyclopedia Britannica. https://www.britannica.com/topic/Likert-Scale Kuzon, W. M., Urbanchek, M. G. & McCabe, S. (1996). The seven deadly sins of statistical analysis. Annals of Plastic Surgery, 37, 265–272. https://doi.org/10.1097/00000637- 199609000-00006 Lee, J. A. & Soutar, G. N. (2010). Is Schwartz’s value survey an interval scale, and does it really matter? Journal of Cross-Cultural Psychology, 41(1), 76– 86. https://doi. org/10.1177/0022022109348920 Lim, H.-E. (2008). The use of different happiness rating scales: bias and comparison problem? Social Indicators Research, 87, 259–267. https://doi.org/10.1007/s11205-007- 9171-x Marcus-Roberts, H. M. & Roberts, F. S. (1987). Meaningless statistics. Journal of Educational Statistics, 12, 383–394. https://doi.org/10.2307/1165056 Markus, K. A. & Borsboom, D. (2012). The cat came back: evaluating arguments against psychological measurement. Theory & Psychol, 22(4), 452–466. https://doi. org/10.1177/0959354310381155 Michell, J. (1990). An Introduction to the Logic of Psychological Measurement. ErlbaumAssociates. Munshi, J. (2014). A method for constructing Likert scales. Social Science Research Network. https://doi.org/10.2139/ssrn.2419366 Sheng, Y. & Sheng, Z. (2012). Is coefficient alpha robust to non-normal data? Frontiers in Psychology, 3(34), 1–13. https://doi.org/10.3389/fpstg.2012.00034 Šimkovic, M. & Träuble, B. (2019). Robustness of statistical methods when measure is affected by ceiling and/or floor effect. PloS one, 14(8), 1–47. https://doi.org/10.1371/ journal.pone.0220889 Simms, L. J., Zelazny, K., Williams, T. F. & Bernstein, L. (2019). Does the number of response options matter? Psychometric perspectives using personality questionnaire data. Psychological Assessment, 31(4), 557–566. https://doi.org/10.1037/pas0000648 Snell, E. (1964). A Scaling Procedure for Ordered Categorical Data. Biometrics, 20(3), 592–607. https://doi.org/10.2307/2528498 Uyumaz, G. & Sırgancı, G. (2021). Determining the Factors Affecting the Psychological Distance Between Categories in the Rating Scale. International Journal of Contemporary Educational Research, 8(3), 178–190. https://doi.org/10.33200/ijcer.858599 Wu, Ch.-H. (2007). An Empirical Study on the Transformation of Likert scale Data to Numerical Scores. Applied Mathematical Sciences, 1(58), 2851–2862. https://doi. org/10.12988/ams Yusoff, R. & Janor, R. M. (2014). Generation of an Interval Metric Scale to Measure Attitude. SAGE Open, 4(1), 1–16. https://doi.org/10.1177/2158244013516768
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dc.rights.spa.fl_str_mv	Derechos de autor 2022 CULTURA EDUCACIÓN Y SOCIEDAD
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dc.publisher.spa.fl_str_mv	Corporación Universidad de la Costa
dc.publisher.place.spa.fl_str_mv	Colombia
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spelling	Atribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)Derechos de autor 2022 CULTURA EDUCACIÓN Y SOCIEDADhttps://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Chakrabartty, Satyendra Nathb25b2afc0d71b590991838088c48bde66002023-01-28T03:14:29Z2023-01-28T03:14:29Z2022-11-29Chakrabartty, S. (2022). Equidistant Likert as weighted sum of Response Categories. Cultura, Educación y Sociedad, 14(1), 75–92. DOI: http://dx.doi.org/10.17981/cultedusoc.14.1.2023.042145-9258https://hdl.handle.net/11323/983510.17981/cultedusoc.14.1.2023.042389-7724Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/Introduction: Addition of scores of Likert items may not be meaningful since equidistant property is not satisfied. This implies computation of mean, standard deviation, correlation, regression and Cronbach alpha using sum of item variances and test variance could be problematic. Objective: Avoiding limitation of summative Likert scores by transforming raw item scores to continuous monotonic scores satisfying equidistant property and evaluate the methods with respect to desired properties and testing normality of transformed test scores. Methodology: The methodological paper gives three methods of transforming discrete, ordinal item scores to continuous scores by weighted sum where weights consider frequencies of different response-categories of different items and generate continuous data satisfying equidistant and monotonic properties. Results and discussions: All the proposed methods avoided major limitations of summative Likert scores, generates continuous data satisfying equidistant and monotonic properties. The method based on frequencies of response-categories for different items (Method 3) passed the normality test unlike the Method 1 and Method 2. Normally distributed transformed scores in Method 3 facilitate undertaking analysis under parametric set up. Conclusions: Proposed methods having high correlations with summative Likert scores, retained similar factor structure and provides reconciliation to the debate on ordinal vs. interval nature of data generated from a Likert questionnaire. Considering the theoretical advantages, the Method 3 is recommended for scoring Likert items primarily due to Normal distribution of individual scores facilitating meaningfulness of operations and to undertake parametric statistical analysisIntroducción: La suma de puntajes de elementos de Likert puede no ser significativa ya que no se cumple la propiedad de equidistancia. Esto implica que el cálculo de la media, la desviación estándar, la correlación, la regresión y el alfa de Cronbach utilizando la suma de las varianzas de los elementos y la varianza de la prueba podría ser problemático. Objetivo: Evitar la limitación de las puntuaciones de Likert sumativas transformando las puntuaciones de los ítems sin procesar en puntuaciones monotónicas continuas que satisfagan la propiedad equidistante y evalúen los métodos con respecto a las propiedades deseadas y prueben la normalidad de las puntuaciones de las pruebas transformadas. Metodología: El documento metodológico proporciona tres métodos para transformar puntajes discretos y ordinales de ítems en puntajes continuos por suma ponderada donde los pesos consideran frecuencias de diferentes categorías de respuesta de diferentes ítems y generan datos continuos que satisfacen propiedades equidistantes y monótonas. Resultados y discusión: Todos los métodos propuestos evitaron las principales limitaciones de las puntuaciones de Likert sumativas, generando datos continuos que satisfacen las propiedades equidistantes y monótonas. El método basado en frecuencias de categorías de respuesta para diferentes ítems (Método 3) pasó la prueba de normalidad a diferencia del Método 1 y el Método 2. Las puntuaciones transformadas normalmente distribuidas en el Método 3 facilitan la realización de análisis bajo una configuración paramétrica. Conclusiones: Los métodos propuestos que tienen altas correlaciones con las puntuaciones de Likert sumativas, conservan una estructura factorial similar y brindan reconciliación al debate sobre la naturaleza ordinal frente a la de intervalo de los datos generados a partir de un cuestionario de Likert. Teniendo en cuenta las ventajas teóricas, se recomienda el Método 3 para puntuar elementos de Likert principalmente debido a la distribución normal de las puntuaciones individuales que facilita la significatividad de las operaciones y para realizar análisis estadísticos paramétricos.18 páginasapplication/pdfengCorporación Universidad de la CostaColombiahttps://revistascientificas.cuc.edu.co/culturaeducacionysociedad/article/view/3915Equidistant likert as weighted sum of response categoriesLikert equidistante como suma ponderada de categorías de respuestaArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/version/c_970fb48d4fbd8a85Cultura Educación y SociedadArvidsson, R. (2019). On the use of ordinal scoring scales in social life cycle assessment. The International Journal of Life Cycle Assessment, 24(3), 604–606. https://doi. org/10.1007/s11367-018-1557-2Barua, A. (2013). Methods for Decision–making in Survey Questionnaires Based on Likert Scale. Journal of Asian Scientific Research, 3(1), 35–38. https://archive.aessweb.com/ index.php/5003/article/view/3446Bürkner, P.C. & Vuorre, M. (2019). Ordinal Regression Models in Psychology: A Tutorial. Advances in Methods and Practices in Psychological Science, 2(1), 77–101. https://doi. org/10.1177/2515245918823199Carifio, J. & Perla, R. (2007). Ten Common Misunderstandings, Misconceptions, Persistent Myths and Urban Legends about Likert Scales and Likert Response Formats and their Antidotes. Journal of Social Sciences, 3, 106–116. http://dx.doi.org/10.3844/ jssp.2007.106.116Chakrabartty, S. N. (2021). Optimum number of Response Categories. Current Psychology, 104(1), 1–15. https://doi.org/10.1007/s12144-021-01866-6Dawes, J. (2007). Do data characteristics change according to the number of scale points used? International Journal of Market Research, 50(1), 61–77. https://doi. org/10.1177/147078530805000106cFlora, D. B. & Curran, P. J. (2004). An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis with Ordinal Data. Psychological Methods, 9(4), 466–491. https://doi.org/10.1037/1082-989X.9.4.466Granberg-Rademacker, J. S. (2010). An Algorithm for Converting Ordinal Scale Measurement Data to Interval/Ratio Scale. Educational and Psychological Measurement, 70(1), 74–90. https://doi.org/10.1177/0013164409344532Harwell, M. R. & Gatti, G. G. (2001). Rescaling ordinal data to interval data in educational research. Review of Educational Research, 71, 105–131. https://doi. org/10.3102/00346543071001105Hinne, M. (2013). Additive conjoint measurement and the resistance toward falsifiability in psychology. Frontiers in Psychology, 4(1), 1–4. https://doi.org/10.3389/fpsyg.2013.00246Huiping, W. & Leung, S-O. (2017). Can Likert Scales be Treated as Interval Scales?—A Simulation Study. Journal of Social Service Research, 43(4), 527–532. https://doi.org/ 10.1080/01488376.2017.1329775Jamieson, S. (2005, Aug. 11). Likert scale. Encyclopedia Britannica. https://www.britannica.com/topic/Likert-ScaleKuzon, W. M., Urbanchek, M. G. & McCabe, S. (1996). The seven deadly sins of statistical analysis. Annals of Plastic Surgery, 37, 265–272. https://doi.org/10.1097/00000637- 199609000-00006Lee, J. A. & Soutar, G. N. (2010). Is Schwartz’s value survey an interval scale, and does it really matter? Journal of Cross-Cultural Psychology, 41(1), 76– 86. https://doi. org/10.1177/0022022109348920Lim, H.-E. (2008). The use of different happiness rating scales: bias and comparison problem? Social Indicators Research, 87, 259–267. https://doi.org/10.1007/s11205-007- 9171-xMarcus-Roberts, H. M. & Roberts, F. S. (1987). Meaningless statistics. Journal of Educational Statistics, 12, 383–394. https://doi.org/10.2307/1165056Markus, K. A. & Borsboom, D. (2012). The cat came back: evaluating arguments against psychological measurement. Theory & Psychol, 22(4), 452–466. https://doi. org/10.1177/0959354310381155Michell, J. (1990). An Introduction to the Logic of Psychological Measurement. ErlbaumAssociates.Munshi, J. (2014). A method for constructing Likert scales. Social Science Research Network. https://doi.org/10.2139/ssrn.2419366Sheng, Y. & Sheng, Z. (2012). Is coefficient alpha robust to non-normal data? Frontiers in Psychology, 3(34), 1–13. https://doi.org/10.3389/fpstg.2012.00034Šimkovic, M. & Träuble, B. (2019). Robustness of statistical methods when measure is affected by ceiling and/or floor effect. PloS one, 14(8), 1–47. https://doi.org/10.1371/ journal.pone.0220889Simms, L. J., Zelazny, K., Williams, T. F. & Bernstein, L. (2019). Does the number of response options matter? Psychometric perspectives using personality questionnaire data. Psychological Assessment, 31(4), 557–566. https://doi.org/10.1037/pas0000648Snell, E. (1964). A Scaling Procedure for Ordered Categorical Data. Biometrics, 20(3), 592–607. https://doi.org/10.2307/2528498Uyumaz, G. & Sırgancı, G. (2021). Determining the Factors Affecting the Psychological Distance Between Categories in the Rating Scale. International Journal of Contemporary Educational Research, 8(3), 178–190. https://doi.org/10.33200/ijcer.858599Wu, Ch.-H. (2007). An Empirical Study on the Transformation of Likert scale Data to Numerical Scores. Applied Mathematical Sciences, 1(58), 2851–2862. https://doi. org/10.12988/amsYusoff, R. & Janor, R. M. (2014). Generation of an Interval Metric Scale to Measure Attitude. 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corporada en las Obras Colectivas.

b.	Distribuir copias o fonogramas de las Obras, exhibirlas públicamente, ejecutarlas públicamente y/o ponerlas a disposición pública, incluyéndolas como incorporadas en Obras Colectivas, según corresponda.

c.	Distribuir copias de las Obras Derivadas que se generen, exhibirlas públicamente, ejecutarlas públicamente y/o ponerlas a disposición pública.
Los derechos mencionados anteriormente pueden ser ejercidos en todos los medios y formatos, actualmente conocidos o que se inventen en el futuro. Los derechos antes mencionados incluyen el derecho a realizar dichas modificaciones en la medida que sean técnicamente necesarias para ejercer los derechos en otro medio o formatos, pero de otra manera usted no está autorizado para realizar obras derivadas. Todos los derechos no otorgados expresamente por el Licenciante quedan por este medio reservados, incluyendo pero sin limitarse a aquellos que se mencionan en las secciones 4(d) y 4(e).

4. Restricciones.
La licencia otorgada en la anterior Sección 3 está expresamente sujeta y limitada por las siguientes restricciones:

a.	Usted puede distribuir, exhibir públicamente, ejecutar públicamente, o poner a disposición pública la Obra sólo bajo las condiciones de esta Licencia, y Usted debe incluir una copia de esta licencia o del Identificador Universal de Recursos de la misma con cada copia de la Obra que distribuya, exhiba públicamente, ejecute públicamente o ponga a disposición pública. No es posible ofrecer o imponer ninguna condición sobre la Obra que altere o limite las condiciones de esta Licencia o el ejercicio de los derechos de los destinatarios otorgados en este documento. No es posible sublicenciar la Obra. Usted debe mantener intactos todos los avisos que hagan referencia a esta Licencia y a la cláusula de limitación de garantías. Usted no puede distribuir, exhibir públicamente, ejecutar públicamente, o poner a disposición pública la Obra con alguna medida tecnológica que controle el acceso o la utilización de ella de una forma que sea inconsistente con las condiciones de esta Licencia. Lo anterior se aplica a la Obra incorporada a una Obra Colectiva, pero esto no exige que la Obra Colectiva aparte de la obra misma quede sujeta a las condiciones de esta Licencia. Si Usted crea una Obra Colectiva, previo aviso de cualquier Licenciante debe, en la medida de lo posible, eliminar de la Obra Colectiva cualquier referencia a dicho Licenciante o al Autor Original, según lo solicitado por el Licenciante y conforme lo exige la cláusula 4(c).

b.	Usted no puede ejercer ninguno de los derechos que le han sido otorgados en la Sección 3 precedente de modo que estén principalmente destinados o directamente dirigidos a conseguir un provecho comercial o una compensación monetaria privada. El intercambio de la Obra por otras obras protegidas por derechos de autor, ya sea a través de un sistema para compartir archivos digitales (digital file-sharing) o de cualquier otra manera no será considerado como estar destinado principalmente o dirigido directamente a conseguir un provecho comercial o una compensación monetaria privada, siempre que no se realice un pago mediante una compensación monetaria en relación con el intercambio de obras protegidas por el derecho de autor.

c.	Si usted distribuye, exhibe públicamente, ejecuta públicamente o ejecuta públicamente en forma digital la Obra o cualquier Obra Derivada u Obra Colectiva, Usted debe mantener intacta toda la información de derecho de autor de la Obra y proporcionar, de forma razonable según el medio o manera que Usted esté utilizando: (i) el nombre del Autor Original si está provisto (o seudónimo, si fuere aplicable), y/o (ii) el nombre de la parte o las partes que el Autor Original y/o el Licenciante hubieren designado para la atribución (v.g., un instituto patrocinador, editorial, publicación) en la información de los derechos de autor del Licenciante, términos de servicios o de otras formas razonables; el título de la Obra si está provisto; en la medida de lo razonablemente factible y, si está provisto, el Identificador Uniforme de Recursos (Uniform Resource Identifier) que el Licenciante especifica para ser asociado con la Obra, salvo que tal URI no se refiera a la nota sobre los derechos de autor o a la información sobre el licenciamiento de la Obra; y en el caso de una Obra Derivada, atribuir el crédito identificando el uso de la Obra en la Obra Derivada (v.g., "Traducción Francesa de la Obra del Autor Original," o "Guión Cinematográfico basado en la Obra original del Autor Original"). Tal crédito puede ser implementado de cualquier forma razonable; en el caso, sin embargo, de Obras Derivadas u Obras Colectivas, tal crédito aparecerá, como mínimo, donde aparece el crédito de cualquier otro autor comparable y de una manera, al menos, tan destacada como el crédito de otro autor comparable.

d.	Para evitar toda confusión, el Licenciante aclara que, cuando la obra es una composición musical:

i.	Regalías por interpretación y ejecución bajo licencias generales. El Licenciante se reserva el derecho exclusivo de autorizar la ejecución pública o la ejecución pública digital de la obra y de recolectar, sea individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, SAYCO), las regalías por la ejecución pública o por la ejecución pública digital de la obra (por ejemplo Webcast) licenciada bajo licencias generales, si la interpretación o ejecución de la obra está primordialmente orientada por o dirigida a la obtención de una ventaja comercial o una compensación monetaria privada.

ii.	Regalías por Fonogramas. El Licenciante se reserva el derecho exclusivo de recolectar, individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, los consagrados por la SAYCO), una agencia de derechos musicales o algún agente designado, las regalías por cualquier fonograma que Usted cree a partir de la obra (“versión cover”) y distribuya, en los términos del régimen de derechos de autor, si la creación o distribución de esa versión cover está primordialmente destinada o dirigida a obtener una ventaja comercial o una compensación monetaria privada.

e.	Gestión de Derechos de Autor sobre Interpretaciones y Ejecuciones Digitales (WebCasting). Para evitar toda confusión, el Licenciante aclara que, cuando la obra sea un fonograma, el Licenciante se reserva el derecho exclusivo de autorizar la ejecución pública digital de la obra (por ejemplo, webcast) y de recolectar, individualmente o a través de una sociedad de gestión colectiva de derechos de autor y derechos conexos (por ejemplo, ACINPRO), las regalías por la ejecución pública digital de la obra (por ejemplo, webcast), sujeta a las disposiciones aplicables del régimen de Derecho de Autor, si esta ejecución pública digital está primordialmente dirigida a obtener una ventaja comercial o una compensación monetaria privada.

5. Representaciones, Garantías y Limitaciones de Responsabilidad.
A MENOS QUE LAS PARTES LO ACORDARAN DE OTRA FORMA POR ESCRITO, EL LICENCIANTE OFRECE LA OBRA (EN EL ESTADO EN EL QUE SE ENCUENTRA) “TAL CUAL”, SIN BRINDAR GARANTÍAS DE CLASE ALGUNA RESPECTO DE LA OBRA, YA SEA EXPRESA, IMPLÍCITA, LEGAL O CUALQUIERA OTRA, INCLUYENDO, SIN LIMITARSE A ELLAS, GARANTÍAS DE TITULARIDAD, COMERCIABILIDAD, ADAPTABILIDAD O ADECUACIÓN A PROPÓSITO DETERMINADO, AUSENCIA DE INFRACCIÓN, DE AUSENCIA DE DEFECTOS LATENTES O DE OTRO TIPO, O LA PRESENCIA O AUSENCIA DE ERRORES, SEAN O NO DESCUBRIBLES (PUEDAN O NO SER ESTOS DESCUBIERTOS). ALGUNAS JURISDICCIONES NO PERMITEN LA EXCLUSIÓN DE GARANTÍAS IMPLÍCITAS, EN CUYO CASO ESTA EXCLUSIÓN PUEDE NO APLICARSE A USTED.

6. Limitación de responsabilidad.
A MENOS QUE LO EXIJA EXPRESAMENTE LA LEY APLICABLE, EL LICENCIANTE NO SERÁ RESPONSABLE ANTE USTED POR DAÑO ALGUNO, SEA POR RESPONSABILIDAD EXTRACONTRACTUAL, PRECONTRACTUAL O CONTRACTUAL, OBJETIVA O SUBJETIVA, SE TRATE DE DAÑOS MORALES O PATRIMONIALES, DIRECTOS O INDIRECTOS, PREVISTOS O IMPREVISTOS PRODUCIDOS POR EL USO DE ESTA LICENCIA O DE LA OBRA, AUN CUANDO EL LICENCIANTE HAYA SIDO ADVERTIDO DE LA POSIBILIDAD DE DICHOS DAÑOS. ALGUNAS LEYES NO PERMITEN LA EXCLUSIÓN DE CIERTA RESPONSABILIDAD, EN CUYO CASO ESTA EXCLUSIÓN PUEDE NO APLICARSE A USTED.

7. Término.

a.	Esta Licencia y los derechos otorgados en virtud de ella terminarán automáticamente si Usted infringe alguna condición establecida en ella. Sin embargo, los individuos o entidades que han recibido Obras Derivadas o Colectivas de Usted de conformidad con esta Licencia, no verán terminadas sus licencias, siempre que estos individuos o entidades sigan cumpliendo íntegramente las condiciones de estas licencias. Las Secciones 1, 2, 5, 6, 7, y 8 subsistirán a cualquier terminación de esta Licencia.

b.	Sujeta a las condiciones y términos anteriores, la licencia otorgada aquí es perpetua (durante el período de vigencia de los derechos de autor de la obra). No obstante lo anterior, el Licenciante se reserva el derecho a publicar y/o estrenar la Obra bajo condiciones de licencia diferentes o a dejar de distribuirla en los términos de esta Licencia en cualquier momento; en el entendido, sin embargo, que esa elección no servirá para revocar esta licencia o que deba ser otorgada , bajo los términos de esta licencia), y esta licencia continuará en pleno vigor y efecto a menos que sea terminada como se expresa atrás. La Licencia revocada continuará siendo plenamente vigente y efectiva si no se le da término en las condiciones indicadas anteriormente.

8. Varios.

a.	Cada vez que Usted distribuya o ponga a disposición pública la Obra o una Obra Colectiva, el Licenciante ofrecerá al destinatario una licencia en los mismos términos y condiciones que la licencia otorgada a Usted bajo esta Licencia.

b.	Si alguna disposición de esta Licencia resulta invalidada o no exigible, según la legislación vigente, esto no afectará ni la validez ni la aplicabilidad del resto de condiciones de esta Licencia y, sin acción adicional por parte de los sujetos de este acuerdo, aquélla se entenderá reformada lo mínimo necesario para hacer que dicha disposición sea válida y exigible.

c.	Ningún término o disposición de esta Licencia se estimará renunciada y ninguna violación de ella será consentida a menos que esa renuncia o consentimiento sea otorgado por escrito y firmado por la parte que renuncie o consienta.

d.	Esta Licencia refleja el acuerdo pleno entre las partes respecto a la Obra aquí licenciada. No hay arreglos, acuerdos o declaraciones respecto a la Obra que no estén especificados en este documento. El Licenciante no se verá limitado por ninguna disposición adicional que pueda surgir en alguna comunicación emanada de Usted. Esta Licencia no puede ser modificada sin el consentimiento mutuo por escrito del Licenciante y Usted.


Equidistant likert as weighted sum of response categories

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