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Calculating a Factorial ANOVA From Means and Standard Deviations

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Abstract

In this article, I present an alternative formula for the calculation of a factorial analysis of variance (ANOVA), which requires only the mean, standard deviation, and size for each cell of the design, rather than the individual scores. This new method allows a modern hand-held calculator to do most of the work, while still giving students the educational experience of working directly with data. An example is given, in which the new method is applied to a published table of data from a two-way unbalanced ANOVA design. I argue that the new formula is not just easier to use than the traditional (raw-score) calculation formula (especially when dealing with higher order factorial designs), but that it is a better teaching tool, and it conveniently allows estimation of effect sizes from means and standard deviations even when the original authors do not present the corresponding F ratios. Although it seems that instructors are moving away from teaching ANOVA calculation in favor of focusing on the interpretation of computer output, I propose that students will learn more about the structure of ANOVA from using my new method than by avoiding hand calculation entirely (using only statistical software), or by performing calculations with the traditional method.
... SOs were analysed using a two-way analysis of variance (ANOVA) test with the type of sounds as a factor. These ANOVA tests were computed from the summary statistics taken from the figures, as the original data were lost (Larson, 1992;Cohen, 2002). If significant, we performed post hoc Tukey honesty tests. ...
Article
Recent studies have shown that slow oscillations (SOs) can be driven by rhythmic auditory stimulation, which deepens slow‐wave sleep (SWS) and improves memory and the immune‐supportive hormonal milieu related to this sleep stage. While different attempts have been made to optimise the driving of the SOs by changing the number of click stimulations, no study has yet investigated the impact of applying more than five clicks in a row. Likewise, the importance of the type of sounds in eliciting brain responses is presently unclear. In a study of 12 healthy young participants (10 females; aged 18–26 years), we applied an established closed‐loop stimulation method, which delivered sequences of 10 pink noises, 10 pure sounds (B note of 247 Hz), 10 pronounced “a” vowels, 10 sham, 10 variable sounds, and 10 “oddball” sounds on the up phase of the endogenous SOs. By analysing area under the curve, amplitude, and event related potentials, we explored whether the nature of the sound had a differential effect on driving SOs. We showed that every stimulus in a 10‐click sequence, induces a SO response. Interestingly, all three types of sounds that we tested triggered SOs. However, pink noise elicited a more pronounced response compared to the other sounds, which was explained by a broader topographical recruitment of brain areas. Our data further suggest that varying the sounds may partially counteract habituation.
... A two-sample t-test was also used to compare cassiosome nematocyst abundances across populations. Factorial ANOVAs using means and standard deviations (Cohen 2002) were performed with the rpsychi package v0.8 (Okumura 2012) using R v4.0.2 (R Core Team 2013) in RStudio v1.3.1073 (RStudio Team 2020) to verify the nesting of C. ornata nematocysts within previously reported Cassiopea cnidomes (Gamero-Mora et al. 2022; Ames et al. 2020;Heins et al. 2015). ...
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This contribution investigates phenotypic plasticity in Cassiopea ornata Haeckel, 1880 from Guam, Micronesia. We collected C. ornata from two distinct habitats and used DNA barcoding for species identification. With this, we were able to document intraspecific phenotypic variation between populations that is likely reflective of distinct ecotypes rather than species-specific disparities. In particular, macromorphological characters, such as vesicle shapes and sizes, have been used as characters to discriminate among species of Cassiopea varied between populations. In addition, we uncovered differences in cassiosome structure and composition between populations that suggest differences in trophic modes across populations. Conducting a meta-analysis of a comprehensive cnidome dataset, we show that nematocysts may provide important information for species delineation and identification in Cassiopea, a suite of characters not fully exploited thus far. We interpret differences in vesicle and cassiosome morphology in conjunction with nematocyst size disparities as a reflection of environment-mediated shifts in trophic strategy (photo-autotrophy versus heterotrophy). Given the interest in Cassiopea as a model organism, the observations presented herein lay out a roadmap for studies that aim at linking environmental heterogeneity to phenotypic plasticity.
... Wert drückt die Variation aus, die im Datensatz stecken würde, wäre der jeweilige Faktor oder die jeweilige Interaktion nicht im Modell (B. H.Cohen, 2002;Rasch et al., 2010).Aus diesem partiellen Wert kann mit Formel 6.19 zusätzlich die Effektstärke nach Cohen (f ) berechnet werden. Laut J.Cohen (1988) wird bei Effektstärken ab 0.1 von einem schwachen, ab 0.25 von einem mittleren und ab 0.4 von einem starken Effekt gesprochen. ...
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Das Ziel dieser Arbeit war neue Eingangsdaten für die Landoberflächenbeschreibung des regionalen Klimamodells REMO zu finden und ins Modell zu integrieren, um die Vorhersagequalität des Modells zu verbessern. Die neuen Daten wurden so in das Modell eingebaut, dass die bisherigen Daten weiterhin als Option verfügbar sind. Dadurch kann überprüft werden, ob und in welchem Umfang sich die von jedem Klimamodell benötigten Rahmendaten auf Modellergebnisse auswirken. Im Zuge der Arbeit wurden viele unterschiedliche Daten und Methoden zur Generierung neuer Parameter miteinander verglichen, denn neben dem Ersetzen der konstanten Eingangswerte für verschiedene Oberflächenparameter und den damit verbundenen Änderungen wurden als zusätzliche Verbesserung auch Veränderungen an der Parametrisierung des Bodens speziell in Hinblick auf die Bodentemperaturen in REMO vorgenommen. Im Rahmen dieser Arbeit wurden die durch die verschiedenen Änderungen ausgelösten Auswirkungen für das CORDEX-Gebiet EUR-44 mit einer Auflösung von ca. 50km und für das in dem darin eingebetteten neu definierten Deutschlandgebiet GER-11 mit einer Auflösung von ca. 12km getestet sowie alle Änderungen anhand von verschiedenen Beobachtungsdatensätzen validiert. Die vorgenommenen Arbeiten gliederten sich in drei Hauptteile. Der erste Teil bestand in dem vom eigentlichen Klimamodell unabhängigen Vergleich der verschiedenen Eingangsdaten auf unterschiedlichen Auflösungen und deren Performanz in allen Teilen der Erde, wobei ein besonderer Fokus auf der Qualität in den späteren Modellgebieten lag. Unter Berücksichtigung der Faktoren, wie einer globalen Verfügbarkeit der Daten, einer verbesserten räumlichen Auflösung und einer kostenlosen Nutzung der Daten sowie verschiedener Validationsergebnissen von anderen Studien, wurden in dieser Arbeit vier neue Topographiedatensätze (SRTM, ALOS, TANDEM und ASTER) und drei neue Bodendatensätze (FAOn, Soilgrid und HWSD) für die Verwendung im Präprozess von REMO aufbereitet und miteinander sowie mit den bisher in REMO verwendeten Daten verglichen. Auf Grundlage dieser Vergleichsstudien schieden bei den Topographiedaten die verwendeten Datensatz-Versionen von SRTM, ALOS und TANDEM für die in dieser Arbeit durchgeführten REMO-Läufe aus. Bei den neuen Bodendatensätzen wurde ausgenutzt, dass diese verschiedenen Bodeneigenschaften für unterschiedliche Tiefen als Karten zur Verfügung stellen. In REMO wurden bisher alle benötigten Bodenparameter abhängig von fünf verschiedenen Bodentexturklassen und einer zusätzlichen Torfklasse ausgewiesen und als konstant über die gesamte Modellbodensäule (bis ca. 10m) angenommen. Im zweiten Teil wurden auf Basis der im ersten Teil ausgewählten neuen Datensätze und den neu verfügbaren Bodenvariablen verschiedene Sensitivitätsstudien über das Beispieljahr 2000 durchgeführt. Dabei wurden verschiedene neue Parametrisierungen für die bisher aus der Textur abgeleiteten Bodenvariablen und die Parametrisierung von weiteren hydrologischen und thermalen Bodeneigenschaften verglichen. Ferner wurde aufgrund der neuen nicht über die Tiefe konstanten Bodeneigenschaften eine neue numerische Methode zur Berechnung der Bodentemperaturen der fünf Schichten in REMO getestet, welche wiederum andere Anpassungen erforderte. Der Test und die Auswahl der verschiedenen Datensatz- und Parametrisierungsversionen auf die Modellperformanz wurde in drei Experimentpläne unterteilt. Im ersten Plan wurden die Auswirkungen der ausgewählten Topographie- und Bodendatensätze überprüft. Der zweite Plan behandelte die Unterschiede der verschiedenen Parametrisierungsarten der Bodenvariablen hinsichtlich der verwendeten Variablen zur Berechnung der Bodeneigenschaften, der über die Tiefe variablen oder konstanten Eigenschaften und der verwendeten Berechnungsmethode der Bodentemperaturänderungen. Durch die Erkenntnisse aus diesen beiden Experimentplänen, die für beide Untersuchungsgebiete durchgeführt wurden, ergaben sich im dritten Plan weitere Parametrisierungsänderungen. Alle Änderungen dieses dritten Experimentplans wurden sukzessiv getestet, sodass der paarweise Vergleich von zwei aufeinanderfolgenden Modellläufen die Auswirkungen der Neuerung im jeweils zweiten Lauf widerspiegelt. Der letzte Teil der Arbeit bestand aus der Analyse von fünf längeren Modellläufen (2000-2018), die zur Überprüfung der Ergebnisse aus den Sensitivitätsstudien sowie zur Einschätzung der Performanz in weiteren teilweise extremen atmosphärischen Bedingungen durchgeführt wurden. Hierfür wurden die bisherige Modellversion von REMO (id01) für die beiden Untersuchungsgebiete EUR-44 und GER-11 als Referenzläufe, zwei aufgrund der Vergleichsergebnisse von Experimentplan 3 selektierte Modellversionen (id06 und id15a für GER-11) sowie die finale Version (id18a für GER-11), die alle vorgenommenen Änderungen dieser Arbeit enthält, ausgewählt. Es stellte sich heraus, dass sowohl die neuen Topographiedaten als auch die neuen Bodendaten große Differenzen zu den bisherigen Daten in REMO haben. Zudem änderten sich die von diesen konstanten Eingangsdaten abgeleiteten Hilfsvariablen je nach verwendeter Parametrisierung sehr deutlich. Dies war besonders gut anhand der Bodenparameter zu erkennen. Sowohl die räumliche Verteilung als auch der Wertebereich der verschiedenen Modellversionen unterschieden sich stark. Eine Einschätzung der Qualität der resultierenden Parameter wurde jedoch dadurch erschwert, dass auch die verschiedenen zur Validierung herangezogenen Bodendatensätze für diese Parameter deutlich voneinander abweichen. Die finale Modellversion id18a ähnelte trotz der umfassenden Änderungen in den meisten Variablen den Ergebnissen der bisherigen REMO-Version. Je nach zeitlicher und räumlicher Aggregation sowie unterschiedlichen Regionen und Jahreszeiten wurden leichte Verbesserungen, aber auch leichte Verschlechterungen im Vergleich zu den klimatologischen Validationsdaten festgestellt. Größere Veränderungen im Vergleich zur bisherigen Modellversion konnten in den tieferen Bodenschichten aufgezeigt werden, welche allerdings aufgrund von fehlenden Validationsdaten nicht beurteilt werden konnten. Für alle 2m-Temperaturen konnte eine tendenzielle leichte Erwärmung im Vergleich zum bisherigen Modelllauf beobachtet werden, was sich einerseits negativ auf die ohnehin durchschnittlich zu hohe Minimumtemperatur, aber andererseits positiv auf die bisher zu niedrige Maximumtemperatur des Modells in den betrachteten Gebieten auswirkte. Im Niederschlagssignal und in den 10m-Windvariablen konnten keine signifikanten Änderungen nachgewiesen werden, obwohl die neue Topographie an manchen Stellen im Modellgebiet deutlich von der bisherigen abweicht. Des Weiteren variierte das Ranking der verschiedenen Modellversionen jeweils nach dem angewendeten Qualitätsindex. Um diese Ergebnisse besser einordnen zu können, muss berücksichtigt werden, dass die neuen Daten für Modellgebiete mit 50 bzw. 12km räumlicher Auflösung und der damit verbundenen hydrostatischen Modellversion getestet wurden. Zudem sind vor allem in Fall der Topographie die bisher enthaltenen GTOPO-Daten (1km Auflösung) für die Aggregation auf diese gröbere Modellauflösung geeignet. Die bisherigen Bodendaten stoßen jedoch mit 50km Auflösung bereits an ihre Grenzen. Zusätzlich ist zu beachten, dass nicht nur die Mittelwerte dieser Daten, sondern auch deren Subgrid-Variabilität als Variablen im Modell für verschiedene Parametrisierungen verwendet werden. Daher ist es essentiell, dass die Eingangsdaten eine deutlich höhere Auflösung bereitstellen als die zur Modellierung definierte Auflösung. Für lokale Klimasimulationen mit Auflösungen im niedrigen Kilometerbereich spielen auch die Vertikalbewegungen (nicht-hydrostatische Modellversion) eine wichtige Rolle, die stark von der Topographie sowie deren horizontaler und vertikaler Änderungsrate beeinflusst werden, was die in dieser Arbeit eingebauten wesentlich höher aufgelösten Daten für die zukünftige Weiterentwicklung von REMO wertvoll machen kann.
... Violence Risk Scale -Sexual Offense version (VRS-SO; Olver et al., 2007) Olver (2018) reported that of the 913 cases in the norms, 755 cases were specifically from high intensity treatment programs. In order to directly analyze whether these cases are significantly different than typical SVP cases, I obtained the descriptive statistics for SVP cases with Static-99R and VRS-SO scores at Sand Ridge (Kelley, 2018) as well as the descriptive statistics from the three samples reported in Olver (2018). 1 A one-way analysis of variance (ANOVA) using sample sizes, means, and standard deviations was used to examine differences between the four high risk samples (Cohen, 2002). This approach was found to be preferable to comparing an average of the VRS-SO sample means to the SVP sample mean since there was found to be significant differences in variance between the three VRS-SO samples. ...
Article
In many parts of the United States, individuals can be civilly committed as Sexually Violent Persons (SVP) to a secure treatment center based on their history of sexual offenses, current mental disorder, and current risk for sexual recidivism. While the specific criteria vary between jurisdictions, SVP civil commitment is indefinite, and periodic examinations occur to determine if ongoing commitment is necessary. Release recommendations may be made in part based on patients’ treatment progress. Therefore, incorporating treatment change into periodic risk assessments is an important role of the SVP evaluator. The current paper sought to explore the benefits of using an actuarial tool within SVP populations to measure decreased sexual recidivism risk as a result of treatment change. Specific discussion of the use of the Violence Risk Scale – Sexual Offense version (Olver et al., 2007, https://doi.org/10.1037/1040-3590.19.3.318 ) is provided.
... Statistical calculations were made using R statistical software (https://www.r-project.org/) using libraries rpsychi for calculation of two-way ANOVA (Cohen 2002;Huck and Malgady 1978) and ggplot2 for visualisation of the results. Differences of confidence intervals in the plots were evaluated according to Cumming (2012). ...
Article
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Monitoring of pollution in the vicinity of roads connected to winter road maintenance is one of the important tools for optimising winter maintenance technology and reducing its environmental impact. The aim of this study was to determine the relationship between winter road maintenance and the increased concentration of sodium ion to characterise the harm caused by the de-icing agents on selected types of individual components grown in the Norway spruce ecosystem. The model area is located in the immediate vicinity of the D1 motorway connecting Prague and Brno (Czech Republic), at 103 km. The area is thus exposed to long-term contamination from automobile transport, and the monitoring was carried out for 3 consecutive years. A clear effect of the de-icing agents on conifers near the road has been demonstrated by the symptoms of salt damage visually observed in close proximity to the road (at a sampling distance of 5 m). The needles of these spruce trees also showed increased sodium concentrations, regardless of the age of the needles. The study also confirms that sodium accumulates in all selected components of the analysed ecosystem (moos, humus, soil). The sodium concentration has been found to decrease with increasing distance from the road for all of the components.
... 1 Horvath and Reid (1971) 2. Hunter and Ash (1973) 3. Slowik and Buckley (1975) 4. Wicklander and Hunter (1975) 5. Raskin, Kircher, Honts, and Horoitz (1988) 6. Barland, Honts, and Barger (1989) 7. Podlesny and Truslow (1993) 8. Krapohl and Norris (2000) 9. Senter (2003) 10. Senter and Dollins (2003) Two-way unbalanced ANOVAs were calculated using the method described by Cohen (2002) to test the level of significance of differences among results using grand total and subtotal scores for criterion guilty and criterion innocent groups, including decision accuracy, sensitivity, specificity, false-positive and false-negative errors, and inconclusive results. Unbalanced ANOVAs were used due to difference in study sample sizes. ...
... The summarized data for each species were used rather than individual values as the data for individuals were rarely available, and it has been shown that the conclusions and effect sizes made from summary data are very similar to those made with individual data (Steinberg et al., 1997;Tudur Smith et al., 2016). Two-way analysis of variance (ANOVA) for summary data (Cohen 2002) were used to look for statistical differences and interactions between and within foraging mode (benthic and meso vs. epipelagic) and morphology group (sea lion vs. fur seal). To implement the ANOVAs, the mean, standard deviation and sample size were included in the formula and implemented using the function ind.twoway.second() in the R package rpshychi (Okumura and Okumura 2012). ...
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Physiology places constraints on an animal’s ability to forage and those unable to adapt to changing conditions may face increased challenges to reproduce and survive. As the global marine environment continues to change, small, air-breathing, endothermic marine predators such as otariids (fur seals and sea lions) and particularly females, who are constrained by central place foraging during breeding, may experience increased difficulties in successfully obtaining adequate food resources. We explored whether physiological limits of female otariids may be innately related to body morphology (fur seals vs sea lions) and/or dictate foraging strategies (epipelagic vs mesopelagic or benthic). We conducted a systematic review of the increased body of literature since the original reviews of Costa et al. (When does physiology limit the foraging behaviour of freely diving mammals? Int Congr Ser 2004;1275:359–366) and Arnould and Costa (Sea lions in drag, fur seals incognito: insights from the otariid deviants. In Sea Lions of the World Fairbanks. Alaska Sea Grant College Program, Alaska, USA, pp. 309–324, 2006) on behavioural (dive duration and depth) and physiological (total body oxygen stores and diving metabolic rates) parameters. We estimated calculated aerobic dive limit (cADL—estimated duration of aerobic dives) for species and used simulations to predict the proportion of dives that exceeded the cADL. We tested whether body morphology or foraging strategy was the primary predictor of these behavioural and physiological characteristics. We found that the foraging strategy compared to morphology was a better predictor of most parameters, including whether a species was more likely to exceed their cADL during a dive and the ratio of dive time to cADL. This suggests that benthic and mesopelagic divers are more likely to be foraging at their physiological capacity. For species operating near their physiological capacity (regularly exceeding their cADL), the ability to switch strategies is limited as the cost of foraging deeper and longer is disproportionally high, unless it is accompanied by physiological adaptations. It is proposed that some otariids may not have the ability to switch foraging strategies and so be unable adapt to a changing oceanic ecosystem.
... The mean of each measure (ASSR amplitudes, phase coherence, and LI) was obtained based on the entire group of participants without resampling. All statistical analyses were based on the mean, estimated standard deviation, and the number of participants, rather than on individual scores (Cohen, 2002;Nagy, 2013) using custom scripts in MATLAB R2016b (MathWorks). ...
Article
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Speech understanding problems are highly prevalent in the aging population, even when hearing sensitivity is clinically normal. These difficulties are attributed to changes in central temporal processing with age and can potentially be captured by age-related changes in neural generators. The aim of this study is to investigate age-related changes in a wide range of neural generators during temporal processing in middle-aged and older persons with normal audiometric thresholds. A minimum-norm imaging technique is employed to reconstruct cortical and subcortical neural generators of temporal processing for different acoustic modulations. The results indicate that for relatively slow modulations (<50 Hz), the response strength of neural sources is higher in older adults than in younger ones, while the phase-locking does not change. For faster modulations (80 Hz), both the response strength and the phase-locking of neural sources are reduced in older adults compared to younger ones. These age-related changes in temporal envelope processing of slow and fast acoustic modulations are possibly due to loss of functional inhibition, which is accompanied by aging. Both cortical (primary and non-primary) and subcortical neural generators demonstrate similar age-related changes in response strength and phase-locking. Hemispheric asymmetry is also altered in older adults compared to younger ones. Alterations depend on the modulation frequency and side of stimulation. The current findings at source level could have important implications for the understanding of age-related changes in auditory temporal processing and for developing advanced rehabilitation strategies to address speech understanding difficulties in the aging population.
... The ind.twoway.second function in the rpsychi package uses the means, SDs, and cell sizes to recalculate the two-way ANOVAs using an unweighted-means solution as described in Cohen (2002). We were unsure how the Food and Brand Lab had calculated their two-way ANOVAs, but since they cited an article about SPSS in Article 1, we speculated that they may have used this software. ...
Preprint
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We previously reported over 150 inconsistencies in a series of four articles (the "pizza papers") from the Cornell Food and Brand Lab that described a study of eating habits at an all-you-can-eat pizza buffet. The lab's initial response led us to investigate more of their work, and our investigation has now identified issues with at least 45 publications from this lab. Perhaps because of the growing media attention, Cornell and the lab have released a statement concerning the pizza papers, which included a response to the inconsistencies, along with data and code. Many of the inconsistencies were identified with the new technique of granularity testing, and this case has the highest density of granularity inconsistencies that we know of. This is also the first time a data set has been made public after granularity concerns were raised, making it a highly suitable case study for showing the accuracy and potential of this technique. It is also important that a third party audit the lab's response, given the continuing investigation of misconduct and presumably future reports and data releases. Our careful inspection of the data set suggests no evidence of fabrication, but we found the lab's report confusing, incomplete, and error prone. In addition, we found the number of missing, unusual, and logically impossible responses in the data set highly concerning. Unfortunately, given the unsound theory, poor methodology, questionable data, and countless errors, we find it remarkable that these four papers were published and recommend retraction of all four papers.
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Monitoring of pollution in the vicinity of roads connected to winter road maintenance is one of the important tools for optimising winter maintenance technology and reducing its environmental impact. The aim of this study was to determine the relationship between winter road maintenance and the increased concentration of sodium ion to characterize the harm caused by the de-icing agents on selected types of individual components grown in the Norway Spruce ecosystem. The model area is located in the immediate vicinity of the D1 motorway connecting Prague and Brno (Czech Republic), at 103rd kilometer. The area is thus exposed to long-term contamination from automobile transport and the monitoring was carried out for three consecutive years. A clear effect of the de-icing agents on conifers near the road has been demonstrated by the symptoms of salt damage visually observed in close proximity to the road (at a sampling distance of 5 m). The needles of these spruce trees also showed increased sodium concentrations, regardless of the age of the needles. The study also confirms that sodium accumulates in all selected components of the analysed ecosystem (moos, humus, soil). The sodium concentration has been found to decrease with increasing distance from the road for all of the components.
Article
Social connectedness and its relationship with anxiety, self-esteem, and social identity was explored in the lives of women. Social connectedness was negatively related to trait anxiety and made a larger unique contribution to trait anxiety than social support or collective self-esteem. Women with high connectedness also reported greater social identification in high, as compared with low, cohesion conditions. Women with low connectedness exhibited no difference in either condition. Social connectedness was also positively related to state self-esteem across both conditions but did not have an effect on state anxiety. Future research in gender and cultural differences, self-evaluation process, and intervention strategies are discussed in light of the findings.