Linear trend analysis of time series is standard procedure in many scientific disciplines. If the number of data is large, a trend may be statistically significant even if data are scattered far from the trend line. This study introduces and tests a quality criterion for time trends referred to as statistical meaningfulness, which is a stricter quality criterion for trends than high statistical significance. The time series is divided into intervals and interval mean values are calculated. Thereafter, r2 and p values are calculated from regressions concerning time and interval mean values. If r2≥0.65 at p≤0.05 in any of these regressions, then the trend is regarded as statistically meaningful. Out of ten investigated time series from different scientific disciplines, five displayed statistically meaningful trends. A Microsoft Excel application (add-in) was developed which can perform statistical meaningfulness tests and which may increase the operationality of the test. The presented method for distinguishing statistically meaningful trends should be reasonably uncomplicated for researchers with basic statistics skills and may thus be useful for determining which trends are worth analysing further, for instance with respect to causal factors. The method can also be used for determining which segments of a time trend may be particularly worthwhile to focus on.

Performance Measure Definition: Average Call Processing Interval

Model performance evaluation: The Mean Winkler Interval score (MWIS)

We can assess the overall performance of a regression model that produces prediction intervals by using the mean Winkler Interval score [1,2,3] which, for an individual interval, is given by:

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where \(y\) is the true value, \(u\) it the upper prediction interval, \(l\) is the lower prediction interval, and \(\alpha\) is (1-coverage). For example, for 90% coverage, \(\alpha = 0.1\). Note that the Winkler Interval score constitutes a proper scoring rule [2,3].

• [1] Robert L. Winkler "*A Decision-Theoretic Approach to Interval Estimation*", Journal of the American Statistical Association, volume 67, pp. 187-191 (1972)
• [2] Tilmann Gneiting and Adrian E Raftery "*Strictly Proper Scoring Rules, Prediction, and Estimation*", Journal of the American Statistical Association, volume 102, pp. 359-378 (2007) (Section 6.2)
• [3] Jonas R. Brehmer, and Tilmann Gneiting "*Scoring interval forecasts: Equal-tailed, shortest, and modal interval*", Bernoulli volume 27 pp. 1993-2010 (2021)

Python code: Usage example

Attach this dataset to a notebook, then:

import sys
sys.path.append('/kaggle/input/winkler-interval-score-metric/')
import MWIS_metric
help(MWIS_metric.score)

MWIS,coverage = MWIS_metric.score(predictions["y_true"],predictions["lower"],predictions["upper"],alpha)
print(f"Local MWI score   ",round(MWIS,3))
print("Predictions coverage  ", round(coverage*100,1),"%")

A comment on Tressoldi et al's article on journals' impact factor and statistical quality (PLOS ONE 8(2), e56180, 2013, doi:10.1371/journal.pone.0056180) on the author's page at Frontiers in Psychology's Loop profiles.

Group data (mean and 95% confidence interval) for pain and function outcome measures.

Nonparametric estimates obtained with our method and parametric estimates of SI features (mean , standard deviation , median , and 95th quantile ) for different publicly available serial interval datasets. Values in round brackets correspond to confidence intervals for our method and confidence or credible intervals for parametric methods. The third column indicates the sample size. NR: Not Reported. The symbol * indicates that information was obtained by contacting the corresponding author of the article listed in the data source column.

This dataset identifies all regions in which the full 95% confidence interval is between 22 and 32 �C for all 12 months. The sea surface temperature data includes the mean sea surface temperature per month, the standard deviation and the number of observations used to calculate the mean. Based on these values, the 95% upper and lower confidence levels about the mean for each month have been generated.

This metric, uses the mean percent FRID to a measure of the extent to which contemporary fires (i.e., since 1908) are burning at frequencies similar to the frequencies that occurred prior to Euro-American settlement, with the mean reference FRI binned into another basis for comparison. Mean PFRID is a metric of fire return interval departure (FRID), and measures the departure of current FRI from reference mean FRI in percent.

This dataset is about: (Table 3 core) Interval-mean bedding directions based on core-scan data of core CRP-3. Please consult parent dataset @ https://doi.org/10.1594/PANGAEA.485006 for more information.

This video lecture and slide set presents a pragmatic statistical philosophy, including both frequentist and Bayesian ideas as well as providing careful definitions of inference, hypothesis testing, and P values.Latest slide set with video, MMED 2017:- 'Dushoff-StatsPhilosophy.pdf'- 'Dushoff-Intro to Statistical Philosophy.mp4'Latest slide set, MMED 2018:'DushoffStatisticalPhilosophyMMED2018.pdf'

Full report of statistics, including 95% CI intervals calculated both by parametric and non-parametric means corresponding to S4 Data.

Performance Measure Definition: Trauma Alert Scene Interval

Implementation of Generic and consistent confidence and credible regions. Christian Bartels (2015) figshare. http://dx.doi.org/10.6084/m9.figshare.1528163

A generic, consistent, efficient and exact method is proposed for set selection. The method is generic in that its definition and implementation uses only the likelihood function. The method is consistent in that the same criterion is used to select confidence and credible sets making the two kinds of sets consistent even though the two sets may differ since they answer different questions. The method is exact in that no approximations are used except numerical integration which can be made as exact as needed by investing computational resources. The method is comparatively efficient and requires computational resources comparable to what is needed for a Bayesian analysis and may be more efficient than bootstrap of maximum likelihood estimates as it avoids repeated minimizations of randomly perturbed data. Central to the proposed approach are the use of (1) reference priors (e.g., Bernardo, 2005), (2) pointwise mutual information as test statistics and (3) importance sampling to efficiently evaluate series of related statistical integrals (e.g., Schafer, 2009). These central pieces are expected to be useful to address statistical questions beyond set selection.

Full report of statistics, including 95% CI intervals calculated both by parametric and non-parametric means corresponding to S1 Data.

Supporting documentation on code lists, subject definitions, data accuracy, and statistical testing can be found on the American Community Survey website in the Data and Documentation section...Sample size and data quality measures (including coverage rates, allocation rates, and response rates) can be found on the American Community Survey website in the Methodology section..Although the American Community Survey (ACS) produces population, demographic and housing unit estimates, it is the Census Bureau''s Population Estimates Program that produces and disseminates the official estimates of the population for the nation, states, counties, cities and towns and estimates of housing units for states and counties..Explanation of Symbols:An ''**'' entry in the margin of error column indicates that either no sample observations or too few sample observations were available to compute a standard error and thus the margin of error. A statistical test is not appropriate..An ''-'' entry in the estimate column indicates that either no sample observations or too few sample observations were available to compute an estimate, or a ratio of medians cannot be calculated because one or both of the median estimates falls in the lowest interval or upper interval of an open-ended distribution..An ''-'' following a median estimate means the median falls in the lowest interval of an open-ended distribution..An ''+'' following a median estimate means the median falls in the upper interval of an open-ended distribution..An ''***'' entry in the margin of error column indicates that the median falls in the lowest interval or upper interval of an open-ended distribution. A statistical test is not appropriate..An ''*****'' entry in the margin of error column indicates that the estimate is controlled. A statistical test for sampling variability is not appropriate. .An ''N'' entry in the estimate and margin of error columns indicates that data for this geographic area cannot be displayed because the number of sample cases is too small..An ''(X)'' means that the estimate is not applicable or not available..Estimates of urban and rural population, housing units, and characteristics reflect boundaries of urban areas defined based on Census 2010 data. As a result, data for urban and rural areas from the ACS do not necessarily reflect the results of ongoing urbanization..While the 2010-2014 American Community Survey (ACS) data generally reflect the February 2013 Office of Management and Budget (OMB) definitions of metropolitan and micropolitan statistical areas; in certain instances the names, codes, and boundaries of the principal cities shown in ACS tables may differ from the OMB definitions due to differences in the effective dates of the geographic entities..Occupation codes are 4-digit codes and are based on Standard Occupational Classification 2010..Data are based on a sample and are subject to sampling variability. The degree of uncertainty for an estimate arising from sampling variability is represented through the use of a margin of error. The value shown here is the 90 percent margin of error. The margin of error can be interpreted roughly as providing a 90 percent probability that the interval defined by the estimate minus the margin of error and the estimate plus the margin of error (the lower and upper confidence bounds) contains the true value. In addition to sampling variability, the ACS estimates are subject to nonsampling error (for a discussion of nonsampling variability, see Accuracy of the Data). The effect of nonsampling error is not represented in these tables..Source: U.S. Census Bureau, 2010-2014 American Community Survey 5-Year Estimates

Performance Measure Definition: Stroke Alert Call-to-Door Interval

This dataset is about: (Table 3 BHTV) Interval-mean bedding directions based on borehole televiewer data of core CRP-3. Please consult parent dataset @ https://doi.org/10.1594/PANGAEA.485006 for more information.

This dataset identifies all regions in which the full 95% confidence interval is greater than 0.5 mg/m3 that were combined for the months available in each hemisphere for the blue mussel. The chlorophyll 2 data includes the mean chlorophyll 2 level per month, the standard deviation and the number of observations used to calculate the mean. Based on these values, the 95% upper and lower confidence levels about the mean for each month have been generated.

Performance Measure Definition: STEMI Alert Call-to-Door Interval

Performance Measure Definition: Trauma Alert Call-to-Door Interval