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have written on stationarity, unit root testing, co-integration, and related issues (a summary of some of the works in this area can be found in an information paper by the Royal
Swedish Academy of Sciences (2003)); and Ho-Trieu & Tucker (1990) have written on logarithmic time trends with results indicating linear time trends are special cases of
1639: = 0.00006. Incidentally, it could be reasonably argued that as age is a natural continuously variable index, it should not be categorized into decades, and an effect of age and serum trypsin is sought by correlation (assuming the raw data is available). A further example is of a substance measured at four time points in different groups:
190:
135:
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Research results by mathematicians, statisticians, econometricians, and economists have been published in response to those questions. For example, detailed notes on the meaning of linear time trends in the regression model are given in
Cameron (2005); Granger, Engle, and many other econometricians
1374:
The estimated coefficient associated with a linear trend variable such as time is interpreted as a measure of the impact of a number of unknown or known but immeasurable factors on the dependent variable over one unit of time. Strictly speaking, this interpretation is applicable for the estimation
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studies often seek to determine a link between sets of data, such as of a clinical or scientific metric in three different diseases. But data may also be linked in time (such as change in the effect of a drug from baseline, to month 1, to month 2), or by an external factor that may or may not be
1534:
to the independent variable (such as cyclic influences), the use of least-squares estimation of the trend is not valid. Also, where the variations are significantly larger than the resulting straight line trend, the choice of start and end points can significantly change the result. That is, the
1529:
with a normal distribution. Real data (for example, climate data) may not fulfill these criteria. This is important, as it makes an enormous difference to the ease with which the statistics can be analyzed so as to extract maximum information from the data series. If there are other non-linear
1600:
on the degree of pain, or increasing doses of different strengths of a drug on a measurable index, i.e. a dose - response effect) to change in direct order as the effect develops. Suppose the mean level of cholesterol before and after the prescription of a statin falls from 5.6
1614:, depending on the nature of the data. Nevertheless, because the groups are ordered, a standard ANOVA is inappropriate. Should the cholesterol fall from 5.4 to 4.1 to 3.7, there is a clear linear trend. The same principle may be applied to the effects of allele/
1625:
The mathematics of linear trend estimation is a variant of the standard ANOVA, giving different information, and would be the most appropriate test if the researchers hypothesize a trend effect in their test statistic. One example is levels of serum
1461:°C (by coincidence, about the same value as the interannual variation). Hence, the trend is statistically different from 0. However, as noted elsewhere, this time series doesn't conform to the assumptions necessary for least-squares to be valid.
1630:
in six groups of subjects ordered by age decade (10â19 years up to 60â69 years). Levels of trypsin (ng/mL) rise in a direct linear trend of 128, 152, 194, 207, 215, 218 (data from Altman). Unsurprisingly, a 'standard' ANOVA gives
1711: = 0.012. However, should the data have been collected at four time points in the same individuals, linear trend estimation would be inappropriate, and a two-way (repeated measures) ANOVA would have been applied.
1539:. Statistical inferences (tests for the presence of a trend, confidence intervals for the trend, etc.) are invalid unless departures from the standard assumptions are properly accounted for, for example, as follows:
1609:
would most likely find a significant fall at one and two months, but the fall is not linear. Furthermore, a post-hoc test may be required. An alternative test may be a repeated measures (two way) ANOVA or
1016:, meaning that those data points are effectively less certain), then this can be taken into account during the least-squares fitting by weighting each point by the inverse of the variance of that point.
1363:
The use of a linear trend line has been the subject of criticism, leading to a search for alternative approaches to avoid its use in model estimation. One of the alternative approaches involves
1481:. Black = unfiltered data; red = data averaged every 10 points; blue = data averaged every 100 points. All have the same trend, but more filtering leads to higher
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determined by the researcher and/or their subject (such as no pain, mild pain, moderate pain, or severe pain). In these cases, one would expect the effect test statistic (e.g., influence of a
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952:. The least-squares method assumes the errors are independently distributed with a normal distribution. If this is not the case, hypothesis tests about the unknown parameters
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gathered tends to increase or decrease over time or is influenced by changes in an external factor. Linear trend estimation essentially creates a straight line on a
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1360:, then the estimated trend is deemed significantly different from zero at that significance level, and the null hypothesis of a zero underlying trend is rejected.
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It is harder to see a trend in a noisy time series. For example, if the true series is 0, 1, 2, 3, all plus some independent normally distributed "noise"
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1446:
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45:
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time frame only. Outside of this time frame, it cannot be determined how these immeasurable factors behave both qualitatively and quantitatively.
1567:: taking first (or occasionally second) differences of the data, with the level of differencing being identified through various unit root tests.
308:
with the dependent variable (typically the measured data) on the vertical axis and the independent variable (often time) on the horizontal axis.
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to the variance of the dependent variable. It says what fraction of the variance of the data is explained by the fitted trend line. It does
2360:
2060:
153:
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628:, the difference at each data point is squared, and then added together, giving the "sum of squares" measurement of error. The values of
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1968:
Kungl. Vetenskapsakademien (2003). "Time-series econometrics: Cointegration and autoregressive conditional heteroskedasticity".
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Thus far, the data have been assumed to consist of the trend plus noise, with the noise at each data point being
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1327:, the distribution of calculated trends is to be expected from random (trendless) data. If the estimated trend,
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1707: = 0.091, because the overall variance exceeds the means, whereas linear trend estimation gives
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To analyze a (time) series of data, it can be assumed that it may be represented as trend plus noise:
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Once the "noise" of the series is known, the significance of the trend can be assessed by making the
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1575:, the linear trend in data can be estimated by using the 'tslm' function of the 'forecast' package.
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at baseline to 3.4 mmol/L at one month and to 3.7 mmol/L at two months. Given sufficient power, an
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1884:
Bianchi, M.; Boyle, M.; Hollingsworth, D. (1999). "A comparison of methods for trend estimation".
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Advanced
Information on the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel
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of the trend line (see graph); the statistical significance of the trend is determined by its
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Ho-Trieu, N. L.; Tucker, J. (1990). "Another note on the use of a logarithmic time trend".
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1323:, is not different from 0. From the above discussion of trends in random data with known
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1269:'s from the residuals â this is often the only way of estimating the variance of the
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Commonly, where only a single time series exists to be analyzed, the variance of the
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in nucleotides XX, XY, YY are in fact a trend of no Y's, one Y, and then two Y's.
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2004:
1828:"IPCC Third Assessment Report â Climate Change 2001 â Complete online versions"
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1039:'s is estimated by fitting a trend to obtain the estimated parameter values
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fit is a common method to fit a straight line through the data. This method
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This formula first calculates the difference between the observed data
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1602:
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1978:
1907:
Cameron, S. (2005). "Making
Regression Analysis More Useful, II".
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Dependence: autocorrelated time series might be modelled using
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derived from the data parameterize the simple linear estimator
304:
that can be chosen to fit the data. The simplest function is a
1911:. Maidenhead: McGraw Hill Higher Education. pp. 171â198.
1635: < 0.0001, whereas linear trend estimation gives
183:
128:
66:
25:
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Non-normal distribution for errors: in the simplest cases, a
1012:'s all have the same distribution, but if not (if some have
911:. If one can reject the null hypothesis that the errors are
1926:
Chatfield, C. (1993). "Calculating
Interval Forecasts".
1527:
independent and identically distributed random variables
204:
1776:"Making Regression More Useful II: Dummies and Trends"
1497:), which is 1 minus the ratio of the variance of the
1405:, and a sample series of length 50 is given, then if
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Autoregressive conditional heteroskedasticity (ARCH)
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while making little difference to the fitted trend.
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The least-squares fitting process produces a value,
1457:°C over 140 years, with 95% confidence limits of 0.2
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may be too technical for most readers to understand
16:
Statistical technique to aid interpretation of data
1356:, is larger than the critical value for a certain
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1315:
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1230:
1187:
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1445:record of the past 140 years as presented by the
1158:{\displaystyle {\hat {y}}={\hat {a}}t+{\hat {b}}}
741:{\displaystyle {\hat {y}}={\hat {a}}x+{\hat {b}}}
444:are chosen to minimize the sum of squared errors
319:the sum of the squared errors in the data series
1427:100, the trend will probably be visible; but if
3499:Multivariate adaptive regression splines (MARS)
1979:"Self-similarity of high-order moving averages"
2024:. London: Chapman and Hall. pp. 212â220.
1949:Review of Marketing and Agricultural Economics
1550:Non-constant variance: in the simplest cases,
1438:10000, the trend will be buried in the noise.
2054:
386:observed for those points in time, values of
8:
935:
922:
1977:Arianos, S.; Carbone, A.; Turk, C. (2011).
1928:Journal of Business and Economic Statistics
1477:Illustration of the effect of filtering on
284:that models the general direction that the
60:Learn how and when to remove these messages
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2022:Practical Statistics for Medical Research
1441:Consider a concrete example, such as the
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992:may be inaccurate. It is simplest if the
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245:Learn how and when to remove this message
227:Learn how and when to remove this message
211:, without removing the technical details.
172:Learn how and when to remove this message
117:Learn how and when to remove this message
1972:. The Royal Swedish Academy of Sciences.
1641:
1449:. The interannual variation is about 0.2
621:{\displaystyle ({\hat {a}}t+{\hat {b}})}
80:This article includes a list of general
19:For broader coverage of this topic, see
1800:"The Royal Swedish Academy of Sciences"
1767:
748:. The term "trend" refers to the slope
4025:KaplanâMeier estimator (product limit)
1513:. Often, filtering a series increases
4375:Regression with time series structure
1242:, and estimating the variance of the
209:make it understandable to non-experts
7:
4335:
4035:Accelerated failure time (AFT) model
1858:Forecasting: principles and practice
1851:
1849:
1822:
1820:
1545:autoregressive moving average models
1100:thus allowing the predicted values
272:patterns, or trends, occur when the
4347:
3630:Analysis of variance (ANOVA, anova)
1703:This is a clear trend. ANOVA gives
1416:0.1, the trend will be obvious; if
3725:CochranâMantelâHaenszel statistics
2351:Pearson product-moment correlation
1618:, where it could be argued that a
1371:technique in econometric studies.
837:{\displaystyle y_{t}=at+b+e_{t}\,}
537:{\displaystyle \sum _{t}\left^{2}}
144:tone or style may not reflect the
86:it lacks sufficient corresponding
14:
915:, then the non-stationary series
41:This article has multiple issues.
4346:
4334:
4322:
4309:
4308:
777:in the least squares estimator.
339:. Given a set of points in time
188:
154:guide to writing better articles
133:
71:
30:
3984:Least-squares spectral analysis
1741:Least-squares spectral analysis
1168:to be subtracted from the data
49:or discuss these issues on the
2965:Mean-unbiased minimum-variance
1940:10.1080/07350015.1993.10509938
1620:single-nucleotide polymorphism
1453:°C, and the trend is about 0.6
1340:
1231:{\displaystyle {\hat {e}}_{t}}
1216:
1149:
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887:are unknown constants and the
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292:Fitting a trend: Least-squares
1:
4278:Geographic information system
3494:Simultaneous equations models
3461:Coefficient of determination
3072:Uniformly most powerful test
1607:ANOVA (analysis of variance)
907:'s are randomly distributed
4030:Proportional hazards models
3974:Spectral density estimation
3956:Vector autoregression (VAR)
3390:Maximum posterior estimator
2622:Randomized controlled trial
1093:{\displaystyle {\hat {b}},}
4401:
3790:Multivariate distributions
2210:Average absolute deviation
2005:10.1103/physreve.84.046113
1443:global surface temperature
1349:{\displaystyle {\hat {a}}}
1061:{\displaystyle {\hat {a}}}
770:{\displaystyle {\hat {a}}}
679:{\displaystyle {\hat {b}}}
650:{\displaystyle {\hat {a}}}
437:{\displaystyle {\hat {b}}}
408:{\displaystyle {\hat {a}}}
264:technique used to analyze
18:
4304:
4107:
4094:
3778:Structural equation model
3686:
3661:
3432:
3408:
3140:
3114:Score/Lagrange multiplier
2720:
2707:
2529:Sample size determination
2490:
2477:
2107:
2094:
2076:
1887:Applied Economics Letters
941:{\displaystyle \{y_{t}\}}
300:, there are a variety of
4273:Environmental statistics
3795:Elliptical distributions
3588:Generalized linear model
3517:Simple linear regression
3287:HodgesâLehmann estimator
2744:Probability distribution
2653:Stochastic approximation
2215:Coefficient of variation
1559:generalized linear model
1535:model is mathematically
1507:statistical significance
4380:Statistical forecasting
3933:Cross-correlation (XCF)
3541:Non-standard predictors
2975:LehmannâScheffĂ© theorem
2648:Adaptive clinical trial
1900:10.1080/135048599353726
1579:Trends in clinical data
1199:the data), leaving the
781:Data as trend and noise
258:Linear trend estimation
148:used on Knowledge (XXG)
101:more precise citations.
4329:Mathematics portal
4150:Engineering statistics
4058:NelsonâAalen estimator
3635:Analysis of covariance
3522:Ordinary least squares
3446:Pearson product-moment
2850:Statistical functional
2761:Empirical distribution
2594:Controlled experiments
2323:Frequency distribution
2101:Descriptive statistics
1961:10.22004/ag.econ.12288
1552:weighted least squares
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152:See Knowledge (XXG)'s
4245:Population statistics
4187:System identification
3921:Autocorrelation (ACF)
3849:Exponential smoothing
3763:Discriminant analysis
3758:Canonical correlation
3622:Partition of variance
3484:Regression validation
3328:(JonckheereâTerpstra)
3227:Likelihood-ratio test
2916:Frequentist inference
2828:Locationâscale family
2749:Sampling distribution
2714:Statistical inference
2681:Cross-sectional study
2668:Observational studies
2627:Randomized experiment
2456:Stem-and-leaf display
2258:Central limit theorem
2020:Altman, D.G. (1991).
1485:of fitted trend line.
1476:
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1289:{\displaystyle e_{t}}
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1262:{\displaystyle e_{t}}
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379:{\displaystyle y_{t}}
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4168:Probabilistic design
3753:Principal components
3596:Exponential families
3548:Nonlinear regression
3527:General linear model
3489:Mixed effects models
3479:Errors and residuals
3456:Confounding variable
3358:Bayesian probability
3336:Van der Waerden test
3326:Ordered alternative
3091:Multiple comparisons
2970:RaoâBlackwellization
2933:Estimating equations
2889:Statistical distance
2607:Factorial experiment
2140:Arithmetic-Geometric
1834:on November 20, 2009
1561:might be applicable.
1530:effects that have a
1331:
1307:
1273:
1246:
1206:
1172:
1107:
1072:
1043:
1023:
996:
976:
956:
919:
891:
871:
851:
792:
752:
690:
661:
632:
579:
552:
451:
419:
390:
363:
343:
323:
4240:Official statistics
4163:Methods engineering
3844:Seasonal adjustment
3612:Poisson regressions
3532:Bayesian regression
3471:Regression analysis
3451:Partial correlation
3423:Regression analysis
3022:Prediction interval
3017:Likelihood interval
3007:Confidence interval
2999:Interval estimation
2960:Unbiased estimators
2778:Model specification
2658:Up-and-down designs
2346:Partial correlation
2302:Index of dispersion
2220:Interquartile range
1997:2011PhRvE..84d6113A
1861:. 20 September 2014
1756:Regression analysis
1751:Prediction interval
1643:
1469:-squared) and trend
4260:Spatial statistics
4140:Medical statistics
4040:First hitting time
3994:Whittle likelihood
3645:Degrees of freedom
3640:Multivariate ANOVA
3573:Heteroscedasticity
3385:Bayesian estimator
3350:Bayesian inference
3199:KolmogorovâSmirnov
3084:Randomization test
3054:Testing hypotheses
3027:Tolerance interval
2938:Maximum likelihood
2833:Exponential family
2766:Density estimation
2726:Statistical theory
2686:Natural experiment
2632:Scientific control
2549:Survey methodology
2235:Standard deviation
1642:
1616:genotype frequency
1487:
1397:standard deviation
1358:significance level
1346:
1313:
1286:
1259:
1228:
1185:
1155:
1090:
1058:
1029:
1002:
982:
962:
938:
897:
877:
857:
834:
767:
738:
676:
647:
618:
565:
534:
463:
434:
405:
376:
349:
329:
4362:
4361:
4300:
4299:
4296:
4295:
4235:National accounts
4205:Actuarial science
4197:Social statistics
4090:
4089:
4086:
4085:
4082:
4081:
4017:Survival function
4002:
4001:
3864:Granger causality
3705:Contingency table
3680:Survival analysis
3657:
3656:
3653:
3652:
3509:Linear regression
3404:
3403:
3400:
3399:
3375:Credible interval
3344:
3343:
3127:
3126:
2943:Method of moments
2812:Parametric family
2773:Statistical model
2703:
2702:
2699:
2698:
2617:Random assignment
2539:Statistical power
2473:
2472:
2469:
2468:
2318:Contingency table
2288:
2287:
2155:Generalized/power
1984:Physical Review E
1701:
1700:
1465:Goodness of fit (
1387:Noisy time series
1343:
1316:{\displaystyle a}
1219:
1152:
1134:
1119:
1084:
1055:
1032:{\displaystyle e}
1005:{\displaystyle e}
985:{\displaystyle b}
965:{\displaystyle a}
900:{\displaystyle e}
880:{\displaystyle b}
860:{\displaystyle a}
764:
735:
717:
702:
673:
644:
612:
594:
575:and the estimate
515:
497:
454:
431:
402:
352:{\displaystyle t}
332:{\displaystyle y}
255:
254:
247:
237:
236:
229:
182:
181:
174:
146:encyclopedic tone
127:
126:
119:
64:
4392:
4385:Change detection
4350:
4349:
4338:
4337:
4327:
4326:
4312:
4311:
4215:Crime statistics
4109:
4096:
4013:
3979:Fourier analysis
3966:Frequency domain
3946:
3893:
3859:Structural break
3819:
3768:Cluster analysis
3715:Log-linear model
3688:
3663:
3604:
3578:Homoscedasticity
3434:
3410:
3329:
3321:
3313:
3312:(KruskalâWallis)
3297:
3282:
3237:Cross validation
3222:
3204:AndersonâDarling
3151:
3138:
3109:Likelihood-ratio
3101:Parametric tests
3079:Permutation test
3062:1- & 2-tails
2953:Minimum distance
2925:Point estimation
2921:
2872:Optimal decision
2823:
2722:
2709:
2691:Quasi-experiment
2641:Adaptive designs
2492:
2479:
2356:Rank correlation
2118:
2109:
2096:
2063:
2056:
2049:
2040:
2035:
2016:
1973:
1964:
1943:
1922:
1903:
1871:
1870:
1868:
1866:
1853:
1844:
1843:
1841:
1839:
1830:. Archived from
1824:
1815:
1814:
1812:
1810:
1805:. 8 October 2003
1804:
1796:
1790:
1789:
1787:
1785:
1780:
1772:
1644:
1460:
1456:
1452:
1437:
1433:
1426:
1422:
1415:
1411:
1401:
1355:
1353:
1352:
1347:
1345:
1344:
1336:
1322:
1320:
1319:
1314:
1303:that the trend,
1295:
1293:
1292:
1287:
1285:
1284:
1268:
1266:
1265:
1260:
1258:
1257:
1237:
1235:
1234:
1229:
1227:
1226:
1221:
1220:
1212:
1194:
1192:
1191:
1186:
1184:
1183:
1164:
1162:
1161:
1156:
1154:
1153:
1145:
1136:
1135:
1127:
1121:
1120:
1112:
1099:
1097:
1096:
1091:
1086:
1085:
1077:
1067:
1065:
1064:
1059:
1057:
1056:
1048:
1038:
1036:
1035:
1030:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
971:
969:
968:
963:
950:trend-stationary
947:
945:
944:
939:
934:
933:
906:
904:
903:
898:
886:
884:
883:
878:
866:
864:
863:
858:
843:
841:
840:
835:
832:
831:
804:
803:
776:
774:
773:
768:
766:
765:
757:
747:
745:
744:
739:
737:
736:
728:
719:
718:
710:
704:
703:
695:
685:
683:
682:
677:
675:
674:
666:
656:
654:
653:
648:
646:
645:
637:
627:
625:
624:
619:
614:
613:
605:
596:
595:
587:
574:
572:
571:
566:
564:
563:
543:
541:
540:
535:
533:
532:
527:
523:
522:
518:
517:
516:
508:
499:
498:
490:
479:
478:
462:
443:
441:
440:
435:
433:
432:
424:
414:
412:
411:
406:
404:
403:
395:
385:
383:
382:
377:
375:
374:
359:and data values
358:
356:
355:
350:
338:
336:
335:
330:
250:
243:
232:
225:
221:
218:
212:
192:
191:
184:
177:
170:
166:
163:
157:
156:for suggestions.
137:
136:
129:
122:
115:
111:
108:
102:
97:this article by
88:inline citations
75:
74:
67:
56:
34:
33:
26:
4400:
4399:
4395:
4394:
4393:
4391:
4390:
4389:
4365:
4364:
4363:
4358:
4321:
4292:
4254:
4191:
4177:quality control
4144:
4126:Clinical trials
4103:
4078:
4062:
4050:Hazard function
4044:
3998:
3960:
3944:
3907:
3903:BreuschâGodfrey
3891:
3868:
3808:
3783:Factor analysis
3729:
3710:Graphical model
3682:
3649:
3616:
3602:
3582:
3536:
3503:
3465:
3428:
3427:
3396:
3340:
3327:
3319:
3311:
3295:
3280:
3259:Rank statistics
3253:
3232:Model selection
3220:
3178:Goodness of fit
3172:
3149:
3123:
3095:
3048:
2993:
2982:Median unbiased
2910:
2821:
2754:Order statistic
2716:
2695:
2662:
2636:
2588:
2543:
2486:
2484:Data collection
2465:
2377:
2332:
2306:
2284:
2244:
2196:
2113:Continuous data
2103:
2090:
2072:
2067:
2032:
2019:
1976:
1967:
1946:
1925:
1919:
1906:
1883:
1880:
1875:
1874:
1864:
1862:
1855:
1854:
1847:
1837:
1835:
1826:
1825:
1818:
1808:
1806:
1802:
1798:
1797:
1793:
1783:
1781:
1778:
1774:
1773:
1769:
1764:
1717:
1581:
1523:
1521:Advanced models
1471:
1458:
1454:
1450:
1435:
1431:
1424:
1420:
1413:
1409:
1399:
1389:
1329:
1328:
1305:
1304:
1301:null hypothesis
1276:
1271:
1270:
1249:
1244:
1243:
1209:
1204:
1203:
1175:
1170:
1169:
1105:
1104:
1070:
1069:
1041:
1040:
1021:
1020:
1014:higher variance
994:
993:
974:
973:
954:
953:
925:
917:
916:
889:
888:
869:
868:
849:
848:
823:
795:
790:
789:
783:
750:
749:
688:
687:
659:
658:
630:
629:
577:
576:
555:
550:
549:
487:
483:
470:
469:
465:
464:
449:
448:
417:
416:
388:
387:
366:
361:
360:
341:
340:
321:
320:
296:Given a set of
294:
251:
240:
239:
238:
233:
222:
216:
213:
205:help improve it
202:
193:
189:
178:
167:
161:
158:
151:
142:This article's
138:
134:
123:
112:
106:
103:
93:Please help to
92:
76:
72:
35:
31:
24:
17:
12:
11:
5:
4398:
4396:
4388:
4387:
4382:
4377:
4367:
4366:
4360:
4359:
4357:
4356:
4344:
4332:
4318:
4305:
4302:
4301:
4298:
4297:
4294:
4293:
4291:
4290:
4285:
4280:
4275:
4270:
4264:
4262:
4256:
4255:
4253:
4252:
4247:
4242:
4237:
4232:
4227:
4222:
4217:
4212:
4207:
4201:
4199:
4193:
4192:
4190:
4189:
4184:
4179:
4170:
4165:
4160:
4154:
4152:
4146:
4145:
4143:
4142:
4137:
4132:
4123:
4121:Bioinformatics
4117:
4115:
4105:
4104:
4099:
4092:
4091:
4088:
4087:
4084:
4083:
4080:
4079:
4077:
4076:
4070:
4068:
4064:
4063:
4061:
4060:
4054:
4052:
4046:
4045:
4043:
4042:
4037:
4032:
4027:
4021:
4019:
4010:
4004:
4003:
4000:
3999:
3997:
3996:
3991:
3986:
3981:
3976:
3970:
3968:
3962:
3961:
3959:
3958:
3953:
3948:
3940:
3935:
3930:
3929:
3928:
3926:partial (PACF)
3917:
3915:
3909:
3908:
3906:
3905:
3900:
3895:
3887:
3882:
3876:
3874:
3873:Specific tests
3870:
3869:
3867:
3866:
3861:
3856:
3851:
3846:
3841:
3836:
3831:
3825:
3823:
3816:
3810:
3809:
3807:
3806:
3805:
3804:
3803:
3802:
3787:
3786:
3785:
3775:
3773:Classification
3770:
3765:
3760:
3755:
3750:
3745:
3739:
3737:
3731:
3730:
3728:
3727:
3722:
3720:McNemar's test
3717:
3712:
3707:
3702:
3696:
3694:
3684:
3683:
3666:
3659:
3658:
3655:
3654:
3651:
3650:
3648:
3647:
3642:
3637:
3632:
3626:
3624:
3618:
3617:
3615:
3614:
3598:
3592:
3590:
3584:
3583:
3581:
3580:
3575:
3570:
3565:
3560:
3558:Semiparametric
3555:
3550:
3544:
3542:
3538:
3537:
3535:
3534:
3529:
3524:
3519:
3513:
3511:
3505:
3504:
3502:
3501:
3496:
3491:
3486:
3481:
3475:
3473:
3467:
3466:
3464:
3463:
3458:
3453:
3448:
3442:
3440:
3430:
3429:
3426:
3425:
3420:
3414:
3413:
3406:
3405:
3402:
3401:
3398:
3397:
3395:
3394:
3393:
3392:
3382:
3377:
3372:
3371:
3370:
3365:
3354:
3352:
3346:
3345:
3342:
3341:
3339:
3338:
3333:
3332:
3331:
3323:
3315:
3299:
3296:(MannâWhitney)
3291:
3290:
3289:
3276:
3275:
3274:
3263:
3261:
3255:
3254:
3252:
3251:
3250:
3249:
3244:
3239:
3229:
3224:
3221:(ShapiroâWilk)
3216:
3211:
3206:
3201:
3196:
3188:
3182:
3180:
3174:
3173:
3171:
3170:
3162:
3153:
3141:
3135:
3133:Specific tests
3129:
3128:
3125:
3124:
3122:
3121:
3116:
3111:
3105:
3103:
3097:
3096:
3094:
3093:
3088:
3087:
3086:
3076:
3075:
3074:
3064:
3058:
3056:
3050:
3049:
3047:
3046:
3045:
3044:
3039:
3029:
3024:
3019:
3014:
3009:
3003:
3001:
2995:
2994:
2992:
2991:
2986:
2985:
2984:
2979:
2978:
2977:
2972:
2957:
2956:
2955:
2950:
2945:
2940:
2929:
2927:
2918:
2912:
2911:
2909:
2908:
2903:
2898:
2897:
2896:
2886:
2881:
2880:
2879:
2869:
2868:
2867:
2862:
2857:
2847:
2842:
2837:
2836:
2835:
2830:
2825:
2809:
2808:
2807:
2802:
2797:
2787:
2786:
2785:
2780:
2770:
2769:
2768:
2758:
2757:
2756:
2746:
2741:
2736:
2730:
2728:
2718:
2717:
2712:
2705:
2704:
2701:
2700:
2697:
2696:
2694:
2693:
2688:
2683:
2678:
2672:
2670:
2664:
2663:
2661:
2660:
2655:
2650:
2644:
2642:
2638:
2637:
2635:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2598:
2596:
2590:
2589:
2587:
2586:
2584:Standard error
2581:
2576:
2571:
2570:
2569:
2564:
2553:
2551:
2545:
2544:
2542:
2541:
2536:
2531:
2526:
2521:
2516:
2514:Optimal design
2511:
2506:
2500:
2498:
2488:
2487:
2482:
2475:
2474:
2471:
2470:
2467:
2466:
2464:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2398:
2393:
2387:
2385:
2379:
2378:
2376:
2375:
2370:
2369:
2368:
2363:
2353:
2348:
2342:
2340:
2334:
2333:
2331:
2330:
2325:
2320:
2314:
2312:
2311:Summary tables
2308:
2307:
2305:
2304:
2298:
2296:
2290:
2289:
2286:
2285:
2283:
2282:
2281:
2280:
2275:
2270:
2260:
2254:
2252:
2246:
2245:
2243:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2206:
2204:
2198:
2197:
2195:
2194:
2189:
2184:
2183:
2182:
2177:
2172:
2167:
2162:
2157:
2152:
2147:
2145:Contraharmonic
2142:
2137:
2126:
2124:
2115:
2105:
2104:
2099:
2092:
2091:
2089:
2088:
2083:
2077:
2074:
2073:
2068:
2066:
2065:
2058:
2051:
2043:
2037:
2036:
2030:
2017:
1974:
1965:
1944:
1934:(2): 121â135.
1923:
1917:
1904:
1894:(2): 103â109.
1879:
1876:
1873:
1872:
1845:
1816:
1791:
1766:
1765:
1763:
1760:
1759:
1758:
1753:
1748:
1743:
1738:
1733:
1728:
1723:
1716:
1713:
1699:
1698:
1695:
1692:
1688:
1687:
1684:
1681:
1677:
1676:
1673:
1670:
1666:
1665:
1662:
1659:
1655:
1654:
1651:
1648:
1580:
1577:
1569:
1568:
1562:
1555:
1554:might be used.
1548:
1522:
1519:
1505:relate to the
1470:
1463:
1388:
1385:
1367:tests and the
1342:
1339:
1312:
1283:
1279:
1256:
1252:
1240:detrended data
1225:
1218:
1215:
1182:
1178:
1166:
1165:
1151:
1148:
1142:
1139:
1133:
1130:
1124:
1118:
1115:
1089:
1083:
1080:
1054:
1051:
1028:
1001:
981:
961:
937:
932:
928:
924:
913:non-stationary
896:
876:
856:
845:
844:
830:
826:
822:
819:
816:
813:
810:
807:
802:
798:
782:
779:
763:
760:
734:
731:
725:
722:
716:
713:
707:
701:
698:
672:
669:
643:
640:
617:
611:
608:
602:
599:
593:
590:
584:
562:
558:
546:
545:
531:
526:
521:
514:
511:
505:
502:
496:
493:
486:
482:
477:
473:
468:
461:
457:
430:
427:
401:
398:
373:
369:
348:
328:
293:
290:
288:is heading.
253:
252:
235:
234:
196:
194:
187:
180:
179:
162:September 2023
141:
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132:
125:
124:
79:
77:
70:
65:
39:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
4397:
4386:
4383:
4381:
4378:
4376:
4373:
4372:
4370:
4355:
4354:
4345:
4343:
4342:
4333:
4331:
4330:
4325:
4319:
4317:
4316:
4307:
4306:
4303:
4289:
4286:
4284:
4283:Geostatistics
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4265:
4263:
4261:
4257:
4251:
4250:Psychometrics
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
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4200:
4198:
4194:
4188:
4185:
4183:
4180:
4178:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4155:
4153:
4151:
4147:
4141:
4138:
4136:
4133:
4131:
4127:
4124:
4122:
4119:
4118:
4116:
4114:
4113:Biostatistics
4110:
4106:
4102:
4097:
4093:
4075:
4074:Log-rank test
4072:
4071:
4069:
4065:
4059:
4056:
4055:
4053:
4051:
4047:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4022:
4020:
4018:
4014:
4011:
4009:
4005:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3971:
3969:
3967:
3963:
3957:
3954:
3952:
3949:
3947:
3945:(BoxâJenkins)
3941:
3939:
3936:
3934:
3931:
3927:
3924:
3923:
3922:
3919:
3918:
3916:
3914:
3910:
3904:
3901:
3899:
3898:DurbinâWatson
3896:
3894:
3888:
3886:
3883:
3881:
3880:DickeyâFuller
3878:
3877:
3875:
3871:
3865:
3862:
3860:
3857:
3855:
3854:Cointegration
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3829:Decomposition
3827:
3826:
3824:
3820:
3817:
3815:
3811:
3801:
3798:
3797:
3796:
3793:
3792:
3791:
3788:
3784:
3781:
3780:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3740:
3738:
3736:
3732:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3700:Cohen's kappa
3698:
3697:
3695:
3693:
3689:
3685:
3681:
3677:
3673:
3669:
3664:
3660:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3627:
3625:
3623:
3619:
3613:
3609:
3605:
3599:
3597:
3594:
3593:
3591:
3589:
3585:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3553:Nonparametric
3551:
3549:
3546:
3545:
3543:
3539:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3514:
3512:
3510:
3506:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3476:
3474:
3472:
3468:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3443:
3441:
3439:
3435:
3431:
3424:
3421:
3419:
3416:
3415:
3411:
3407:
3391:
3388:
3387:
3386:
3383:
3381:
3378:
3376:
3373:
3369:
3366:
3364:
3361:
3360:
3359:
3356:
3355:
3353:
3351:
3347:
3337:
3334:
3330:
3324:
3322:
3316:
3314:
3308:
3307:
3306:
3303:
3302:Nonparametric
3300:
3298:
3292:
3288:
3285:
3284:
3283:
3277:
3273:
3272:Sample median
3270:
3269:
3268:
3265:
3264:
3262:
3260:
3256:
3248:
3245:
3243:
3240:
3238:
3235:
3234:
3233:
3230:
3228:
3225:
3223:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3193:
3189:
3187:
3184:
3183:
3181:
3179:
3175:
3169:
3167:
3163:
3161:
3159:
3154:
3152:
3147:
3143:
3142:
3139:
3136:
3134:
3130:
3120:
3117:
3115:
3112:
3110:
3107:
3106:
3104:
3102:
3098:
3092:
3089:
3085:
3082:
3081:
3080:
3077:
3073:
3070:
3069:
3068:
3065:
3063:
3060:
3059:
3057:
3055:
3051:
3043:
3040:
3038:
3035:
3034:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3004:
3002:
3000:
2996:
2990:
2987:
2983:
2980:
2976:
2973:
2971:
2968:
2967:
2966:
2963:
2962:
2961:
2958:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2935:
2934:
2931:
2930:
2928:
2926:
2922:
2919:
2917:
2913:
2907:
2904:
2902:
2899:
2895:
2892:
2891:
2890:
2887:
2885:
2882:
2878:
2877:loss function
2875:
2874:
2873:
2870:
2866:
2863:
2861:
2858:
2856:
2853:
2852:
2851:
2848:
2846:
2843:
2841:
2838:
2834:
2831:
2829:
2826:
2824:
2818:
2815:
2814:
2813:
2810:
2806:
2803:
2801:
2798:
2796:
2793:
2792:
2791:
2788:
2784:
2781:
2779:
2776:
2775:
2774:
2771:
2767:
2764:
2763:
2762:
2759:
2755:
2752:
2751:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2731:
2729:
2727:
2723:
2719:
2715:
2710:
2706:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2673:
2671:
2669:
2665:
2659:
2656:
2654:
2651:
2649:
2646:
2645:
2643:
2639:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2599:
2597:
2595:
2591:
2585:
2582:
2580:
2579:Questionnaire
2577:
2575:
2572:
2568:
2565:
2563:
2560:
2559:
2558:
2555:
2554:
2552:
2550:
2546:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2501:
2499:
2497:
2493:
2489:
2485:
2480:
2476:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2406:Control chart
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2388:
2386:
2384:
2380:
2374:
2371:
2367:
2364:
2362:
2359:
2358:
2357:
2354:
2352:
2349:
2347:
2344:
2343:
2341:
2339:
2335:
2329:
2326:
2324:
2321:
2319:
2316:
2315:
2313:
2309:
2303:
2300:
2299:
2297:
2295:
2291:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2264:
2261:
2259:
2256:
2255:
2253:
2251:
2247:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2207:
2205:
2203:
2199:
2193:
2190:
2188:
2185:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2132:
2131:
2128:
2127:
2125:
2123:
2119:
2116:
2114:
2110:
2106:
2102:
2097:
2093:
2087:
2084:
2082:
2079:
2078:
2075:
2071:
2064:
2059:
2057:
2052:
2050:
2045:
2044:
2041:
2033:
2027:
2023:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1991:(4): 046113.
1990:
1986:
1985:
1980:
1975:
1971:
1966:
1962:
1958:
1954:
1950:
1945:
1941:
1937:
1933:
1929:
1924:
1920:
1914:
1910:
1905:
1901:
1897:
1893:
1889:
1888:
1882:
1881:
1877:
1860:
1859:
1852:
1850:
1846:
1833:
1829:
1823:
1821:
1817:
1801:
1795:
1792:
1777:
1771:
1768:
1761:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1736:Least squares
1734:
1732:
1729:
1727:
1726:Extrapolation
1724:
1722:
1719:
1718:
1714:
1712:
1710:
1706:
1696:
1693:
1690:
1689:
1685:
1682:
1679:
1678:
1674:
1671:
1668:
1667:
1663:
1660:
1657:
1656:
1652:
1649:
1646:
1645:
1640:
1638:
1634:
1629:
1623:
1621:
1617:
1613:
1612:Friedman test
1608:
1604:
1599:
1595:
1592:on levels of
1591:
1586:
1578:
1576:
1574:
1566:
1563:
1560:
1556:
1553:
1549:
1546:
1542:
1541:
1540:
1538:
1533:
1528:
1520:
1518:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1484:
1480:
1475:
1468:
1464:
1462:
1448:
1444:
1439:
1430:
1419:
1408:
1404:
1398:
1394:
1386:
1384:
1382:
1376:
1372:
1370:
1369:cointegration
1366:
1361:
1359:
1337:
1326:
1310:
1302:
1297:
1281:
1277:
1254:
1250:
1241:
1223:
1213:
1202:
1198:
1180:
1176:
1146:
1140:
1137:
1128:
1122:
1113:
1103:
1102:
1101:
1087:
1078:
1049:
1026:
1017:
1015:
999:
979:
959:
951:
930:
926:
914:
910:
894:
874:
854:
828:
824:
820:
817:
814:
811:
808:
805:
800:
796:
788:
787:
786:
780:
778:
758:
729:
723:
720:
711:
705:
696:
667:
638:
606:
600:
597:
588:
560:
556:
529:
524:
519:
509:
503:
500:
491:
484:
480:
475:
471:
466:
459:
455:
447:
446:
445:
425:
396:
371:
367:
346:
326:
318:
314:
313:least-squares
309:
307:
306:straight line
303:
299:
291:
289:
287:
283:
279:
275:
271:
267:
263:
259:
249:
246:
231:
228:
220:
217:December 2023
210:
206:
200:
197:This article
195:
186:
185:
176:
173:
165:
155:
149:
147:
140:
131:
130:
121:
118:
110:
100:
96:
90:
89:
83:
78:
69:
68:
63:
61:
54:
53:
48:
47:
42:
37:
28:
27:
22:
21:Curve fitting
4351:
4339:
4320:
4313:
4225:Econometrics
4175: /
4158:Chemometrics
4135:Epidemiology
4128: /
4101:Applications
3943:ARIMA model
3890:Q-statistic
3839:Stationarity
3735:Multivariate
3678: /
3674: /
3672:Multivariate
3670: /
3610: /
3606: /
3380:Bayes factor
3279:Signed rank
3191:
3165:
3157:
3145:
2840:Completeness
2676:Cohort study
2574:Opinion poll
2509:Missing data
2496:Study design
2451:Scatter plot
2373:Scatter plot
2366:Spearman's Ï
2328:Grouped data
2021:
1988:
1982:
1969:
1955:(1): 89â90.
1952:
1948:
1931:
1927:
1909:Econometrics
1908:
1891:
1885:
1863:. Retrieved
1857:
1836:. Retrieved
1832:the original
1807:. Retrieved
1794:
1782:. Retrieved
1770:
1746:Line fitting
1708:
1704:
1702:
1636:
1632:
1624:
1583:Medical and
1582:
1570:
1537:misspecified
1524:
1514:
1502:
1494:
1488:
1482:
1478:
1466:
1440:
1428:
1417:
1406:
1402:
1392:
1390:
1377:
1373:
1362:
1298:
1239:
1196:
1167:
1018:
846:
784:
547:
310:
295:
257:
256:
241:
223:
214:
198:
168:
159:
143:
113:
104:
85:
57:
50:
44:
43:Please help
40:
4353:WikiProject
4268:Cartography
4230:Jurimetrics
4182:Reliability
3913:Time domain
3892:(LjungâBox)
3814:Time-series
3692:Categorical
3676:Time-series
3668:Categorical
3603:(Bernoulli)
3438:Correlation
3418:Correlation
3214:JarqueâBera
3186:Chi-squared
2948:M-estimator
2901:Asymptotics
2845:Sufficiency
2612:Interaction
2524:Replication
2504:Effect size
2461:Violin plot
2441:Radar chart
2421:Forest plot
2411:Correlogram
2361:Kendall's Ï
2031:041227630-5
1731:Forecasting
1594:cholesterol
1532:correlation
1511:t-statistic
274:information
262:statistical
99:introducing
4369:Categories
4220:Demography
3938:ARMA model
3743:Regression
3320:(Friedman)
3281:(Wilcoxon)
3219:Normality
3209:Lilliefors
3156:Student's
3032:Resampling
2906:Robustness
2894:divergence
2884:Efficiency
2822:(monotone)
2817:Likelihood
2734:Population
2567:Stratified
2519:Population
2338:Dependence
2294:Count data
2225:Percentile
2202:Dispersion
2135:Arithmetic
2070:Statistics
1918:0077104285
1878:References
1721:Estimation
1585:biomedical
1197:detrending
948:is called
268:patterns.
82:references
46:improve it
3601:Logistic
3368:posterior
3294:Rank sum
3042:Jackknife
3037:Bootstrap
2855:Bootstrap
2790:Parameter
2739:Statistic
2534:Statistic
2446:Run chart
2431:Pie chart
2426:Histogram
2416:Fan chart
2391:Bar chart
2273:L-moments
2160:Geometric
1598:analgesic
1565:Unit root
1499:residuals
1491:r-squared
1365:unit root
1341:^
1217:^
1201:residuals
1150:^
1132:^
1117:^
1082:^
1053:^
762:^
733:^
715:^
700:^
671:^
642:^
610:^
592:^
513:^
495:^
481:−
456:∑
429:^
400:^
317:minimizes
302:functions
107:July 2019
52:talk page
4315:Category
4008:Survival
3885:Johansen
3608:Binomial
3563:Isotonic
3150:(normal)
2795:location
2602:Blocking
2557:Sampling
2436:QâQ plot
2401:Box plot
2383:Graphics
2278:Skewness
2268:Kurtosis
2240:Variance
2170:Heronian
2165:Harmonic
2013:22181233
1838:June 17,
1809:June 17,
1784:June 17,
1715:See also
1325:variance
4341:Commons
4288:Kriging
4173:Process
4130:studies
3989:Wavelet
3822:General
2989:Plug-in
2783:L space
2562:Cluster
2263:Moments
2081:Outline
1993:Bibcode
1865:May 17,
1628:trypsin
1238:as the
203:Please
95:improve
4210:Census
3800:Normal
3748:Manova
3568:Robust
3318:2-way
3310:1-way
3148:-test
2819:
2396:Biplot
2187:Median
2180:Lehmer
2122:Center
2028:
2011:
1915:
1603:mmol/L
1590:statin
1459:
1455:
1451:
1436:
1432:
1425:
1421:
1414:
1410:
1400:
1381:cycles
1195:(thus
909:errors
847:where
84:, but
3834:Trend
3363:prior
3305:anova
3194:-test
3168:-test
3160:-test
3067:Power
3012:Pivot
2805:shape
2800:scale
2250:Shape
2230:Range
2175:Heinz
2150:Cubic
2086:Index
1803:(PDF)
1779:(PDF)
1762:Notes
1697:0.79
1694:2.40
1686:0.66
1683:2.22
1675:0.75
1672:1.94
1664:0.56
1650:mean
1596:, an
278:graph
260:is a
4067:Test
3267:Sign
3119:Wald
2192:Mode
2130:Mean
2026:ISBN
2009:PMID
1913:ISBN
1867:2015
1840:2012
1811:2012
1786:2012
1661:1.6
1447:IPCC
1296:'s.
1068:and
972:and
867:and
657:and
415:and
311:The
298:data
286:data
282:data
270:Data
266:data
3247:BIC
3242:AIC
2001:doi
1957:doi
1936:doi
1896:doi
1653:SD
1571:In
1503:not
1395:of
280:of
207:to
4371::
2007:.
1999:.
1989:84
1987:.
1981:.
1953:58
1951:.
1932:11
1930:.
1890:.
1848:^
1819:^
1691:4
1680:3
1669:2
1658:1
1647:#
1383:.
55:.
3192:G
3166:F
3158:t
3146:Z
2865:V
2860:U
2062:e
2055:t
2048:v
2034:.
2015:.
2003::
1995::
1963:.
1959::
1942:.
1938::
1921:.
1902:.
1898::
1892:6
1869:.
1842:.
1813:.
1788:.
1709:p
1705:p
1637:p
1633:p
1573:R
1547:.
1515:r
1495:r
1493:(
1483:r
1479:r
1467:r
1434:=
1429:E
1423:=
1418:E
1412:=
1407:E
1403:E
1393:e
1338:a
1311:a
1282:t
1278:e
1255:t
1251:e
1224:t
1214:e
1181:t
1177:y
1147:b
1141:+
1138:t
1129:a
1123:=
1114:y
1088:,
1079:b
1050:a
1027:e
1000:e
980:b
960:a
936:}
931:t
927:y
923:{
895:e
875:b
855:a
829:t
825:e
821:+
818:b
815:+
812:t
809:a
806:=
801:t
797:y
759:a
730:b
724:+
721:x
712:a
706:=
697:y
668:b
639:a
616:)
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601:+
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589:a
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561:t
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544:.
530:2
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467:[
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327:y
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230:)
224:(
219:)
215:(
201:.
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164:)
160:(
150:.
120:)
114:(
109:)
105:(
91:.
62:)
58:(
23:.
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