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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
1650: = 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:
201:
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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
1385:
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
1545:
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
1540:
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
1611:
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
1625:, 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/
1636:
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
1472:°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.
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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
1722: = 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.
1550:. 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:
1620:
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
1027:, 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.
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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
1492:. 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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963:. 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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1371:, 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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time frame only. Outside of this time frame, it cannot be determined how these immeasurable factors behave both qualitatively and quantitatively.
1578:: taking first (or occasionally second) differences of the data, with the level of differencing being identified through various unit root tests.
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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
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105:
639:, 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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1979:
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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1338:, the distribution of calculated trends is to be expected from random (trendless) data. If the estimated trend,
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1718: = 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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1586:, 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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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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1334:, is not different from 0. From the above discussion of trends in random data with known
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1280:'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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2015:
1839:"IPCC Third Assessment Report â Climate Change 2001 â Complete online versions"
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1050:'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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1971:
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This formula first calculates the difference between the observed data
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1600:
1989:
1918:
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
315:
that can be chosen to fit the data. The simplest function is a
1922:. Maidenhead: McGraw Hill Higher Education. pp. 171â198.
1646: < 0.0001, whereas linear trend estimation gives
194:
139:
77:
36:
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Non-normal distribution for errors: in the simplest cases, a
1023:'s all have the same distribution, but if not (if some have
922:. If one can reject the null hypothesis that the errors are
1937:
Chatfield, C. (1993). "Calculating
Interval Forecasts".
1538:
independent and identically distributed random variables
215:
1787:"Making Regression More Useful II: Dummies and Trends"
1508:), which is 1 minus the ratio of the variance of the
1416:, 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.
1500:
The least-squares fitting process produces a value,
1468:°C over 140 years, with 95% confidence limits of 0.2
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4017:
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may be too technical for most readers to understand
27:
Statistical technique to aid interpretation of data
1367:, is larger than the critical value for a certain
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1456:record of the past 140 years as presented by the
1169:{\displaystyle {\hat {y}}={\hat {a}}t+{\hat {b}}}
752:{\displaystyle {\hat {y}}={\hat {a}}x+{\hat {b}}}
455:are chosen to minimize the sum of squared errors
330:the sum of the squared errors in the data series
1438:100, the trend will probably be visible; but if
3510:Multivariate adaptive regression splines (MARS)
1990:"Self-similarity of high-order moving averages"
2035:. London: Chapman and Hall. pp. 212â220.
1960:Review of Marketing and Agricultural Economics
1561:Non-constant variance: in the simplest cases,
1449:10000, the trend will be buried in the noise.
2065:
397:observed for those points in time, values of
8:
946:
933:
1988:Arianos, S.; Carbone, A.; Turk, C. (2011).
1939:Journal of Business and Economic Statistics
1488:Illustration of the effect of filtering on
295:that models the general direction that the
71:Learn how and when to remove these messages
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2033:Practical Statistics for Medical Research
1452:Consider a concrete example, such as the
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1003:may be inaccurate. It is simplest if the
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256:Learn how and when to remove this message
238:Learn how and when to remove this message
222:, without removing the technical details.
183:Learn how and when to remove this message
128:Learn how and when to remove this message
1983:. The Royal Swedish Academy of Sciences.
1652:
1460:. The interannual variation is about 0.2
632:{\displaystyle ({\hat {a}}t+{\hat {b}})}
91:This article includes a list of general
30:For broader coverage of this topic, see
1811:"The Royal Swedish Academy of Sciences"
1778:
759:. The term "trend" refers to the slope
4036:KaplanâMeier estimator (product limit)
1524:. Often, filtering a series increases
4386:Regression with time series structure
1253:, and estimating the variance of the
220:make it understandable to non-experts
7:
4346:
4046:Accelerated failure time (AFT) model
1869:Forecasting: principles and practice
1862:
1860:
1833:
1831:
1556:autoregressive moving average models
1111:thus allowing the predicted values
283:patterns, or trends, occur when the
4358:
3641:Analysis of variance (ANOVA, anova)
1714:This is a clear trend. ANOVA gives
1427:0.1, the trend will be obvious; if
3736:CochranâMantelâHaenszel statistics
2362:Pearson product-moment correlation
1629:, where it could be argued that a
1382:technique in econometric studies.
848:{\displaystyle y_{t}=at+b+e_{t}\,}
548:{\displaystyle \sum _{t}\left^{2}}
155:tone or style may not reflect the
97:it lacks sufficient corresponding
25:
926:, then the non-stationary series
52:This article has multiple issues.
4357:
4345:
4333:
4320:
4319:
788:in the least squares estimator.
350:. Given a set of points in time
199:
165:guide to writing better articles
144:
82:
41:
3995:Least-squares spectral analysis
1752:Least-squares spectral analysis
1179:to be subtracted from the data
60:or discuss these issues on the
2976:Mean-unbiased minimum-variance
1951:10.1080/07350015.1993.10509938
1631:single-nucleotide polymorphism
1464:°C, and the trend is about 0.6
1351:
1242:{\displaystyle {\hat {e}}_{t}}
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1160:
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898:are unknown constants and the
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303:Fitting a trend: Least-squares
1:
4289:Geographic information system
3505:Simultaneous equations models
3472:Coefficient of determination
3083:Uniformly most powerful test
1618:ANOVA (analysis of variance)
918:'s are randomly distributed
4041:Proportional hazards models
3985:Spectral density estimation
3967:Vector autoregression (VAR)
3401:Maximum posterior estimator
2633:Randomized controlled trial
1104:{\displaystyle {\hat {b}},}
4412:
3801:Multivariate distributions
2221:Average absolute deviation
2016:10.1103/physreve.84.046113
1454:global surface temperature
1360:{\displaystyle {\hat {a}}}
1072:{\displaystyle {\hat {a}}}
781:{\displaystyle {\hat {a}}}
690:{\displaystyle {\hat {b}}}
661:{\displaystyle {\hat {a}}}
448:{\displaystyle {\hat {b}}}
419:{\displaystyle {\hat {a}}}
275:technique used to analyze
29:
4315:
4118:
4105:
3789:Structural equation model
3697:
3672:
3443:
3419:
3151:
3125:Score/Lagrange multiplier
2731:
2718:
2540:Sample size determination
2501:
2488:
2118:
2105:
2087:
1898:Applied Economics Letters
952:{\displaystyle \{y_{t}\}}
311:, there are a variety of
4284:Environmental statistics
3806:Elliptical distributions
3599:Generalized linear model
3528:Simple linear regression
3298:HodgesâLehmann estimator
2755:Probability distribution
2664:Stochastic approximation
2226:Coefficient of variation
1570:generalized linear model
1546:model is mathematically
1518:statistical significance
4391:Statistical forecasting
3944:Cross-correlation (XCF)
3552:Non-standard predictors
2986:LehmannâScheffĂ© theorem
2659:Adaptive clinical trial
1911:10.1080/135048599353726
1590:Trends in clinical data
1210:the data), leaving the
792:Data as trend and noise
269:Linear trend estimation
159:used on Knowledge (XXG)
112:more precise citations.
4340:Mathematics portal
4161:Engineering statistics
4069:NelsonâAalen estimator
3646:Analysis of covariance
3533:Ordinary least squares
3457:Pearson product-moment
2861:Statistical functional
2772:Empirical distribution
2605:Controlled experiments
2334:Frequency distribution
2112:Descriptive statistics
1972:10.22004/ag.econ.12288
1563:weighted least squares
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163:See Knowledge (XXG)'s
4256:Population statistics
4198:System identification
3932:Autocorrelation (ACF)
3860:Exponential smoothing
3774:Discriminant analysis
3769:Canonical correlation
3633:Partition of variance
3495:Regression validation
3339:(JonckheereâTerpstra)
3238:Likelihood-ratio test
2927:Frequentist inference
2839:Locationâscale family
2760:Sampling distribution
2725:Statistical inference
2692:Cross-sectional study
2679:Observational studies
2638:Randomized experiment
2467:Stem-and-leaf display
2269:Central limit theorem
2031:Altman, D.G. (1991).
1496:of fitted trend line.
1487:
1362:
1329:
1302:
1300:{\displaystyle e_{t}}
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1273:{\displaystyle e_{t}}
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1199:{\displaystyle y_{t}}
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390:{\displaystyle y_{t}}
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345:
4179:Probabilistic design
3764:Principal components
3607:Exponential families
3559:Nonlinear regression
3538:General linear model
3500:Mixed effects models
3490:Errors and residuals
3467:Confounding variable
3369:Bayesian probability
3347:Van der Waerden test
3337:Ordered alternative
3102:Multiple comparisons
2981:RaoâBlackwellization
2944:Estimating equations
2900:Statistical distance
2618:Factorial experiment
2151:Arithmetic-Geometric
1845:on November 20, 2009
1572:might be applicable.
1541:effects that have a
1342:
1318:
1284:
1257:
1217:
1183:
1118:
1083:
1054:
1034:
1007:
987:
967:
930:
902:
882:
862:
803:
763:
701:
672:
643:
590:
563:
462:
430:
401:
374:
354:
334:
4251:Official statistics
4174:Methods engineering
3855:Seasonal adjustment
3623:Poisson regressions
3543:Bayesian regression
3482:Regression analysis
3462:Partial correlation
3434:Regression analysis
3033:Prediction interval
3028:Likelihood interval
3018:Confidence interval
3010:Interval estimation
2971:Unbiased estimators
2789:Model specification
2669:Up-and-down designs
2357:Partial correlation
2313:Index of dispersion
2231:Interquartile range
2008:2011PhRvE..84d6113A
1872:. 20 September 2014
1767:Regression analysis
1762:Prediction interval
1654:
1480:-squared) and trend
4271:Spatial statistics
4151:Medical statistics
4051:First hitting time
4005:Whittle likelihood
3656:Degrees of freedom
3651:Multivariate ANOVA
3584:Heteroscedasticity
3396:Bayesian estimator
3361:Bayesian inference
3210:KolmogorovâSmirnov
3095:Randomization test
3065:Testing hypotheses
3038:Tolerance interval
2949:Maximum likelihood
2844:Exponential family
2777:Density estimation
2737:Statistical theory
2697:Natural experiment
2643:Scientific control
2560:Survey methodology
2246:Standard deviation
1653:
1627:genotype frequency
1498:
1408:standard deviation
1369:significance level
1357:
1324:
1297:
1270:
1239:
1196:
1166:
1101:
1069:
1040:
1013:
993:
973:
949:
908:
888:
868:
845:
778:
749:
687:
658:
629:
576:
545:
474:
445:
416:
387:
360:
340:
4373:
4372:
4311:
4310:
4307:
4306:
4246:National accounts
4216:Actuarial science
4208:Social statistics
4101:
4100:
4097:
4096:
4093:
4092:
4028:Survival function
4013:
4012:
3875:Granger causality
3716:Contingency table
3691:Survival analysis
3668:
3667:
3664:
3663:
3520:Linear regression
3415:
3414:
3411:
3410:
3386:Credible interval
3355:
3354:
3138:
3137:
2954:Method of moments
2823:Parametric family
2784:Statistical model
2714:
2713:
2710:
2709:
2628:Random assignment
2550:Statistical power
2484:
2483:
2480:
2479:
2329:Contingency table
2299:
2298:
2166:Generalized/power
1995:Physical Review E
1712:
1711:
1476:Goodness of fit (
1398:Noisy time series
1354:
1327:{\displaystyle a}
1230:
1163:
1145:
1130:
1095:
1066:
1043:{\displaystyle e}
1016:{\displaystyle e}
996:{\displaystyle b}
976:{\displaystyle a}
911:{\displaystyle e}
891:{\displaystyle b}
871:{\displaystyle a}
775:
746:
728:
713:
684:
655:
623:
605:
586:and the estimate
526:
508:
465:
442:
413:
363:{\displaystyle t}
343:{\displaystyle y}
266:
265:
258:
248:
247:
240:
193:
192:
185:
157:encyclopedic tone
138:
137:
130:
75:
16:(Redirected from
4403:
4396:Change detection
4361:
4360:
4349:
4348:
4338:
4337:
4323:
4322:
4226:Crime statistics
4120:
4107:
4024:
3990:Fourier analysis
3977:Frequency domain
3957:
3904:
3870:Structural break
3830:
3779:Cluster analysis
3726:Log-linear model
3699:
3674:
3615:
3589:Homoscedasticity
3445:
3421:
3340:
3332:
3324:
3323:(KruskalâWallis)
3308:
3293:
3248:Cross validation
3233:
3215:AndersonâDarling
3162:
3149:
3120:Likelihood-ratio
3112:Parametric tests
3090:Permutation test
3073:1- & 2-tails
2964:Minimum distance
2936:Point estimation
2932:
2883:Optimal decision
2834:
2733:
2720:
2702:Quasi-experiment
2652:Adaptive designs
2503:
2490:
2367:Rank correlation
2129:
2120:
2107:
2074:
2067:
2060:
2051:
2046:
2027:
1984:
1975:
1954:
1933:
1914:
1882:
1881:
1879:
1877:
1864:
1855:
1854:
1852:
1850:
1841:. Archived from
1835:
1826:
1825:
1823:
1821:
1816:. 8 October 2003
1815:
1807:
1801:
1800:
1798:
1796:
1791:
1783:
1655:
1471:
1467:
1463:
1448:
1444:
1437:
1433:
1426:
1422:
1412:
1366:
1364:
1363:
1358:
1356:
1355:
1347:
1333:
1331:
1330:
1325:
1314:that the trend,
1306:
1304:
1303:
1298:
1296:
1295:
1279:
1277:
1276:
1271:
1269:
1268:
1248:
1246:
1245:
1240:
1238:
1237:
1232:
1231:
1223:
1205:
1203:
1202:
1197:
1195:
1194:
1175:
1173:
1172:
1167:
1165:
1164:
1156:
1147:
1146:
1138:
1132:
1131:
1123:
1110:
1108:
1107:
1102:
1097:
1096:
1088:
1078:
1076:
1075:
1070:
1068:
1067:
1059:
1049:
1047:
1046:
1041:
1022:
1020:
1019:
1014:
1002:
1000:
999:
994:
982:
980:
979:
974:
961:trend-stationary
958:
956:
955:
950:
945:
944:
917:
915:
914:
909:
897:
895:
894:
889:
877:
875:
874:
869:
854:
852:
851:
846:
843:
842:
815:
814:
787:
785:
784:
779:
777:
776:
768:
758:
756:
755:
750:
748:
747:
739:
730:
729:
721:
715:
714:
706:
696:
694:
693:
688:
686:
685:
677:
667:
665:
664:
659:
657:
656:
648:
638:
636:
635:
630:
625:
624:
616:
607:
606:
598:
585:
583:
582:
577:
575:
574:
554:
552:
551:
546:
544:
543:
538:
534:
533:
529:
528:
527:
519:
510:
509:
501:
490:
489:
473:
454:
452:
451:
446:
444:
443:
435:
425:
423:
422:
417:
415:
414:
406:
396:
394:
393:
388:
386:
385:
370:and data values
369:
367:
366:
361:
349:
347:
346:
341:
261:
254:
243:
236:
232:
229:
223:
203:
202:
195:
188:
181:
177:
174:
168:
167:for suggestions.
148:
147:
140:
133:
126:
122:
119:
113:
108:this article by
99:inline citations
86:
85:
78:
67:
45:
44:
37:
21:
18:Trend estimation
4411:
4410:
4406:
4405:
4404:
4402:
4401:
4400:
4376:
4375:
4374:
4369:
4332:
4303:
4265:
4202:
4188:quality control
4155:
4137:Clinical trials
4114:
4089:
4073:
4061:Hazard function
4055:
4009:
3971:
3955:
3918:
3914:BreuschâGodfrey
3902:
3879:
3819:
3794:Factor analysis
3740:
3721:Graphical model
3693:
3660:
3627:
3613:
3593:
3547:
3514:
3476:
3439:
3438:
3407:
3351:
3338:
3330:
3322:
3306:
3291:
3270:Rank statistics
3264:
3243:Model selection
3231:
3189:Goodness of fit
3183:
3160:
3134:
3106:
3059:
3004:
2993:Median unbiased
2921:
2832:
2765:Order statistic
2727:
2706:
2673:
2647:
2599:
2554:
2497:
2495:Data collection
2476:
2388:
2343:
2317:
2295:
2255:
2207:
2124:Continuous data
2114:
2101:
2083:
2078:
2043:
2030:
1987:
1978:
1957:
1936:
1930:
1917:
1894:
1891:
1886:
1885:
1875:
1873:
1866:
1865:
1858:
1848:
1846:
1837:
1836:
1829:
1819:
1817:
1813:
1809:
1808:
1804:
1794:
1792:
1789:
1785:
1784:
1780:
1775:
1728:
1592:
1534:
1532:Advanced models
1482:
1469:
1465:
1461:
1446:
1442:
1435:
1431:
1424:
1420:
1410:
1400:
1340:
1339:
1316:
1315:
1312:null hypothesis
1287:
1282:
1281:
1260:
1255:
1254:
1220:
1215:
1214:
1186:
1181:
1180:
1116:
1115:
1081:
1080:
1052:
1051:
1032:
1031:
1025:higher variance
1005:
1004:
985:
984:
965:
964:
936:
928:
927:
900:
899:
880:
879:
860:
859:
834:
806:
801:
800:
794:
761:
760:
699:
698:
670:
669:
641:
640:
588:
587:
566:
561:
560:
498:
494:
481:
480:
476:
475:
460:
459:
428:
427:
399:
398:
377:
372:
371:
352:
351:
332:
331:
307:Given a set of
305:
262:
251:
250:
249:
244:
233:
227:
224:
216:help improve it
213:
204:
200:
189:
178:
172:
169:
162:
153:This article's
149:
145:
134:
123:
117:
114:
104:Please help to
103:
87:
83:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4409:
4407:
4399:
4398:
4393:
4388:
4378:
4377:
4371:
4370:
4368:
4367:
4355:
4343:
4329:
4316:
4313:
4312:
4309:
4308:
4305:
4304:
4302:
4301:
4296:
4291:
4286:
4281:
4275:
4273:
4267:
4266:
4264:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4228:
4223:
4218:
4212:
4210:
4204:
4203:
4201:
4200:
4195:
4190:
4181:
4176:
4171:
4165:
4163:
4157:
4156:
4154:
4153:
4148:
4143:
4134:
4132:Bioinformatics
4128:
4126:
4116:
4115:
4110:
4103:
4102:
4099:
4098:
4095:
4094:
4091:
4090:
4088:
4087:
4081:
4079:
4075:
4074:
4072:
4071:
4065:
4063:
4057:
4056:
4054:
4053:
4048:
4043:
4038:
4032:
4030:
4021:
4015:
4014:
4011:
4010:
4008:
4007:
4002:
3997:
3992:
3987:
3981:
3979:
3973:
3972:
3970:
3969:
3964:
3959:
3951:
3946:
3941:
3940:
3939:
3937:partial (PACF)
3928:
3926:
3920:
3919:
3917:
3916:
3911:
3906:
3898:
3893:
3887:
3885:
3884:Specific tests
3881:
3880:
3878:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3842:
3836:
3834:
3827:
3821:
3820:
3818:
3817:
3816:
3815:
3814:
3813:
3798:
3797:
3796:
3786:
3784:Classification
3781:
3776:
3771:
3766:
3761:
3756:
3750:
3748:
3742:
3741:
3739:
3738:
3733:
3731:McNemar's test
3728:
3723:
3718:
3713:
3707:
3705:
3695:
3694:
3677:
3670:
3669:
3666:
3665:
3662:
3661:
3659:
3658:
3653:
3648:
3643:
3637:
3635:
3629:
3628:
3626:
3625:
3609:
3603:
3601:
3595:
3594:
3592:
3591:
3586:
3581:
3576:
3571:
3569:Semiparametric
3566:
3561:
3555:
3553:
3549:
3548:
3546:
3545:
3540:
3535:
3530:
3524:
3522:
3516:
3515:
3513:
3512:
3507:
3502:
3497:
3492:
3486:
3484:
3478:
3477:
3475:
3474:
3469:
3464:
3459:
3453:
3451:
3441:
3440:
3437:
3436:
3431:
3425:
3424:
3417:
3416:
3413:
3412:
3409:
3408:
3406:
3405:
3404:
3403:
3393:
3388:
3383:
3382:
3381:
3376:
3365:
3363:
3357:
3356:
3353:
3352:
3350:
3349:
3344:
3343:
3342:
3334:
3326:
3310:
3307:(MannâWhitney)
3302:
3301:
3300:
3287:
3286:
3285:
3274:
3272:
3266:
3265:
3263:
3262:
3261:
3260:
3255:
3250:
3240:
3235:
3232:(ShapiroâWilk)
3227:
3222:
3217:
3212:
3207:
3199:
3193:
3191:
3185:
3184:
3182:
3181:
3173:
3164:
3152:
3146:
3144:Specific tests
3140:
3139:
3136:
3135:
3133:
3132:
3127:
3122:
3116:
3114:
3108:
3107:
3105:
3104:
3099:
3098:
3097:
3087:
3086:
3085:
3075:
3069:
3067:
3061:
3060:
3058:
3057:
3056:
3055:
3050:
3040:
3035:
3030:
3025:
3020:
3014:
3012:
3006:
3005:
3003:
3002:
2997:
2996:
2995:
2990:
2989:
2988:
2983:
2968:
2967:
2966:
2961:
2956:
2951:
2940:
2938:
2929:
2923:
2922:
2920:
2919:
2914:
2909:
2908:
2907:
2897:
2892:
2891:
2890:
2880:
2879:
2878:
2873:
2868:
2858:
2853:
2848:
2847:
2846:
2841:
2836:
2820:
2819:
2818:
2813:
2808:
2798:
2797:
2796:
2791:
2781:
2780:
2779:
2769:
2768:
2767:
2757:
2752:
2747:
2741:
2739:
2729:
2728:
2723:
2716:
2715:
2712:
2711:
2708:
2707:
2705:
2704:
2699:
2694:
2689:
2683:
2681:
2675:
2674:
2672:
2671:
2666:
2661:
2655:
2653:
2649:
2648:
2646:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2609:
2607:
2601:
2600:
2598:
2597:
2595:Standard error
2592:
2587:
2582:
2581:
2580:
2575:
2564:
2562:
2556:
2555:
2553:
2552:
2547:
2542:
2537:
2532:
2527:
2525:Optimal design
2522:
2517:
2511:
2509:
2499:
2498:
2493:
2486:
2485:
2482:
2481:
2478:
2477:
2475:
2474:
2469:
2464:
2459:
2454:
2449:
2444:
2439:
2434:
2429:
2424:
2419:
2414:
2409:
2404:
2398:
2396:
2390:
2389:
2387:
2386:
2381:
2380:
2379:
2374:
2364:
2359:
2353:
2351:
2345:
2344:
2342:
2341:
2336:
2331:
2325:
2323:
2322:Summary tables
2319:
2318:
2316:
2315:
2309:
2307:
2301:
2300:
2297:
2296:
2294:
2293:
2292:
2291:
2286:
2281:
2271:
2265:
2263:
2257:
2256:
2254:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2217:
2215:
2209:
2208:
2206:
2205:
2200:
2195:
2194:
2193:
2188:
2183:
2178:
2173:
2168:
2163:
2158:
2156:Contraharmonic
2153:
2148:
2137:
2135:
2126:
2116:
2115:
2110:
2103:
2102:
2100:
2099:
2094:
2088:
2085:
2084:
2079:
2077:
2076:
2069:
2062:
2054:
2048:
2047:
2041:
2028:
1985:
1976:
1955:
1945:(2): 121â135.
1934:
1928:
1915:
1905:(2): 103â109.
1890:
1887:
1884:
1883:
1856:
1827:
1802:
1777:
1776:
1774:
1771:
1770:
1769:
1764:
1759:
1754:
1749:
1744:
1739:
1734:
1727:
1724:
1710:
1709:
1706:
1703:
1699:
1698:
1695:
1692:
1688:
1687:
1684:
1681:
1677:
1676:
1673:
1670:
1666:
1665:
1662:
1659:
1591:
1588:
1580:
1579:
1573:
1566:
1565:might be used.
1559:
1533:
1530:
1516:relate to the
1481:
1474:
1399:
1396:
1378:tests and the
1353:
1350:
1323:
1294:
1290:
1267:
1263:
1251:detrended data
1236:
1229:
1226:
1193:
1189:
1177:
1176:
1162:
1159:
1153:
1150:
1144:
1141:
1135:
1129:
1126:
1100:
1094:
1091:
1065:
1062:
1039:
1012:
992:
972:
948:
943:
939:
935:
924:non-stationary
907:
887:
867:
856:
855:
841:
837:
833:
830:
827:
824:
821:
818:
813:
809:
793:
790:
774:
771:
745:
742:
736:
733:
727:
724:
718:
712:
709:
683:
680:
654:
651:
628:
622:
619:
613:
610:
604:
601:
595:
573:
569:
557:
556:
542:
537:
532:
525:
522:
516:
513:
507:
504:
497:
493:
488:
484:
479:
472:
468:
441:
438:
412:
409:
384:
380:
359:
339:
304:
301:
299:is heading.
264:
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245:
207:
205:
198:
191:
190:
173:September 2023
152:
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136:
135:
90:
88:
81:
76:
50:
49:
47:
40:
26:
24:
14:
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10:
9:
6:
4:
3:
2:
4408:
4397:
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4389:
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4381:
4366:
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4353:
4344:
4342:
4341:
4336:
4330:
4328:
4327:
4318:
4317:
4314:
4300:
4297:
4295:
4294:Geostatistics
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4276:
4274:
4272:
4268:
4262:
4261:Psychometrics
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
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4214:
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4211:
4209:
4205:
4199:
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4185:
4182:
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4177:
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4158:
4152:
4149:
4147:
4144:
4142:
4138:
4135:
4133:
4130:
4129:
4127:
4125:
4124:Biostatistics
4121:
4117:
4113:
4108:
4104:
4086:
4085:Log-rank test
4083:
4082:
4080:
4076:
4070:
4067:
4066:
4064:
4062:
4058:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4031:
4029:
4025:
4022:
4020:
4016:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3982:
3980:
3978:
3974:
3968:
3965:
3963:
3960:
3958:
3956:(BoxâJenkins)
3952:
3950:
3947:
3945:
3942:
3938:
3935:
3934:
3933:
3930:
3929:
3927:
3925:
3921:
3915:
3912:
3910:
3909:DurbinâWatson
3907:
3905:
3899:
3897:
3894:
3892:
3891:DickeyâFuller
3889:
3888:
3886:
3882:
3876:
3873:
3871:
3868:
3866:
3865:Cointegration
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3840:Decomposition
3838:
3837:
3835:
3831:
3828:
3826:
3822:
3812:
3809:
3808:
3807:
3804:
3803:
3802:
3799:
3795:
3792:
3791:
3790:
3787:
3785:
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3765:
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3760:
3757:
3755:
3752:
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3749:
3747:
3743:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3711:Cohen's kappa
3709:
3708:
3706:
3704:
3700:
3696:
3692:
3688:
3684:
3680:
3675:
3671:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3638:
3636:
3634:
3630:
3624:
3620:
3616:
3610:
3608:
3605:
3604:
3602:
3600:
3596:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3564:Nonparametric
3562:
3560:
3557:
3556:
3554:
3550:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3525:
3523:
3521:
3517:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3487:
3485:
3483:
3479:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3454:
3452:
3450:
3446:
3442:
3435:
3432:
3430:
3427:
3426:
3422:
3418:
3402:
3399:
3398:
3397:
3394:
3392:
3389:
3387:
3384:
3380:
3377:
3375:
3372:
3371:
3370:
3367:
3366:
3364:
3362:
3358:
3348:
3345:
3341:
3335:
3333:
3327:
3325:
3319:
3318:
3317:
3314:
3313:Nonparametric
3311:
3309:
3303:
3299:
3296:
3295:
3294:
3288:
3284:
3283:Sample median
3281:
3280:
3279:
3276:
3275:
3273:
3271:
3267:
3259:
3256:
3254:
3251:
3249:
3246:
3245:
3244:
3241:
3239:
3236:
3234:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3204:
3200:
3198:
3195:
3194:
3192:
3190:
3186:
3180:
3178:
3174:
3172:
3170:
3165:
3163:
3158:
3154:
3153:
3150:
3147:
3145:
3141:
3131:
3128:
3126:
3123:
3121:
3118:
3117:
3115:
3113:
3109:
3103:
3100:
3096:
3093:
3092:
3091:
3088:
3084:
3081:
3080:
3079:
3076:
3074:
3071:
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3068:
3066:
3062:
3054:
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3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3015:
3013:
3011:
3007:
3001:
2998:
2994:
2991:
2987:
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2982:
2979:
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2977:
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2969:
2965:
2962:
2960:
2957:
2955:
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2945:
2942:
2941:
2939:
2937:
2933:
2930:
2928:
2924:
2918:
2915:
2913:
2910:
2906:
2903:
2902:
2901:
2898:
2896:
2893:
2889:
2888:loss function
2886:
2885:
2884:
2881:
2877:
2874:
2872:
2869:
2867:
2864:
2863:
2862:
2859:
2857:
2854:
2852:
2849:
2845:
2842:
2840:
2837:
2835:
2829:
2826:
2825:
2824:
2821:
2817:
2814:
2812:
2809:
2807:
2804:
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2802:
2799:
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2787:
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2775:
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2770:
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2726:
2721:
2717:
2703:
2700:
2698:
2695:
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2690:
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2676:
2670:
2667:
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2644:
2641:
2639:
2636:
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2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2610:
2608:
2606:
2602:
2596:
2593:
2591:
2590:Questionnaire
2588:
2586:
2583:
2579:
2576:
2574:
2571:
2570:
2569:
2566:
2565:
2563:
2561:
2557:
2551:
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2533:
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2528:
2526:
2523:
2521:
2518:
2516:
2513:
2512:
2510:
2508:
2504:
2500:
2496:
2491:
2487:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2417:Control chart
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2399:
2397:
2395:
2391:
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2239:
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2227:
2224:
2222:
2219:
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2210:
2204:
2201:
2199:
2196:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2143:
2142:
2139:
2138:
2136:
2134:
2130:
2127:
2125:
2121:
2117:
2113:
2108:
2104:
2098:
2095:
2093:
2090:
2089:
2086:
2082:
2075:
2070:
2068:
2063:
2061:
2056:
2055:
2052:
2044:
2038:
2034:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2002:(4): 046113.
2001:
1997:
1996:
1991:
1986:
1982:
1977:
1973:
1969:
1965:
1961:
1956:
1952:
1948:
1944:
1940:
1935:
1931:
1925:
1921:
1916:
1912:
1908:
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1900:
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1888:
1871:
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1863:
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1844:
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1812:
1806:
1803:
1788:
1782:
1779:
1772:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1747:Least squares
1745:
1743:
1740:
1738:
1737:Extrapolation
1735:
1733:
1730:
1729:
1725:
1723:
1721:
1717:
1707:
1704:
1701:
1700:
1696:
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1679:
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1668:
1667:
1663:
1660:
1657:
1656:
1651:
1649:
1645:
1640:
1634:
1632:
1628:
1624:
1623:Friedman test
1619:
1615:
1610:
1606:
1603:on levels of
1602:
1597:
1589:
1587:
1585:
1577:
1574:
1571:
1567:
1564:
1560:
1557:
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1552:
1551:
1549:
1544:
1539:
1531:
1529:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1495:
1491:
1486:
1479:
1475:
1473:
1459:
1455:
1450:
1441:
1430:
1419:
1415:
1409:
1405:
1397:
1395:
1393:
1387:
1383:
1381:
1380:cointegration
1377:
1372:
1370:
1348:
1337:
1321:
1313:
1308:
1292:
1288:
1265:
1261:
1252:
1234:
1224:
1213:
1209:
1191:
1187:
1157:
1151:
1148:
1139:
1133:
1124:
1114:
1113:
1112:
1098:
1089:
1060:
1037:
1028:
1026:
1010:
990:
970:
962:
941:
937:
925:
921:
905:
885:
865:
839:
835:
831:
828:
825:
822:
819:
816:
811:
807:
799:
798:
797:
791:
789:
769:
740:
734:
731:
722:
716:
707:
678:
649:
617:
611:
608:
599:
571:
567:
540:
535:
530:
520:
514:
511:
502:
495:
491:
486:
482:
477:
470:
466:
458:
457:
456:
436:
407:
382:
378:
357:
337:
329:
325:
324:least-squares
320:
318:
317:straight line
314:
310:
302:
300:
298:
294:
290:
286:
282:
278:
274:
270:
260:
257:
242:
239:
231:
228:December 2023
221:
217:
211:
208:This article
206:
197:
196:
187:
184:
176:
166:
160:
158:
151:
142:
141:
132:
129:
121:
111:
107:
101:
100:
94:
89:
80:
79:
74:
72:
65:
64:
59:
58:
53:
48:
39:
38:
33:
32:Curve fitting
19:
4362:
4350:
4331:
4324:
4236:Econometrics
4186: /
4169:Chemometrics
4146:Epidemiology
4139: /
4112:Applications
3954:ARIMA model
3901:Q-statistic
3850:Stationarity
3746:Multivariate
3689: /
3685: /
3683:Multivariate
3681: /
3621: /
3617: /
3391:Bayes factor
3290:Signed rank
3202:
3176:
3168:
3156:
2851:Completeness
2687:Cohort study
2585:Opinion poll
2520:Missing data
2507:Study design
2462:Scatter plot
2384:Scatter plot
2377:Spearman's Ï
2339:Grouped data
2032:
1999:
1993:
1980:
1966:(1): 89â90.
1963:
1959:
1942:
1938:
1920:Econometrics
1919:
1902:
1896:
1874:. Retrieved
1868:
1847:. Retrieved
1843:the original
1818:. Retrieved
1805:
1793:. Retrieved
1781:
1757:Line fitting
1719:
1715:
1713:
1647:
1643:
1635:
1594:Medical and
1593:
1581:
1548:misspecified
1535:
1525:
1513:
1505:
1499:
1493:
1489:
1477:
1451:
1439:
1428:
1417:
1413:
1403:
1401:
1388:
1384:
1373:
1309:
1250:
1207:
1178:
1029:
857:
795:
558:
321:
306:
268:
267:
252:
234:
225:
209:
179:
170:
154:
124:
115:
96:
68:
61:
55:
54:Please help
51:
4364:WikiProject
4279:Cartography
4241:Jurimetrics
4193:Reliability
3924:Time domain
3903:(LjungâBox)
3825:Time-series
3703:Categorical
3687:Time-series
3679:Categorical
3614:(Bernoulli)
3449:Correlation
3429:Correlation
3225:JarqueâBera
3197:Chi-squared
2959:M-estimator
2912:Asymptotics
2856:Sufficiency
2623:Interaction
2535:Replication
2515:Effect size
2472:Violin plot
2452:Radar chart
2432:Forest plot
2422:Correlogram
2372:Kendall's Ï
2042:041227630-5
1742:Forecasting
1605:cholesterol
1543:correlation
1522:t-statistic
285:information
273:statistical
110:introducing
4380:Categories
4231:Demography
3949:ARMA model
3754:Regression
3331:(Friedman)
3292:(Wilcoxon)
3230:Normality
3220:Lilliefors
3167:Student's
3043:Resampling
2917:Robustness
2905:divergence
2895:Efficiency
2833:(monotone)
2828:Likelihood
2745:Population
2578:Stratified
2530:Population
2349:Dependence
2305:Count data
2236:Percentile
2213:Dispersion
2146:Arithmetic
2081:Statistics
1929:0077104285
1889:References
1732:Estimation
1596:biomedical
1208:detrending
959:is called
279:patterns.
93:references
57:improve it
3612:Logistic
3379:posterior
3305:Rank sum
3053:Jackknife
3048:Bootstrap
2866:Bootstrap
2801:Parameter
2750:Statistic
2545:Statistic
2457:Run chart
2442:Pie chart
2437:Histogram
2427:Fan chart
2402:Bar chart
2284:L-moments
2171:Geometric
1609:analgesic
1576:Unit root
1510:residuals
1502:r-squared
1376:unit root
1352:^
1228:^
1212:residuals
1161:^
1143:^
1128:^
1093:^
1064:^
773:^
744:^
726:^
711:^
682:^
653:^
621:^
603:^
524:^
506:^
492:−
467:∑
440:^
411:^
328:minimizes
313:functions
118:July 2019
63:talk page
4326:Category
4019:Survival
3896:Johansen
3619:Binomial
3574:Isotonic
3161:(normal)
2806:location
2613:Blocking
2568:Sampling
2447:QâQ plot
2412:Box plot
2394:Graphics
2289:Skewness
2279:Kurtosis
2251:Variance
2181:Heronian
2176:Harmonic
2024:22181233
1849:June 17,
1820:June 17,
1795:June 17,
1726:See also
1336:variance
4352:Commons
4299:Kriging
4184:Process
4141:studies
4000:Wavelet
3833:General
3000:Plug-in
2794:L space
2573:Cluster
2274:Moments
2092:Outline
2004:Bibcode
1876:May 17,
1639:trypsin
1249:as the
214:Please
106:improve
4221:Census
3811:Normal
3759:Manova
3579:Robust
3329:2-way
3321:1-way
3159:-test
2830:
2407:Biplot
2198:Median
2191:Lehmer
2133:Center
2039:
2022:
1926:
1614:mmol/L
1601:statin
1470:
1466:
1462:
1447:
1443:
1436:
1432:
1425:
1421:
1411:
1392:cycles
1206:(thus
920:errors
858:where
95:, but
3845:Trend
3374:prior
3316:anova
3205:-test
3179:-test
3171:-test
3078:Power
3023:Pivot
2816:shape
2811:scale
2261:Shape
2241:Range
2186:Heinz
2161:Cubic
2097:Index
1814:(PDF)
1790:(PDF)
1773:Notes
1708:0.79
1705:2.40
1697:0.66
1694:2.22
1686:0.75
1683:1.94
1675:0.56
1661:mean
1607:, an
289:graph
271:is a
4078:Test
3278:Sign
3130:Wald
2203:Mode
2141:Mean
2037:ISBN
2020:PMID
1924:ISBN
1878:2015
1851:2012
1822:2012
1797:2012
1672:1.6
1458:IPCC
1307:'s.
1079:and
983:and
878:and
668:and
426:and
322:The
309:data
297:data
293:data
281:Data
277:data
3258:BIC
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