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Text of the page (random words):
is large enough the law of large numbers and the central limit theorem become applicable and then the distribution of the estimators is approximately normal under this assumption all formulas derived in the previous section remain valid with the only exception that the quantile t n 2 of student s t distribution is replaced with the quantile q of the standard normal distribution occasionally the fraction 1 n 2 is replaced with 1 n when n is large such a change does not alter the results appreciably numerical example edit see also ordinary least squares example and linear least squares example this data set gives average masses for women as a function of their height in a sample of american women of age 30 39 although the ols article argues that it would be more appropriate to run a quadratic regression for this data the simple linear regression model is applied here instead height m x i 1 47 1 50 1 52 1 55 1 57 1 60 1 63 1 65 1 68 1 70 1 73 1 75 1 78 1 80 1 83 mass kg y i 52 21 53 12 54 48 55 84 57 20 58 57 59 93 61 29 63 11 64 47 66 28 68 10 69 92 72 19 74 46 i displaystyle i x i displaystyle x_ i y i displaystyle y_ i x i 2 displaystyle x_ i 2 x i y i displaystyle x_ i y_ i y i 2 displaystyle y_ i 2 1 1 47 52 21 2 1609 76 7487 2725 8841 2 1 50 53 12 2 2500 79 6800 2821 7344 3 1 52 54 48 2 3104 82 8096 2968 0704 4 1 55 55 84 2 4025 86 5520 3118 1056 5 1 57 57 20 2 4649 89 8040 3271 8400 6 1 60 58 57 2 5600 93 7120 3430 4449 7 1 63 59 93 2 6569 97 6859 3591 6049 8 1 65 61 29 2 7225 101 1285 3756 4641 9 1 68 63 11 2 8224 106 0248 3982 8721 10 1 70 64 47 2 8900 109 5990 4156 3809 11 1 73 66 28 2 9929 114 6644 4393 0384 12 1 75 68 10 3 0625 119 1750 4637 6100 13 1 78 69 92 3 1684 124 4576 4888 8064 14 1 80 72 19 3 2400 129 9420 5211 3961 15 1 83 74 46 3 3489 136 2618 5544 2916 σ displaystyle sigma 24 76 931 17 41 0532 1548 2453 58498 5439 there are n 15 points in this data set hand calculations would be started by finding the following five sums s x i x i 24 76 s y i y i 931 17 s x x i x i 2 41 0532 s y y i y i 2 58498 5439 s x y i x i y i 1548 2453 displaystyle begin aligned s_ x sum _ i x_ i 24 76 qquad s_ y sum _ i y_ i 931 17 5pt s_ xx sum _ i x_ i 2 41 0532 s_ yy sum _ i y_ i 2 58498 5439 5pt s_ xy sum _ i x_ i y_ i 1548 2453 end aligned these quantities would be used to calculate the estimates of the regression coefficients and their standard errors β n s x y s x s y n s x x s x 2 61 272 α 1 n s y β 1 n s x 39 062 s ε 2 1 n n 2 n s y y s y 2 β 2 n s x x s x 2 0 5762 s β 2 n s ε 2 n s x x s x 2 3 1539 s α 2 s β 2 1 n s x x 8 63185 displaystyle begin aligned widehat beta frac ns_ xy s_ x s_ y ns_ xx s_ x 2 61 272 8pt widehat alpha frac 1 n s_ y widehat beta frac 1 n s_ x 39 062 8pt s_ varepsilon 2 frac 1 n n 2 left ns_ yy s_ y 2 widehat beta 2 ns_ xx s_ x 2 right 0 5762 8pt s_ widehat beta 2 frac ns_ varepsilon 2 ns_ xx s_ x 2 3 1539 8pt s_ widehat alpha 2 s_ widehat beta 2 frac 1 n s_ xx 8 63185 end aligned graph of points and linear least squares lines in the simple linear regression numerical example the 0 975 quantile of student s t distribution with 13 degrees of freedom is t 13 2 1604 and thus the 95 confidence intervals for α and β are α α t 13 s α 45 4 32 7 β β t 13 s β 57 4 65 1 displaystyle begin aligned alpha in widehat alpha mp t_ 13 s_ widehat alpha 45 4 32 7 5pt beta in widehat beta mp t_ 13 s_ widehat beta 57 4 65 1 end aligned the product moment correlation coefficient might also be calculated r n s x y s x s y n s x x s x 2 n s y y s y 2 0 9946 displaystyle widehat r frac ns_ xy s_ x s_ y sqrt ns_ xx s_ x 2 ns_ yy s_ y 2 0 9946 alternatives edit calculating the parameters of a linear model by minimizing the squared error in slr there is an underlying assumption that only the dependent variable contains measurement error if the explanatory variable is also measured with error then simple regression is not appropriate for estimating the underlying relationship because it will be biased due to regression dilution other estimation methods that can be used in place of ordinary least squares include least absolute deviations minimizing the sum of absolute values of residuals and the theil sen estimator which chooses a line whose slope is the median of the slopes determined by pairs of sample points deming regression total least squares also finds a line that fits a set of two dimensional sample points but unlike ordinary least squares least absolute deviations and median slope regression it is not really an instance of simple linear regression because it does not separate the coordinates into one dependent and one independent variable and could potentially return a vertical line as its fit can lead to a model that attempts to fit the outliers more than the data line fitting edit this section is an excerpt from line fitting edit this page is a primary topic and an article should be written about it one or more editors believe it holds the title of a broad concept article the article may be written here or drafted elsewhere first related titles should be described here while unrelated titles should be moved to simple linear regression disambiguation relevant discussion may be found on the talk page may 2019 line fitting is the process of constructing a straight line that has the best fit to a series of data points several methods exist considering vertical distance simple linear regression resistance to outliers robust simple linear regression perpendicular distance orthogonal regression this is not scale invariant i e changing the measurement units leads to a different line weighted geometric distance deming regression scale invariant approach major axis regression this allows for measurement error in both variables and gives an equivalent equation if the measurement units are altered simple linear regression without the intercept term single regressor edit sometimes it is appropriate to force the regression line to pass through the origin because x and y are assumed to be proportional for the model without the intercept term y βx the ols estimator for β simplifies to β i 1 n x i y i i 1 n x i 2 x y x 2 displaystyle widehat beta frac sum _ i 1 n x_ i y_ i sum _ i 1 n x_ i 2 frac overline xy overline x 2 substituting x h y k in place of x y gives the regression through h k β i 1 n x i h y i k i 1 n x i h 2 x h y k x h 2 x y k x h y h k x 2 2 h x h 2 x y x y x h y k x 2 x 2 x h 2 cov x y x h y k var x x h 2 displaystyle begin aligned widehat beta frac sum _ i 1 n x_ i h y_ i k sum _ i 1 n x_ i h 2 frac overline x h y k overline x h 2 6pt frac overline xy k bar x h bar y hk overline x 2 2h bar x h 2 6pt frac overline xy bar x bar y bar x h bar y k overline x 2 bar x 2 bar x h 2 6pt frac operatorname cov x y bar x h bar y k operatorname var x bar x h 2 end aligned where cov and var refer to the covariance and variance of the sample data uncorrected for bias the last form above demonstrates how moving the line away from the center of mass of the data points affects the slope see also edit design matrix simple linear regression linear trend estimation linear segmented regression proofs involving ordinary least squares derivation of all formulas used in this article in general multidimensional case newey west estimator references edit seltman howard j 2008 09 08 experimental design and analysis pdf p 227 statistical sampling and regression simple linear regression columbia university retrieved 2016 10 17 when one independent variable is used in a regression it is called a simple regression lane david m introduction to statistics pdf p 462 zou kh tuncali k silverman sg 2003 correlation and simple linear regression radiology 227 3 617 22 doi 10 1148 radiol 2273011499 issn 0033 8419 oclc 110941167 pmid 12773666 altman naomi krzywinski martin 2015 simple linear regression nature methods 12 11 999 1000 doi 10 1038 nmeth 3627 issn 1548 7091 oclc 5912005539 pmid 26824102 s2cid 261269711 kenney j f and keeping e s 1962 linear regression and correlation ch 15 in mathematics of statistics pt 1 3rd ed princeton nj van nostrand pp 252 285 1 2 muthukrishnan gowri 17 jun 2018 maths behind polynomial regression muthukrishnan maths behind polynomial regression retrieved 30 jan 2024 mathematics of polynomial regression polynomial regression a php regression class numeracy maths and statistics academic skills kit newcastle university simple linear regression retrieved 30 jan 2024 valliant richard jill a dever and frauke kreuter practical tools for designing and weighting survey samples new york springer 2013 draper n r smith h 1998 applied regression analysis 3rd ed john wiley isbn 0 471 17082 8 casella g and berger r l 2002 statistical inference 2nd edition cengage isbn 978 0 534 24312 8 pp 558 559 external links edit wolfram mathworld s explanation of least squares fitting and how to calculate it mathematics of simple regression robert nau duke university v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio score lagrange multiplier wald specific tests z test normal student s t test f test goodness of fit chi squared g test kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk likelihood ratio test model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova anova analysis of covariance multivariate anova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics chemometrics methods engineering probabilistic design process quality control reliability system identification social statistics actuarial science census crime statistics demography econometrics jurimetrics national accounts official statistics population statistics psychometrics spatial statistics cartography environmental statistics ...
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