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~ . CONTENTS PTeface_........._._........__...,_..._._._ Abstract......_......,.......__._......__.. Introductioll_______________________________ DistributioDS__ ____ __ ___ __ _ _ __ _ _ ___ _ _ _ ___ _ __ Cumulative distributions~ _ _ _ ____________ Stat~ticalinference_________________________ Correlation and regression_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Serial correlatioD__ _____________________ Regression methods_ un __ _ _ __ _ ___ _ _ _ _ ___ _ u Regression models_ _ ____________________ 1ransfonnatioDS________________________ Simple linear regression___ __ _ _ ____ _ _ _ __ __ Multiple linear regressioD_ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Regression computation using "e" multi- pliem___..............___.........._ Regressions ha:ving various numbers of inde- pendentvariables_____________________ Use of digital computers..________________ Application of the regression method______ . Regression methods-Continued Graphical regressioll____________ ____ __ ___ Graphical multiple regressioD..________ _ __~ Graphical multiple regression when the independent variables are highly corre- lated between themselves____________~_ Choice of graphical or analytical method for multiple regression_____ _______________ Determining equations of graphical relations_ __ General methodsh______________________ Definition of equatiollS_ _________________ Othertoom................................. Analysis of variance_________________n__ Analysis of covarianoo__ _ n __ __ _ __ __ _ n n Multivariate ana1ysis___~n__ ____ _ ___ n__ Characteristics of hydrologic data_____________ Effects of data characteristics on analysis_ Outliers............................... Selected references____~_ ______ ________ ______ Page III 1 1 1 3 4 6 9 9 9 10 11 12 15 18 18 18 FIGURES 1. Histogram, or frequency distribution, of 1,000 tree-ring indices__n_n__~_n 2. Probability density curve of 1;000 tree~ring indices______n___n_______~__ 3. Probability density curve and its cumulative form_______n____n__~__n_~ 4-8. Diagrams showing- 4. Normal distribution_______ ______ ____ _____ _ _ ________ _______ _____ 5. Distribution of means of samples from a normal distribution________ 6. Distribution of variances of samples__n _ _ _ __ __ n n n _ __ n n _ __ n ~ 7. Hypothetical sampling distribution of means______________________ 8. Normal distribution of plotted points about the regression line_ _ _ ___ Graph showing plot used in demonstrating the effect of sample" rangb on computed correlation coefficient___ __________ ______ ____ _ ____ ____~~ ___ " Equations and graphs of some common ' regression mode1s_ _ __ ____ ______ ~_~ Data plotted on natural and log scales showing the achievement of equal variance about the regression line by use of the log transformationn__~n Plot of data from table 2 showing computed regression line_ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Equations and graphs of three models based on the plotted data_ _ _ _ _ _ _ _ _ __ Graphs showing four possible outcomes of plotting Y against X _ __ n _ n ____ Plot showing the two regression lines and the structural line _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Graph showing method of estimating the standard error of a graphical re- ~ion-----___________________________________________________~__ Example of graphical multiple regression~_____________n___n_n______u Example of graphical multiple regression using arithmetic scalesnn_____n~ Graphical regression using highly correlated independent variables_________~ Graphical regression in which one variable is used twice________nun_un Graph showing multiple linear regression by the method of residuals_ _ _ _ _ _ __ Graph showing coaxial graphical multiple regression______________________ 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19, 20. . 2l. 22, Page 2 2 3 4 4 5 6 7 8 10 11 12 20 21 21 22 23 25 25 28 30 31 V Page 21 22 24 26 27 28 29 31 31 33 35 36 36 37 38