Skip to Main Content
 

Global Search Box

 
 
 

ETD Abstract Container

Abstract Header

A New Approach to ANOVA Methods for Autocorrelated Data

Liu, Gang
http://orcid.org/0000-0003-2151-1072
http://rave.ohiolink.edu/etdc/view?acc_num=toledo1461226897

Abstract Details

2016, Doctor of Philosophy, University of Toledo, Mathematics.
First, we introduce some time series models: autoregressive models (AR), periodic autoregressive models (PAR), and some prevalent statistical analysis, including the One-way ANOVA Analysis, Yule-Walker Estimation, Tukey's Honest Significance Difference Test, Likelihood Ratio Method, Spectral Domain Method, and Time Series Bootstrapping Method. These statistical techniques are illustrated by some examples. Second, we reexamine ANOVA problems for autocorrelated data. Using linear prediction techniques for stationary time series, a new test statistic that assesses a null hypothesis of equal means is proposed and investigated. Our test statistic mimics the classical $F$-type ratio form used with independent data, but substitutes estimated prediction residuals in for the errors. This simple tactic departs from past studies that adjust the quadratic forms in the numerator and denominator in the $F$ ratio for autocorrelation. The advantages are that our statistic retains the classical null hypothesis $F$ distribution (now as a limit) with the customary degrees of freedom. The statistic is shown to perform well in simulations. Asymptotic proofs are given in the case of autoregressive random errors; applications to NBA sports and stocks are supplied. Third, we would like to test whether or not a time series has equal seasonal means. The problem is especially prevalent in environmental series, where most time-ordered data display some periodic structure. This dissertation takes a new look at the problem, proposing statistics in both the time and frequency domains. In the time domain, we propose a new statistic with an ANOVA form that is based upon the one-step-ahead prediction errors of the series. This statistic inherits many features of the $F$ distribution arising in classic one-way ANOVA analyses. Its asymptotic distribution is established when model parameters must be estimated. In the frequency domain, a statistic that modifies Fisher's classical test for a seasonal mean superimposed upon independent and identically distributed Gaussian noise is presented. The performance and comparison of these statistics are studied via simulation. Implementation of the methods is simple as the statistics are constructed from sample means, autocovariances, and periodograms of the series. Applications to average monthly air temperatures, death counts from lung disease and a data set of monthly temperatures from Spokane Washington are given. Fourth, we reexamine ANOVA problems for autocorrelated data with unequal seasonal variance. In temperature series, most time-ordered data display some periodic structures, and different variations in each period. Both the proposed ANOVA test and the proposed Welch test are considered here, and the proposed Welch test will follow an adjusted $F$ distribution. The performance and comparison of the classic ANOVA test, the proposed ANOVA test, the proposed Welch test and the log-likelihood ratio test are studied via simulation. Application to monthly temperatures (Spokane Washington) based on the proposed Welch test is provided. Finally, a R function "\textbf{GAR}" is programmed for testing the seasonal means of the autocorrelated data under an autoregressive model. The results will exhibit the estimates of the parameters, including the best fitted order, the estimates of the white noise variance, the estimates of the white noise mean, the proposed ANOVA test statistic with its p-value, the log-likelihood value under both the null hypothesis and alternative hypothesis, the value of log-likelihood ratio and the proposed spectral domain test statistic with its p-value.
Qin Shao (Advisor)
Donald B. White (Committee Member)
Geoffrey K. Martin (Committee Member)
Rong Liu (Committee Member)
126 p.
Mathematics
ANOVA, Autocorrelation, Autoregression, F-statistic, Time Series, Periodicity, Seasonal means

Recommended Citations

Citations

  • Liu, G. (2016). A New Approach to ANOVA Methods for Autocorrelated Data [Doctoral dissertation, University of Toledo]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1461226897

    APA Style (7th edition)

  • Liu, Gang. A New Approach to ANOVA Methods for Autocorrelated Data. 2016. University of Toledo, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=toledo1461226897.

    MLA Style (8th edition)

  • Liu, Gang. "A New Approach to ANOVA Methods for Autocorrelated Data." Doctoral dissertation, University of Toledo, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1461226897

    Chicago Manual of Style (17th edition)

toledo1461226897
2,049
© 2015, some rights reserved.Creative Commons License A New Approach to ANOVA Methods for Autocorrelated Data by Gang Liu is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Toledo and OhioLINK.