This thesis considers both univariate and multivariate approaches to the analysis of a set of repeated-measures data. Since repeated measures on the same subject are correlated over time, the usual analysis of variance assumption of independence is often violated. The models in this thesis demonstrate different approaches to the analysis of repeated-measures data, and highlight their advantages and disadvantages.
Milk from two groups of lactating cows, one group vaccinated, the other not, was analysed every month after calving for eight months in order to measure the amount of bacteria in the milk. The primary goal of the experiment was to determine if a vaccine developed by the Royal Melbourne Institute of Technology’s Biology Department led to a significant decrease in mean bacteria production per litre of milk produced compared to the control group.
A univariate model suitable for repeated measures data was initially tried, with mean bacteria production in the treatment group not significantly different from the control group (p < 0.68).
The multivariate approach to repeated measures, profile analysis, yielded similar results for treatment effects (p < 0.68), while meeting the necessary assumptions for multivariate analysis.
Finally, a generalised multivariate analysis of variance was carried out in order to fit polynomial growth curves for both the control and the vaccinated groups and to test if the growth curves were equal for the two groups. It was found that a slope-intercept model was adequate to describe both growth curves and that the growth curve for the treatment group did not differ significantly from that of the control group (p < 0.11).
Table of Contents
1. INTRODUCTION
2. EXPLORING THE DATA
3. TIME BY TIME ANOVA
4. UNIVARIATE APPROACH TO REPEATED MEASURES
4.1 Repeated Measures ANOVA
4.2 Testing For Compound Symmetry
5. MULTIVARIATE APPROACH TO REPEATED MEASURES
5.1 Profile Analysis
5.2 Assumptions of Profile Analysis
5.3 Testing For Multivariate Normality
5.4 Testing For The Equality of Covariance Matrices
5.5 Hypothesis Tests for Profile Analysis
6. THE GENERALISED MULTIVARIATE ANALYSIS OF VARIANCE
6.1 Growth Curves
6.2 Hypothesis Tests For Growth Curves
6.3 Testing Polynomial Adequacy
6.4 Testing For The Equality of Growth Curves
7. CONCLUSION
Objectives and Topics
This thesis examines and compares different statistical methodologies for the analysis of longitudinal repeated-measures data, specifically focusing on milk bacteria cell counts in cows. The primary research goal is to determine if a vaccine developed by the Royal Melbourne Institute of Technology results in a statistically significant reduction in bacteria production compared to a control group over an eight-month period.
- Univariate repeated measures ANOVA techniques and their limitations.
- Multivariate approaches, including profile analysis, to handle correlated observations.
- Generalised Multivariate Analysis of Variance (GMANOVA) for modeling growth curves.
- Evaluation of statistical assumptions such as compound symmetry and multivariate normality.
- Comparison of various statistical tests for parallelism and equality of growth patterns.
Excerpt from the Book
1. Introduction
Milk from two groups of lactating cows, one group vaccinated, the other not, was analysed every month after calving for eight months in order to measure the amount of bacteria in the milk. The primary goal of the experiment was to determine if a vaccine developed by the Royal Melbourne Institute of Technology’s Biology Department led to a significant decrease in mean bacteria production compared to the control group.
Experiments such as this fit into the family of designs known in the literature as repeated measures data, longitudinal models, or growth curves. Data from these models generally arise whenever more than two observations of the same variable are made on an individual subject or experimental unit. These models are especially common in biology, agriculture, and medicine and most often occur when observations on a group of subjects are repeated over a period of time.
Repeated measures data such as this require somewhat different statistical treatment than normal because the observations are not independent. This lack of independence lies at the core of repeated measures analysis, and is what differentiates it from the more commonly used statistical analyses. The implications of a lack of independence within a subject’s responses are serious, as will be explained in Chapter 4. For the data collected by the RMIT biology department, this implies that the amount of bacteria found in a litre of a cow’s milk will, at any given time, be correlated with the amount of bacteria found in that cow’s milk at subsequent or preceding times. In addition, the correlation between the amount of bacteria produced at different times also tends to be stronger the shorter the time interval. In other words, the amount of bacteria produced per litre of milk is more dependent on the amount of bacteria produced one month ago than the amount of bacteria produced five months ago.
Correlation between observations is usually present in these types of experiments. Nearby plots in a field trial are usually more similar than plots further apart.
Summary of Chapters
1. INTRODUCTION: Provides an overview of the experimental study on milk bacteria and introduces the challenges inherent in analyzing repeated-measures data.
2. EXPLORING THE DATA: Discusses the importance of graphical representation in identifying patterns and potential outliers in the dataset before formal analysis.
3. TIME BY TIME ANOVA: Introduces the simple time-by-time ANOVA approach, highlights its computational simplicity, and explains its major limitations regarding correlated responses.
4. UNIVARIATE APPROACH TO REPEATED MEASURES: Examines the univariate repeated measures ANOVA model, its dependency assumptions, and the need for data transformation.
5. MULTIVARIATE APPROACH TO REPEATED MEASURES: Discusses profile analysis as a robust alternative, focusing on assumptions, testing procedures, and hypothesis testing.
6. THE GENERALISED MULTIVARIATE ANALYSIS OF VARIANCE: Details the GMANOVA model for fitting growth curves and testing the adequacy of polynomial models.
7. CONCLUSION: Synthesizes the findings from the different statistical approaches and offers final insights on the effectiveness of the vaccine.
Keywords
Repeated measures, Longitudinal data, ANOVA, MANOVA, GMANOVA, Profile analysis, Growth curves, Compound symmetry, Bacteria production, Vaccine efficacy, Statistical modeling, Multivariate normality, Covariance matrices, Hypothesis testing, Longitudinal models
Frequently Asked Questions
What is the fundamental objective of this thesis?
The thesis aims to identify and apply appropriate statistical methods to analyze repeated-measures data, specifically testing whether a vaccine significantly reduces bacteria levels in cows compared to a control group.
What are the central themes of the research?
The research centers on longitudinal data analysis, addressing the challenges of correlated observations, testing model assumptions, and applying both univariate and multivariate statistical techniques.
What is the primary research question?
The central question is whether the vaccine developed by the RMIT Biology Department leads to a significant decrease in mean bacteria production per litre of milk over eight months.
Which scientific methods are employed?
The study utilizes time-by-time ANOVA, univariate repeated measures ANOVA, multivariate profile analysis, and the Generalised Multivariate Analysis of Variance (GMANOVA) for growth curve fitting.
What topics are covered in the main body?
The main body systematically explores the data, discusses ANOVA limitations, details the transition to multivariate methods, explains GMANOVA procedures, and concludes with an evaluation of the vaccine's efficacy.
Which keywords best characterize this work?
Key terms include repeated measures, longitudinal data, ANOVA, GMANOVA, profile analysis, and growth curves.
Why is the independence assumption a problem in this study?
Because cell counts from the same cow are measured over time, they are inherently correlated; standard ANOVA assumes independence, which is violated here, necessitating specialized models.
What did the profile analysis conclude regarding the vaccine?
The profile analysis, like the univariate ANOVA, found no significant interaction between group membership and time, indicating the vaccine did not have a significant effect on bacteria production over the observation period.
Why was a log-transformation applied to the cell count data?
The raw data were highly skewed and showed increased variance with higher fitted values; the loge transformation improved symmetry and minimized the impact of outliers.
What is the advantage of using GMANOVA over simpler regression models?
GMANOVA allows for the fitting of growth curves while accounting for the complex covariance structure of longitudinal data, which standard linear regression fails to do.
- Citation du texte
- Tony Wragg (Auteur), 1999, Univariate and Multivariate Methods for the Analysis of Repeated Measures Data, Munich, GRIN Verlag, https://www.grin.com/document/110642
