ANOVA Analysis of Variance – One Way Ing. Elena Mielcová, Ph.D. STATISTICAL DATA PROCESSING/NPSTZ ANOVA: Analysis of Variance – One Factor OUTLINE OF THE LECTURE 1.One-way ANOVA 2. 2.Sources of Variability 3. 3. ANOVA Hypothesis 4. 4. ANOVA Test 5. 5. Statistical Software: ANOVA Tests 6. 6.Correlation Ratio 7. 7. 1. ANOVA: Analysis of Variance – One Factor ANOVA •ANOVA = Analysis of Variance •ANOVA –one of the most frequently used statistical procedures in marketing as well as other areas of data analysis. –this method enables one to assess the potential influence of a qualitative or quantitative variable on another quantitative variable. •Example of ANOVA use: –to evaluate effects of different forms of a promotional campaign on the sales of a product. In this case, different promotional campaigns resent different categories of the observed qualitative variable = promotional campaign. The sales are then the quantitative variable in question. ANOVA: Analysis of Variance – One Factor ANOVA Effects of factors: –potential effect can be expressed mathematically in such a way that the expression analyses whether a change in the level of the qualitative/quantitative variable changes the population mean of the other observed quantitative variable. – In this sense, ANOVA tests if there are any differences among the population means of the quantitative variable. –ANOVA is based on decomposition of what is called the total variability of the observed variable. –Depending on how many main sources or factors appear in the decomposition, we talk about one-way ANOVA, two-way ANOVA and so on. ANOVA: Analysis of Variance – One Factor One – Way ANOVA ANOVA: Analysis of Variance – One Factor One – Way ANOVA •Statistical test •The main principle of the analysis of variance is to decompose the total variability of the observed variable. •The total variability, measured by the sum of squared deviations of the individual values of the variable from their average, is divided by the decomposition into a part that reflects a variability within the samples and a part which reflects a variability between the samples. ANOVA: Analysis of Variance – One Factor Sources of Variability ANOVA: Analysis of Variance – One Factor Sources of Variability ANOVA: Analysis of Variance – One Factor Sources of Variability ANOVA: Analysis of Variance – One Factor Sources of Variability ANOVA: Analysis of Variance – One Factor ANOVA Hypothesis •Analysis of variance is a statistical test. –Therefore, we work with a pair of hypotheses: a null hypothesis and an alternative hypothesis. •ANOVA has its conditions under which it was derived: – The method assumes that each of the k random samples comes from a normal distribution, and the distributions have the same variance. –Also, the samples were drawn independently of each other. In analysis of variance, more than two samples are usually worked with, and ANOVA: Analysis of Variance – One Factor ANOVA Hypothesis ANOVA: Analysis of Variance – One Factor ANOVA – Test Criterion ANOVA: Analysis of Variance – One Factor ANOVA – Critical Value ANOVA: Analysis of Variance – One Factor ANOVA - Results ANOVA: Analysis of Variance – One Factor ANOVA - Results •If the test confirms that the factor X affects Y, we may ask which population means are different. • •It can be the case that only population means are different, while all the other population means are the same. • •There are methods that try to answer this question, one of them being devised by Scheffé and one by Tukey. ANOVA: Analysis of Variance – One Factor ANOVA – Alternative way of calculation of SS •These formulas are more convenient if the variabilities are to be calculated on a calculator: ANOVA: Analysis of Variance – One Factor ANOVA: Typical computer outcome: Typical computer outcome of one-way ANOVA analysis consists of an ANOVA table with these components: Result can be seen from p-value or from comparison of calculated test criterion with provided critical value. ANOVA: Analysis of Variance – One Factor Example •The following table contains data obtained through several independent random samplings. The observed factor is the number of octanes used to describe the quality of car fuel (90, 91, 95, 98 octanes are usually available). Thus, the factor is monitored at four possible levels. For each of the levels, five car drivers using the fuel of the corresponding quality were randomly selected. In this case, all samples have the same size, which is not required for one-way ANOVA. We want to know whether the quality of the fuel affects fuel consumption (car mileage). ANOVA: Analysis of Variance – One Factor Example •Data: ANOVA: Analysis of Variance – One Factor Example - Solution •Calculation via Excel – Data Analysis tool: • • ANOVA: Analysis of Variance – One Factor Example - Solution •In the dialogue window, it is necessary to insert as the Input Range a reference to the area of the Excel spreadsheet that contains the data samples to be worked with in ANOVA: • ANOVA: Analysis of Variance – One Factor Example - Solution •Excel results are given in a form of table: • • • • • • • •Since the test criterion is greater than critical value, we reject the null hypothesis that fuel quality has no effect on car mileage. •In other words, it seems the factor does have an influence on car mileage. ANOVA: Analysis of Variance – One Factor A Measure of Dependence ANOVA: Analysis of Variance – One Factor A Measure of Dependence •Determination ratio P2 can take on any value from interval [0,1]. • •The stronger the dependence of Y on X, the closer the characteristic is to one, and the closer the among-group sum of squares is to the total sum of squares (total variability). Under such condition the within-group variability approaches zero. • •The closer the determination ratio is to zero, the smaller the part of the total variability which is accounted for by the among-group variability. In this case, the dependence of Y on X is weak. INTRODUCTION – Estimation and Hypothesis Testing – Parametric Tests G:\KLIENTI\OVX\2008-06-SLU-DesignManual\2008-10-DM\2008-11-04-Stavba01\final03\export\kolecka.wmf Reading List •GUJARATI, D., 2009 . Essentials of Econometrics – Appendix C •TOŠENOVSKÝ, F., 2014. Statistical Methods for Economists – Chapter 6 INTRODUCTION – Estimation and Hypothesis Testing – Parametric Tests G:\KLIENTI\OVX\2008-06-SLU-DesignManual\2008-10-DM\2008-11-04-Stavba01\final03\export\kolecka.wmf Next Lecture: •Two-Way ANOVA and Latin Squares • • • • THANK YOU