One way Analysis of Variance (ANOVA)

Transcription

1 One way Analysis of Variance (ANOVA) Esra Akdeniz March 22nd, 2016

2 Introduction Test hypothesis concerning one population mean. Test hypothesis concerning two population means What if we want to compare more than two population means?

3 Analysis of Variance In this lecture we will learn how to test the equality of 3 or more population means. Assume we have k populations. Then H 0 and H A are defined as... Assumptions: The populations are normally distributed with mean µ i, i = 1,..., k. The variances are equal, that is σ 2 1 = σ2 2 =... = σ2 k.

4 One Way? Variance? One way indicates that there is a single factor or characteristic that distinguishes the various populations from each other.

5 One Way? Variance? One way indicates that there is a single factor or characteristic that distinguishes the various populations from each other. Analysis of variance we are studying the spread/variation/dispersion within a data set. Two measures of variability can be measured: the variation within a group: how different individual values are from their population means. the variation between the groups: how different each population mean is from the overall mean.

6 One Way? Variance? One way indicates that there is a single factor or characteristic that distinguishes the various populations from each other. Analysis of variance we are studying the spread/variation/dispersion within a data set. Two measures of variability can be measured: the variation within a group: how different individual values are from their population means. the variation between the groups: how different each population mean is from the overall mean. IF the variability within the k different populations is small relative to the variability among their respective means, this suggests that the population means are in fact different.

7 Within-group variability: s 2 W = MSE = (n1 1)2 s (n 2 1) 2 s (n k 1) 2 s 2 k n k Between-group variability: sb 2 = MSTr = n1( x1 x)2 + n 2( x 2 x) n k ( x k x) 2, k 1 where x is the overall mean =...

8 F Test The test statistic in a single factor (one way) ANOVA is F = MSTr MSE This follows an F-distribution with degrees of freedom (k 1), (n k).

9 F distribution It has 2 degrees of freedom: one for the numerator, one for the denominator. F distribution cannot get negative values. It is skewed to the right.

10 Example The accompanying data resulted from a study investigating the effects of carbon monoxide exposure on individuals with coronary heart disease, the FEV distributions of patients associated with each of the three medical centers make up distinct normal populations. Assume the variances from these populations are equal. Center Forced expiratory volume Johns Hopkins Rancho Los Amigos St. Louis Test the hypothesis that the population means for three mixtures are equal.

11 Solution

12 Example A study was conducted to follow three groups of overweight males for a period of one year. The first group decreased their energy intake by dieting but did not participate in an exercise program. The second group exercised regularly but did not alter their eating habits. The third group changed neither diet nor their level of physical activity. At the end of a year, total change in body weight was measured for each individual. Assuming all these populations are coming from a normal distribution and the variances are equal, among these three populations, is there any evidence of a difference in mean change in body weight? Group n x s kg. 3.7kg kg. 3.9kg kg. 3.7kg.

13 Multiple Comparison Procedures Post hoc Tests What if we reject H 0 in the ANOVA procedure? We can be more specific about the differences among population means. Bonferroni correction. Tukey s test, also known as the Tukey range test, Tukey method, Tukey s honest significance test, Tukey s HSD (honest significant difference) test, or the Tukey Kramer method.

14 Nonparametric Test: Kruskal Wallis Use Kruskal Wallis test if you don t have normally distributed populations.

15 Parametric versus Nonparametric Advantages of Nonparametric Procedures: Do not require normality. Computationally faster. Can also be used for ordinal data. Less sensitive to extreme values. Disadvantages of Nonparametric Procedures: If the assumptions of parametric tests are satisfied, nonparametric tests are less powerful. Don t use every information about the distribution, since we are using ranks rather than the actual observations.