Chapter 14 – 1 Chapter 14: Analysis of Variance Understanding Analysis of Variance The Structure of Hypothesis Testing with ANOVA Decomposition of SST Assessing the Relationship Between Variables SPSS Applications Reading the Research Literature

Chapter 14 – 2 Examples Using two sample t test, we could compare means between two groups, but how about more than two group? The differences on the mean GPA among racial groups. Is there any racial discrimination on annual income?

Chapter 14 – 3 ANOVA Analysis of Variance (ANOVA) - An inferential statistics technique designed to test for significant relationship between two variables in two or more samples. The logic is the same as in t-tests, just extended to independent variables with two or more samples.

Chapter 14 – 4 Understanding Analysis of Variance One-way ANOVA – An analysis of variance procedure using one dependent and one independent variable. ANOVAs examine the variation between samples, as well as the variation within a single sample.

Chapter 14 – 5 The Structure of Hypothesis Testing with ANOVA Assumptions: (1) Independent random samples are used. Our choice of sample members from one population has no effect on the choice of members from subsequent populations. (2) The dependent variable is measured at the interval-ratio level. Some researchers, however, do apply ANOVA to ordinal level measurements.

Chapter 14 – 6 The Structure of Hypothesis Testing with ANOVA Assumptions: (3) The population is normally distributed. Though we generally cannot confirm whether the populations are normal, we must assume that the population is normally distributed in order to continue with the analysis. (4) The population variances are equal.

Chapter 14 – 7 Stating the Research and Null Hypotheses H 1 : At least one mean is different from the others. H 0 : μ 1 = μ 2 = μ 3 = μ 4 =…

Chapter 14 – 8 The logic Why might sample means differ? 1.Group Membership 2.Differences not due to group membership The fundamental technique is a partitioning of the total sum of squares errors into between-group error and within-group error.

Chapter 14 – 9 The logic

Chapter 14 – 10

Chapter 14 – 11 Within-Group Sum of Squares n j = the number of cases in a sample (j represents the number of different groups) = each individual score in a sample = the mean of a group This tells us the variations within our groups; it also tells us the amount of unexplained variance.

Chapter 14 – 12 Between-Group Sum of Squares n j = the number of cases in a sample (j represents the number of different samples) = the mean of a sample = the overall mean This tells us the differences between the groups

Chapter 14 – 13 Decomposition of Variance

Chapter 14 – 14 The Structure of Hypothesis Testing with ANOVA Total Sum of Squares

Chapter 14 – 15 The Structure of Hypothesis Testing with ANOVA Mean Square Between An estimate of the between-group variance obtained by dividing the between-group sum of squares by its degrees of freedom. Mean square between (MS B ) = SS B /df b where df b = degrees of freedom between df b = j – 1 j = number of categories

Chapter 14 – 16 The Structure of Hypothesis Testing with ANOVA Mean Square Within An estimate of the within-group variance obtained by dividing the within-group sum of squares by its degrees of freedom. Mean square within (MS W )= SS W /df w where df w = degrees of freedom within df w = N – j N = total number of cases j = number of categories

Chapter 14 – 17 The F test It is simply the ratio of the two variance estimates: Df B = j - 1 Df W = N - j Df T = N - 1

Chapter 14 – 18 Definitions F ratio (F statistic) – Used in an analysis of variance, the F statistic represents the ratio of between-group variance to within-group variance F obtained – The test statistic computed by the ratio for between-group to within-group variance. F critical – The F score associated with a particular alpha level and degrees of freedom. This F score marks the beginning of the region of rejection for our null hypothesis.

Chapter 14 – 19 df b df w

Chapter 14 – 20 Example: Obtained vs. Critical F Since the obtained F is beyond the critical F value, we reject the Null hypothesis of no difference

Chapter 14 – 21 Example Research Question Does the presence of others influence helping behavior? Hypotheses In SymbolsIn Words H0H0 μ 1 = μ 2 = μ 3 The presence of others does not influence helping. H1H1 Not H 0 The presence of others does influence helping. Or at least one of means does not equal to others.

Chapter 14 – 22 The Data # of seconds it took for folks to help # people present

Chapter 14 – 23

Chapter 14 – 24 Example Assumptions 1) The subjects are sampled randomly. 2) The groups are independent. 3) The population variances are equal. 4) The population distribution of the DV is normal in shape.

Chapter 14 – 25

Chapter 14 – 26 Example Decision Reject H 0 and conclude that the presence of others does influence helping. Or at least one of means does not equal to others. SourceSSdfMSFp Between <.05 Within Total

Chapter 14 – 27 SPSS Example: Bush’s Job Approval

Chapter 14 – 28 SPSS Example: Clinton’s Job Approval

Chapter 14 – 29 Exercise A manager wishes to determine whether the mean times required to complete a certain task differ for the three levels of employee training. He randomly selected 6 employees with each of the three levels of training (Beginner, Intermediate and Advanced).

Chapter 14 – 30 Data iadvmedbeg

Chapter 14 – 31 Reading the Research Literature

Chapter 14 – 32 Reading the Research Literature