1. ANOVA Analysis of Variance A Short Introduction by Brad Morantz

2. Example Have data for miss distances for 3 types of weather: –Clear and sunny –Rain –Fog The question: –Does the weather have effect on miss distances? –Are the population means for each condition equal (within allowable tolerance)? In statistics talk, are all means equal?

3. The Problem There is variability in the system. –Each time a missile is fired –Many variables: wind, brightness of sun, countermeasures, precipitation, much more Expect to get different values each time How can we tell if certain factors actually are causing a difference? –Each repetition is different –How do we know when some variance is too much –How do we know if a certain factor is having an affect

4. The Solution ANOVA = ANalysis Of VAriance This is for a single dependent variable Can also be ‘blocked’ to control other things, called noise reducing –For example, to group flights by distance or over time –Need more data/observations to do this Must be of comparable variance Can also be used for two factor test –e.g velocity and weather

5. Overview & Purpose Null Hypothesis H 0 is that all means are equal (population means as estimated by sample means) μ 1 = μ 2 =.... = μ n If we reject the null, it signifies that we could not prove that all are equal within allotted variability in system Does NOT mean that all are different Use another test (Tukey’s HSD) to see which one(s) is/are different

6. Components SSE is sum of square error SST is total sum of squares SST = SSTreatment + SSError MST = SST/(k-1) MSE = SSE/(N-k) F = MST/MSE The test criteria to reject or fail to reject null hypothesis k = number of treatments N = number of observations

7. Interpretation Program will usually give critical value –Depending on specified allowed tail If F value is more than critical value –Then reject null hypothesis If F value is less than critical value –Then Fail to reject the null hypothesis Check to make sure that variances are approximately equal/close Look at graph of data –is it approximately bell shaped?

8. Blocked ANOVA Variance and noise reducing technique Use when there are more than one factor –Ex. Day of the week has affect –Ex. Type of launch aircraft Would still allow to see if weather had affect Requires more observations

9. Manova Multivariate Analysis of Variance –When there are two or more dependent variables Need specialized high power (read expensive) software

10. Limitations Assumes that all of the data is approximately normally distributed Assumes that all of the data has about the same variance Is only concerned with the estimates of the population means that were calculated from the samples

11. Example F critical given as 3.35 and the calculated value is 5.32 so we reject the null hypothesis that all means are equal. Note that the variances are close The P value is the probability of obtaining a result at least as extreme as a given data point, under the null hypothesis. Note that the P value is.011 which indicates that if we had chosen an alpha of.01, the null would not be rejected. These value are made up values

12. References Applied Linear Statistical Models, Neter, Kutner, Nachstein, & Wasserman Multivariate Datta Analysis, Hair, Anderson, Tatham, & Grablowsky Most statistics books