ANOVAAnalysis of Variance

A Short Introduction byBrad Morantz

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?

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

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

Overview & Purpose

• Null Hypothesis H0 is that all means are equal (population means as estimated by sample means)

μ1 = μ2 = . . . . = μn

• 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

Components

• SSE is sum of square error• SST is total sum of squaresSST = SSTreatment + SSError

• MST = SST/(k-1)• MSE = SSE/(N-k)• F = MST/MSEThe test criteria to reject or fail to reject null hypothesis

• k = number of treatments• N = number of observations

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?

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

Manova

• Multivariate Analysis of Variance– When there are two or more dependent

variables

• Need specialized high power (read expensive) software

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

Example

Day Night Fog Anova: Single Factor8 7 66 4 3 SUMMARY4 5 4 Groups Count Sum Average Variance7 6 5 Day 10 72 7.2 2.49 5 4 Night 10 60 6 2.2222228 6 5 Fog 10 50 5 2.2222227 4 38 8 79 7 6 ANOVA6 8 7 Source of Variation SS df MS F P-value F crit

Between Groups 24.26667 2 12.13333 5.318182 0.01129 3.354131Within Groups 61.6 27 2.281481

Total 85.86667 29

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

References

• Applied Linear Statistical Models, Neter, Kutner, Nachstein, & Wasserman

• Multivariate Datta Analysis, Hair, Anderson, Tatham, & Grablowsky

• Most statistics books