Unequal Variance and ANOVA ©2005 Dr. B. C. Paul

ANOVA Assumptions ANOVA assumes the populations sampled in each class are normally distributed Also assumes that the variance of those distributions is the same  How can we check the homogeity of variance assumption?

Our Story Quincy’s company has been outsourcing American jobs and wishes to know if there are differences between where his factories are located, what shift the workers are on, and the number of rejected widgets.  Quincy gathers data on the rejection rate at his three factories using 20 weekly rates each taken at random from the record.

Quincy’s data set Quincy’s favorite plant is in Yucatan Mexico (where he is forced to spend much of his time on visits at the beach) – he labels this plant #1  His day shift he calls 1  His night shift he calls 2 Quincy’s second plant is in Peoria, Illinois. The plant has sentimental value to him since he used to go there with his grandfather, but the workers there are uppity and want things like fair wages and safe working conditions – he labels this plant #2 Quincy’s third plant is in Malaysia. Quincy does not wish to discuss what he does during visits to this plant.

Quincy Enters His Data Set Rejection rate/ 1000 Widgets. Day or Night Shift Which Plant

Quincy Scratches His Head What kind of test should he do? There are too many plants and shifts here to try to do flocks of T tests. Then he remembers ANOVA  His variables plant shift and plant are really categories The numerical values are arbitrary and not even ordered  Only his # of rejects is ordered  With two classes to divide by – shift and plant – Quincy decides to do a two way ANOVA

Reaching for his trusty SPSS program Quincy Begins Quincy clicks Analyze to pull down The menu. He highlights general linear model To bring up the pop-out menu He then highlights and clicks Multivariate.

Quincy Picks # of rejects for his variable and shift and plant as his fixed factors.

This Time Quincy Looks at Options Quincy wants to see his means Displayed But most important he wants To use Lavene’s test to check That the variance is the same For all the plants and shifts.

Quincy clicks continue and OK and the program is off to the races It starts with the Kill Joy special of the Day. Levene’s test says there is no Way the variance is the same for all The categories.

Then it gives a result it has already called into question. The most powerful effect Was which plant (Oh – Nuts outsourcing does Seem to change reject rate) Shift was the next most Important. But the response to shift work Also varies by where the Plant is located.

Importance of the controlling variables About 71% of the variation Can be traced to which plant And which shift is working.

Shift Effects The 95% confidence intervals for The mean of Day Shift and Night Shift do not come close to Overlapping. The values suggest night shift Messes up more than day shift.

This Really Sucks For Quincy’s foreign plant the 95% confidence intervals overlap So Quincy may not be able to be Certain the foreign plants are Different – However – his U.S. plant has a Distinctly lower rejection rate from The foreign plants. Quincy’s outsourcing has impacted Rejection rate.

Moving on to Plant and Shift Interactions. The U.S. plant on either day shift Or night shift has much lower Rejection than any of the foreign plants We also cannot be sure that there is an effect of Shift in the U.S. operations.

Continued Inspection On night shift the screw-up rate is similar for Malaysia and Mexico On Day shift, however, Malaysia shows better Performance than Mexico

Usefulness of Means Displays Can see that the plots of individual classes with confidence intervals can give you an at a glimpse view of which differences specifically are driving the test results Bad news is those confidence intervals used a common standard error which Levene’s test proved was wrong.

What Happens When Assumptions Fail? We have a very clear cut result from our data, but unfortunately its legitimacy has been called into question The more good information we have the closer we can peg our answers  We saw that adding samples improved the certainty of our conclusions  We saw with the Brehens-Fisher T test that when we lost homogeity of variance we were less sure of our values than before  We know we have lost something here also General models and methods don’t help us know exactly how much and don’t tell us what to do about it. We have been warned of an error and we know the direction that the error will push our answers.

So Are the U.S. Plants Better Or Not? With the high certainty levels here and fairly good sample size we probably can still be sure about our U.S. plants  If things were close we simply would not know for sure with the analysis and methods we have so far.

What about the homogeneous error variance in regression? In many regression analysis the largest source for poorly distributed error variance is lack of fit of the model  We already know how to bring in other variables and look for non-linear effects If variance still changes it is often that larger numbers have greater variance.  Some sort of data normalization can help this