1. Two Way ANOVA ©2005 Dr. B. C. Paul

2. ANOVA Application ANOVA allows us to review data and determine whether a particular effect is changing our results ANOVA allows us to review data and determine whether a particular effect is changing our results We tried the Red Rooster Carburetor on different types of cars and found that the type of car used made little difference compared to other things. We tried the Red Rooster Carburetor on different types of cars and found that the type of car used made little difference compared to other things. Sometimes we are interested in more than one possible cause Sometimes we are interested in more than one possible cause Remember too that we are only determining whether an effect is important relative to other things Remember too that we are only determining whether an effect is important relative to other things A big unaccounted for variable can mask everything else A big unaccounted for variable can mask everything else

3. Fuel Economy Improvement of Red Rooster Does not appear to be car dependent How about Driver Dependent How about Driver Dependent We could keep on going and testing one effect at a time We could keep on going and testing one effect at a time That could get pretty long That could get pretty long Remember too that any effect not accounted for is treated as a random variation Remember too that any effect not accounted for is treated as a random variation Random variation goes in the denominator of the F test Random variation goes in the denominator of the F test The larger the “random” variation the harder it is to see a major effect The larger the “random” variation the harder it is to see a major effect One Solution is ANOVA on more than one variable at a time One Solution is ANOVA on more than one variable at a time

4. Our Case Suppose we had our 10 drivers drive each of the first four cars four times, twice without the RR carburetor and twice with. Suppose we had our 10 drivers drive each of the first four cars four times, twice without the RR carburetor and twice with. Suppose we paired the results before and after and calculated % improvement Suppose we paired the results before and after and calculated % improvement The scatter in the data can now be viewed as The scatter in the data can now be viewed as Total Scatter = Difference by type of car + Difference by Driver + Difference due to interactions of Driver and Car + Everything else in the universe that we didn’t account for. Total Scatter = Difference by type of car + Difference by Driver + Difference due to interactions of Driver and Car + Everything else in the universe that we didn’t account for. Statistician’s Language Statistician’s Language SS Total =SS Treatment 1 +SS Treatement 2 +SS Interaction +SS error SS Total =SS Treatment 1 +SS Treatement 2 +SS Interaction +SS error

5. Enter Our Data in SPSS We have our Gas Mileage Improvement Column For each value we enter a number To tell which of the 4 cars was Tested and which of the 10 drivers Was operating the vehicle.

6. Oh- By the Way How Did You Get Those Cute Variable Names Up Top? Click on the Variable View Tab at the bottom.

7. The Result Can Enter Variable Name Can also enter width and Decimal places

8. Oh Yes We Were About to Run a Two Way ANOVA Go to Analyze and Click the Pull down menu Highlight General Linear Model On the pop down menu. A menu Opens to the side Highlight Univariate and Click

9. The Two ANOVA Menu Comes Up Highlight Your Dependent Variable (ie – in this case improvement in gas Mileage) Prepare to click on the Arrow to move it to the Dependent variable box.

10. Identify Your Fixed Factors Highlight Auto Click the arrow to move to Fixed factor. Next do the same with Driver We have now indicated that our gas mileage Improvement is believed to depend on the type Of car and who is driving it.

11. When You Are Done, Click Ok OK

12. Two Way ANOVA Comes Up

13. Looking at My Results We have effects for Auto type, Driver, and Interaction Between the auto and driver

14. As before the computer calculates the square of the values in each cell, calculates degrees of freedom and then divides the sum of squares by the degrees of freedom to get the mean square

15. We Divide our Mean Square for Treatments and interaction by the mean square error. The resulting statistic has a F distribution.

16. Check the Significance of Our Results For Driver our F score is 1155.782 which is over 99.9% Significant – English Translation “There’s not a snowballs Chance in Hell that the gas mileage improvement does not Depend on the driver. We will make sure our advertisement says “Individual Results may vary”

17. Checking Up on Auto Auto effect is 17.3% significant – ie there is almost A 20% chance of producing these numbers when The type of car makes no difference. Probably cannot reject the null hypothesis – we simply cannot Conclude with confidence that the type of car made a difference.

18. What Happened to Auto When we did our one way ANOVA we had no significance When we did our one way ANOVA we had no significance F value was something like 0.11 F value was something like 0.11 Now the number is making us wonder even though we can’t prove anything Now the number is making us wonder even though we can’t prove anything Answer is we pulled other variables out of the “error” category Answer is we pulled other variables out of the “error” category We know that the driver made a huge difference and yet we were calling that random We know that the driver made a huge difference and yet we were calling that random Not accounting for other effects can mask things that might be significant otherwise Not accounting for other effects can mask things that might be significant otherwise

19. Interaction There is a definite car and driver effect, ie the same driver will Achieve different results in different types of cars.

20. Assumptions We Made Our results and random error distributions were all normal Our results and random error distributions were all normal The error variance (unaccounted for errors) were the same for each category or car and driver The error variance (unaccounted for errors) were the same for each category or car and driver