Psychology 210 Statistics Lab #9: Two-Way Analysis of Variance Today we’re going to do a couple of factorial analyses of variance. The goal is mostly to get you familiar with the idiosyncrasies of ANOVA output from SPSS. In addition, you’ll get some valuable practice interpreting interactions. First, launch SPSS and open the data file called filters.sav located at the class web page: http://people.whitman.edu/~herbrawt/classes/210/psych210.html Last week’s Analyze Æ Compare Means Æ Oneway command is a simple way to run analysis of variance, but as the name indicates, is only relevant when you have a single independent variable. ANOVA with more than one independent variable (2-way, 3-way or 56-way) makes use of a different command, Analyze Æ General Linear Model Æ Univariate. Though it might initially sound weird to call a factorial ANOVA “univariate”, the term refers to the use of a single dependent variable. You can select as many independent variables as you like. The following window should appear:The data we’ll be playing with today are from a statement by Texaco, Inc. to the Air and water Pollution Subcommittee of the Senate Public Works Committee on June 26, 1973. Mr. John McKinley, President of Texaco, cited the Octel filter, developed by Associated Octel Company as an effective reducer of air pollution. The committee had raised questions about the effects of pollution filters on vehicle performance, fuel consumption, exhaust gas back pressure, and silencing. A young congressional aide in particular (let’s call him R. Nader. No, that’s too obvious - Ralph N is better) is concerned with noise pollution and wants to make sure that the Octel filter does not produce higher noise levels than the standard filter. The data presented contain the necessary data to answer his questions: noise output (in decibels) measured in cars with and without the Octel filter. Also in the file are 2 other variables of interest: size of the car (small, medium or large) and where the measurement was taken from (left side or right side) For our first example, I’ve selected noise output as the dependent variable, and both filter type and side of car as independent variables. Thus, the output should provide us with information about 4 groups, in the form of two main effects and an interaction. Here’s what the output will probably look like: The relevant lines for our purposes are those labeled type, side, type*side, and Error. Ignore those lines labeled Corrected Model, Intercept, Total, and Corrected Total. The former two are not quite relevant yet (perhaps later in the semester) and the latter two, while they have familiar names, can be misleading. Fortunately, this is inconsequential, since the 3 relevant F-ratios are all provided in the correct format.Personally, I like to clean things up by creating my own ANOVA Table, like this: Source SS df MS F p TYPE 1056.250 1 1056.250 1.174 n.s. SIDE .694 1 .694 .001 n.s. T x S 17.361 1 17.361 .019 n.s. Within 28800.000 32 900.000 Total 29874.305 35 ***Note: Calculate the bottom factor (Total) by summing the other components ***Note2: SSError on an SPSS output screen is the same thing that we’ve been calling SSWithin in class. ***Note3: if you like to check your homework computations on SPSS, don’t be surprised if SStotal doesn’t match your calculation. The line labeled SStotal includes the other components alluded to above (corrected model and intercept). SScorrected total doesn’t, but does apply a correction procedure if there are unequal numbers in each group. The take-home message is, if you have the same number of scores in each group, look at SScorrected total. If not, simply add the rest of the variance components to get the familiar SStotal. Maddening, eh? Now, let’s interpret the statistics!... In this case, we have 3 nonsignificant F ratios. Not very exciting as an exercise, but that’s the way a lot of analyses turn out. If I were writing about the results, I might say something like the following: “There was no reliable difference in the amount of noise produced by the two types of pollution filters, F(1,32) = 1.174, p > .05. Furthermore, the distribution of noise within a car seems to be uniform, F(1,32) = .001, p > .05. And finally, there was no interaction between filter type and location within the car, indicating that the two filter types did not produce different noise distributions, F (1,32) = .019, p > .05.”Ok, now that that’s all straight, let’s delve a little deeper. Adding more factors to an ANOVA often results in a greater likelihood of significant results. This is because providing more information allows you to explain more of the variance – and makes it easier to predict the dependent variable. Example: Imagine you were trying to predict the weight of junior high students. If I told you a student’s height, you’d be able to predict that student’s weight to a particular level of accuracy. Now, if I told you their height and their sex, your predictions would be even more accurate. And finally, if I told you their height, sex and what grade they’re in, you’d be even more accurate. Each time I added a factor, accuracy of prediction is increased. The same logic applies to ANOVA. Let’s test this by adding in that third factor and venturing into the world of the 3-way ANOVA! Still looking at noise output, let’now include all 3 factors: filter type, side, and size. Select Analyze Æ General Linear Model Æ Univariate and select the appropriate variables. Since car size has more than 2 levels, I might want to run a post-hoc analysis like we did last week. I can do so by clicking (you guessed it) Post-Hoc, and selecting the Tukey option, as shown below (you need to send the size variable over before it will allow you to select Tukey. I could select filter type and side as well, but since that variable has only two values, the option tells me nothing I didn’t already know from the F test. Click Continue to lock in this option and move on.If you’re like me, you prefer to have a visual summary of complex things like ANOVA. Click on plots to get a nice plot of the results. I put size on the horizontal axis, with type represented as separate lines,