Psychology 210 Statistical Methods Spring, 2006 Statistics Lab # 8 As always, reboot your computer and launch SPSS. This week, we’ll be tackling the famous and/or infamous ANalysis Of VAriance, or ANOVA. ANOVA is probably the most widely used statistical method in all of psychology, primarily because of its flexibility. As a psychologist, you’ll come to know, appreciate, and yes, even love all of its wonderful intricacies. To start, let’s revisit the data on singers from the confidence intervals lab (on the web page as “singers.sav”) http://people.whitman.edu/~herbrawt/classes/210/psych210.html The essence of an ANOVA is the F-ratio, which can be defined as the variability between groups, relative to the variability within groups. If this sounds familiar, that’s because it’s conceptually identical to the t-ratio. If you ponder for a moment what the numerator and denominator of the t-ratio really mean, it should make sense. iancegroupswithiniancegroupsbetweenFvarvar−−= XSXtµ−= Let’s jump right in and start by running a quick two-group ANOVA. Select Analyze Æ Compare Means Æ One-Way ANOVA. The “One-Way” means simply that we’ve got one independent (grouping) variable. Later, we’ll consider what to do with more than one independent variable. All that’s left it to specify a pair of variables to analyze… say, Sex (male or female) and Height. Send them to the appropriate boxes in the ANOVA window, as shown below. The only real constraint is that the Dependent variable should be an interval or ratio variable, and the Factor should be discrete. This should allow us to see if male and female singers are on the average, different heights. Click OK before the anticipation overwhelms you.The output will be a simple looking table. The important values to consider are those listed under F and sig. These are the F-ratio and significance level - the probability that any observed difference might be due to random chance, rather than a systematic effect. You can interpret them just like you would interpret a t-test. If the probability is less than .05, we can reject the null hypothesis that the means are really the same. If you’re wondering what the big deal is because this doesn’t tell us anything we didn’t already know how to do with a t-test, give yourself 1,000 imaginary points and a congratulatory handshake. What then, is the advantage to running an ANOVA, if the results are identical to a t-test? To answer this, consider what would happen if one wanted to compare several means: You might decide to run multiple t-tests (i.e., 3 comparisons for 3 means, 6 for 4 means, and so on). This has two unpleasant consequences. 1) Running t-tests over and over is not particularly fun nor is it time-efficient. 2) The probability of making a type-1 error increases with each statistical test. ANOVA avoids both of these problems by incorporating all of the comparisons into a single statistical test (an “omnibus test”, to throw around a little unwieldy jargon). In comparing two means, like we did above, there is no meaningful difference between this and a t-test. However, if we wished to compare more than two groups, we would be wise to use an ANOVA. Let’s do this by comparing the height of people who sing the various voice parts. Again, select Analyze Æ Compare Means Æ One-Way ANOVA. This time, select voice as your factor, and recall that it divides the sample into 4 groups (soprano, alto, tenor, bass). Before continuing, click on the Post-Hoc button, and select the Tukey option (shown to the left). Now click Continue and OK to see the results.The first table looks the same as it did before. However, we’ll need to interpret its output a little bit differently this time. The probability labeled sig is actually the probability that there is one or more significant differences among the means included in the test. For our example, this would be the probability that at least one of the 6 comparisons was statistically significant. Unfortunately, we still don’t know which, and the basic ANOVA doesn’t tell us (though in the two-group example above it should be logically obvious). To determine which groups are indeed different, we would run a post-hoc, or a posteriori test like Tukey’s HSD (which we already asked for). The results of this analysis are displayed on the table marked Multiple Comparisons, below the ANOVA table. This table (also reproduced on the next page) shows comparisons for every pair of means, along with the difference, standard error, and significance level. Those comparisons with a significance level of less than .05 are statistically reliable and are marked with an asterisk for convenience. It is worth noting that this alpha level is adjusted to account for the added number of comparisons; it is lowered so that the total probability of a type 1 error, across all tests is .05 (as opposed to the probability of a type 1 error for each test being .05). Notice that if you did not get an F-ratio of less than .05, there’s no reason to run a post-hoc analysis, since you already know that there will be no reliable differences between any of the groups. As you can see, we have several significant differences. Take a moment to look at them until you see the pattern. It boils down to this: Sopranos and altos are different from tenors and basses. The pattern is more readily seen in the next table (marked homogeneous subsets), which separates groups into subsets that highlight these statistical differencesBased on the statistical differences above, it shows us that all voices in subset 1 are different from those in subset 2. In this case, it makes the pattern a bit more obvious, but be careful - It is possible to have significant differences within a subset. For example, imagine a 5th voice part that is significantly different from sopranos, altos and tenors, but is not significantly different from basses. It would be included in subset 2, since it’s different from each and every voice in subset 1, but not every voice in subset 2. The take-home message is to remember to consider each individual comparison in the larger table.Summary for future reference: One-Way ANOVA: Select Analyze Æ Compare Means Æ One-Way ANOVA Tukey’s HSD post-hoc option: Select Post-Hoc, then Tukey Lab Assignment: As Bob Dylan once sang, “Cuckoo is a pretty bird”. Behavioral ethologists