One-way ANOVA, II 9.07 4/22/2004 Your schedule of coming weeks • • Two-way ANOVA, parts I and II. • No class on Tuesday, 5/4 Thursday class: TA’s talk about statistical learning Today: One-way ANOVA, part II Next week: One-way ANOVA HW due Thursday Week of May 4 Teacher out of town all week and other uses for statistics outside of the scope of this course, and do a bit of a review. Two-way ANOVA HW due Thursday (last HW!) Review from last time • p-value P dftotSStotTotal MSwndfwnSSwn FobtMSbndfbnSSbnBetween FMean square dfSum of squares Source Review from last time MS = SS/df Fobt = MSbn / MSwn obt to Fcrit bn, dfwn) ( ) df,)( 2 2 =−=∑ ∑ tot tot tottot N x xSS df,)( 1 222 =−=−= ∑ ∑ =i iiiibn NMMmnSS df,)( 1 1 2 =−=−= ∑∑ = =i in j iijbntotwn mxSSSSSS Create ANOVA table Within Compare F , df = (df1 -N 1 -k condits # m n k -N condits # 1Relationship to proportion of variance accounted for bn / MStot = proportion of variance accounted for by the systematic effect wn / MStot = proportion of variance not accounted for • obt is: forproportion=obtF •MS•MSSo, another way of looking at Ffor accounted not variance of proportion accounted variance of • • significant effect of x-values on y-values Relationship to correlation analysis Similarly, in ANOVA we are not asking whether there is a difference between a particular pair of conditions (“x-values”) Instead, we are testing whether there is a Relationship to correlation analysis• In correlation analysis, we weren’t interested in whether there was a difference in y-values between a particular pair of x-values0123456789012345Glasses of orange juice per dayDoctor's visits per yearRelationship to correlation analysis• Instead, we were interested in whether there was a significant relationship between the x-values and y-values0123456789012345Glasses of orange juice per dayDoctor's visits per year• there may not be a natural ordering to the conditions (x-values) • relationship is linear (as in correlation • ANOVA: 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 Relationship to correlation analysis The main difference between correlation analysis and ANOVA is that in ANOVA So, one shouldn’t expect that the analysis), since it’s not clear what this would mean ANOVA vs. correlation analysis Condition # Response Relationship between ANOVA and • pair of x-values, we look for an overall effect of the x-values on the y-values by its y-value The tests for ANOVA and • • ANOVA: • forproportion=obtF )1( )2( 2 22 r Nrtobt − − = not accounted for correlation analysis In both cases, since we’re not interested in the difference in mean between a particular looking at how much knowing the x-value of a data point reduces your uncertainty in correlation analysis are very similar In fact, some people treat correlation analysis as a form of ANOVA Correlation: for accounted not variance of proportion accounted variance of Proportion of variance accounted for Proportion of variance 3Post-hoc comparisons So, suppose you get a significant • • – A>B>C • post-hoc comparison to results from a one-way ANOVA If there are only two conditions, you know that there’s a significant difference in the means of those two conditions If there are more than two conditions, you just know that there’s at least one significant difference between the means for each condition A=B>C A>B=C B=A>C=B We need to do a determine which means differ significantly Post-hoc comparisons • • (Post-hoc comparison factoids • obt • – Type I error – – than planned comparisons, i.e. tests that were plannedWe’ll talk about these next. There are a whole bunch of these Your handout covers two of them – Tukey’s HSD (very popular) – Fisher’s protected t-test also called Fisher’s LSD) Require significant FMain idea behind these tests: Before we talked about how multiple comparisons are problematic because they inflate the experiment-wise These tests are designed to adjust Type I error by considering all possible pairwise comparisons As a result, post-hoc comparisons are less powerful before the data was collected. 4• H Significant Difference • Requires that the ni• α, wni • Tukey’s HSD Tukey’s onestly ’s in all levels of the factor (all conditions) are equal The basic idea in Tukey’s HSD Find out how big the difference between two means needs to be in order for the difference to be significant – This difference depends upon your desired the amount of “noise”, MS , the number of means to compare, and n– Critical difference = HSD Compare all differences in mean to HSD, to determine which are significant n MS qHSD wnαα= ααq statistic based on alpha, dfWk, the number of groups. wnMS Is the mean square within groups. n • difficulty affect performance on a math test? m3=3m2=6m1=8 n3=5n2=5n1=5 Σx2=55Σx2=220Σx2=354 Σx=15Σx=30Σx=40 2107 528 484 3612 149 Level 3:Level 2: medium Level 1: easy Tukey’s HSD is the Type I error rate (.05). Is a value from a table of the studentized range , and Is the number of people in each group. Back to our example from last time Does perceived difficult 5p<0.05 P 14147.33Total 7.001284.00 4.5231.67263.33Between FMean square dfSum of squares Source 1. Get qα statistic table • 5.404.964.260.01 4.153.733.060.05 13 5.505.044.320.01 4.203.773.080.05 12 5.625.144.390.01 4.263.823.110.05 11 432α k=# means being compareddfwn ANOVA table for this example Within from the studentized range Table looks like this: In our case, k=3, dfwn=12, and take α=0.05 • a = 3.77 5.404.964.260.01 4.153.733.060.05 13 5.505.044.320.01 4.203.773.080.05 12 5.625.144.390.01 4.263.823.110.05 11 432α k=# means being compareddfwn 2. Compute the HSD • a= 3.77, MSwn = 7, and n = 5, so n MS qHSD wnαα=So, it looks like qFrom the preceding slides, qHSD = 3.77 sqrt(7/5) = 4.46 6to the HSD • α – – m3=3m2=6m1=8 Level 3:Level 2: medium Level 1: easy 2.0 3.0 5.0 Results of the ANOVA and Tukey’s HSD • instruction on performance on the math test (F(2, 12) = 4.52, p<0.05). Post-hoc would be difficult (M=3), p<0.05. Aside: a message from your TA’s • not • • Fisher’s LSD (protected t-test) L S Difference not i• • 3. Compare each difference in mean Any differences > HSD are significant at the level These are absolute differences – ignore the sign HSD = 4.46, so the difference between the easy and difficult condition is significant, but neither of