One-way ANOVA: round, narrow, wide

Transcription

1 5/4/2009 9:19:18 AM Retrieving project from file: 'C:\DOCUMENTS AND SETTINGS\BOB S\DESKTOP\RJS\COURSES\MTAB\FIRSTBASE.MPJ' ========================================================================== This is an analysis of data on running times by three methods of rounding first base in baseball. See Hollander and Wolfe, 1973 (or later edition). The methods are round out, narrow angle, and wide angle. Each of 22 different runners performed roundings of first base using each of the three methods, and the running times were recorded. Here is the data as entered into Minitab. The 2 nd column gives the times, with the 1 st column giving the runner for that time and the 3 rd column giving which method was used for that time. Columns 4, 5, and 6 give this same information in unstacked form for each of the 22 runners. runner times subs round narrow wide round round round round round round round round round round round round round round round round round round round round round round narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow narrow wide wide

2 wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide wide The null hypothesis is that the average running times for each method are the same. As we discussed in class, this design calls for a two-way ANOVA, thus eliminating the effect of the factor runner. However, for purposes of later comparison, let us see what one-way ANOVA gives when we treat this data as three independent samples of size 22 corresponding to the three methods. That is, ignoring the information that each runner involves has run each of the three methods. MTB > AOVOneway 'round' 'narrow' 'wide'. One-way ANOVA: round, narrow, wide Source DF SS MS F P Factor Error Total S = R-Sq = 2.03% R-Sq(adj) = 0.00% Level N Mean StDev round ( * ) narrow ( * ) wide ( * ) = We see that the null hypothesis is accepted with p-value of The three associated confidence intervals for the true means heavily overlap.

3 Next we carry out a nonparametric rank-based version of one-way ANOVA. MTB > Kruskal-Wallis 'times' 'subs'. Kruskal-Wallis Test: times versus subs Kruskal-Wallis Test on times subs N Median Ave Rank Z narrow round wide Overall H = 1.84 DF = 2 P = H = 1.85 DF = 2 P = (adjusted for ties) Again, the null hypothesis is accepted. Now let us carry out a two-way ANOVA. MTB > Twoway 'times' 'runner' 'subs'; SUBC> Means 'runner' 'subs'. Two-way ANOVA: times versus runner, subs Source DF SS MS F P runner subs Error Total S = R-Sq = 93.23% R-Sq(adj) = 89.53% There are now two null hypotheses: (1) that runners all have the same average running time, and (2) that the three methods all have the same running time. The above analysis shows that both hypotheses are rejected with very low p- values.

4 Below we see the individual CIs for the average running times for the 22 runners. Note that some pairs of intervals are nonoverlapping. This supports the conclusion that the average running times are not the same for all the runners. Runner 14 is fastest on average, runner 22 is slowest. runner Mean (-*--) (-*--) (--*-) (-*--) (-*--) (-*--) (--*-) (--*-) (-*--) (--*-) (--*-) (--*-) (--*-) (--*--) (--*-) (-*--) (-*--) (-*--) (--*--) (--*-) (--*-) (-*--) Next we look at the individual CIs for the average running times for the three methods. We see that some pairs of intervals are nonoverlapping, indicating that the mean running times are not the same for the three methods. Based on means, the wide angle method outperforms the other two methods, which are very close to each other, and this finding has statistical significance, although it is based on a normality assumption. subs Mean narrow ( * ) round ( * ) wide ( * )

5 Finally, we perform a nonparametric rank-based version of two-way ANOVA. MTB > Friedman 'times' 'subs' 'runner'. Friedman Test: times versus subs blocked by runner S = DF = 2 P = S = DF = 2 P = (adjusted for ties) Sum of subs N Est Median Ranks narrow round wide Grand median = Again, both null hypotheses are rejected. We see again that the wide angle method outperforms the other two methods, this time based on medians, and that this finding has statistical significance, this time without requiring a normality assumption. HAPPY RUNNING!