independent variable. Again we tested this assumption by running the Shapiro- Wilk test on each of the group's dierence scores, revealing that the assumption of normality need not be rejected (p > 0.5, see Table 12 in Appendix A). The one-way ANOVA further requires that the assumption of homogeneity of vari- ances holds between the independent groups. This assumption was not rejected as Levene's Test of Equality of Error Variance (p = .093) failed to reach signif- icance.
5.2.3 Research Question 1 core results
With preliminary checks completed the one-way between-groups ANCOVA was conducted, results are shown in Table 13 in Appendix A. After adjusting for pre-test Response Times there was no signicant dierence found between the three intervention groups on post-test RT, F (2,36) = .695, p = 0.506. The Covariate, pre-test RT, was signicantly related to the participants' post-test RTs, F (1,36) = 21.5, p < .001.
The one-way ANOVA on dierence scores' results are shown in Table 14, Appendix A. No statistically signicant dierences were found between groups on their dierence scores between pre- and post-test Response Times (p = .066).
5.3 Research Question 2 - Does over-rotation of Tetris
Zoid Type Kirsh and Maglio Standard Tetris Expected Avg.
L-Shape 1.80 1.86 1.50
T-Shape 1.70 1.68 1.50
Z-Shape 0.70 0.70 0.50
Line 0.58 0.54 0.50
Square 0.02 0.01 0.00
Table 7: Approximate average rotations per zoid type in Kirsh and Maglio (1994) compared with average rotations per zoid type for the Standard tetris group.
Following Kirsh and Maglio's demonstration of over-rotation, we examined our Standard Tetris group's in-game data in a similar manner. Table 7 shows the actual average number of 90◦ rotations during our subjects' training period. It is clear that our Standard Tetris group shows the same pattern of over-rotation as Kirsh and Maglio's subjects, that is, according to their criteria our subjects seem to be over-rotating their zoids.
The second aim of our study was to investigate whether the Standard ex- perimental group's average number of rotations changed signicantly over the course of their training. This analysis was accomplished through the use of a se- ries of paired sample t-tests comparing the average number of rotations per zoid type for the Standard Tetris group's rst ten games with the average number of rotations per zoid type for their last ten games.
5.3.1 Data preparation and testing Parametric assumptions
As has been mentioned, the data for the present analysis were drawn from the rst and last ten games of Tetris played by subjects allocated to the Standard Tetris experimental group. For every episode the total number of successful rotations were calculated. These were then used to calculate each subjects' average number of rotations per zoid type at the beginning (rst 10 games) and end (last 10 games) of their training period, yielding the dataset shown in table 8.
Taking our data from the rst and last 10 games was justied by the fact that, although the total number of games our Standard Tetris subjects played varied considerably (average = 59.77, SD = 10.88), each of them played at least 10 games in both their rst and last training sessions.23 Limiting our analysis
23The Modied Tetris group, on the other hand, played an average of 61.92 games (SD = 29.61). It is important to note that subjects in both groups were, on average, exposed to
to these 20 games at the extremes of the training period ensured that we only analysed data drawn from the rst and last training sessions.
Line-Shape T-Shape Z-Shape L-Shape Square-Shape
Subj.No First Last First Last First Last First Last First Last
21 0.6181 0.5536 1.4898 1.5404 0.6578 0.5691 1.9323 1.9171 0.0078 0.0162 22 0.5124 0.5319 1.9912 2.7108 0.7679 0.5414 2.2333 2.4677 0.0081 0.0000 24 0.4792 0.6308 1.12 1.5680 0.5810 0.7793 1.5359 1.9245 0.0085 0.0072 28 0.5385 0.5625 1.8583 1.2222 0.6107 0.3731 1.8667 1.4848 0.0000 0.0000 29 0.4712 0.5462 1.9846 2.0398 0.7506 0.7224 1.9052 1.9594 0.0000 0.0061 31 0.4714 0.4649 1.8736 1.3760 1.5221 0.8807 2.0313 1.6471 0.0556 0.0101 34 0.6429 0.5294 1.6282 1.6963 0.6708 0.6216 1.7114 1.8460 0.0460 0.0466 39 0.5395 0.6529 1.7023 2.0516 0.9280 0.7394 1.8373 2.0393 0.0395 0.0091 43 0.4091 0.4405 1.2459 1.3452 0.7607 0.6627 1.1008 1.5385 0.0364 0.0682 46 0.3134 0.4235 0.5217 0.9552 0.5635 0.6310 1.0432 1.1086 0.0179 0.0125 47 0.5729 0.4444 1.3239 1.2813 0.7891 0.7027 2.0671 1.7059 0.0250 0.0000 90 0.5574 0.5101 1.4400 1.3964 0.5423 0.6424 1.2252 1.7332 0.0152 0.0000 92 0.5189 0.5362 1.7636 1.5068 0.8889 0.8480 2.2000 2.0448 0.0108 0.0000
Table 8: Average number of rotations per zoid type for the Standard tetris group's rst and last 10 tetris games.
Although we do not include the Square-shape zoid data in the following anal- ysis there is not nearly enough data for any serious analysis it is interesting to note that at least some of the time subjects attempted to rotate these zoids even though it has no practical eect in the game.
Paired sample t-tests require that three assumptions about the data hold, rstly, that the data is normally distributed, secondly, that the dierences be- tween the two scores obtained for each subject be normally distributed, and, nally, that variances are equal.
Table 16 in Appendix A shows the results of our tests assessing the assump- tion that our samples, and dierences between scores, are normally distributed.
Note that the assumption of normality is violated for the variable FIRST_10_Z - that is, it is violated for the beginning average rotations for the Z-shaped zoid data. Examining the data in Table 8 reveals that subject 31's average number of rotations for Z-shape zoids is almost three standard deviations greater than the mean.24 It may be possible to motivate excluding this data-point from our
more than the estimated 38 games (footnote 11 above) of Tetris required to approximate the number of MR trials within which individuals have been shown to reach asymptotic levels of MR performance (Kail and Park, 1990).
24More importantly, given that our analysis is conducted using paired samples t-tests, the
analysis for a number of reasons. We might, for instance, use the fact that this subject does not seem to be an outlier with regards to average rotation on other zoid types to argue that this data-point should be excluded. However, given that we are interested in over-rotation it is important for us not to sim- ply exclude any case of over-rotation from our dataset. We therefore present a number of dierent analyses for the Z-shape zoid's beginning and ending av- erage rotation data. Firstly, for completeness, we present but do not discuss a paired sample t-test on the non-normal dataset including subject 31's beginning score. Secondly, we created a new variable FIRST_10_Z_SANS_OUTLIER that, as the name suggests, excludes subject 31's data (see Table 10). This was then used on a separate paired sample t-test comparing beginning and ending Z-shape average rotations (see Table 10). Finally, we ran a non-parametric al- ternative to the paired samples t-test, namely the Wilcoxon Signed-rank test (see Note 25). The assumption of equality of variances was tested using a series of Levene's tests, none of which reached signicance (all p > .05).
Subj. No. Beginning Average Score End Average Score Dierence Score
21 243712 492220 248508
22 149146 97674 -51472
24 131545 187609 56064
28 400928 92874 -308054
29 541200 1534793 993593
31 45520 147169 101649
34 143429 1044740 901311
39 349717 1319748 970031
43 24931 132214 107283
46 9584 28178 18594
47 119000 63950 -55050
90 64200 986325 992125
92 216534 254350 37816
Table 9: Average beginning and ending tetris scores
5.3.2 Research Question 2 core results
With parametric assumptions in place, a series of paired sample t-tests were run in order to compare the Standard Tetris group's average number of rotations at
dierence between subject 31's beginning (1.5221) and ending (0.8807) average rotations for Z-shapes is more than three standard deviations greater than the mean paired dierence between the rest of the dataset (see Table 10).
the beginning and end of their training period, the results of which are shown in Table 10. 25 From these results we see that there were no statistically signicant changes in the average number of rotations for any zoid type from the beginning to the end of training (all p > .05).
Paired Dierences
Condence95%
interval
Mean Std. Deviation Std. Error Mean Lower Upper t df Sig. (2-tailed) FIRST_10_LINE -
LAST_10_LINE -.0140000 .0861766 .0239011 -.0660760 .0380760 -.586 12 .569 FIRST_10_T - LAST_10_T -.0574538 .3794234 .1052331 -.2867371 .1718294 -.546 12 .595 FIRST_10_Z - LAST_10_Z .1015077 .2050396 .0568678 -.0223965 .2254119 1.785 12 .100 FIRST_10_L - LAST_10_L -.0559385 .3067704 .0850828 -.2413180 .1294410 -.657 12 .523 FIRST_10_Z_SANS_OUTLIER
- LAST_10_Z .0565167 .1309826 .0378114 -.0267057 .1397390 1.495 11 .163 Table 10: Paired sample t-tests comparing beginning of training average number
of rotations with end of training average number of rotations per zoid type.
In order to test whether our subjects' Tetris ability improved over the train- ing period a Wilcoxon Signed-rank test was used to analyse the average scores for their rst and last ten games. This revealed a statistically signicant in- crease in Tetris scores (Z = -2.062, p = .039) from the beginning to the end of training, with a large eect size (r = .57).