[From Bruce Abbott (950116.1730 EST)]
The THREECV1 program has been revised to save the reference positions (1st
line) and the 3600 lines of data, which now consist of the three cursor
positions and the mouse ("handle") position. What appears below is an
analysis of the first 1800 data observations using Minitab. C1, C2, and C3
are the three cursor positions; H is the handle position.
UNIVARIATE DESCRIPTIVE STATS
N MEAN MEDIAN TRMEAN STDEV SEMEAN
C1 1800 310.63 309.00 312.07 70.04 1.65
C2 1800 345.58 328.00 344.47 63.40 1.49
C3 1800 319.31 320.00 319.35 5.28 0.12
H 1800 -9.46 -22.00 -11.54 60.18 1.42
Note the means and medians of the three cursor positions. The reference
position was 319. Which cursor do you suppose I was controlling? (Note also
the standard deviations: the controlled cursor would be the one having the
smallest SD.)
BASIC CORRELATIONS
C1 C2 C3
C2 0.105
C3 0.177 0.039
H 0.527 0.783 0.035
The low correlations in the positions of the three cursors emerged despite the
common influence of the mouse on all three. This is due to the fact that each
cursor was being affected by an independent disturbance. The moderately high
correlations between H and C1 and between H and C2 would be 1.0 were it not
for the disturbances, which weaken the relationship. The nearly zero
correlation between H and C3 emerges because H is opposing the disturbance on
C3. Perfect opposition would keep C3 constant, yielding zero correlation with
H.
SCATTERPLOTS
I was surprised at the usuaual look of many of the scatterplots. In normal
scatterplots the points tend to form a more-or-less oval shape, and each
univariate distribution would look more-or-less normal. It is obvious from
these plots that the points are not independent, random values from a
bivariate population distribution. Such plots should set off warning bells
for anyone performing linear regression or correlation analyses of these
relationships. The "shapes" traced out by the points were so unusual that I
gave each plot a name to suggest its appearance:
Correlation of C1 and H = 0.527: "THE PITCH FORK"
H - 3+
- *8+5++6
- *953+*+
80+ 6+373**
- *469++4
- *+2*34
- 3+2 5++64*
- 2+38883++++ *2+5+9*
0+ 5**2 +++ ++++6
- +3+4 6 +++ *59++++++
- ++++ *56465696432++5 568377+++4
- 488++++425942 788+7 28+
- 686 8992
-80+ +32++4244++++
- +++8+4+5+
···
-
-
--------+---------+---------+---------+---------+--------C1
200 250 300 350 400
Correlation of C2 and H = 0.783: "THE X-CHROMOSOME"
H - 3+9
- 3669++67+++
- +**38+* *4922
80+ 8**2 54542
- +++3 23337
- **76+6**
- 2++8** +
- **2+22+93* *+6++38
0+ *847+4+6** 456* 22*448+69+6
- +++ *3+697+98+2 334924+*
- 795 7++++9 23++4+++32
- 4+5+*6++5+73969
- +47+*
-80+ *+++++
- ++++++
-
-
+---------+---------+---------+---------+---------+------C2
240 280 320 360 400 440
Correlation of C3 and H = 0.035: "LOWER MICHIGAN"
H - 2* 29* *2 24 4
- 49 ++5 95 63 *57 69 43 *35 6*
- *4 444 35 22 325 * *4 9*3
80+ * 4*2 ** * * 4+ 5 *
- 342 57 33 *33 * * 333 98 6
- 4 3** 35 23 2* ** 2 **2 *
- *4 626 55 2* 22* 22 26 222 +5 5
- * 23 22 22 ++8 ++ ++ ++9 73 23 7
0+ 23 +8 622 * 5+ +++ ++ +5
- * 65 64 +2* +7 ++ +++ 6
- * 33 +++ ++ ++ +++ ++ 7
- * 6+ +++ 76 56 7+6 2* 33 236 37 35 *
- 42 22 *3 4** +5 *4 24 *
-80+ 5*3 *3 44 2+* 3+ ++ 643 59 59 +3*
- 3 422 4* 42 6+5 ++ ++ 85* 97 +7
-
-
----+---------+---------+---------+---------+---------+--C3
301.0 308.0 315.0 322.0 329.0 336.0
Deviations computed as D[j,i] = C[j,i] - H[i]:
Correlation of D1 and H = -0.364: "THE GOLF CART"
H - +
- ++++
- + +
80+ +3 *+
- + ++
- + 8
- 2+5 ++8
- ++8+ 8++97 +4++
0+ 72 2++++2 ++
- 3+ 6 +++95 4+++++++
- +++8 286656565+++7 2776777++++
- 277777+989796 788877 5+
- *9+ +8
-80+ + 6+5++9++
- ++++++8
-
-
------+---------+---------+---------+---------+---------+D1
200 250 300 350 400 450
Correlation of H and D2 = -0.258: "THE FOX"
H - +
- 9++9+976+9++
- 8++5 +
80+ + 3+
- +++95 7+
- 35 *+78
- 2++8 +*
- 37452 7+3 8+++
0+ ++986654 357* 2346669++6
- ++998 8+9658868566* 33 7+
- *667* 49+9++*++++ 5+++9+++
- ++ 9++768+6786
- ++
-80+ +++
- +++6+
-
-
--------+---------+---------+---------+---------+--------D2
300 330 360 390 420
Correlation of D3 and H = -0.996: "THE SWORD"
H - +
- +++878
- 899++82
80+ 47876
- 689+++
- 6+556*
- 4++++9+
- * 6+++4
0+ 7+++++*
- 5+++++
- ++++++
- ++++++4
- 88++9
-80+ 2++++++
- 4++++
-
-
--------+---------+---------+---------+---------+--------D3
200 250 300 350 400
The above, of course, is the only legitimate relationship on which to apply a
regression analysis. It shows that the handle position was a strong, inverse
linear function of the disturbance value.
REGRESSING HANDLE POSITION ON DISTURBANCE 3
This analysis is based on the relationship depected immediately above.
The regression equation is
H = 318 - 0.995 D3
Predictor Coef Stdev t-ratio p
Constant 317.783 0.690 460.57 0.000
D3 -0.995344 0.002064 -482.16 0.000
s = 5.274 R-sq = 99.2% R-sq(adj) = 99.2%
The analysis estimates the reference at 317.783, with handle position being
equal to -0.995 times the disturbance. The actual values are 319 and -1.0.
The reference estimate is a little worse than our earlier one based on the
mean of C3 (319.31), reflecting the effect of "regression toward the mean."
Thus the mean of the C3 distribution provides a superior estimate of the
reference value than does the intercept of the regression equation.
RESIDUAL PLOT (RESIDUALS AS A FUNCTION OF TIME-ORDER)
The "moving average chart" below actually displays the averages of consecutive
blocks of 40 points over the 1800 point observation period. The residuals are
what's left over after the "influence" of the disturbance is removed from the
handle position, and represent the estimated control error. When plotted
against time-order of observation, the residuals show systematic variation.
These probably represent mostly dynamic effects such as time lags,
undershoots, and overshoots.
Residual Plot
-
M -
o - +
v - +
i 7.5+
n - +
g - + + + +
- + + ++ +
A -----------+-+----+-----+-----------------+-+-UCL=1.610
v 0.0+-----------+--+-+-+++-------+----+---++---+--X=0.0
e -------------------------+-------+------------LCL=-1.610
r - + + + + + +
a - + + +
g -++ + +
e -7.5+
-
- +
-
+---------+---------+---------+---------+---------+
0 10 20 30 40 50
Sample Number
By the way, the "UCL" and "LC" boundaries are the "upper and lower control
limits." This chart is produced by Minitab as part of its suite of process
control charts. A series of consecutive points falling outside the control
limits is supposed to evoke concern that the process being monitored is "out
of control," meaning that the quantity being monitored is falling outside of
specs. Judging from the chart, I was "out of control" quite often....
NORMAL QUANTILE PLOT OF RESIDUALS (ASSESSMENT OF NORMALITY)
Despite the systematic variation of residuals over time, the normal quantile
plot below shows that the distribution of the residuals is well-represented by
a normal curve (indicated by a straight-line relationship below):
C7 -
- *
- 4
2.5+ 574 *
- +4+2+2
- ++++7
- ++++
- +++++
0.0+ ++++++
- ++++++
- +++++
- 6+++++
- +9++5
-2.5+ 4+3
- 4
- *
-
+---------+---------+---------+---------+---------+------Resids
-18.0 -12.0 -6.0 0.0 6.0 12.0
I did quite a bit more "playing around" with the data, but these were some of
the results I found most interesting.
The PCT analysis on these first 1800 observations yielded the same cursor-
handle correlations as reported by Minitab. The model fit gave k = 0.0991
with an RMS error of 4.0 pixels and a correlation between model and data of
0.9978.
Regards,
Bruce