1.4. EDA Case Studies
1.4.2. Case Studies
1.4.2.5. Beam Deflections
1.4.2.5.2. |
Test Underlying Assumptions |
- Determine if the univariate model:
is appropriate and valid.
- Determine if the typical underlying assumptions for an
"in control" measurement process are valid. These assumptions
are:
- random drawings;
- from a fixed distribution;
- with the distribution having a fixed location; and
- the distribution having a fixed scale.
- Determine if the confidence interval
is appropriate and valid where s is the standard deviation of the original data.
- The run sequence plot (upper left) indicates that the data do not have any significant shifts in location or scale over time.
- The lag plot (upper right) shows that the data are not random. The lag plot further indicates the presence of a few outliers.
- When the randomness assumption is thus seriously violated, the histogram (lower left) and normal probability plot (lower right) are ignored since determining the distribution of the data is only meaningful when the data are random.
We need to develop a better model. Non-random data can frequently be modeled using time series mehtodology. Specifically, the circular pattern in the lag plot indicates that a sinusoidal model might be appropriate. The sinusoidal model will be developed in the next section.
We have drawn some lines and boxes on the plot to better isolate
the outliers. The following output helps identify the points that
are generating the outliers on the lag plot.
****************************************************
** print y index xplot yplot subset yplot > 250 **
****************************************************
VARIABLES--Y INDEX XPLOT YPLOT
300.00 158.00 -506.00 300.00
****************************************************
** print y index xplot yplot subset xplot > 250 **
****************************************************
VARIABLES--Y INDEX XPLOT YPLOT
201.00 157.00 300.00 201.00
********************************************************
** print y index xplot yplot subset yplot -100 to 0
subset xplot -100 to 0 **
********************************************************
VARIABLES--Y INDEX XPLOT YPLOT
-35.00 3.00 -15.00 -35.00
*********************************************************
** print y index xplot yplot subset yplot 100 to 200
subset xplot 100 to 200 **
*********************************************************
VARIABLES--Y INDEX XPLOT YPLOT
141.00 5.00 115.00 141.00
That is, the third, fifth, and 158th points appear to be outliers.
This autocorrelation plot shows a distinct cyclic pattern. As with the lag plot, this suggests a sinusoidal model.
This spectral plot shows a single dominant peak at a frequency of 0.3. This frequency of 0.3 will be used in fitting the sinusoidal model in the next section.
SUMMARY
NUMBER OF OBSERVATIONS = 200
***********************************************************************
* LOCATION MEASURES * DISPERSION MEASURES *
***********************************************************************
* MIDRANGE = -0.1395000E+03 * RANGE = 0.8790000E+03 *
* MEAN = -0.1774350E+03 * STAND. DEV. = 0.2773322E+03 *
* MIDMEAN = -0.1797600E+03 * AV. AB. DEV. = 0.2492250E+03 *
* MEDIAN = -0.1620000E+03 * MINIMUM = -0.5790000E+03 *
* = * LOWER QUART. = -0.4510000E+03 *
* = * LOWER HINGE = -0.4530000E+03 *
* = * UPPER HINGE = 0.9400000E+02 *
* = * UPPER QUART. = 0.9300000E+02 *
* = * MAXIMUM = 0.3000000E+03 *
***********************************************************************
* RANDOMNESS MEASURES * DISTRIBUTIONAL MEASURES *
***********************************************************************
* AUTOCO COEF = -0.3073048E+00 * ST. 3RD MOM. = -0.5010057E-01 *
* = 0.0000000E+00 * ST. 4TH MOM. = 0.1503684E+01 *
* = 0.0000000E+00 * ST. WILK-SHA = -0.1883372E+02 *
* = * UNIFORM PPCC = 0.9925535E+00 *
* = * NORMAL PPCC = 0.9540811E+00 *
* = * TUK -.5 PPCC = 0.7313794E+00 *
* = * CAUCHY PPCC = 0.4408355E+00 *
***********************************************************************
LEAST SQUARES MULTILINEAR FIT
SAMPLE SIZE N = 200
NUMBER OF VARIABLES = 1
NO REPLICATION CASE
PARAMETER ESTIMATES (APPROX. ST. DEV.) T VALUE
1 A0 -178.175 ( 39.47 ) -4.514
2 A1 X 0.736593E-02 (0.3405 ) 0.2163E-01
RESIDUAL STANDARD DEVIATION = 278.0313
RESIDUAL DEGREES OF FREEDOM = 198
The slope parameter, A1, has a
t value of 0.022 which
is statistically not significant. This indicates that the slope can
in fact be considered zero.
LEVENE F-TEST FOR SHIFT IN VARIATION
(ASSUMPTION: NORMALITY)
1. STATISTICS
NUMBER OF OBSERVATIONS = 200
NUMBER OF GROUPS = 4
LEVENE F TEST STATISTIC = 0.9378599E-01
FOR LEVENE TEST STATISTIC
0 % POINT = 0.0000000E+00
50 % POINT = 0.7914120
75 % POINT = 1.380357
90 % POINT = 2.111936
95 % POINT = 2.650676
99 % POINT = 3.883083
99.9 % POINT = 5.638597
3.659895 % Point: 0.9378599E-01
3. CONCLUSION (AT THE 5% LEVEL):
THERE IS NO SHIFT IN VARIATION.
THUS: HOMOGENEOUS WITH RESPECT TO VARIATION.
In this case, the Levene test indicates that the standard
deviations are significantly different in the 4 intervals
since the test statistic of 13.2 is greater than the 95%
critical value of 2.6. Therefore we conclude that the scale
is not constant.
RUNS UP
STATISTIC = NUMBER OF RUNS UP
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 63.0 104.2083 10.2792 -4.01
2 34.0 45.7167 5.2996 -2.21
3 17.0 13.1292 3.2297 1.20
4 4.0 2.8563 1.6351 0.70
5 1.0 0.5037 0.7045 0.70
6 5.0 0.0749 0.2733 18.02
7 1.0 0.0097 0.0982 10.08
8 1.0 0.0011 0.0331 30.15
9 0.0 0.0001 0.0106 -0.01
10 1.0 0.0000 0.0032 311.40
STATISTIC = NUMBER OF RUNS UP
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 127.0 166.5000 6.6546 -5.94
2 64.0 62.2917 4.4454 0.38
3 30.0 16.5750 3.4338 3.91
4 13.0 3.4458 1.7786 5.37
5 9.0 0.5895 0.7609 11.05
6 8.0 0.0858 0.2924 27.06
7 3.0 0.0109 0.1042 28.67
8 2.0 0.0012 0.0349 57.21
9 1.0 0.0001 0.0111 90.14
10 1.0 0.0000 0.0034 298.08
RUNS DOWN
STATISTIC = NUMBER OF RUNS DOWN
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 69.0 104.2083 10.2792 -3.43
2 32.0 45.7167 5.2996 -2.59
3 11.0 13.1292 3.2297 -0.66
4 6.0 2.8563 1.6351 1.92
5 5.0 0.5037 0.7045 6.38
6 2.0 0.0749 0.2733 7.04
7 2.0 0.0097 0.0982 20.26
8 0.0 0.0011 0.0331 -0.03
9 0.0 0.0001 0.0106 -0.01
10 0.0 0.0000 0.0032 0.00
STATISTIC = NUMBER OF RUNS DOWN
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 127.0 166.5000 6.6546 -5.94
2 58.0 62.2917 4.4454 -0.97
3 26.0 16.5750 3.4338 2.74
4 15.0 3.4458 1.7786 6.50
5 9.0 0.5895 0.7609 11.05
6 4.0 0.0858 0.2924 13.38
7 2.0 0.0109 0.1042 19.08
8 0.0 0.0012 0.0349 -0.03
9 0.0 0.0001 0.0111 -0.01
10 0.0 0.0000 0.0034 0.00
RUNS TOTAL = RUNS UP + RUNS DOWN
STATISTIC = NUMBER OF RUNS TOTAL
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 132.0 208.4167 14.5370 -5.26
2 66.0 91.4333 7.4947 -3.39
3 28.0 26.2583 4.5674 0.38
4 10.0 5.7127 2.3123 1.85
5 6.0 1.0074 0.9963 5.01
6 7.0 0.1498 0.3866 17.72
7 3.0 0.0193 0.1389 21.46
8 1.0 0.0022 0.0468 21.30
9 0.0 0.0002 0.0150 -0.01
10 1.0 0.0000 0.0045 220.19
STATISTIC = NUMBER OF RUNS TOTAL
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 254.0 333.0000 9.4110 -8.39
2 122.0 124.5833 6.2868 -0.41
3 56.0 33.1500 4.8561 4.71
4 28.0 6.8917 2.5154 8.39
5 18.0 1.1790 1.0761 15.63
6 12.0 0.1716 0.4136 28.60
7 5.0 0.0217 0.1474 33.77
8 2.0 0.0024 0.0494 40.43
9 1.0 0.0002 0.0157 63.73
10 1.0 0.0000 0.0047 210.77
LENGTH OF THE LONGEST RUN UP = 10
LENGTH OF THE LONGEST RUN DOWN = 7
LENGTH OF THE LONGEST RUN UP OR DOWN = 10
NUMBER OF POSITIVE DIFFERENCES = 258
NUMBER OF NEGATIVE DIFFERENCES = 241
NUMBER OF ZERO DIFFERENCES = 0
Values in the column labeled "Z" greater than 1.96 or less than
-1.96 are statistically significant at the 5% level.
Numerous values in this column are much larger than +/-1.96, so
we conclude that the data are not random.
