1.4. EDA Case Studies
1.4.2. Case Studies
1.4.2.4. Josephson Junction Cryothermometry
1.4.2.4.3. |
Quantitative Output and Interpretation |
SUMMARY
NUMBER OF OBSERVATIONS = 700
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* LOCATION MEASURES * DISPERSION MEASURES *
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* MIDRANGE = 0.2898500E+04 * RANGE = 0.7000000E+01 *
* MEAN = 0.2898562E+04 * STAND. DEV. = 0.1304969E+01 *
* MIDMEAN = 0.2898363E+04 * AV. AB. DEV. = 0.1058571E+01 *
* MEDIAN = 0.2899000E+04 * MINIMUM = 0.2895000E+04 *
* = * LOWER QUART. = 0.2898000E+04 *
* = * LOWER HINGE = 0.2898000E+04 *
* = * UPPER HINGE = 0.2899000E+04 *
* = * UPPER QUART. = 0.2899000E+04 *
* = * MAXIMUM = 0.2902000E+04 *
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* RANDOMNESS MEASURES * DISTRIBUTIONAL MEASURES *
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* AUTOCO COEF = 0.3148023E+00 * ST. 3RD MOM. = 0.6580265E-02 *
* = 0.0000000E+00 * ST. 4TH MOM. = 0.2825334E+01 *
* = 0.0000000E+00 * ST. WILK-SHA = -0.2272378E+02 *
* = * UNIFORM PPCC = 0.9554127E+00 *
* = * NORMAL PPCC = 0.9748405E+00 *
* = * TUK -.5 PPCC = 0.7935873E+00 *
* = * CAUCHY PPCC = 0.4231319E+00 *
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LEAST SQUARES MULTILINEAR FIT
SAMPLE SIZE N = 700
NUMBER OF VARIABLES = 1
NO REPLICATION CASE
PARAMETER ESTIMATES (APPROX. ST. DEV.) T VALUE
1 A0 2898.19 (0.9745E-01) 0.2974E+05
2 A1 X 0.107075E-02 (0.2409E-03) 4.445
RESIDUAL STANDARD DEVIATION = 1.287802
RESIDUAL DEGREES OF FREEDOM = 698
The slope parameter, A1, has a
t value of 2.1 which is
statistically significant (the critical value is 1.98). However, the
value of the slope is 0.0011. Given that the slope is nearly zero,
the assumption of constant location is not seriously violated
even though it is (just barely) statistically significant.
LEVENE F-TEST FOR SHIFT IN VARIATION
(ASSUMPTION: NORMALITY)
1. STATISTICS
NUMBER OF OBSERVATIONS = 700
NUMBER OF GROUPS = 4
LEVENE F TEST STATISTIC = 1.432365
FOR LEVENE TEST STATISTIC
0 % POINT = 0.000000
50 % POINT = 0.7894323
75 % POINT = 1.372513
90 % POINT = 2.091688
95 % POINT = 2.617726
99 % POINT = 3.809943
99.9 % POINT = 5.482234
76.79006 % Point: 1.432365
3. CONCLUSION (AT THE 5% LEVEL):
THERE IS NO SHIFT IN VARIATION.
THUS THE GROUPS ARE HOMOGENEOUS WITH RESPECT TO VARIATION.
Since the Levene test statistic value of 1.43 is less than the
95% critical value of 2.67, we conclude that the standard
deviations are not significantly different in the 4 intervals.
Another check is an autocorrelation plot that shows the autocorrelations for various lags. Confidence bands can be plotted at the 95% and 99% confidence levels. Points outside this band indicate statistically significant values (lag 0 is always 1). Dataplot generated the following autocorrelation plot.
The lag 1 autocorrelation, which is generally the one of most interest, is 0.31. The critical values at the 5% level of significance are -0.087 and 0.087. This indicates that the lag 1 autocorrelation is statistically significant, so there is some evidence for non-randomness.
A common test for randomness is the
runs test.
RUNS UP
STATISTIC = NUMBER OF RUNS UP
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 102.0 145.8750 12.1665 -3.61
2 48.0 64.0500 6.2731 -2.56
3 23.0 18.4069 3.8239 1.20
4 11.0 4.0071 1.9366 3.61
5 4.0 0.7071 0.8347 3.95
6 2.0 0.1052 0.3240 5.85
7 2.0 0.0136 0.1164 17.06
8 0.0 0.0015 0.0393 -0.04
9 0.0 0.0002 0.0125 -0.01
10 0.0 0.0000 0.0038 0.00
STATISTIC = NUMBER OF RUNS UP
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 192.0 233.1667 7.8779 -5.23
2 90.0 87.2917 5.2610 0.51
3 42.0 23.2417 4.0657 4.61
4 19.0 4.8347 2.1067 6.72
5 8.0 0.8276 0.9016 7.96
6 4.0 0.1205 0.3466 11.19
7 2.0 0.0153 0.1236 16.06
8 0.0 0.0017 0.0414 -0.04
9 0.0 0.0002 0.0132 -0.01
10 0.0 0.0000 0.0040 0.00
RUNS DOWN
STATISTIC = NUMBER OF RUNS DOWN
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 106.0 145.8750 12.1665 -3.28
2 47.0 64.0500 6.2731 -2.72
3 24.0 18.4069 3.8239 1.46
4 8.0 4.0071 1.9366 2.06
5 4.0 0.7071 0.8347 3.95
6 3.0 0.1052 0.3240 8.94
7 0.0 0.0136 0.1164 -0.12
8 0.0 0.0015 0.0393 -0.04
9 0.0 0.0002 0.0125 -0.01
10 0.0 0.0000 0.0038 0.00
STATISTIC = NUMBER OF RUNS DOWN
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 192.0 233.1667 7.8779 -5.23
2 86.0 87.2917 5.2610 -0.25
3 39.0 23.2417 4.0657 3.88
4 15.0 4.8347 2.1067 4.83
5 7.0 0.8276 0.9016 6.85
6 3.0 0.1205 0.3466 8.31
7 0.0 0.0153 0.1236 -0.12
8 0.0 0.0017 0.0414 -0.04
9 0.0 0.0002 0.0132 -0.01
10 0.0 0.0000 0.0040 0.00
RUNS TOTAL = RUNS UP + RUNS DOWN
STATISTIC = NUMBER OF RUNS TOTAL
OF LENGTH EXACTLY I
I STAT EXP(STAT) SD(STAT) Z
1 208.0 291.7500 17.2060 -4.87
2 95.0 128.1000 8.8716 -3.73
3 47.0 36.8139 5.4079 1.88
4 19.0 8.0143 2.7387 4.01
5 8.0 1.4141 1.1805 5.58
6 5.0 0.2105 0.4582 10.45
7 2.0 0.0271 0.1647 11.98
8 0.0 0.0031 0.0556 -0.06
9 0.0 0.0003 0.0177 -0.02
10 0.0 0.0000 0.0054 -0.01
STATISTIC = NUMBER OF RUNS TOTAL
OF LENGTH I OR MORE
I STAT EXP(STAT) SD(STAT) Z
1 384.0 466.3333 11.1410 -7.39
2 176.0 174.5833 7.4402 0.19
3 81.0 46.4833 5.7498 6.00
4 34.0 9.6694 2.9794 8.17
5 15.0 1.6552 1.2751 10.47
6 7.0 0.2410 0.4902 13.79
7 2.0 0.0306 0.1748 11.27
8 0.0 0.0034 0.0586 -0.06
9 0.0 0.0003 0.0186 -0.02
10 0.0 0.0000 0.0056 -0.01
LENGTH OF THE LONGEST RUN UP = 7
LENGTH OF THE LONGEST RUN DOWN = 6
LENGTH OF THE LONGEST RUN UP OR DOWN = 7
NUMBER OF POSITIVE DIFFERENCES = 262
NUMBER OF NEGATIVE DIFFERENCES = 258
NUMBER OF ZERO DIFFERENCES = 179
Values in the column labeled "Z" greater than 1.96 or less than
-1.96 are statistically significant at the 5% level.
The runs test indicates some mild non-randomness.
Although the runs test and lag 1 autocorrelation indicate some mild non-randomness, it is not sufficient to reject the Yi = C + Ei model. At least part of the non-randomness can be explained by the discrete nature of the data.
A quantitative enhancement to the probability plot is the correlation coefficient of the points on the probability plot. For this data set the correlation coefficient is 0.975. Since this is less than the critical value of 0.987 (this is a tabulated value), the normality assumption is rejected.
Chi-square and
Kolmogorov-Smirnov
goodness-of-fit tests are
alternative methods for assessing distributional adequacy.
The Wilk-Shapiro
and Anderson-Darling
tests can be used to test for normality. Dataplot generates the
following output for the Anderson-Darling normality test.
ANDERSON-DARLING 1-SAMPLE TEST
THAT THE DATA CAME FROM A NORMAL DISTRIBUTION
1. STATISTICS:
NUMBER OF OBSERVATIONS = 700
MEAN = 2898.562
STANDARD DEVIATION = 1.304969
ANDERSON-DARLING TEST STATISTIC VALUE = 16.76349
ADJUSTED TEST STATISTIC VALUE = 16.85843
2. CRITICAL VALUES:
90 % POINT = 0.6560000
95 % POINT = 0.7870000
97.5 % POINT = 0.9180000
99 % POINT = 1.092000
3. CONCLUSION (AT THE 5% LEVEL):
THE DATA DO NOT COME FROM A NORMAL DISTRIBUTION.
The Anderson-Darling test rejects the normality
assumption because the test statistic, 16.76, is greater than the
99% critical value 1.092.
Although the data are not strictly normal, the violation of the normality assumption is not severe enough to conclude that the Yi = C + Ei model is unreasonable. At least part of the non-normality can be explained by the discrete nature of the data.
GRUBBS TEST FOR OUTLIERS
(ASSUMPTION: NORMALITY)
1. STATISTICS:
NUMBER OF OBSERVATIONS = 700
MINIMUM = 2895.000
MEAN = 2898.562
MAXIMUM = 2902.000
STANDARD DEVIATION = 1.304969
GRUBBS TEST STATISTIC = 2.729201
2. PERCENT POINTS OF THE REFERENCE DISTRIBUTION
FOR GRUBBS TEST STATISTIC
0 % POINT = 0.000000
50 % POINT = 3.371397
75 % POINT = 3.554906
90 % POINT = 3.784969
95 % POINT = 3.950619
97.5 % POINT = 4.109569
99 % POINT = 4.311552
100 % POINT = 26.41972
3. CONCLUSION (AT THE 5% LEVEL):
THERE ARE NO OUTLIERS.
For this data set, Grubbs' test does not detect any outliers at
the 10%, 5%, and 1% significance levels.
Analysis for Josephson Junction Cryothermometry Data
1: Sample Size = 700
2: Location
Mean = 2898.562
Standard Deviation of Mean = 0.049323
95% Confidence Interval for Mean = (2898.465,2898.658)
Drift with respect to location? = YES
(Further analysis indicates that
the drift, while statistically
significant, is not practically
significant)
3: Variation
Standard Deviation = 1.30497
95% Confidence Interval for SD = (1.240007,1.377169)
Drift with respect to variation?
(based on Levene's test on quarters
of the data) = NO
4: Distribution
Normal PPCC = 0.97484
Data are Normal?
(as measured by Normal PPCC) = NO
5: Randomness
Autocorrelation = 0.314802
Data are Random?
(as measured by autocorrelation) = NO
6: Statistical Control
(i.e., no drift in location or scale,
data are random, distribution is
fixed, here we are testing only for
fixed normal)
Data Set is in Statistical Control? = NO
Note: Although we have violations of
the assumptions, they are mild enough,
and at least partially explained by the
discrete nature of the data, so we may model
the data as if it were in statistical
control
7: Outliers?
(as determined by Grubbs test) = NO
