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1. Exploratory Data Analysis
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 Statistics As a first step in the analysis, common summary statistics were computed from the data. 
      Sample size  = 700
      Mean         =   2898.562
      Median       =   2899.000  
      Minimum      =   2895.000  
      Maximum      =   2902.000  
      Range        =      7.000  
      Stan. Dev.   =      1.305
Because of the discrete nature of the data, we also compute the normal PPCC. 
      Normal PPCC = 0.97484
Location One way to quantify a change in location over time is to fit a straight line to the data using an index variable as the independent variable in the regression. For our data, we assume that data are in sequential run order and that the data were collected at equally spaced time intervals. In our regression, we use the index variable X = 1, 2, ..., N, where N is the number of observations. If there is no significant drift in the location over time, the slope parameter should be zero. 
      Coefficient     Estimate     Stan. Error      t-Value
          B0         2.898E+03       9.745E-02    29739.288  
          B1         1.071E-03       2.409e-04        4.445
 
      Residual Standard Deviation = 1.288
      Residual Degrees of Freedom = 698
The slope parameter, B1, has a t value of 4.445 which is statistically significant (the critical value is 1.96). However, the value of the slope is 1.071E-03. Given that the slope is nearly zero, the assumption of constant location is not seriously violated even though it is statistically significant.
Variation One simple way to detect a change in variation is with a Bartlett test after dividing the data set into several equal-sized intervals. However, the Bartlett test is not robust for non-normality. Since the nature of the data (a few distinct points repeated many times) makes the normality assumption questionable, we use the alternative Levene test. In particular, we use the Levene test based on the median rather the mean. The choice of the number of intervals is somewhat arbitrary, although values of four or eight are reasonable. We will divide our data into four intervals. 
      H0:  σ12 = σ22 = σ32 = σ42 
      Ha:  At least one σi2 is not equal to the others.

      Test statistic:  W = 1.43
      Degrees of freedom:  k - 1 = 3
      Significance level:  α = 0.05
      Critical value:  Fα,k-1,N-k = 2.618
      Critical region:  Reject H0 if W > 2.618
Since the Levene test statistic value of 1.43 is less than the 95 % critical value of 2.618, we conclude that the variances are not significantly different in the four intervals.
Randomness There are many ways in which data can be non-random. However, most common forms of non-randomness can be detected with a few simple tests. The lag plot in the previous section is a simple graphical technique.

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).

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. 

      H0:  the sequence was produced in a random manner
      Ha:  the sequence was not produced in a random manner  

      Test statistic:  Z = -13.4162
      Significance level:  α = 0.05
      Critical value:  Z1-α/2 = 1.96 
      Critical region:  Reject H0 if |Z| > 1.96
The runs test indicates 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.

Distributional Analysis Probability plots are a graphical test for assessing if a particular distribution provides an adequate fit to a data set.

A quantitative enhancement to the probability plot is the correlation coefficient of the points on the probability plot, or PPCC. For this data set the PPCC based on a normal distribution is 0.975. Since the PPCC 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. The results of the Anderson-Darling test follow. 

      H0:  the data are normally distributed
      Ha:  the data are not normally distributed

      Adjusted test statistic:  A2 = 16.858
      Significance level:  α = 0.05
      Critical value:  0.787
      Critical region:  Reject H0 if A2 > 0.787
The Anderson-Darling test rejects the normality assumption because the test statistic, 16.858, is greater than the 95 % critical value 0.787.

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.

Outlier Analysis A test for outliers is the Grubbs test. 
      H0:  there are no outliers in the data
      Ha:  the maximum value is an outlier

      Test statistic:  G = 2.729201
      Significance level:  α = 0.05
      Critical value for a one-tailed test:  3.950619
      Critical region:  Reject H0 if G > 3.950619
For this data set, Grubbs' test does not detect any outliers at the 0.05 significance level.
Model Although the randomness and normality assumptions were mildly violated, we conclude that a reasonable model for the data is:
    \( Y_{i} = 2898.7 + E_{i} \)
In addition, a 95 % confidence interval for the mean value is (2898.515, 2898.928).
Univariate Report It is sometimes useful and convenient to summarize the above results in a report. 
 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
    Normal Anderson-Darling               = 16.7634
    Data are Normal?
      (as tested by Normal PPCC)          = NO
      (as tested by Anderson-Darling)     = 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
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