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1. Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies Uniform Random Numbers

Quantitative Output and Interpretation

Summary Statistics As a first step in the analysis, common summary statistics are computed for the data.
      Sample size  = 500
      Mean         =   0.5078304
      Median       =   0.5183650
      Minimum      =   0.0024900  
      Maximum      =   0.9970800  
      Range        =   0.9945900
      Stan. Dev.   =   0.2943252
Because the graphs of the data indicate the data may not be normally distributed, we also compute two other statistics for the data, the normal PPCC and the uniform PPCC.
      Normal PPCC  =   0.9771602 
      Uniform PPCC =   0.9995682 
The uniform probability plot correlation coefficient (PPCC) value is larger than the normal PPCC value. This is evidence that the uniform distribution fits these data better than does a normal distribution.
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        0.522923        0.2638E-01    19.82
          B1       -0.602478E-04    0.9125E-04    -0.66
      Residual Standard Deviation = 0.2944917
      Residual Degrees of Freedom = 498  
The t-value of the slope parameter, -0.66, is smaller than the critical value of t0.975,498 = 1.96. Thus, we conclude that the slope is not different from zero at the 0.05 significance level.
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 we know this data set is not approximated well by the normal distribution, 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 = 0.07983
      Degrees of freedom:  k - 1 = 3
      Significance level:  α = 0.05
      Critical value:  Fα,k-1,N-k = 2.623
      Critical region:  Reject H0 if W > 2.623
In this case, the Levene test indicates 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 including the lag plot shown on the previous page.

Another check is an autocorrelation plot that shows the autocorrelations for various lags. Confidence bands can be plotted using 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 0.03. The critical values at the 5 % significance level are -0.087 and 0.087. This indicates that the lag 1 autocorrelation is not statistically significant, so there is no evidence of 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 = 0.2686
      Significance level:  α = 0.05
      Critical value:  Z1-α/2 = 1.96 
      Critical region:  Reject H0 if |Z| > 1.96 
The runs test fails to reject the null hypothesis that the data were produced in a random manner.
Distributional Analysis Probability plots are a graphical test of assessing whether 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 0.977. 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 = 5.765
      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 value of the test statistic, 5.765, is larger than the critical value of 0.787 at the 0.05 significance level.
Model Based on the graphical and quantitative analysis, we use the model
    Yi = C + Ei
where C is estimated by the mid-range and the uncertainty interval for C is based on a bootstrap analysis. Specifically,
    C = 0.499
    95% confidence limit for C = (0.497,0.503)
Univariate Report It is sometimes useful and convenient to summarize the above results in a report.
 Analysis for 500 uniform random numbers
 1: Sample Size                           = 500
 2: Location
    Mean                                  = 0.50783
    Standard Deviation of Mean            = 0.013163
    95% Confidence Interval for Mean      = (0.48197,0.533692)
    Drift with respect to location?       = NO
 3: Variation
    Standard Deviation                    = 0.294326
    95% Confidence Interval for SD        = (0.277144,0.313796)
    Drift with respect to variation?
    (based on Levene's test on quarters
    of the data)                          = NO
 4: Distribution
    Normal PPCC                           = 0.9771602
    Normal Anderson-Darling               = 5.7198390
    Data are Normal?
      (as tested by Normal PPCC)          = NO
      (as tested by Anderson-Darling)     = NO
    Uniform PPCC                          = 0.9995683
    Uniform Anderson-Darling              = 0.9082221
    Data are Uniform?
      (as tested by Uniform PPCC)         = YES
      (as tested by Anderson-Darling)     = YES

 5: Randomness
    Autocorrelation                       = -0.03099
    Data are Random?
      (as measured by autocorrelation)    = YES
 6: Statistical Control
    (i.e., no drift in location or scale,
    data is random, distribution is 
    fixed, here we are testing only for
    fixed uniform)
    Data Set is in Statistical Control?   = YES
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