Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
22.214.171.124. Josephson Junction Cryothermometry
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. = 1.305Because of the discrete nature of the data, we also compute the normal PPCC.
Normal PPCC = 0.97484
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 = 698The 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.
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
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 > 2.618Since 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.
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).
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| > 1.96The 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.
Probability plots are a graphical test for assessing if a
particular distribution provides an adequate fit to a data
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 > 0.787The 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.
A test for outliers is the Grubbs
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 > 3.950619For this data set, Grubbs' test does not detect any outliers at the 0.05 significance level.
Although the randomness and normality assumptions were
mildly violated, we conclude that a reasonable model for the
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