Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
220.127.116.11. Standard Resistor
As a first step in the analysis, common summary statistics are
computed from the data.
Sample size = 1000 Mean = 28.01634 Median = 28.02910 Minimum = 27.82800 Maximum = 28.11850 Range = 0.29050 Stan. Dev. = 0.06349
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 27.9114 0.1209E-02 0.2309E+05 B1 0.20967E-03 0.2092E-05 100.2 Residual Standard Deviation = 0.1909796E-01 Residual Degrees of Freedom = 998The slope parameter, B1, has a t value of 100.2 which is statistically significant. The value of the slope parameter estimate is 0.00021. Although this number is nearly zero, we need to take into account that the original scale of the data is from about 27.8 to 28.2. In this case, we conclude that there is a drift in location.
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 normality assumption
is questionable for these data,
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 = 140.85 Degrees of freedom: k - 1 = 3 Significance level: α = 0.05 Critical value: Fα,k-1,N-k = 2.614 Critical region: Reject H0 if W > 2.614In this case, since the Levene test statistic value of 140.85 is greater than the 5 % significance level critical value of 2.614, we conclude that there is significant evidence of nonconstant variation.
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 4-plot in the previous
section is a simple graphical technique.
One 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 greatest interest, is 0.97. The critical values at the 5 % significance level are -0.062 and 0.062. This indicates that the lag 1 autocorrelation is statistically significant, so there is strong 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 = -30.5629 Significance level: α = 0.05 Critical value: Z1-α/2 = 1.96 Critical region: Reject H0 if |Z| > 1.96Because the test statistic is outside of the critical region, we reject the null hypothesis and conclude that the data are not random.
|Distributional Analysis||Since we rejected the randomness assumption, the distributional tests are not meaningful. Therefore, these quantitative tests are omitted. Since the Grubbs' test for outliers also assumes the approximate normality of the data, we omit Grubbs' test as well.|
It is sometimes useful and convenient to summarize the above
results in a report.
Analysis for resistor case study 1: Sample Size = 1000 2: Location Mean = 28.01635 Standard Deviation of Mean = 0.002008 95% Confidence Interval for Mean = (28.0124,28.02029) Drift with respect to location? = NO 3: Variation Standard Deviation = 0.063495 95% Confidence Interval for SD = (0.060829,0.066407) Change in variation? (based on Levene's test on quarters of the data) = YES 4: Randomness Autocorrelation = 0.972158 Data Are Random? (as measured by autocorrelation) = NO 5: Distribution Distributional test omitted due to non-randomness of the data 6: Statistical Control (i.e., no drift in location or scale, data are random, distribution is fixed) Data Set is in Statistical Control? = NO 7: Outliers? (Grubbs' test omitted due to non-randomness of the data)