Module Five: Outlier Detection for One Sample Case

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Module Five Outlier Detection for One Sample
Case In module Four, we discuss methods for
detecting normality of a response variable, and
ways of dealing with extremes, if exist. In this
unit, we will discuss methods, both numerical and
modern graphical methods for detecting extremes.
We start with one variable case inter-laboratory
testing studies, and extend to two-sample cases
in Module Six
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Detecting outliers for one variable case Consider
the TAPPI inter-laboratory testing study, there
were 87 labs participated the study to test the
Sample GR35. The data reported were the lab
averages. NOTE it is often the case that each
lab test the same sample twice or more for
investigating the within-lab variability as well
as between lab variability. Before an adequate
analysis of within and between lab variability,
it is critical the testing procedure for each lab
is standardized and the testing process is under
statistical control. If there are very unusual
testing results found, one should look for
possible causes, and decide to either keep or
delete the outliers for further analysis.
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• First thing to do in detecting outliers
• The detection of outliers is usually a
  preliminary analysis to ensure the reliability of
  the data. Before conducting any numerical or
  graphical approaches, it is a common practice to
  do the following for identifying obvious mistakes
  from sampling or testing
• A quick visual check through the data values to
  identify obvious typos or impossible data values
  based on the context of the study, for example, a
  miss of one decimal place, or a data values that
  are completely out of the possible range of the
  testing results.
• A quick computation of descriptive statistics
  provides minimum and maximum data values that
  help us quickly check typos or impossible data
  values as well.
• Once these are done, we can apply numerical and
  graphical methods to investigate not-so-obvious
  outliers.

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• Graphical and Numerical methods for Detecting
  Outliers
• The use of Empirical Rule for identifying
  outliers
• Empirical Rule When the distribution of the data
  is mound-shaped, if a data value is outside two
  s.d. of the mean, we may say it is a possible
  outlier (extreme), since there is only about 2.5
  of chance to be lower than or higher than the
  mean.

Note We replace m by and s by s for the
Empirical rule, since (m,s) are not known, and we
estimate them by sample information.
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2. Box-Plot for detecting outliers A popular
graphical tool for detecting outliers of a
variable is the Box-Plot
m
Q3(1.5)IQR
Q1-(1.5)IQR
Q1-3IQR
Q33IQR
Q1
Q3
Where, Inter-Quartile , IQR Q3-Q1, the range of
the middle 50 of the data values. NOTE the s
are possible outliers. and are very likely
outliers. These data values are very far away
from the center (mean and median).
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Revisit the blood pressure data for 15-20 years
old young adults
Likely outlier
Possible outliers
170
Sample mean
Median
Possible outliers
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• Box plot of the systolic blood pressure shows
  that 210 is a likely outlier, and several others
  are possible outliers (such as 170, and 70).
• This plot can be done by hand easily. Minitab
  also has this plot. Here are steps of
  constructing box plot using Minitab
• Go to Graph Menu, choose Box Plot.
• In the Dialog box, enter the variable name in Y.
  If we want to conduct two box plots based on the
  gender, then, enter Gender in X.
• In the Data Display, one can add more displays
  than the default ones by add the row 3, say, Mean
  Symbol for displaying sample mean on the plot.
• Annotation allows to show outlier values, mean
  and median on the plot.
• Frame allows to display more than one plot on the
  same page.

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Box Plots of Blood Pressure, Comparing Male and
Female
The likely outlier, 210 is a Male. The
distribution for Male is somewhat
skewed-to-right. However, excluding 210, it will
be pretty much symmetric. However, there are some
potential extremes on either end. The
distribution shape for Female is approximately
symmetric, and therefore, we can assume Normality
for Female.
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Hands-on activity Use the Inter-laboratory
testing data the TAPPI data to construct a box
plot for GR-Lab35-Mean variable and
GR-Lab35-Mean-1 variable. And identify the likely
outliers from each variable.
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3. Numerical Methods for Detecting Outliers a.
Studentized Residuals (as known as
CPVs(Comparative Performance Values),
h-statistics in the literature of
Inter-laboratory testing studies). b. Deleted
Studentized Residuals Consider the TAPPI lab data
of sample GR35 Using the notations y1, y2, y3,
., yn to represent the n data values, one from
each lab. When the same testing procedure is
applied and each lab process is under statistical
control, the expected testing result should be
the same. We will use the notation m for the
expected measurement.
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A Simple model for describing the one sample
testing As we demonstrated in the 2 cm drawing
activity, there is always some uncertainties
above and below the true measurement, and that if
there is no special causes or systematic bias,
the deviations between each labs testing result,
ei yi m, should behave at a random fashion.
This suggests that each testing result can be
expressed in the following model yi m ei
for i 1,2,3, ., n labs This describe the
expected situation in one sample testing. We then
use the observed lab testing data to estimate the
expected testing result and to investigate the
random deviation. By using the sample data, this
is what we have yi ei . is the
average of all included labs (as known as grand
mean). ei is what we call residual. And we also
see that average of eis is zero.
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• If the testing result , yi from a lab is likely
  an outlier, its corresponding ei will be far
  away from the average, 0. Therefore, one can use
  the residual to detect labs with extreme testing
  result.
• In stead of using the residual, ei, itself (the
  value depends on the measurement units), we use
  some standardized form of ei to detect outliers,
  so that, it will not be measurement dependent.
• A classical one is Standardized ei (as known as
  CPV as well as h-statistics in inter-laboratory
  studies)
• How to compute standardized ei for each lab?
• Compute , the grand mean of all included
  labs.
• Compute ei yi
• Compute the between-lab variance, s2 and standard
  deviation, s
• s2 and
• 4. Standardized ei ei/s

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• How to use standardized residual (CPV or
  h-statistic) to detect outliers?
• A quick rule
• If standardized residual gt 2 or lt -2 then it is a
  possible outlier. Since, based on the normal
  probability, there is approximately only 2.5 of
  chance to have a standardized residual gt 2 or lt
  -2, respectively.
• If standardized residual gt 2.6 or lt -2.6, then,
  it is a likely outlier. There is approximately
  only 0.5 of chance to be gt 2.6 or lt -2.6,
  respectively.
• NOTE 2.0 and 2.6 are values from the
  Z-distribution, N(0,1).

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• A more precise rule
• Standardized residual gt t(.025, n-1) or lt
  -t(.025, n-1), then, it is a probable outlier.
• Standardized residual gt t(.005, n-1) or lt
  -t(.005, n-1), then, it is a likely outlier.
• NOTE t(a, n-1) is a value of t-distribution. The
  standardized residual follows a t-distribution
  with degrees of freedom n-1 in this case.
  t-distribution is very similar to Z-distribution.
  T depends on sample size. When sample size is
  larger, t is eventually the same as Z.

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• A more sensitive measure for detecting outliers
• Deleted Standardized Residual, dj.
• The steps for computing this measurement
• Delete the jth case,
• then compute and residual ei(j) yi -
  for every case, including the jth case.
• Compute and s(j) using the (n-1)
  residuals, excluding jth case.
• Compute the deleted standardized residual, dj
  ej(j)/s(j)
• Repeat the steps 1-3 for cases j 1,2,3 ., n.
• Since the Deleted Standardized residual for the
  jth observation estimates all quantities with
  this observation deleted from the data set, the
  jth observation cannot influence these estimates.
  Therefore, unusual Y values clearly stand out. It
  is more sensitive than the classical standardized
  residual.

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How to use the Deleted Standardized Residual to
detect outliers? The same quick rule as the
standardized residual applies here. However, if
we are to be more precise, we need to use the
t-distribution. In applying the t-distribution,
the degrees of freedom is now (n-2). For most of
applications, the rule QUICK RULE is sufficient.
Unless the sample size n is very small. A common
wisdom is that n lt 30 is small. However, for
practical reason in outlier detection, it is
appropriate to consider n lt 20 to be small, and
that the t-distribution should be applied. The
key issue after detecting the outliers is to find
out the possible causes of these outliers.
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The h-plot for Inter-laboratory Testing The
h-plot plots the CPV values on a two dimensional
plot with a center line and upper and lower
limits along the X-axis. The X-axis is the Lab
ID. The CPV values of replications within each
lab, if existed, are grouped together. The Y-axis
is the standardized (or deleted studentized
residuals). An example is given in the following
2
0
-2
1 2 3 4 5 6 7 8 9 10
11 12
One may use the more precise t-values for the
upper and lower bounds In this plot, there are
12 labs. Each lab has two replications. The
length of each line is the standardized residual
(h-value or CPV) or deleted studentized residual.
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• The h-plot is a graphical view of the
  standardized residuals or deleted studentized
  residuals. The same plot is not available in
  Minitab. However, Minitab does provide all needed
  numerical measurements. We can create a similar
  graph using Minitab as well.
• The outlier detection using residuals is a very
  useful tool. In the above case, we consider the
  simplest model that describe one sample data as
  y m e. This model assumes
• Each lab is similar in its operation,
• The testing procedure is standardized,
• The operators have similar quality,
• The testing material is similar.
• If any of these assumptions is seriously
  violated, this model is not adequate. A more
  complicated model should be considered. The
  outliers detection should not be applied to
  response variable directly if we know in advance
  the violation of these assumptions.

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• Use Minitab to compute numerical measurements for
  conducting outlier detection for one sample case
• NOTE This process involves a lot of
  computations. We do not do this by hand. Here is
  the steps of using Minitab to compute residuals,
  standardized residuals, and deleted standardized
  residual.
• The TAPPI study is used for demonstration here.
• Create a column of 1s, say, in C7
• a. Go to Calc, choose Make Patterned Data,
  select Arbitrary Set of Numbers, in the Dialog
  box, enter C7 to store the data, enter 1 in the
  Arbitrary set of Numbers, List each value 87
  times, the sample size, and List the whole
  sequence 1 times.
• Go to Stat, choose Regression, then select
  Regression.
• In the Dialog box, enter the response variable,
  say C5, and enter predictors C7, the column with
  all 1.
• Click on Options, and deselect Fit Intercept.

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• Steps- Continued
• Click on Storage, and select Residuals,
  Standardized Residuals, Deleted Studentized
  Residuals, and Fits. Each of these will appear as
  a column is the worksheet.
• Residuals is named RESI1,
• Standardized Residual is named SRES1,
• Deleted Studentized Residual is namedTRES1
• The Fitted Value is named FITS1. In the one
  sample case, this is exactly the Grand Mean of
  all included labs.
• The number at the end of each variable will
  increase by one, such as RESI2, SRES2, for
  additional storage in the later analysis.
• We can change the variable names as we wish.

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• There are two additional selections in the
  Regression Procedure Graphs, Results.
• Click on Graphs, it allows you to conduct
  graphical detection of these residuals. Choose
  some graphs as you wish to see. For example, one
  may choose Standardized choose Normal Plot of
  Residuals to conduct a normal probability plot
  for standardized residuals.
• The Graphs will appear in the graph window.
• 7. Click on Results, it allows to choose the
  amount of computer output as needed. The last one
  gives the most extensive output.
• The results will appear in the Session Window.

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• Use Minitab to construct the h-plot
• Since Minitab does not have the same plot as
  h-plot shown before, I will demonstrate how to
  use other procedure to construct a plot that is
  similar to the h-plot using the TAPPI data.
• Go to Stat, choose Control Charts, then select
  Individuals.
• In the Dialog box, enter SRES1 into the
  Variable box (or any variable of interest such as
  deleted studentized residuals.
• Enter 0 for Historical Mean. This will be the
  center line on the plot.
• There are five additional selections and three
  graph editing selections. Leave Test and Estimate
  as default.
• Click on S-Limit selection, and enter 2 for
  upper sigma limit and 2 for lower sigma limit.
  You can also change the line color and line type.
• Click on Stamp selection, enter C1 as the Tick
  Labels. This will define the ticks on the X-axis
  using the laboratory names.
• Click on Options selection, you can change the
  symbol attributes and connection line attributes.

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• Case Example TAPPI Inter-laboratory Study
• Lets start with the SAMPLE GR35.
• A quick eye-checking immediately suggest the
  following cases are clear outliers, and they are
  removed from the outlier detection analysis
  immediately
• U3438 Lab mean 80.55 , U3531 Lab mean
  85.75
• Now, we follow the procedure described above to
  compute the standardized residuals and deleted
  studentized residuals using the remaining data
  and normal plot analysis.
• The unusual observations are Unusual Observations
• Lab Code GR35-Lab Fit SE Fit
  Residual St Resid
• U2415 1.00 76.0630 77.5273
  0.0652 -1.4643 -2.45R
• U3154 1.00 79.5500 77.5273
  0.0652 2.0227 3.39R
• U3185 1.00 79.1000 77.5273
  0.0652 1.5727 2.63R
• U3216 1.00 79.1620 77.5273
  0.0652 1.6347 2.74R
• U3249 1.00 76.2630 77.5273
  0.0652 -1.2643 -2.12R
• U3292 1.00 79.1380 77.5273
  0.0652 1.6107 2.70R
• U3334 1.00 78.7750 77.5273
  0.0652 1.2477 2.09R

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The normal probability plot and Normality test
for the Standardized Residuals
The pattern does not follow a straight line well.
The Normality Test suggests the lab testing
results clearly do not follow normal.
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• The quick rule is used to detect the outliers in
  this case, since the sample size is large.
• Both standardized residuals and deleted
  studentized residuals give the same group of
  unusual labs.
• These labs of which the testing results are found
  unusual will be notified. Further analysis is
  then taken to find out if there are any special
  causes or reasons for these unusual lab results.
• NOTE, the result using one sample detection
  technique is somewhat different from the
  two-sample plot approach. Since some labs which
  do not show outliers from this sample may show
  outliers when testing another sample. This is one
  reason why we should also conduct two-sample
  plots.

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This is created by Minitab. It is not quite the
same as the h-plot. It does the same function as
the h-plot and more. The mark 1 is the lab
which is over 3, a definite outlier. The labs
outside the upper and lower limit of 2 are
considered as outlier. One can choose to use
different upper and lower bounds.
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Hands-on Activity Detect labs which result
outliers in testing Sample GR 36 of the TAPPI
study.
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• Use of Basic Quality Control Chart Techniques for
  monitoring laboratory performances
• Quality Control charts were originally developed
  to monitor the mean shift and and the variation
  changes along the time domain in manufacturing
  process. For the inter-laboratory performance of
  testing a given material, we can apply the same
  charting method to monitor the performance of
  laboratories based on two measurements
• laboratory measurement means and
• within-lab measurement variations.
• The control charts to be discussed are called

Example A study of a chromatographic method was
conducted for determining malathion. Ten labs
participated in the study each lab received a
subsample of a technical grade malathion (Tech),
two wettable powders (25 WP and 50 WP), and an
emulsifiable concentrate (58 EC), and a dust.
Each participant also received an internally
tested standard of malathion (99.1) along with
the analytical method. (Wernimont, 1985).
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Row lab Rep WP25 WP50 1 1
1 26.17 50.76 2 1 2 26.22
50.67 3 1 3 25.85 50.81 4
1 4 25.80 50.72 5 2 1 26.44
50.82 6 2 2 26.57 50.90 7
2 3 25.80 51.04 8 2 4
26.06 50.96 9 3 1 26.95 52.53
10 3 2 26.91 52.54 11 3 3
26.98 52.55 12 3 4 26.91
52.47 13 5 1 26.23 50.20 14
5 2 26.00 50.47 15 5 3 26.22
50.39 16 5 4 26.18 50.43 17
6 1 25.45 51.65 18 6 2
25.62 51.67
Row lab Rep WP25 WP50 19 6
3 27.01 51.72 20 6 4 25.72
52.07 21 7 1 26.14 50.53 22
7 2 26.78 50.75 23 7 3 26.04
49.99 24 7 4 25.97 50.92 25
8 1 25.70 50.00 26 8 2
25.90 50.30 27 8 3 25.80 50.50
28 8 4 25.70 50.60 29 9 1
26.13 50.26 30 9 2 26.13
50.36 31 9 3 25.91 50.97 32
9 4 25.86 50.44 33 10 1 26.22
50.23 34 10 2 26.20 50.27 35
10 3 25.84 50.29 36 10 4
25.84 49.97
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Construction of
Consider the above Malathion testing study. Ten
labs particilated in the study. Each Lab tested
material WP50 for four replications. Lab 4 was
excluded since it did not complete the testing.
Lab ID Rep1 Rep2 Rep3 Rep4 Sample mean, Range,
1 x11 x12 x13 x14 R1
2 x21 x22 x23 x24 R2
3
5
6
7
8
9
10 X10,1 X10,2 X10,3 X10,4 R10
Average
Range Largest Smallest in each Lab.
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An X-bar chart is to monitor the laboratory mean.
If labs are consistent, then, the average of each
lab should be close. If all of them the equal,
then, the grand average is the same of lab
average. If lab averages are very different (that
is some lab systematic biases exist), then there
will have deviation between grant mean and lab
mean. This provides the basis of the X-bar chart.
The lab averages are then plotted along the lab
order. The multiple 3 is applied commonly in
process control. Under the normality assumption,
there is 99.7 of chance the lab sample mean
should be within the interval. As the chart
indicates, we need to estimate the grand mean and
SE of lab mean. Since range is usually easier to
compute, the estimate of the population variance
and, hence the SE of lab mean can also be
estimated, using the distribution of Range.
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The expected value of Range E(R) d2sx , where
d2 depends on sample size (in the lab testing
case, it is the of replications conducted by
each lab. The values of d2 will be provided in
the class. Therefore, the estimate of sx is given
by And the SE of sample mean is
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Analyzing the malathion data the WP50 variable

• X-bar chart suggests that there exists a very
  large mean differences among labs. This is an
  indication of systematic lab bias. When comparing
  with the standard proportion of 50, Lab 3 shows
  much higher lab average than others. Some
  attention to Lab 3 should be taken.
• R-chart indicates, in general, no lab has
  dramatically high within-lab variation. However,
  Lab 7 has somewhat higher within-lab variation.

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Analyzing the Malathion Data 25 Variable
X-bar chart for the WP25 variable also show that
Lab 3 has a significantly high lab average. A
closer check is necessary. The R-chart indicates
the within-lab variation exceeds the upper limit.
A review of Lab 6 for special causes would be
recommended.
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• Some General Comments of applying the control
  charts for monitoring laboratory means and
  within-lab variations
• This X-bar, R-chart technique is valid under the
  assumptions
• The response variable follows a normal
  distribution.
• The same or very similar material is tested by
  every participated lab.
• The operation of each lab is independent of
  others.
• In most laboratory studies,
• condition (3) is usually satisfied.
• Condition (2) may be satisfied if the preparation
  and distribution of material and the time period
  of conducting the lab testing is within a
  reasonable time period.
• If there are more than one material tested by
  participated labs, we can conduct a series of
  control charts to monitor each material. There
  are also multivariate control charts that can be
  applied to monitor more than one material at a
  time and take into account the laboratory
  systematic biases into account.
• The Youdens two-sample plots can be applied (to
  be discussed later) to diagnose the lab
  performance based on two samples at a time.

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Other Control Charts that may be useful for
monitoring inter-laboratory testing study
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How to use Minitab to conduct control chart
analysis?
Constructing X-bar and R-charts is
straightforward even by hand. However, Minitab
can do the charting and much much more for us.
There are steps are constructing the X-bar and
R-charts
1. GO to Stat, choose Control Charts, select
Xbar-R 2. In the dialog box, depending on the
data arrangement in the worksheet. If response is
in one column and lab in another column, enter
response and lab id columns into single column
and sub-group size. 3. There are four
selections. We have shown these before. Click on
Stamp selection, and enter the column that
consists of the correct Lab ID or Name . The
correct ID or Lab Name will show on the X-ticks
for easier reading.
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• Hands-on Activity
• Analyze the other variables in the Malathion
  data, and draw your final conclusion about the
  lab consistency with regards to
• Lab averages,
• Within-lab variations

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