Download Multiple Sample Comparison SnapStat

STATGRAPHICS – Rev. 1/10/2005
Multiple Sample Comparison SnapStat
Summary
The Multiple Sample Comparison SnapStat creates a one-page summary that compares two or
more independent samples of variable data. It includes tests to determine whether or not there are
significant differences between the means and/or standard deviations of the populations from
which the samples were taken. In addition, the data is displayed graphically using a multiple
scatterplot, a multiple box-and-whisker plot, a means plot, and an ANOM plot. The calculations
performed are a subset of those performed in the Multiple Sample Comparison procedure.
However, the output is preformatted to fit on a single page.
Sample StatFolio: multsamsnapstat.sgp
Sample Data:
The file pulse rates.sf6 contains the results of an experiment reported by Milliken and Johnson
(1992) in which 78 workers were assigned at random to six groups. Each group was given a
work task to perform, and pulse rates were measured after each individual had worked on his
assigned task for one hour. After several individuals dropped out of the study, the final data
were:
Task 1
27
31
26
32
39
37
38
39
30
28
27
27
34
Task 2
29
28
37
24
35
40
40
31
30
25
29
25
Task 3
34
36
34
41
30
44
44
32
32
31
Task 4
34
34
43
44
40
47
34
31
45
28
Task 5
28
28
26
35
31
30
34
34
26
20
41
21
Task 6
28
26
29
25
35
34
37
28
21
28
26
The final n = 68 measurements have been arranged in q = 6 columns, one for each group of
subjects.
Alternatively, the data could have been arranged in a table with all of the pulse rates in a single
column, together with a column identifying which task the subject was given. A portion of such a
file is shown below:
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STATGRAPHICS – Rev. 1/10/2005
Subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
…
Pulse Rate
27
31
26
32
39
37
38
39
30
28
27
27
34
29
28
37
24
35
40
40
31
30
25
29
25
34
…
Task
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
…
Either data structure can be analyzed by the Multiple Sample Comparison SnapStat procedure. If
the same data is to be used in other procedures such as the General Linear Models procedure, it
should be structured in the second manner.
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Data Input
When the Multiple Sample Comparison SnapStat is selected from the main menu, the first dialog
box displayed asks you to specify the format in which the data has been entered:
•
Multiple Data Columns: indicates that each sample has been placed into a separate column.
•
Data and Code Columns: indicates that all observations have been placed into a single
column, with a second column indicating which sample each observation belongs to.
•
Sample Statistics: indicates that the original observations are not available. However, the
sample sizes, sample means, and sample standard deviations have been placed into 3
columns of the data sheet. In this case, some options will not be available.
Multiple Data Columns
If the data have been placed in separate columns for each sample, the column names must be
entered on the second dialog box:
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•
STATGRAPHICS – Rev. 1/10/2005
Samples: two or more numeric columns containing the observations, one column for each
sample.
•
Select: subset selection.
Data and Code Columns
If the data from all samples have been placed into a single column, then enter the name of that
column and the column containing the group identifiers:
•
Data: numeric column containing the observations from all samples.
•
Level codes: numeric or non-numeric column containing an identifier for the sample
corresponding to each data value.
•
Select: subset selection.
Sample Statistics
If the original observations are not available but the means and standard deviations of each
sample are known, enter the sample statistics into separate columns of the datasheet:
Task
1
2
3
4
5
6
Size
13
12
10
10
12
11
Mean
31.9231
31.0833
35.8000
38.0000
29.5000
28.8182
Standard Deviation
4.95751
5.66422
5.30827
6.59966
6.00757
4.75012
Then complete the second dialog box as shown below:
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•
Sample Means: numeric column containing the means of each sample.
•
Sample Standard Deviations: numeric column containing the standard deviations of each
sample.
•
Sample Sizes: numeric column containing the sizes of each sample.
•
Sample Labels: optional column containing labels for each sample.
•
Select: subset selection.
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Output
The output from the SnapStat consists of a single page pf graphs and numerical statistics.
SnapStat: Multiple Sample Comparison
Count
13
12
10
10
12
11
68
Mean
31.9231
31.0833
35.8
38
29.5
28.8182
32.3088
Sigma
4.95751
5.66422
5.30827
6.59966
6.00757
4.75012
6.24203
50
45
response
Sample
Task 1
Task 2
Task 3
Task 4
Task 5
Task 6
Scatterplot
40
35
30
Box-and-Whisker Plot
Task 2
ANOVA Table
Sum of
Source
Squares
Between
694.439
Within
1916.08
Total
2610.51
Task 3
P-Value = 0.0015
Task 1
Task 4
Task 6
Task 5
Task 4
Task 3
Task 1
20
Task 2
25
Mean
Square
138.888
30.9045
Df
5
62
67
F-Ratio
4.49
Variance Check
Levene's: 0.641611
P-Value = 0.6688
Task 5
Task 6
20
25
30
35 40
45
50
response
Means Plot
Analysis of Means Plot
With 95% Decision Limits
39
38
37
CL=32.31
35
LDL=28.24
Mean
35
32
UDL=36.38
33
 2005 by StatPoint, Inc.
Task 6
Task 5
Task 4
Task 3
Task 2
Task 6
Task 5
Task 4
27
Task 3
26
Task 2
29
Task 1
31
29
Task 1
Mean
With 95.0 Percent LSD Intervals
41
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Summary Statistics (top left)
The top left section of the output displays summary statistics for each sample of observations.
The table includes:
1. Count: the number of observations in each sample, nj.
2. Mean: the average pulse rate for the subjects given each of the 6 tasks, Y j .
3. Sigma: the standard deviations of each sample, sj.
Note that the group assigned to Task 4 has the highest mean and standard deviation.
Scatterplot (top right)
The top right section displays the observations within each group. It seems to suggest that pulse
rates are somewhat higher for subjects assigned to tasks 3 and 4.
Box-and-Whisker Plot (left center)
The left center section of the output displays a multiple box-and-whisker plot. Box-and-whisker
plots are constructed in the following manner:
•
A box is drawn extending from the lower quartile of the sample to the upper quartile.
This is the interval covered by the middle 50% of the data values when sorted from
smallest to largest.
•
A vertical line is drawn at the median (the middle value).
•
If requested, a plus sign is placed at the location of the sample mean.
•
Whiskers are drawn from the edges of the box to the largest and smallest data values,
unless there are values unusually far away from the box (which Tukey calls outside
points). Outside points, which are points more than 1.5 times the interquartile range
(box width) above or below the box, are indicated by point symbols. Any points
more than 3 times the interquartile range above or below the box are called far
outside points, and are indicated by point symbols with plus signs superimposed on
top of them. If outside points are present, the whiskers are drawn to the largest and
smallest data values which are not outside points.
In the sample data, the variability appears to be similar within each sample, although the
locations show some differences. There are no outside points.
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ANOVA Table and Variance Check (right center)
The right center section contains both an analysis of variance table and a variance check.
ANOVA Table
This table divides the overall variability among the n measurements into two components:
1. A “within groups” component, which measures the variability among pulse rates of
subjects given the same task.
2. A “between groups” component, which measures the variability among subjects given
different tasks.
Of particular importance is the F-ratio, which tests the hypothesis that the mean response for all
samples is the same. Formally, it tests the null hypothesis
H0: µ1 = µ2 = ... = µq
versus the alternative hypothesis
HA: not all µj equal
If F is sufficiently large, the null hypothesis is rejected.
The statistical significance of the F-ratio is most easily judged by its P-value. If the P-value is
less than 0.05, the null hypothesis of equal means is rejected at the 5% significance level, as in
the current example. This does not imply that every mean is significantly different from every
other mean. It simply implies that the means are not all the same.
Variance Check
One of the assumptions underlying the analysis of variance is that the variances of the
populations from which the samples come are the same. A test is performed to test the
hypotheses:
Null Hypothesis: all σj are equal
Alt. Hypothesis: not all σj are equal
If the P-Value for the test is small (less than 0.05 if operating at the 5% significance level), then
the hypothesis of equal variances is rejected. The test performed depends on the selection on the
ANOVA/Regression tab of the Preferences dialog box, accessible from the Edit menu.
For the pulse rate data, the sample means are significantly different, but the sample variances are
not.
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Means Plot (bottom left)
This plot shows the sample means together with uncertainty intervals. The type of interval
plotted depends on the setting in the ANOVA/Regression tab of the Preferences dialog box,
accessible from the edit menu. The type of intervals that may be selected are:
•
Confidence intervals - displays confidence intervals for the group means using the
pooled within-group standard deviation:
Y j ± tα / 2 , n − q
•
MS within
nj
(1)
LSD intervals - designed to compare any pair of means with the stated confidence level.
The intervals are given by
Yj ±
2M
2
MS within
nj
(2)
where M is defined as in the Multiple Range Tests. This formula also applies to the three
selections below.
•
Tukey HSD Intervals - designed for comparing all pairs of means. The stated
confidence level applies to the entire family of pairwise comparisons.
•
Scheffe Intervals - designed for comparing all contrasts. Not usually relevant here.
•
Bonferroni Intervals - designed for comparing a selected number of contrasts. Tukey’s
intervals are usually tighter.
Analysis of Means Plot (bottom left)
This plot constructs a chart similar to a standard control chart, where each sample mean is
plotted together with a centerline and upper and lower decision limits. The centerline is located
at the grand average of all of the observations Y . The decision limits are located at
Y ± hn −q ,1−α
MS within
nj
 q −1


q


(3)
where h is a critical value obtained from a table of the multivariate t distribution. The chart tests
the null hypothesis that all of the sample means are equal to the grand mean. Any means that fall
outside the decision limits indicate that the corresponding sample differs significantly from that
overall mean.
The advantage of the ANOM plot is that it shows at a glance which means are significantly
different than the average of all the samples. It also does so using a type of chart with which
many engineers and operators are quite familiar. It is easy to see from the above chart that Task
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4 has a significantly higher pulse rate than average, while all of the other task means are within
the decision limits. The procedure is exact if all sample sizes are equal and approximate if they
don’t differ too much.
 2005 by StatPoint, Inc.
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