Transcript presentation - Loyola University New Orleans

Visual EDF Software to Check
the Normality Assumption
Dr. Maria E. Calzada
Dr. Stephen M. Scariano
Loyola University New Orleans
[email protected]
http://www.loyno.edu/~calzada


Problem 7.11 of Moore’s the Basic Practice of Statistics
reads (paraphrased):
Our subjects are 11 people diagnosed as being dependent
on caffeine. Each subject was barred from coffee, colas,
and other substances containing caffeine. Instead they
took capsules containing their normal caffeine intake.
During a different time period, they took placebo
capsules… Table 7.3 contains data on two of several tests
given to the subjects. “Depression” is the score of the
Beck Depression inventory. Higher scores show more
symptoms of depression. “Beats” is the beats per minute
the subject achieved when asked to press a button 200
times as quickly as possible. We are interested in whether
being deprived of caffeine affects these outcomes.
Depressio
n
Depressio
n
(Caffeine)
1
Subject
Beats
Beats
Difference
(Placebo)
Difference
Depressio
n
(Caffeine)
(Placebo)
Beats
5
16
-11
281
201
80
2
5
23
-18
284
262
22
3
4
5
-1
300
283
17
4
3
7
-4
421
290
131
5
8
14
-6
240
259
-19
6
5
24
-19
294
291
3
7
0
6
-6
377
354
23
8
0
3
-3
345
346
-1
9
2
15
-13
303
283
20
10
11
12
-1
340
391
-51
11
1
0
1
408
411
-3

Does this matched pairs study give evidence
that being deprived of caffeine raises
depression scores? Check that the differences
are not strikingly nonnormal.
 Now check the differences in beats per
minute with and without caffeine. You
should hesitate to use the t procedures on
these data. Why?
Histogram
3.5
3.0
2.5
2.0
Frequency
1.5
1.0
Std. Dev = 6.92
.5
Mean = -7.4
N = 11.00
0.0
-20.0
-15.0
-17.5
-10.0
-12.5
-5.0
-7.5
0.0
-2.5
DIFFDEP
Histogram of the difference in depression scores, as
given by SPSS. Are these data normal?
Histogram
5
4
3
Frequency
2
1
Std. Dev = 48.75
Mean = 20.2
N = 11.00
0
-50.0
-25.0
0.0
25.0
50.0
75.0
100.0 125.0
DIFFBEAT
Histogram of the difference in Beats score. Are these data normal?
Normal P-P Plot of DIFFDEP
1.00
.75
Expected Cum Prob
.50
.25
0.00
0.00
.25
.50
.75
1.00
Observed Cum Prob
Normal probability plot for difference in depression data
Normal P-P Plot of DIFFBEAT
1.00
.75
Expected Cum Prob
.50
.25
0.00
0.00
.25
.50
.75
1.00
Observed Cum Prob
Normal probability plot for difference in beats data

While slight departures from normality are
usually inconsequential, substantive
departures from normality can seriously
impair the validity of statistical procedures
framed under the “Normality Assumption.”

The problem is: How can we help our
students distinguish between “slight
departures” and “substantive departures”
from normality.
EDF TESTS FOR NORMALITY

We have written a Visual Basic Program
that implements the Lilliefors test for
normality and the Anderson-Darling Test for
normality.
 We like the visual aspects of the Lilliefors
test.
 We like the power of the Anderson-Darling
Test.
Lilliefors Test
We compute the distances between the data’s
Empirical Distribution Function F n xand the
Cumulative Standard Normal Distribution x.
 If these distances are “not too large” we do not
have evidence to reject the “Normality
Assumption.”
 Upper- percentiles for the largest distance have
been tabulated using Monte-Carlo Simulations.

Anderson-Darling Test

Based on the accumulating square distance
function

2
D
n  
Fn
x
x


x
dF
x


Usually
1

x

x

1 
x


Our program implements the discrete AndersonDarling test statistic developed by Stephens (1974)
to approximate D

.
Conclusions

The Anderson-Darling and Lilliefors tests
are important tools that should be routinely
used alongside probability plots to check
the normality assumption.
 The stand-alone desktop program enhances
the craft of exploratory data analysis.
 Program is available for downloading at
http://www.loyno.edu/~calzada
Some References

David S. Moore. The Basic Practice of Statistics,
Second Edition. Freeman, 2000.
 H. Lilliefors(1969). “On the KolmogorovSmirnov Test for Normality with Mean and
Variance Unknown.” Journal of the American
Statistical Association 64, pp. 387-389.
 M. A. Stephens (1974). “EDF Statistics for
Goodness of Fit and Some Comparisons.” Journal
of the American Statistical Association 69, pp.
730-737.