STAT 651
Lecture 10
Copyright (c) Bani Mallick
1
Topics in Lecture #10



Comparing two population means using
rank tests
Comparing two population variances using
Levene’s test
The effect of outliers
Copyright (c) Bani Mallick
2
Book Sections Covered in Lecture #10



Chapter 6.3 (Wilcoxon Test)
Page 368 (Levene’s test, although it is
called Levine’s test): This is slightly
different from what SPSS does
The material on outliers is from my own
notes
Copyright (c) Bani Mallick
3
Lecture 9 Review: Comparing Two
Populations
X1  X 2

Difference of sample means

The s.d. from repeated sampling is
σ σ
+
n1 n 2
2
1

2
2
You need reasonably large samples from
BOTH populations
Copyright (c) Bani Mallick
4
Lecture 9 Review: Comparing Two
Populations

If you can reasonably believe that the
population sd’s are nearly equal, it is
customary to pick the equal variance
assumption and estimate the common
standard deviation by
sp 
(n1  1)s  (n 2  1)s
n1  n 2  1
2
1
Copyright (c) Bani Mallick
2
2
5
Lecture 9 Review: Comparing Two
Populations

The standard error then of
the value
sp

X1  X 2
is
1 1

n1 n 2
The number of degrees of freedom is
n1  n 2  2
Copyright (c) Bani Mallick
6
Lecture 9 Review: Comparing Two
Populations

A (1a)100% CI for
μ1 -μ 2
±
t
(n
+n
-2)s
α/2
1
2
p
X1  X 2

is
1 1
+
n1 n 2
Note how the sample sizes determine the CI
length
Copyright (c) Bani Mallick
7
Lecture 9 Review: Comparing Two
Populations

The CI can of course be used to test
hypotheses
H0 :μ1 =μ 2 vs Ha :μ1¹μ 2

This is the same as
H0 :μ1 -μ 2 =0 vs Ha :μ1 -μ 2 ¹0

So we just need to check whether 0 is in the
interval, just as we have done
Copyright (c) Bani Mallick
8
Lecture 9 Review: Comparing Two
Populations

Generally, you should make your sample sizes
nearly equal, or at least not wildly unequal.
Consider a total sample size of 100
X1  X2  ta /2 (n1 +n 2 -2)s p



1 1

n1 n 2
1 1

n1 n 2
= 1 if n1 = 1, n2 = 99
= 0.20 if n1 = 50, n2 = 50
Thus, in the former case, your CI would be 5
times longer!
Copyright (c) Bani Mallick
9
Lecture 9 Review: NHANES
Comparison
Mean(Healthy) – Mean(Cancer)

The 95% CI is from 0.0065 to 0.5223
Confidence Interval
0=
Hypothesized
value
0.065
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0.5223
10
Arsenic and Squamous Cell Skin
Cancer


The question is whether arsenic ingestion is
related to squamous call carcinoma
We used the transformation
X = log(0.005 + toe arsenic)
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11
Arsenic and Squamous Cell Skin
Cancer
0. 00
Healthy
Percent
15%
10%
5%
0%
1. 00
Cancer
Percent
15%
10%
5%
0%
-5.00
-3.00
-1.00
1.00
log(0.005 + Toe Ar s e nic Le ve l)
Copyright (c) Bani Mallick
12
Arsenic and Squamous Cell Skin
Cancer
2
1
1
0
log(0.005 + Toe Arsenic)
-1
3
5
7
12
13
14
18
20
2
4
6
9
8
10
11
15
16
17
19
21
806
807
804
805
808
-2
-3
-4
-5
N=
284
524
Cancer
Healthy
Squamous Cell Carcinoma Status
Copyright (c) Bani Mallick
13
Arsenic and Squamous Cell Skin
Cancer, Healthy Cases
Normal Q-Q Plot of log(0.005 + Toe Arsenic)
0
-1
Expected Normal Value
-2
-3
-4
-5
-5
-4
-3
-2
-1
0
Observed Value
Copyright (c) Bani Mallick
14
Arsenic and Squamous Cell Skin
Cancer: Cancer Cases
Normal Q-Q Plot of log(0.005 + Toe Arsenic)
0
-1
Expected Normal Value
-2
-3
-4
-5
-5
-4
-3
-2
-1
0
1
Observed Value
Copyright (c) Bani Mallick
15
Arsenic and Squamous Cell Skin
Cancer

Healthy, s = 0.59, IQR = 0.69

Squamous, s = 0.62, IQR = 0.71

These statistics and box plots indicate that
the two populations do not have vastly
different variability.
Copyright (c) Bani Mallick
16
Arsenic and Squamous Cell Skin
Cancer


Healthy: mean = -2.33, n = 215,
se = 0.040
Squamous: mean = -2.3365, n = 140,
se = 0.052

Mean difference = 0.020, se = 0.066

95% CI= [-0.109, 0.149]

p = 0.76: what does this mean?
Copyright (c) Bani Mallick
17
Arsenic and Squamous Cell Skin
Cancer

Graphs, statistics, CI, p-value, all tell us that
not much seems to be going on!
Copyright (c) Bani Mallick
18
Robust Inference via Rank Tests



Because sample means and standard
deviations are sensitive to outliers, so too are
comparisons of populations based on them
Rank tests form a robust alternative, that can
be used to check the results of t-statistic
inferences
You are looking for major discrepancies, and
then trying to explain them
Copyright (c) Bani Mallick
19
Robust Inference via Rank Tests



Rank tests are very easy to compute, and
SPSS provides them.
Typically called the Wilcoxon rank sum test
The algorithm is to assign ranks to each
observation in the pooled data set

Then apply a t-test to these ranks

Robust because ranks can never get wild
Copyright (c) Bani Mallick
20
Robust Inference via Rank Tests

Here is how data are ranked
Data
#1
#2
Ranks
#1
1

#2
4

-3 7
28 44
22
50
45
55
81
56
2
3
6
10
5
7
8
9
Now run a t-test
Copyright (c) Bani Mallick
21
Robust Inference via Rank Tests



The rank tests give the same answer no
matter whether you take the raw data, their
logarithms or their square roots.
If you have data (raw or transformed) that
pass q-q plots tests, then Wilcoxon and t-test
should have much the same p-values
In this case, you can use the latter to get CI’s
Copyright (c) Bani Mallick
22
Robust Inference via Rank Tests

Differences between rank and t-tests occur
for two reasons generally: outliers and very
non-bell shaped histograms
Copyright (c) Bani Mallick
23
Robust Inference via Rank Tests


In SPSS, you can get Wilcoxon rank sum tests
as follows (SPSS calls them Mann-Whitney
U)
“Analyze”, “Nonparametric Tests”, “2
independent samples”
Copyright (c) Bani Mallick
24
Robust Inference via Rank Tests

Toe Arsenic log(0.005 + Toe Arsenic)

Note how p-values are the same (= 0.468)
a
Test Statistics
Mann-Whitney U
Wilcoxon W
Z
Asymp. Sig. (2-tailed)
log(0.005 +
Toe Arsenic)
14364.500
24234.500
-.725
.468
Toe Arsenic
14364.500
24234.500
-.725
.468
a. Grouping Variable: Squamous Cancer Status
Copyright (c) Bani Mallick
25
Robust Inference via Rank Tests,
NHANES



Saturated Fat p-values: t-test = 0.057 ,
rank test = 0.014
Log(Saturated Fat):
rank test = 0.014
t-test = 0.012,
Note how the transform, which is more bellshaped, agrees more closely with the rank
test!
Copyright (c) Bani Mallick
26
Robust Inference via Rank Tests


An SPSS peculiarity: to do rank tests, you
need to have defined a numeric variable that
categorizes the groups.
The alternative is to convert the data to
numbers and then give value labels.
Copyright (c) Bani Mallick
27
Inference for Equality of Variances




We have described situations that comparing
variability of populations is desired.
Ott and Longnecker (Chapter 7) give methods
for comparing population variances
NEVER USE THESE METHODS
They are notoriously unreliable, affected by
outliers, non-perfectly bell shaped, etc.
Copyright (c) Bani Mallick
28
Inference for Equality of Variances

SPSS uses what is called Levene’s test

From the SPSS Help file (slightly edited)

Levene Test

For each case, it computes the absolute
difference between the value of that case and
its cell mean and performs a t-test on those
absolute differences.
Copyright (c) Bani Mallick
29
Inference for Equality of Variances



Levene Test
For each case, it computes the absolute
difference between the value of that case and
its cell mean and performs a t-test on those
absolute differences.
This is a reasonable test, although I prefer to
use a rank test instead of the t-test
Copyright (c) Bani Mallick
30
Inference for Equality of Variances



I suggest that you supplement the Levene
test with a look at the IQR in boxplots
If you really need to understand scientifically
the question of equality of variance, I suggest
that you consult a bona-fide statistician
I’ll now illustrate Levene’s test using NHANES
(and this is the last time for these data)
Copyright (c) Bani Mallick
31
Inference for Equality of Variances:
Note the outlier
140
120
119
100
80
60
60
118
Saturated Fat
40
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
32
Inference for Equality of Variances
6
5
119
4
3
2
1
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
33
Inference for Equality of Variances

P-value of Levene’s test for Saturated Fat =
0.378

Same P-value, but without the outlier = 0.010

P-value for log(Saturated Fat) = .667


P-Value for Levene’s test for Saturated Fat
when using the rank test instead of the t-test
= 0.039
P-Value for Levene’s test for log(Saturated
Fat) when using the rank test instead of the
t-test = 0.665
Copyright (c) Bani Mallick
34
Inference for Equality of Variances



As you can see, the rank test version of
Levene’s test gives answers much more in
keeping with the box plots
The problem was clearly the outlier, so you
can expect trouble with Levene’s test if there
is a massive outlier
Remember, t-tests have trouble with outliers
Copyright (c) Bani Mallick
35
Inference for Equality of Variances



As you can see, the rank test version of
Levene’s test gives answers much more in
keeping with the box plots
The problem is that it’s a pain to compute the
rank test version in SPSS
However, theory says that the rank test
version is the better, so in exams I’ll give it to
you.
Copyright (c) Bani Mallick
36
The Effect of an Outlier
What will happen to the sample mean
For cancer cases if I remove the
outlier?
140
120
119
100
80
60
60
118
Saturated Fat
40
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
37
The effect of anoutlier
What will happen to the sample
mean for the cancer cases if I
remove the outlier?
It will decrease
140
120
119
100
80
60
60
118
Saturated Fat
40
What will happen to the sample
standard error for the cancer
cases if I remove the outlier?
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
38
The effect of anoutlier
What will happen to the difference
between the sample mean for
healthy cases and the same mean
for cancer cases, if I delete the
outlier?
140
120
119
100
80
60
60
118
Saturated Fat
40
What will happen to the sample
standard error for the cancer
cases if I remove the outlier?
It will decrease
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
39
The effect of anoutlier
What will happen to the difference
between the sample mean for
healthy cases and the same mean
for cancer cases, if I delete the
outlier? It will increase
140
120
119
100
80
60
60
118
Saturated Fat
40
What will happen to the sample
standard error of this difference
if I remove the outlier?
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Copyright (c) Bani Mallick
40
The effect of anoutlier
What will happen to the difference
between the sample mean for
healthy cases and the same mean
for cancer cases, if I delete the
outlier? It will increase
140
120
119
100
80
60
60
118
Saturated Fat
40
What will happen to the sample
standard error of this difference
if I remove the outlier?
It will decrease
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Therefore, what will happen to the p-value if I delete the outlier?
Copyright (c) Bani Mallick
41
The effect of anoutlier
What will happen to the difference
between the sample mean for
healthy cases and the same mean
for cancer cases, if I delete the
outlier? It will increase
140
120
119
100
80
60
60
118
Saturated Fat
40
What will happen to the sample
standard error of this difference
if I remove the outlier?
It will decrease
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Therefore, what will happen to the p-value if I delete the outlier?
It will get smaller
Copyright (c) Bani Mallick
42
The effect of anoutlier
With the outlier, p = 0.057
140
120
119
100
80
60
60
118
Saturated Fat
40
I remove the outlier, p = 0.002
20
0
-20
N=
59
60
Cancer
Healthy
Health Status
Therefore, what will happen to the p-value if I delete the outlier?
It will get smaller
Copyright (c) Bani Mallick
43