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Various statistical analysis methods including descriptive statistics, percentiles, m-estimation, tests of normality, group statistics, levene test of homogeneity of variances, detrended normal plot, and boxplot. It also discusses the conditions for using these methods and their interpretations.
Typology: Study notes
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1
The following notation is used throughout this chapter unless otherwise noted:
Let y 1 < K y (^) m be m distinct ordered observations for the sample and c 1 , K,cm be the corresponding caseweights. Then
cc (^) i ck k
i = = =
∑ 1
cumulative frequency up to and including yi
and
W cc (^) m ck k
m = = = =
∑ 1
total sum of weights.
Minimum and Maximum
min = y 1 , max=y (^) m
Range
range = y (^) m −y 1
Mean (^) b gy
y
c y
i i i
m
∑ 1
Confidence Interval for the Mean
lower bound SE upper bound SE
− −
y t y t
W W
α α
/ , / ,
2 1 2 1
where SE is the standard error.
Median
The median is the 50th percentile, which is calculated by the method requested. The default method is HAVERAGE.
Interquartile Range (IQR)
IQR = 75th percentile − 25th percentile, where the 75th and 25th percentiles are calculated by the method requested for percentiles.
Variance (^) e js^2
s W
ci y (^) i y i
m 2 2 1
=
∑ b^ g
Standard Deviation
s = s^2
5% Trimmed Mean (^) 0.
cc (^) k tc y (^) k W cc (^) k tc y (^) k c yi i i k
k 0 9 1 1 1 2
1 1 0 9 1 1 2 2 1
2
. =^. −^ +^ −^ −^ +
R S
|
T|
U V
|
W|
− e j e j ∑
where k 1 and k 2 satisfy the following conditions
cc (^) k 1 < tc ≤ cc (^) k 1 + 1 , W − cc (^) k 2 < tc ≤ W −cck 2 − 1
and
tc = 0 05. W
Note: If k 1 + 1 = k 2 , then T0 9 (^). =y (^) k 2
There are five methods for computation of percentiles. Let
tc 1 = Wp , tc 2 = (^) bW + (^1) gp
where p is the requested percentile divided by 100, and k 1 and k 2 satisfy
cc tc cc cc tc cc
k k k k
1 1 2 2
1 1 2 1
Then,
g
tc cc c
g tc cc
g
tc cc c
g tc cc
k k
k
k k
k
1
1 1
1 1
2
2 1
2 2
1 1
1
2 2
2
∗
∗
d i
d i
Let x be the pth percentile; the five definitions are as follows:
Waverage (Weighted Average at y (^) tc 1 )
x
y g g y g y g c g y g y g c
k k k k k k k
R
S
||
T
| |
∗ ∗ ∗
∗
∗
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
if if and if and
e j b g
Round (Observation Closest to tc 1 )
If c (^) k 1 + 1 ≥ 1 , then
x
y g y g
k k
R S
|
T|
∗
∗
1 1
1 1 2 1 1 1 2
if if
If c (^) k 1 + 1 < 1 , then
x
y g y g
k k
R S
| T|^ +
1 1
1 1 2 1 1 1 2
if if
Empirical (Empirical Distribution Function)
x
y g y g
k k
R S
|
T|
∗
∗
1 1
1 1 1
if if
and
L dc L W c L W c dc
1 2 3
∗ ∗ ∗ ∗
Then for every i, i = 1 2 3, , , find hi such that
cc (^) hi ≤ Li < cchi+ 1
and
a y a y a c
a y a y a c
y a
i
i h i h i h
i h i h i h
h i
i i i
i i i
i
R
S
| |
T
| |
∗ ∗
∗
∗
∗
1 1
1 1
1
e j b g
if and
if and
if
where
a L cc
a a c
i i h
i
i h
i
i
∗
∗
1
M-Estimation (Robust Location Estimation)
The M-estimator T of location is the solution of
c y T i (^) s
i i
m ΨΨΨΨ F − HG^
I KJ^
=
∑ 0 1
where ΨΨΨΨ is an odd function and s is a measure of the spread.
An alternative form of M-estimation is
c y T s
y T i (^) s
i i i
m F − HG^
I KJ^
F − HG^
I KJ^
=
∑^ ω^0 1
where
u u
b g
After rearranging the above equation, we get
c y y T s
c y T s
i i
i i
m
i i i
= m
F − H G I K J
F^ − H G I K J
=
=
∑
∑
1
1
Therefore, the algorithm to find M-estimators is defined iteratively by
c y y T s
c y T s
k
i i
i k i
m
i i k i
=
=
F − H G I K J
F^ − H G I K J
∑
∑
1
1
1
The algorithm stops when either
Tk (^) + 1 − T (^) k ≤ ε (^) b Tk (^) + 1 +Tkg 2 , where ε = 0 005.
or the number of iterations exceeds 30.
Andrew’s Wave ( c ), c > 0
u
c u
u c
u c
u c
i
i
i i
i
b g =
F H
I K
R
S
||
T
| |
sin if
0 if
Tukey’s Biweight ( c )
u c
u c
u c
i
i i
i
b g =
F H G
I K J ≤
R
S
| |
T
| |
2 2
2 if
if
By default, c = 4.685.
Shapiro-Wilk Statistic ( W )
Since the W statistic is based on the order statistics of the sample, the caseweights have to be restricted to integers. Hence, before W is calculated, all the caseweights are rounded to the closest integer and the series is expanded. Let ci^ ∗^ be the closest integer to ci ; then
cc (^) i c (^) k W cc c k
i s m k k
m ∗ ∗ =
= (^) ∑ = =∑ 1 1
The original series y = (^) ly 1 , K,y (^) mq is expanded to
x = (^) ox 1 , K,x (^) wst
where
x (^) cc x (^) cc y (^) i i m i − i
1 1
Then the W statistic is defined as
a x
x x
i i i
W
i i
W
s
s
F
H
G G
I
K
J J
=
=
∑
∑
1
2
2
1
b g
where
x
x
W
a a
a a
a a
a c m i W
m
i W
c
m
i i
W
s
W
s s
s
s s
s
W W
i i s
i s
s
i
s
s
s s
R
S
| | |
T
| | |
F HG^
I KJ
=
−
1
1
2 2
1 1
2
2
1
2
10
2
2
b g db g i
db g i b g
b g
if
if
where is the c.d.f. of a standard normal distribution
log
K
1
−
i a^ i
Ws
d i
The following formula is used to estimate the critical value Dc for probability
0.1.
b b ac c (^) a
F− − − H
I K
where, if W ≤ 100 ,
a W
b W
c W W
b g
If
2 W > 100
a W b W c
0 98 0 49
. .
The Lilliefors significance p is calculated as follows:
If D (^) a = Dc , p= 01..
If D (^) a > D (^) c , p = exp (^) {aD a^2 + bD (^) a+ c−2 3025851. }.
If D0 2 (^). ≤ D (^) a < Dc, linear interpolation between D0 2. and Dc where D0 2. is the
critical value for probability 0.2 is done.
If D (^) a > D (^) 0 2. , pis reported as > 0 2..
2 This algorithm applies to SPSS 7.0 and later releases. To learn about algorithms for previous releases, call SPSS Technical Support.
Assume that there are k kb ≥ (^2) g combinations of grouping factors. For every
combination i, i = 1 2, , K ,k, let (^) o y (^) i 1 , K,yimit be the sample observations with
the corresponding caseweights (^) oc (^) i 1 , K,cimit.
If a transformation value, a, is given, the spread(s) and level(l) are defined based on the transformed data. Let x be the transformed value of y; for every i = 1 , K, k , j = 1 , K,mi
x
y a y ij
ij ij
= (^) a
R = S
|
T|
ln if otherwise
Then the spread (^) b gsi and the level (^) b gli are respectively defined as the Interquartile
Range and the median of (^) ox (^) i 1 , K,ximit with corresponding caseweights
o^ c (^) i^1 ,^ K,cimit. However, if^ a^ is not specified, the spread and the level are natural logarithms of the Interquartile Range and of the median of the original data. Finally, the slope is the regression coefficient of s on l, which is defined as
l l s s
l l
i i i
k
i i
k
=
=
∑
∑
d ib g
d i
1 2
1
In some situations, the transformations cannot be done. The spread-versus-level plot and Levene statistic will not be produced if:
The significance of La is calculated from the F distribution with degrees of freedom k − 1 and W − k. Groups with zero variance are included in the test.
With the current version of Levene’s test La, the followings can be considered as options in order to obtain robust Levene’s tests:
Median calculation is done by the method requested. The default method is HAVERAGE. Once the ~x (^) i ’s and hence z (^) li(b^ )^ ’s are calculated, apply the formula for La, shown in the section above, to obtain Lb by replacing zil, z (^) i and z with z (^) li(^ b)^ , z (^) i(^ b)^ and z (b^ )^ respectively.
Two significances of Lb are given. One is calculated from a F-distribution with degrees of freedom k - 1 and W - k. Another is calculated from a F- distribution with degrees of freedom k - 1 and v. The value of v is given by:
v =
u
u v
i i
k
i i i
k
=
=
∑
∑
1
2
2
1
where
ui = c (^) il z (^) ilb zib l
mi ( (^ )^ − (^ )) =
∑
2 1
in which
z (^) i(^ b)^ =
c z w
il il
b
l i
mi ( )
=
∑ 1
and
v (^) i = wi - 1.
Once the Ti, .0 9 ’s and hence z (^) il(^ c)^ ’s are calculated, apply the formula of La to obtain Lc by replacing zil, z (^) i and z with z (^) li(^ c)^ , z (^) i(c^ )^ and z (^ c)^ respectively.
The significance of Lc is calculated from a F-distribution with degrees of freedom k - 1 and W - k.
For every distinct observation y (^) i , Riis the rank (the mean of ranks is assigned to ties). The normal score NS (^) i is calculated by
i = i
F H G
I K J
where ΨΨΨΨ −^1 is the inverse of the standard normal cumulative distribution function. The NPPLOT is the plot of (^) b y 1 , NS (^1) g, K, (^) b y (^) m ,NSmg.
The detrended normal plot is the scatterplot of (^) b y 1 , D 1 g, K, (^) b y (^) m ,Dmg, where Di is the difference between the Z-score and normal score, which is defined by
Di = Z (^) i −NSi
and
Dallal, G. E., and Wilkinson, L. 1986. An analytic approximation to the distribution of Lilliefor’s test statistic for normality. The American Statistician, 40 (4): 294–296 (Correction: 41 : 248).
Frigge, M., Hoaglin, D. C., and Iglewicz, B. 1987. Some implementations for the boxplot. In: Computer Science and Statistics Proceedings of the 19 th Symposium on the Interface, R. M. Heiberger and M. Martin, eds. Alexandria, Va.: American Statistical Association.
Glaser, R. E. (1983). Levene’s Robust Test of Homogeneity of Variances. Encyclopedia of Statistical Sciences 4. NY: Wiley, p608-610.
Hoaglin, D. C., Mosteller, F., and Tukey, J. W. 1983. Understanding robust and exploratory data analysis. New York: John Wiley & Sons, Inc.
Hoaglin, D. C., Mosteller, F., and Tukey, J. W. 1985. Exploring data tables, trends, and shapes. New York: John Wiley & Sons, Inc.
Lilliefors, H. W. 1967. On the Kolmogorov-Smirnov tests for normality with mean and variance unknown. Journal of the American Statistical Association, 62: 399 – 402.
Loh, W. Y. (1987). Some Modifications of Levene’s Test of Variance Homogeneity. Journal of Statistical Computation and Simulation, 28, p213-
Tukey, J. W. 1977. Exploratory data analysis. Reading, Mass.: Addison-Wesley.
Velleman, P. F., and Hoaglin, D. C. 1981. Applications, basics, and computing of exploratory data analysis. Boston: Duxbury Press.
