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If a WEIGHT variable is specified, it is used to replicate a case as many times as indicated by the weight value rounded to the nearest integer. If the workspace requirements are exceeded and sampling has been selected, a random sample of cases is chosen for analysis using the algorithm described in SAMPLE. For the RUNS test, if sampling is specified, it is ignored. The tests are described in Siegel (1956).
Note: Before SPSS version 10.1.3, the WEIGHT variable was used to replicate a case as many times as indicated by the integer portion of the weight value. The case then had a probability of additional inclusion equal to the fractional portion of the weight value.
One-Sample Chi-Square Test
Cell Specification
If the (lo, hi) specification is used, each integer value in the lo to hi range is designated a cell. Otherwise, each distinct value encountered is considered a cell.
Observed Frequencies b g O i
If (lo, hi) has been selected, every observed value is truncated to an integer and, if it is in the lo to hi range, it is included in the frequency count for the corresponding cell. Otherwise, a count of the frequency of occurrence of the distinct values is obtained.
Expected Frequencies b EXP i g
If none or EQUAL is specified,
EXP
N
i =^ k
number of observations [in range] number of cells
b g
b g
When the expected values b g E i are specified either as counts, percentages, or
proportions,
EXP E
E
i i N i i
= k
F
H
G
G
G
G
G
I
K
J J J J J =
∑ 1
If there are cells with expected values less than 5, the number of such cells and the minimum expected value are printed. If the number of user-supplied expected frequencies is not equal to the number of cells generated, or if an expected value is less than or equal to zero, the test terminates with an error message.
Chi-Square and Its Degrees of Freedom
2
1 1
=
∑
O EXP
EXP
df k
i i i i
k b g
The significance level is from the chi-square distribution with k − 1 degrees of freedom.
Poisson
meanb g λ =
=
∑ X^ i N i
N
1
The test is not done if, for the uniform, all data are not within the user-specified range or, for the Poisson, the data are not non-negative integers. If the variance of the normal or the mean of the Poisson is zero, the test is also not done.
Calculation of Theoretical Cumulative Distribution Functions
For Uniform
F 0 b X i g = Xi −
min max min
For Poisson
F X e i l
l
l
X (^) i 0 0
b g =
−
=
∑
If λ ≥ 100 000, , the normal approximation is used.
For Normal
F X F X^ X
i (^) S
0 b g^ =^ 0 1F i −
HG^
I
, KJ
where the algorithm for the generation of F 0 1 , b g Z is described in Appendix 1.
Calculation of Differences
For the Uniform and Normal, two differences are computed:
D F X F X
D F X F X i N
i i i i i i
$ − ~ (^) $ , ,
1 0 0 1
b g b g
b g b g K
For the Poisson: 1
D F X^ F X^ X^ i^ N X
D F X F X
i i^ i^ i i
i i i
= −^ −^ −^ >^ =
R
S|
T|
b g b g
b g b g
K
The maximum positive, negative, and absolute differences are printed.
Test Statistic and Significance
The test statistic is
Z = N max (^) i e Di , D ~ i j
The two-tailed probability level is estimated using the first three terms of the Smirnov (1948) formula.
if
if
0 27 1 1 2 506628^9
Z p
Z p (^) Z Q Q Q
. ,.^ e j
where Q = e −^ 1 233701.^ Z^ −^2.
(^1) This algorithm applies to SPSS 7.0 and later releases.
Number of Runs
For each of the data points, in the sequence in the file, the difference
Di = Xi − CUTPOINT
is computed. If Di ≥ 0 , the difference is considered positive, otherwise negative. The number of times the sign changes, that is, Di ≥ 0 and Di + 1 <0 , or Di < 0 and Di + 1 ≥0 , as well as the number of positive (^) d i n (^) p and (^) b g n (^) a signs, are
determined. The number of runs b g R is the number of sign changes plus one.
Significance Level
The sampling distribution of the number of runs b g R is approximately normal
with
r
p a p a
r
p a p a a p p a p a
n n n n
n n n n n n n n n n
2
d i d i d i
The two-sided significance level is based on
Z R^ r r
unless n < 50 ; then
Z
R R
R R
R
c
r r r r r r r
R
S|
T
b g
b g
if if if
Binomial Test
n 1 Number of observations in category 1 n 2 Number of observations in category 2 p (^) Test probability
m min b n 1 , n 2 g
N (^) n (^) 1 + n 2
p ∗^ p^ if^ m^ =^ n^ 1 , 1 −^ p^ if m^ = n 2
Two-tailed exact probability is
0
N
i
p p
N
m
p p i
m F i N^ i^ m N^ m HG
I KJ^
F
H
GG
I
K
JJ −^
F HG
I KJ^
=
∗ ∗ −^ ∗ ∗ − ∑ e^ j^ e^ j
If an approximate probability is reported, the following algorithm is used:
Z n^ Np Np p
Z n^ Np Np p P Zi
1 1 2 1
= +^ −
= −^ −
b g b g
b g probability from standard normal distribution
Then,
P X n P Z P Z P X n P Z
1 2 1 1 1
b g b g b g
b g b g
and the two-tailed approximate probability is
2 P X b ≤ n 1 g − P X b = n 1 g
Test Statistic and Significance Level
If n (^) p + nn ≤ 25 , the exact probability of r or fewer “successes” occurring in n (^) p + nn trials, when p = 0 5. and r = min (^) d n (^) p , nn i, is calculated recursively from the binomial
p X r n^ p^ inn i
r
≤ = F + n^ p^ nn
HG^
I
= KJ
b g (^) ∑ b g 0
If n (^) p + nn > 25 , the significance level is based on the normal approximation
Z
n n n n c n n
p n p n p n
max ,.. .
d i 0 5d^ i 0 5 0 5
A two-tailed significance level is printed.
Wilcoxon Matched-Pairs Signed-Rank Test
Computation of Ranked Differences
For each case, the difference
Di = X (^) i − Yi
is computed, as well as the absolute value of Di. All nonzero absolute differences are then sorted into ascending order, and ranks are assigned. In the case of ties, the average rank is used. The sums of the ranks corresponding to positive differences (^) d S (^) p i and negative differences b g S (^) n are calculated. The average positive rank is
X (^) p = S (^) p np
and the average negative rank is
X (^) n = S (^) n nn
where n (^) p is the number of cases with positive differences and n (^) n the number with negative differences.
Test Statistic and Significance Level
The test statistic is^2
Z
S S n n
n n n t t
p n
j j j
= (^) l
=
∑
min (^) d , i c b g h
b gb g (^) e j
1
where n Number of cases with non-zero differences l Number of ties t (^) j Number of elements in the j -th tie, j = 1, K, l
For large sample sizes the distribution of Z is approximately standard normal. A two-tailed probability level is printed.
Cochran’s Q Test
Computation of Basic Statistics
For each of the N cases, the k variables specified may take on only one of two possible values. If more than two values, or only one, are encountered, a message is printed and the test is not done. The first value encountered is designated a “success” and for each case the number of variables that are “successes” are
(^2) This algorithm applies to SPSS 7.0 and later releases.
2 1 2
=
∑
∑
Nk k C N k
T Nk k
l l
k c b gh b g
e j
where (^) ∑ T is the same as in Kendall’s coefficient of concordance (see Lehmann, 1985, p. 265).
The significance level is from the χ 2 distribution with k − 1 degrees of freedom.
Kendall’s Coefficient of Concordance
N , k , and l are the same as in Friedman, above.
Coefficient of Concordance (W)^4
W F
N k
N k k N k k N T
F HG^
I KJ^
F
H
G G
I
K
J (^1) ∑ J
2 2
b g^2
e j e j
where F = Friedman χ 2 statistic.
T t t l
k
i
N = − = =
∑ ∑ ∑
3 1 1
e j
with t = number of variables tied at each tied rank for each case.
Test Statistic and Significance Level
χ 2 = N k b − 1 g W
(^4) This algorithm applies to SPSS 7.0 and later releases.
The significance level is from the χ 2 distribution with k − 1 degrees of freedom.
The Two-Sample Median Test
Table Construction
If the median value is not specified by the user, the combined data from both samples are sorted and the median calculated.
Md
X X N
X
N N N
R + S| T|
2 2 1 1 2
e j^2
b g
if is even otherwise
where X (^) N is the largest value and X (^) 1 the smallest. The number of cases in each of the two groups which exceed the median are counted. These will be denoted as g 1 and g 2 , and the corresponding sample sizes as n 1 and n 2.
Test Statistic and Significance Level
- If N ≤ 30 , the significance level is from Fisher’s exact test. (See Appendix 5.)
- If N > 30 , the test statistic is
χ (^) c
g n g g n g N N g g n n g g n n
2 1 2 2 2 1 1
2
1 2 1 2 1 2 1 2
b g b g
b gb g
which is distributed as a χ 2 with 1 degree of freedom.
Z U^ n n n n N N
N N T
i i
F − −
H
GG
I
K
∑ JJ
1 2
1 2 3
b g
b g
which is distributed approximately as a standard normal. A two-tailed significance level is printed.
Kolmogorov-Smirnov Two-Sample Test
Calculation of the Empirical Cumulative Distribution Functions and Differences
For each of the two groups separately the data sorted into ascending order, from X (^) 1 to X (^) ni , and the empirical cdf for group i is computed as
F^ $ X
X X
j n X X X X X
i i (^) j j n
b g =
R
S
T
1 1 1
For all of the X (^) j values in the two groups, the difference between the two groups is
D (^) j = F $ 1^ d X (^) j i − F $ 2 (^) d Xj i
where F $ 1 (^) d X (^) j i is the cdf for the group with the larger sample size. The maximum positive, negative, and absolute differences are also computed.
Test Statistic and Level of Significance
The test statistic (Smirnov, 1948) is
Z D n n j j n n j
max 1 2 1 2
and the significance level is calculated using the Smirnov approximation described in the K-S one sample test.
Wald-Wolfowitz Runs Test
Calculation of Number of Runs
All observations from the two samples are pooled and sorted into ascending order. The number of changes in the group numbers corresponding to the ordered data are counted. The number of runs ( R ) is the number of group changes plus one. If there are ties involving observations from the two groups, both the minimum and maximum number of runs possible are calculated.
Significance Level
If n (^) 1 + n 2 , the total sample size, is less than or equal to 30, the one-sided significance level is exactly calculated from
P r R n n n
n r
n r r
R ≤ =
F +
HG^
I
KJ
F
HG^
I
KJ^
F
HG^
I
b g (^) ∑ = KJ
1
1 2
2
when R is even. When R is odd
P r R n n n
n k
n k
n k
n r k
R ≤ =
F +
HG^
I
KJ
F
HG^
I
KJ^
F
HG^
I
KJ^
F
HG^
I
KJ^
F
HG^
I
KJ
L
N
M
O
Q
P
=
b g (^) ∑
1
1 2 1 2 2
where
r = 2 k − 1.
For sample sizes greater than 30, the normal approximation is used (see RUNS test described previously).
K-Sample Median Test
Table Construction
If the median value is not specified by the user, the combined data from all groups are sorted and the median is calculated.
Md
if is even if is odd
R + S| T|
X X N
X N
N N N
2 2 1 1 2
e j^2
b g
where X (^) N is the largest value and X (^) 1 the smallest. The number of cases in each of the groups that exceed the median are counted and the following table is formed. Group 1 2 3 … k LE Md (^) O 11 O 12 O 13 ... O 1 (^) k R 1 GT Md (^) O 21 O 22 O 23 ... O 2 (^) k R 2 n 1 n 2 n 3 ...^ n (^) k N
Test Statistic and Level of Significance
The χ 2 statistic for all nonempty groups is calculated as
χ 2
2 1
2
1
= =
∑ ∑ Oij^ E^ ij^ Eij j i
k d i
where
E
R n ij N = i^ j.
The significance level is from the χ 2 distribution with k − 1 degrees of freedom,
where k is the number of nonempty groups. A message is printed if any cell has an expected value less than one, or more than 20% of the cells have expected values less than five.
Kruskal-Wallis One-Way Analysis of Variance
Computation of Sums of Ranks
Observations from all k nonempty groups are jointly sorted and ranked, with the average rank being assigned in the case of ties. The number of tied scores in a set of ties, t (^) i , is also found, and the sum of Ti = t (^) i^3^ − ti is accumulated. For each group the sum of ranks, Ri , as well as the number of observations, ni , is obtained.
Test Statistic and Level of Significance
The test statistic unadjusted for ties is
H
N N
R (^) i ni N i
k
=
∑
b g 1
b g
where N is the total number of observations. Adjusted for ties, the statistic is
=
∑
H H
Ti N N i
m 1 3 1
e j
where m is the total number of tied sets.
The significance level is based on the χ 2 distribution, with k − 1 degrees of
freedom.
