monotonic relationship of two variables, x and y
TRANSCRIPT
Monotonic relationship of two variables, X and Y
0
4
8
12
16
0 1 2 3 4
Y
X
Deterministic monotonicity
If X growsthen
Ygrows
too
0
4
8
12
16
0 1 2 3 4
Y
X
Stochastic monotonicity
***
*
*
*
*
*
**
*
* *
*
*
*
*
If X growsthenlikely
Ygrows
too
Ss X Y 1 1 35 2 1.5 34 3 2 36 4 3 37 5 7 38 6 10 39
An example
Ss X rank Y rank 1 1 1 35 2 2 1.5 2 34 1 3 2 3 36 3 4 3 4 37 4 5 7 5 38 5 6 10 6 39 6
Rank data separately for X and Y
Spearman-s rank correlation (rS):
Correlation between ranksIn the above example:
r = 0.91, rS = 0.94
DiscordancyConcordancy
+
A
B
C
D X
Y
Concordancy and discordancy
pp
Kendall-s tau
p+: Proportion of concordantpairs in the population
p-: Proportion of discordantpairs in the population
1 +1 If X and Y are independent:
= 0: no stochastic monotonicity = deterministic
monotone decreasing (inreasing) relationship
Features of Kendall’s
p p
p p
A Kendall’s gamma
For discrete X and Y variables
1 +1 If X and Y are independent: = 0 = 0: no stochastic monotonicuty If = 1: p+ = 0
If = +1: p = 0
Features of Kendall’s
Testing the H0: = 0null hypothesis
Sample tau: Kendall’s rank correlation coefficient (r)
Testing stochastic monotonicity = testing the significancy of r
+
A
B
C
D X
Y
Computation of sample tau
++
C+
c = n = 4d = n= 2
r = (4-2) /(4+2)
= 2/6 = 0.33
c = # of concordanciesd = # of discordanciesT = # of total couples
= n(n-1)/2
r = (c - d)/T, = (c - d)/(c+d)
In which cases will r = ?
Formulea of r and
Ss X Y 1 1 35 2 1.5 34 3 2 36 4 3 37 5 7 38 6 10 39
An example
r(p < 0.02);
rS(p < 0.02);
r(p < 0.10);
Comparison of several Comparison of several independent samplesindependent samples
-60
-40
-20
0
20
40
60
80G
SR
-dec
reas
e
Agr1 Agr2 Agr3 Light Verbal
Groups
Normal Person. disorder
Holocaustgroup
0
0.5
1
1.5
2
2.5
Average Rorschach time (min)
Comparison of population means
H0: E(X1) = E(X2) = ... = E(XI)
H0: 1 = 2 = ... = I
One way independent sample ANOVA
SStotal = SSb + SSw
SStotal: Total variability
SSb: Between sample variability
SSw: Within sample variability
Basic identity
Varb = SSb/(I - 1) = SSb/dfb
- Treatment variance
Varw = SSw/(N - I) = SSw/dfw
- Error variance
One-way ANOVA
Test statistic: F = Varb/Varw
Treatment variance
1
)(
1
2
1
I
xxn
I
SSVar
I
iii
bb
Error variance
I
ii
I
iii
ww
df
Vardf
IN
SSVar
1
1
H0: 1 = 2 = ... = I
F = Varb/Varw ~ F-distribution
Assumptions of ANOVA
F F: reject H0 at level
+
Independent samplesNormality of the dependent variable
Variance homogeneity (identical population variances)
Assumptions of ANOVA
Welch test James test Brown-Forsythe test
Robust ANOVA’s
Levene test
O’Brien test
Testing variance homogeneity
Var1 Var2 ... VarI
or (and)
n1 n2 ... nI
Trust in the result of ANOVA
Different sample sizes
Substantially different sample variances
When to apply a robust ANOVA?
Conventional test: Tukey-Kramer test (Tukey’s HSD test)
Robust test: Games-Howell test
Post hoc analyses
Hij: i = j
Nonlinear coefficientof determination
Explained variance: eta2 = SSb/SStotal
Nonlinear correlationcoefficient: eta
SStotal = SSb + SSw
An exampleAn example
Agr1 Agr2 Agr3 Light Verb.
n i 5 4 6 4 4
xi 14.506.75 5.20 -13.45-30.08
s i 29.609.15 6.96 13.11 14.57
Levene test:
F(4, 7) = 0.784 (p > 0.10, n.s.)
O’Brien test:
F(4, 8) = 1.318 (p > 0.10, n.s.)
Testing variance homogeneity
Treatment var.: Varb = 1413.9 Error variance: Varw = 286.2
F(4, 18) = 1413.9/286.2= 4.940**
Nonlinear coeff. of determin.:eta2 = SSb/SStotal = 0.523
Conventional ANOVA
Welch test:W(4, 8) = 5.544*
James test:U = 27.851+
Brown-Forsythe test:BF(4, 9) = 5.103*
Robust ANOVA’s
Tukey-Kramer test: T12= 0.97 T13= 1.28T14= 3.48 T15= 5.55**T23= 0.20 T24= 2.39T25= 4.35* T34= 2.42T35= 4.57* T45= 1.97
Pairwise comparison of means