monotonic relationship of two variables, x and y
TRANSCRIPT
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Monotonic relationship of two variables, X and Y
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Y
X
Deterministic monotonicity
If X growsthen
Ygrows
too
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0 1 2 3 4
Y
X
Stochastic monotonicity
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If X growsthenlikely
Ygrows
too
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Ss X Y 1 1 35 2 1.5 34 3 2 36 4 3 37 5 7 38 6 10 39
An example
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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
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Spearman-s rank correlation (rS):
Correlation between ranksIn the above example:
r = 0.91, rS = 0.94
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DiscordancyConcordancy
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A
B
C
D X
Y
Concordancy and discordancy
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pp
Kendall-s tau
p+: Proportion of concordantpairs in the population
p-: Proportion of discordantpairs in the population
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1 +1 If X and Y are independent:
= 0: no stochastic monotonicity = deterministic
monotone decreasing (inreasing) relationship
Features of Kendall’s
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p p
p p
A Kendall’s gamma
For discrete X and Y variables
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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
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Testing the H0: = 0null hypothesis
Sample tau: Kendall’s rank correlation coefficient (r)
Testing stochastic monotonicity = testing the significancy of r
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+
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
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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
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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);
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Comparison of several Comparison of several independent samplesindependent samples
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-60
-40
-20
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80G
SR
-dec
reas
e
Agr1 Agr2 Agr3 Light Verbal
Groups
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Normal Person. disorder
Holocaustgroup
0
0.5
1
1.5
2
2.5
Average Rorschach time (min)
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Comparison of population means
H0: E(X1) = E(X2) = ... = E(XI)
H0: 1 = 2 = ... = I
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One way independent sample ANOVA
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SStotal = SSb + SSw
SStotal: Total variability
SSb: Between sample variability
SSw: Within sample variability
Basic identity
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Varb = SSb/(I - 1) = SSb/dfb
- Treatment variance
Varw = SSw/(N - I) = SSw/dfw
- Error variance
One-way ANOVA
Test statistic: F = Varb/Varw
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Treatment variance
1
)(
1
2
1
I
xxn
I
SSVar
I
iii
bb
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Error variance
I
ii
I
iii
ww
df
Vardf
IN
SSVar
1
1
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H0: 1 = 2 = ... = I
F = Varb/Varw ~ F-distribution
Assumptions of ANOVA
F F: reject H0 at level
+
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Independent samplesNormality of the dependent variable
Variance homogeneity (identical population variances)
Assumptions of ANOVA
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Welch test James test Brown-Forsythe test
Robust ANOVA’s
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Levene test
O’Brien test
Testing variance homogeneity
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Var1 Var2 ... VarI
or (and)
n1 n2 ... nI
Trust in the result of ANOVA
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Different sample sizes
Substantially different sample variances
When to apply a robust ANOVA?
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Conventional test: Tukey-Kramer test (Tukey’s HSD test)
Robust test: Games-Howell test
Post hoc analyses
Hij: i = j
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Nonlinear coefficientof determination
Explained variance: eta2 = SSb/SStotal
Nonlinear correlationcoefficient: eta
SStotal = SSb + SSw
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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
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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
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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
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Welch test:W(4, 8) = 5.544*
James test:U = 27.851+
Brown-Forsythe test:BF(4, 9) = 5.103*
Robust ANOVA’s
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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