http://www.tufts.edu/~gdallal/sizenotes.htm
*What does pairing really do?
**Estimates of the standard deviation of a Single Measurement
dois temas relacionados com a leitura do artigo ‘Measurement error”
de J Martin Bland e Douglas G Altman,
em Statistics Notes: Measurement error
BMJ 1996;313:744.1
http://www.jerrydallal.com/LHSP/ci_means/paired.htm
Whether data are independent samples or paired, the best estimate of the difference
between population means is the difference between sample means.
When the data are two independent samples of size n with approximately equal
sample standard deviations (sx ≈ sy ≈ s), a 95% confidence interval for the population mean
difference, μx - μy é : x- y, is :
ou μx - μy = (média de X – média de Y) ±1.96 s SQRT(2/n)
Now suppose the data are n paired samples ((Xi,Yi): i=1,..,n) where the sample
standard deviations of the Xs and Ys are roughly equal (sx ≈ sy ≈ s and the correlation
between X and Y is r. A 95% confidence interval for the population mean difference,
μx - μy ou
x-
y,
is:
μx - μy = (média de X – média de Y) ±1.96 s SQRT{2(1-r)/n)}
If the two responses are uncorrelated--that is, if the correlation coefficient is 0--the
pairing is ineffective.
The confidence interval is no shorter than it would have been had the investigators not
taken the trouble to collect paired data. On the other hand, the stronger the correlation, the
narrower the confidence interval and the more effective was the pairing.
This formula also illustrates that pairing can be worse than ineffective. Had the
correlation been negative, the confidence interval would have been longer than it would have
been with independent samples.
Copyright © 2001 Gerard E. Dallal
Last modified: 07/16/2008 00:47:55.
.............................................................................................................................................
2
Estimates of the standard deviation of a Single Measurement
http://www.tufts.edu/~gdallal/sizenotes.htm
Estimating the within group standard deviation, ,
When the Response Is a Difference
When the response being studied is change or a difference, the sample size formulas
require the standard deviation of the difference between measurements, not the standard
deviation of the individual measurements.
It is one thing to estimate the standard deviation of total cholesterol when many individuals
are measure once;
it is quite another
to estimate the standard deviation of the change in cholesterol levels when changes are measured.
One trick that might help: Often a good estimate of the standard deviation of the differences is
unavailable, but we have reasonable estimates of the standard deviation of a single measurement.
The standard deviations of the individual measurements will often be roughly equal. Call that standard
deviation . Then, the standard deviation of the paired differences is equal to
 (2[1-]), se n = 1
μx - μy = (média de X – média de Y) ±1.96 s SQRT{2(1-r)/n)}
ou se n = 1 então vem
z = {(diferença de médias amostrais) – (diferença de médias populacionais)} / (DP das diferenças)
μx - μy = é zero, então, teremos: (média de X – média de Y) ±1.96  SQRT{2(1-r)/1)}
(média de X – média de Y) ±1.96  SQRT{2(1-r)/1)} =
(média de X – média de Y) ±1.96  (2[1-])
where  is the correlation coefficient when the two measurements are plotted against each other.
If the correlation coefficient is a not terribly strong 0.50, the standard deviation of the
differences will be equal to  and (the standard deviation of the differences) gets smaller as the
correlation increases.
……………………………………………………………………………………………
3
Exemplo numérico da relação entre DP das diferenças em relação ao DP das medidas individuais
repetidas = Erro do Método
DP difª = DP individual vezes raiz quadrada de {2 vezes (1-coef. de correlação)}
valores fictícios
Inicial
1ª
leitura
8
9
14
19
22
25
28
32
33
15
Inicial
2ª
leitura
12
15
16
21
26
27
32
36
39
21
Inicial
média
10
12
15
20
24
26
30
34
36
18
Final
1ª
leitura
11
10
17
20
25
28
29
34
37
18
Final
2ª
leitura
15
16
19
24
29
30
33
36
45
22
Final
média
FinalInicial
13
13
18
22
27
29
31
35
41
20
3
1
3
2
3
3
1
1
5
2
Correlations: Inicial; Final
Pearson correlation of Inicial and Final = 0,991
P-Value = 0,0001
...........................................
Paired T-Test and CI: Inicial; Final
Paired T for Inicial - Final
Inicial
Final
Difference
N
10
10
10
Mean
22,50
24,90
-2,400
StDev
9,03
9,33
1,265
T-Test of mean difference = 0 (vs not = 0):
T-Value = -6,00 P-Value = 0,001
Variable
Final-Inicial
N
10
Mean
2,400
StDev
1,265
……………………………………………………………………………………….
DP das diferenças = 1,265 = DP erro medidas individuais (2√(1-r)) é a fórmula
DP erro medidas individuais = 1,265 / [ 2 vezes SQRT (1-0,991) = 1,265 / [ 2 vezes SQRT (0,009)] =
= 1,265 / [ 2 x 0,09486833 ] = 1,265 / 0,189736 = 6,668 valor da fórmula
Statistix 9,1
RM ANOVA 1 fator repetido
Analysis of Variance Table for Dados
Source
R
Tempo
Error R*Grupos
Error
Total
DF
9
1
9
20
39
SS
2354
58
8
860
3280
MS
262
58
1
43
F
P
61,71
0,0001
4
DP erro medidas individuais = raiz quadrada de MS = raiz quadrada de 43 = 6,557 valor obtido via Tabela
ANOVA
conclusão: o DP
medidas individuais
= 6,668 valor da fórmula é “roughly equal” ao valor de 6,557
obtido via Tabela ANOVA
conclusão: é válida essa fórmula apresentada pelo Gerard E. Dallal
The standard deviations of the individual measurements will often be roughly equal. Call that standard
deviation . Then, the standard deviation of the paired differences is equal to
 (2[1-]), se n = 1
..........................................................................................................................................
5
Disposição de entrada com os valores das duas leituras inicial e final, no programa Statistix, para
se obter o DP erro de medidas individuais, via modelo RM ANOVA
Dados
8
9
14
19
22
25
28
32
33
15
12
15
16
21
26
27
32
36
39
21
11
10
17
20
25
28
29
34
37
18
15
16
19
24
29
30
33
36
45
22
Tempo
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Antes
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
Depois
R
(10 pacientes com duas
leituras)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10