Transcript Measurement of Physical Function

Group Comparisons
Part 2
Robert Boudreau, PhD
Co-Director of Methodology Core
PITT-Multidisciplinary Clinical Research Center
for Rheumatic and Musculoskeletal Diseases
Core Director for Biostatistics
Center for Aging and Population Health
Dept. of Epidemiology, GSPH
Flow chart for group
comparisons
Measurements to be compared
continuous
discrete
( binary, nominal, ordinal with few values)
Distribution approx normal
or N ≥ 20?
No
Yes
Non-parametrics
T-tests
Chi-square
Fisher’s Exact
Covered in Part 1
Continuous Distributions
 Normal distribution
Mean
 Standard deviation
(computation, interpretation )
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Empirical Rule: 68%, 95%, 99%
Confidence Intervals, t-distribution
(computation, interpretation )
Empirical rule for interpreting SD
in normal distributions
2 SD
= 95.4%
1.96 SD
= 95%
(exact)
Confidence Interval
for a Population Mean
Mean:
Standard error of the mean*:
* Standard error is general term for standard deviation of some estimator
Degrees of Freedom (DF) = n-1
Example:
n=19, df=18
t “critical value” is wider than normal z (1.96)
because standard deviation (s) is an estimate (data)
/2= 0.025 for 95%CI
 estimation “noise” is accounted for
1.96  Normal dist (limit)
Aflatoxin levels of raw peanut kernels
n= 15
df=14 (=n-1)
16, 22, 23 26, 26, 27, 28, 30, 31, 35, 36, 37, 48, 50, 52
t 0.025,14= 2.145
95% C.I: 32.47 ± 2.145*(10.63/√15)
= 32.47 ± 2.145*2.744
= 32.47 ± 5.89
= 10.63/√15 = 2.744
95% C.I: (26.58, 38.36)  
Outline For Today
Continuous Distributions
 Comparing 2-groups
Two-sample t-test
 pooled and unequal variance versions
 Wilcoxon Rank-Sum (non-parametric)

Next Didactic
 Comparing > 2 groups
ANOVA, Kruskal-Wallis
 Adjusting for Multiple Comparisons
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
Categorical data

Binomial distr, Chi-squares, Fisher’s Exact
2-sample independent t-test
for comparing means of two groups
General Formula: stdev = √ variance
If comparing two estimates from independent samples
 Variance(of difference) = sum of variances

e.g. compare mean TNFα cytokine levels at first visit
in RA (n=20) vs JRA (n=17) patients
Note also: Variance(of sum) = sum of variances (if independent)
2-sample t-test
to compare two groups
Case 1: Equal variances
(i.e. SDs similar, not too
different between groups)
“pooled” variance estimate
df = n1 + n2 – 2
Note: df= (n1-1) + (n2-1)
2-sample t-test
to compare two groups
Case 2: Unequal variances (i.e. SDs very different
between the groups)
denom = Stderr of numerator
D.F = Welch-Satterthwaite equation (best approx df)
Does Cell Phone Use While Driving
Impair Reaction Times?
Sample of 64 students from Univ of Utah
 Randomly assigned: cell phone group or control
=> 32 in each group
 On machine that simulated driving situations:
=> at irregular periods a target flashed red or green
 Participants instructed to hit “brake button” as soon as
possible when they detected red light
 Control group listened to radio or to books-on-tape
 Cell phone group carried on conversation about a
political issue with someone in another room
Outlier
(Milliseconds)
Does Cell Phone Use While Driving Impair
Reaction Times ? (2-sided t-test)
Cell Phone
Control
-------Difference
N
32
32
Mean
585.2
533.7
51.5
SD
89.6
65.3
}
SDs quite
different
(~ 0.05 sec)
= sqrt(89.62/32+65.32/32)=19.6
= 56.685
t = 51.5/19.6 = 2.63, p=0.011
Does Cell Phone Use While Driving Impair
Reaction Times ? (2-sided t-test)
Cell Phone
Control
-------Difference
N
32
32
Mean
585.2
533.7
51.5
SD
89.6
65.3
}
SDs quite
different
(~ 0.05 sec)
t = 51.5/19.6 = 2.63, p=0.011
area=0.011/2=0.0055
Half of area
on each side
area=0.011/2=0.0055
-2.63
2.63
Half of area
on each side
Removing the one high outlier
from cell phone group
Cell Phone
Control
-------Difference
N
31
32
Mean
573.1
533.7
SD
58.9 (“equal
65.3 variances”)
39.4
= 62.69
(pooled var)
df= n1+n2-2 = 61
t = 39.4/(62.69*√(1/31+1/32)) = 2.52 (p=0.015)
Three-dimensional and thermal surface imaging
produces reliable measures of joint shape and
temperature: a potential tool for quantifying arthritis
Steven J Spalding, C Kent Kwoh, Robert Boudreau,
Joseph Enama, Julie Lunich, Daniel Huber, Louis Denes
and Raphael Hirsch
Arthritis Research & Therapy 2008
3-D (Laser) & Thermal Imaging
JIA patient
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9-year-old female with anti-nuclear antigen
(ANA)-negative and rheumatoid factor (RF)negative polyarticular JIA who presented with
left wrist pain, warmth, and swelling.
Proceeded with intra-articular steroid injection
Imaged before procedure, then 5 days after
Reduction in volume of 2 ml (10% decrease)
No significant change in SDI
HDI values changed from 1.9°C prior to the
injection to 1.1°C after
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
T-test (2-sample independent)
or Wilcoxon Rank-Sum (aka Mann-Whitney)
…………...
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
T-test (2-sample independent)
or Wilcoxon Rank-Sum (aka Mann-Whitney)
Control
(n=10)
Arthritis
(n=9)
1.2
1.4
1.1
2.4
1.0
2.3
1.2
2.1
0.6
3.0
0.5
1.1
1.0
1.4
1.0
1.3
1.3
1.1
1.2
Mean
1.01
1.79
SD
0.26
0.70
Median
1.05
1.40
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
T-test (2-sample independent)
“pooled” df = 10+9-2=17
T-Tests
Variable
Method
Variances
HDI
HDI
Pooled
Satterthwaite
Equal
Unequal
DF
t Value
Pr > |t|
17
10.2
3.36
3.23
0.0037
0.0089
Test for Equality of Variances
Variable
Method
HDI
Folded F
Num DF
Den DF
F Value
Pr > F
8
9
6.60
0.0105
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
Wilcoxon Rank-Sum (aka Mann-Whitney)
The idea/motivation:
 Method should work for any distribution
 non-parametric
 Base statistical test on ranks
 rank = order when all data is sorted from
lowest to highest
 each group then gets a “rank sum”
 Won’t be affected by outliers
 Like all statistical tests, p-value is based on
distribution (of difference in rank-sums here)
assuming there is no difference between the groups
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
Wilcoxon Rank-Sum (aka Mann-Whitney)
Base statistical test on ranks
 each group gets a “rank sum”
 Like all statistical tests, p-value is based on distribution
(of difference in rank-sums here)
assuming there is no difference between the groups
 just like shuffling cards
(with only two colors on cards; even if different n’s)
 the critical values are the “extreme” differences in
rank-sums between the two groups
(α = 0.05 => the most extreme 5% of differences )
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
Wilcoxon Rank-Sum (aka Mann-Whitney)
Wilcoxon Scores (Rank Sums) for Variable HDI
Classified by Variable Group
Group
Control
Arthritis
N
10
9
Sum of
Scores
64.50
125.50
Expected
Under H0
100.0
90.0
Std Dev
Under H0
12.172013
12.172013
Average scores were used for ties.
Mean
Score
6.45000
13.94444
HDI (Heat Distribution Index) of MCPs
10 adults controls vs 9 adults with active RA
Wilcoxon Rank-Sum (aka Mann-Whitney)
Wilcoxon Two-Sample Test
Statistic (S)
125.5000
Normal Approximation
Z
One-Sided Pr > Z
Two-Sided Pr > |Z|
2.8754
0.0020
0.0040
t Approximation
One-Sided Pr > Z
Two-Sided Pr > |Z|
0.0050
0.0101
Exact Test
One-Sided Pr >= S
Two-Sided Pr >= |S - Mean|
0.0012
0.0023
Z includes a continuity correction of 0.5.