statistics for biomedical engineering labs
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0555.4180
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List of Experiments
Exp. 1: Bone Mechanics
Exp. 2: Soft Tissue Mechanics
Exp. 3: Pressure Pulse Propagation in Arteries Exp. 4: Pressure Wave Velocity in Arteries
Exp. 5: Mechanical Properties of Bioresorbable
Polymers
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Requirements
Take midterm and final exams.
Attend all 5 laboratory sessions & submit all
background & Protocol and final reports (usepredefined formats).
Sign a safety form.
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Grading
Final grade:60% Experiments
20% Midterm Exam
20% Final Exam
Each experiment:
70% Final Lab Reports20% Initial Background & Protocol lab reports
10% Oral Quiz, Background & Protocol
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Basic Statistical Tools
Advanced Biomechanics Laboratory2011
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Why Statistics?
Easy way to sum up data.
Enables you to compare data from your
experiments.
Quantitatively and qualitatively assesses
relationships in your data.
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What is Statistics?
Allows quantitative analysis and calculation of
the uncertainty magnitude.
Estimation of the measurements uncertainty
after they are made.
Design experiments in an efficient process.
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Sources of Uncertainty Systematic uncertainty - fixed error
Occurs every time a measurement is made under identical conditions.
Limitations and accuracy of the equipment.
Random error Environmental variations.
Limitations of human sensing ability (unavoidable).
Noise in the process, random unstableness in the experimental
system.
Illegitimate error (unacceptable human mistakes)
Sloppy experimental technique.
Erroneous calculation. 8
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Common Statistics
Mean(the average) a measure of where the center of your
distribution lies. It is simply the sum of all observations divided
by the number of observations.
Median(2nd quartile or 50th percentile) the middle value.
50% of values are above and 50% below the median. If N is an
odd number, then the median is the middle value. if N is an
even number, then the median is half way between the two
middle values.
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Common Statistics Cont.
Standard Deviation (SD) a measure of how far theobservations in a sample deviate from the mean. It is
analogous to an average distance (independent of
direction) from the mean.
Variance the square value of SD.
1
)(2
2
n
xxSDVar
i
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Example
Measured data: 1, 4, 4, 4, 5, 18, 20
Mean=
SD =
Median = 4
Variance = SD2=58.33333
63762.7
17
)820()818()85()84(3)81(22222
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201854441
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Normal Distribution
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Normal Distribution & the SD
1 SD from the mean is ~68.2% Interval
2 SD from the mean is ~95.4% Interval
1.96 SD from the mean is 95% Interval
Mean Variance
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Degrees of Freedom (DF)
The number of independent data points available forcalculation.
For example, if we have ten data points (n=10), aftercalculating the mean, only nine will still beindependent. Therefore, there will be only nine (n-1)degrees of freedom for variance estimation.
1
)(2
2
n
xxSDVar
i
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Standard Error of the Mean(SE Mean)
SE Mean is an estimation of the dispersion that you
would observe in the distribution of sample means, if
you continued to take samples of the same size
from the population.
(N number of data points)
NSDSEMean /
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Confidence Intervals for The Mean
A confidence interval for a mean specifies a range of
values within which the unknown population parameter,
in this case the mean, may lie.
We interpret an interval calculated at a 95% level, as we
are 95% confident that the interval contains the true
population mean. We could also say that 95% of all
confidence intervals formed in this manner (from different
samples of the population) will include the true population
mean. 16
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Confidence Intervals for TheMean
In general, you compute the 95% confidence
interval for the mean with the following formula:
Lower limit = M - Z.95
m
Upper limit = M + Z.95m
Z.95 is the number of standard deviations extending
from the mean of a normal distribution required to
contain 0.95 of the area.
m is the standard error of the mean. 17
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Confidence Intervals - Example
Assume that the weights of 10-year old children are
normally distributed with a mean of 90 pounds and a
standard deviation of 36 pounds.
What is the sampling distribution of the mean for a
sample size of 9 (including the 95% confidence
interval for the mean)? 18
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Example Cont.
The sampling distribution of the mean has a mean of 90 and a standard
deviation of 36/3 = 12.
The middle 95% of the distribution is shaded
95% Interval = 1.96 SD
90 - (1.96)(12) = 90-23.52=66.48
90 + (1.96)(12) = 90+23.52= 113.52
If we compute the mean (M) from a sample, and create an interval ranging
from M - 23.52 to M + 23.52, this interval has a 0.95 probability of
containing 90 (the true population mean).
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Confidence Interval t distribution
CI- confidence interval around the mean
n
st
xCI
t is the confidence that the
user wants to develop in the
estimation.
DF 0.95 0.99
2 4.303 9.925
3 3.182 5.841
4 2.776 4.604
5 2.571 4.032
8 2.306 3.355
10 2.228 3.169
Abbreviated t table
CL = Confidence Limit
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Example: Confidence IntervalExcel: TINV=(p,DF)
The set of data for Youngs modulus of a bone
specimen tested in tension is:
xi=(9.75, 10.2, 11.62, 9.43, 10.56, 10.4, 9.98,11.02, 10.19, 10.35) Gpa.
What is the 95% confidence interval of the mean?
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Example Cont.
The probability for 95% confidence isp=0.05
DF=n-1, n=10, DF=9
t = TINV(p,DF)=2.262159
Mean = 10.35
SD = 0.623592
CL = 0.446
CI=10.350.45 22
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Test of Significance
How can we determine whether the data agrees
with the theoretical value or not?
t-test test of significance
snxxttest
/*
For this type of analysis to be valid, both samples we
are comparing should be drawn from the same
population with the same distribution of variance.23
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Example
Does the mean value of the example
agrees with the theoretical value 10.2 GPa?
ttest= (10.35-10.2)100.5/0.632=0.075 < 2.262
=> Theres no significant difference.
|t| < tcr no statistical difference
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t-Test
The t-test assesses whether the means oftwo groups are statisticallydifferentfrom each other.
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Paired t-testis employed when measuringtwo different quantities from the same
specimen, repeating this measurement
across different samples.
Unpaired t-testis employed whenmeasuring two different quantities from
different specimens. The sample size of the
two samples may or may not be equal.
t-Test Cont.
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Hypothesis Testing
The null hypothesis for the test is that allpopulation means (level means) are thesame.
Ho: 1 - 2 = 0
The alternative hypothesis is that one ormore population means differ from theothers.
H1: 1 - 2 0
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How to test the difference betweengroup means for significance?
First step: specify the null hypothesis and an alternativehypothesis.
Second step: choose a significance level. Usually a 0.05 level is chosen.
Third step: compute the t-value.
Fourth step: determine the probability value for the computed t-value
using a t table.
Fifth step: compare the probability value to the significance level. If it is
less than the significance level (0.05), then the effect is significant. If the
effect is significant, the null hypothesis is rejected and vice versa.
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Excel: =TTEST()
2
1,3
p0.05 no statistical difference 29
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Excel:Tools/ Data analysis/ t-test: paired two sample for means
0
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Regression
Regression analysis (curve fitting)- the objective
technique for evaluating the best fit of data.
Linear regression with linear relationship Y=a+bX.The regression line is obtained by minimizing the sum of
the actual data squared deviations from the best-fit line.
The aim is to find coefficients a & b, that minimize the
expression.
n
i
n
i
iiibXaYYY
1 1
22))(()(
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R2-value
Regression correlation coefficients areshown as either R or R2.
For R2: 0.8 1: Very strong
0.6 0.8: Strong
0.4 0.6: Moderate 0.2 0.4: Weak
0.0 0.2: Very Weak
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E l
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Excel:RMB/Add Trendline
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Excel: Tools/ Data analysis/ Regression
a
b
RSquare- closer to 1, better the fit
Standard Error- closer to 0, better the fit34
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Correlation Examplebad experimental design
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Experimental Design
For experimental design, we want to have a wayto estimate the uncertainty inherent in themeasurements, due to the nature of theequipment ormethod used:
Y=A X1X2X3
Y-dependent variable, Xs measured independentvariables, A -constant.
The change in Y due to a change in any of the Xswill be:
DY/Y= DX1/X1 + DX2/X2 + DX3/X3
Y/Y)max = Xi/Xi| 36
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Example: Experimental Design
We want to measure a distance.
The data: 5 sec, 10 m/sec.
The stopwatch we are using allows us to time the
process to the nearest 0.05 sec and the speedometer
allows us to monitor velocity to the nearest 0.1 m/sec.
What is the maximum expected fractional distance
uncertainty and the absolute uncertainty value?
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Experimental Design Cont.
In this example Y is distance, X1 is velocity, & X2 istime. Accordingly, the fractional distance uncertainty
is: (DY/50)max= 0.1/10 + 0.05/5 = 0.02 = 2%
(DY)max = 1m
The maximum expected fractional distance
uncertainty is 2%. The absolute uncertainty value is 1 m.
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Ex mpl : xp im nt l d si n
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Example: experimental design& t Test
Background:
Unique geometrical foot structure.
Stress concentration.
Hypothesis:
The internal compression stress in the bone-soft tissue
interface is higher than the superficial compression stress
in the shoe-heel interface.
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Example: experimental design
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Method:
The two kinds of compression stresses were
measured and computed during gait in the calcaneusregion.
For the purpose of comparison, the peak compressionstress of each gait cycle was collected.
Example: experimental design& t Test Cont.
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Example: experimental design
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Results:
Analysis: Paired, 2 tailed t-test
P = 1.45*10-13 < 0.05 hypothesis supported
Example: experimental design& t Test Cont.
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Some References
http://davidmlane.com/hyperstat
http://www.statsoft.com/textbook/stathome.html
http://helios.bto.ed.ac.uk/bto/statistics/tress4a.html
http://www.stat.yale.edu/Courses/1997-98/101/confint.htm
http://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis
/05_Random_vs_Systematic.html
http://davidmlane.com/hyperstathttp://www.statsoft.com/textbook/stathome.htmlhttp://helios.bto.ed.ac.uk/bto/statistics/tress4a.htmlhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.phys.selu.edu/rhett/plab193/labinfo/Error_Analysis/05_Random_vs_Systematic.htmlhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://www.stat.yale.edu/Courses/1997-98/101/confint.htmhttp://helios.bto.ed.ac.uk/bto/statistics/tress4a.htmlhttp://helios.bto.ed.ac.uk/bto/statistics/tress4a.htmlhttp://helios.bto.ed.ac.uk/bto/statistics/tress4a.htmlhttp://www.statsoft.com/textbook/stathome.htmlhttp://davidmlane.com/hyperstat