statistics for biomedical engineering labs

7/29/2019 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

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)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

statistics for biomedical engineering labs

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