Comparing >2 means with one way ANOVA

41 Pages·2013·5.93 MB·English

by  Jay Kruzic

Institute for Advanced Ceramics, Hamburg University of Technology
November 12, 2012
Materials Science Seminar, October 25th, 2013
Practical Tips for Analyzing
Materials Science Data
Jamie Kruzic
School of Mechanical, Industrial, and Manufacturing Engineering
Oregon State University
Introduction
•  Publication of results is always easier
with rigorous data analysis and
statistical tests of results
–  In bio-related fields (e.g., biomaterials,
biomechanics, etc.) this is absolutely
necessary!
•  In Materials Science we are often
tempted to use the “eyeball test” to
compare data.
–  Only convincing for very large effects
–  Not useful for overlapping data sets,
scattered data, etc.
•  You can have a smoother journal
review process by doing a proper
analysis before submission.
http://www.ifimages.com/
Outline
•  Today we will discuss these common topics with
examples from my research:
–  Comparing two means: the student’s t-test
–  Comparing >2 means: the analysis of variances
(ANOVA)
–  Comparing >2 means: post-hoc tests
–  Regression analysis & proving significant trends
–  Calculating uncertainty from experimental linear
calibrations
http://beyeager.blogspot.com/
•  OSU has Windows software license for
StatGraphics which can handle many of your needs.
–  http://oregonstate.edu/helpdocs/software/
recommended-software/osuware
•  On my Mac I run it using Citrix Receiver.
–  http://engineering.oregonstate.edu/computing/citrix/
steps.html
Statistical Tests
•  This tutorial is not a rigorous lecture on statistical tests.
Rather, the aim is to point out useful tips for material
scientists.
•  Generally a statistical test is used to test a null
hypothesis that implies no effect.
–  e.g., the means of two groups are not different
http://www.xda-developers.com/
•  If the null hypothesis is rejected, than there is a
measurable difference between the groups.
•  Testing the null hypothesis is usually done by calculating
the p-value
–  p = probability of obtaining a given statistical test result
from a data set if null hypothesis is true.
–  Usually p < 0.05 (i.e., 5%) is considered statistically
significant, but cutoff is somewhat arbitrary!
Comparing two means: the student’s t-test
•  In Materials Science we often wish to compare
two material states, processes, etc.
•  Often compare an experimental treatment to a
known control condition:
–  Does heat treating cast alloy for 1hr at 650°C
affect yield strength?
–  Does adding 1% Bi affect the piezoelectric
constant?
•  Or alternatively, compare two methods or
http://www.vias.org/
processes:
–  Do sol-gel and pulse laser deposition give the
same thin film properties?
–  Do two different fracture toughness test methods
produce the same result?
•  If the effect is small, an “eyeball test” will not be
effective at convincing journal reviewers
http://mychinaconnection.com/
Comparing two means: the student’s t-test
•  Student’s t-test is ideal for assessing two
different data sets (not more!).
•  Assumes scatter is data is normally
distributed around mean.
–  Good for random scatter, also assumes
http://projectile.sv.cmu.edu/
variance is equal
–  Not good for systematic scatter (e.g.,
instrument drift with time)
http://projectile.sv.cmu.edu/
•  Null hypothesis is that the difference
between two means = zero
Comparing two means: the student’s t-test
•  Data sets may be paired:
–  Data points in each set are paired
–  E.g., test hardness of each of 10 samples
before and after a heat treatment.
http://blogs.jcvi.org/
•  More commonly in Materials Science data
http://apileofsheep.wordpress.com/
sets are unpaired:
–  Data sets are produced with independent
samples
–  E.g., test dielectric constant for 10
samples made with 1% Bismuth added
and 10 samples without.
Comparing two fracture toughness test methods
•  N = 6 tests with only a notch
•  N = 6 tests with a notch and pre-crack
•  Data sets overlap considerably, are means different?
Kruzic, Kuskowski, Ritchie, J. Biomed Mater A (2005)
Comparing two fracture toughness test methods
Example is perfect for unpaired t-test
Kruzic, Kuskowski, Ritchie, J. Biomed Mater A (2005)
Comparing two fracture toughness test methods
•  p = 0.13 and we cannot reject null hypothesis
•  It is still possible there is a small difference between the means, but
our data cannot prove it.
Kruzic, Kuskowski, Ritchie, J. Biomed Mater A (2005)