Introduction to Statistics

Report
Introduction to Statistics
Harry R. Erwin, PhD
School of Computing and Technology
University of Sunderland
Resources
• Crawley, MJ (2005) Statistics: An Introduction
Using R. Wiley.
• Gentle, J (2002) Elements of Computational
Statistics. Springer.
• Gonick, L., and Woollcott Smith (1993) A Cartoon
Guide to Statistics. HarperResource (for fun).
Who am I?
• Dr. Harry Erwin BS MA PhD MIET MBCS
• My PhD was awarded in bioinformatics. Although my
research interests are in neuroscience, I've had the
coursework and understand current research directions
in computational biology and statistics. I’ve also had
the coursework for a PhD in mathematics.
• I teach computing and neuroscience here at the
University of Sunderland.
Doing Statistics
• Usually you do statistics to explore the structure of
data. The questions you might ask are rather openended. Your understanding is facilitated by a model.
• A model embodies what you currently know about the
data. You can formulate it either as a data-generating
process or a set of rules for processing the data.
• We’ll look at modelling in detail later.
Statistical Models
• Often expressed as a set of equations relating
data elements.
• Can include probability distributions for the
elements. If this is the case, you have a
stochastic model.
• The model should be free to evolve based on
data mining.
Common Stochastic Models
• Parameterized statistical distributions, such as
the normal distribution, binomial distribution, or
the chi-squared distribution.
• Sometimes more complicated, where you might
need to use simulation, resampling, and
visualization to determine the parameters of the
model.
Structure-in-the-data
• Of most interest…, for example:
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Modes
Gaps
Clusters
Symmetry
Shape
Deviations from normality
Visualization
• Multiple views are necessary, particularly for
multivariate data.
• Be able to zoom in on the data as a few points
can obscure the interesting structure.
• Scaling of the axes may be necessary, since our
eyes are not perfect tools for detecting structure.
• Watch out for time-ordered or location-ordered
data, particularly if time or location are not
explicitly reported.
Plots
• Use simple plots to start with.
• Watch for rounded data—shown by horizontal
strata in the data. That often signals other
problems.
• There are a number of plotting tutorials, consult
them.
Statistical Activities
• Data collection (ideally the statistician has a say on
how they are collected)
• Description of a dataset
– Averages
– Spreads
– Extreme points
• Inference within a model or collection of models
• Model selection
How to Do It
• Start by determining what sort of statistical
analysis you will be doing. You need to know:
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Which variable is the response variable?
Which are the explanatory variables?
What kind are the explanatory variables?
What kind of response variable do you have?
• If you have multiple response variables, you need
to do multivariate analysis (more advanced).
Basic Methods
• If all explanatory variables are continuous, plan
on a regression analysis.
• If all explanatory variables are categorical, plan
for an analysis of variance (ANOVA).
• If you have a mix, plan for an analysis of
covariance (ANCOVA)
Effect of the Response Variable
• If the response variable is continuous, then plan on a
normal regression, ANOVA, or ANCOVA.
• If the response variable is a proportion, do a logistic
regression.
• If a count, you need a log linear model.
• If binary, you need a binary logistic analysis
• If time to event or time at death, you will be doing a
survival analysis.
Variation
• You want to understand how the response is
dependent on variation in the explanatory
variables, but you are also interested in lack of
dependence.
• Design the simplest model that explains the data
adequately.
Significance
• You have to determine what the probability of a
false alarm will be—that is, the chance that you
will think something is significant which really
is not.
• Typical values are 5%, 1%, and 0.1%.
• Don’t test every hypothesis. Some will be true
by chance.
Good and Bad Hypotheses
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‘There are vultures in the local park.’
‘There are no vultures in the local park.’
Which is testable?
Discuss…
Answer
• The ‘null hypothesis’ is testable.
• ‘There are no vultures in the local park.’
• You test it by taking measurements and
showing that if the null hypothesis were true,
the chance of those measurements would be
close to zero.
• Discuss further…
Experimental Design
• Replication
– Increases reliability, so be thorough. Often the
answer is ‘30’.
– Discuss why.
• Randomization
– Reduces systematic bias, so do it properly
– Almost never done properly
– Discuss why.
Controls
• “No controls, no conclusions.”
• A ‘control experiment’ is one where you don’t
apply the treatment or don’t enable the part of
your experiment that is supposed to produce the
different outcome.
• You’re comparing the results when the
treatment is applied to the results with no
treatment.
Replication
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Must be independent
Not part of a time series
Not grouped together in space
Of an appropriate spatial scale
Covers the normal variation in initial
conditions.
Error Types
Null hypothesis
actually true
Null hypothesis
actually false
Accept null
hypothesis
Correct
Type II () error
(no paper but no
embarrassment)
(further experiments
can change this)
Reject null
hypothesis
Type I () error
Correct
(can result in a paper (a publishable paper)
you have to
withdraw)
Typical  and  values
• You usually want the probability of rejecting the null
hypothesis () when it is true to be less than 5%.
• You usually want the probability of accepting the null
hypothesis () when it is false to be less than 20%.
• The power of a test is 1- , or greater than 80% in this case.
• Rule of Thumb: the number of replicates to reject the null
hypothesis with probability 80% is about 8s2/d2, where s2 is
the variance in the response and d is the size of the
difference to be detected in a single sample.
Inference
• Strong inference
– A clear hypothesis
– An acceptable test
• Weak inference
– Natural experiments
• Conclusions from natural experiments are
hypotheses. Can still produce good papers.
• Discuss
How Long to Go On?
• To stop the experiment as soon as a pleasing
result is obtained?
• To keep going until the theoretically correct
result is obtained?
• Discuss.
• Gregor Mendel’s experiments with peas.

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