Introduction to Statistics

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Introduction to Statistics

• Biomedical Sciences Degrees Honours Students
• Derek Scott
• d.scott_at_abdn.ac.uk

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Why use statistics?

• Statistics are used to analyse populations and
  predict changes in terms of probability.
• Normally, a representative sample is taken, large
  enough to make likely conclusions about the
  population as a whole.
• Descriptive statistics summarise the data and
  describe the population. These values allow you
  to see how large and how variable the data are.
• Inferential statistics propose null hypothesis
  and endeavour to disprove it. By looking at
  these, you can check for error.

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• When analysing data, you want to make the
  strongest possible conclusion from limited
  amounts of data. To do this, you need to overcome
  2 problems
• Important differences can be obscured by
  biological variability and experimental error.
  This makes it difficult to distinguish real
  differences from random variability.
• The human brain excels at finding patterns, even
  from random data. Our natural inclination
  (especially with our own data) is to conclude
  that any differences are real, and to minimise
  the contribution of random variability.
  Statistical rigor prevents you from making this
  mistake.

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Errors

• Bias or systematic error Data go in a
  predictable direction perhaps due to experimental
  design or human errors. Can remove the errors if
  you identify them.
• Random error Unpredictable errors. Cant get rid
  of these.
• Usually you will quote a measure of error with
  your data (e.g. standard deviation, standard
  error of the mean)
• EXAMPLE The mean height of a student in BM4005
  is 1.71 0.20 (43) metres.

MEAN VALUE
SD or SEM
n, the number of samples
Units!!!
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Independent Sampling 1

• Measure BP in rats, 5 rats per group.
• Measure BP 3 times in each animal.
• You do not have 15 independent measurements,
  since triplicate measurements in each animals
  will be closer to one another than to those in
  other animals.
• You should average values from each rat.
• Now have 5 independent mean values.

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Independent Sampling - 2

• Perform a biochemical test 3 times, each time in
  triplicate.
• Do not have 9 independent values, as an error in
  preparing the reagents for 1 experiment could
  affect all 3 triplicates.
• Average the triplicates, and you have 3
  independent mean values.

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Independent Sampling - 3

• Doing a human exercise study.
• Recruit 10 people from the inner-city, and 10
  people from the countryside.
• Have not independently sampled 20 subjects from
  one population.
• Data from inner-city subjects may be closer to
  each other than to the data from rural subjects.
  You have sampled from 2 populations, and need to
  account for this in your analysis.

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Gaussian (Normal) Distribution

• Data usually follow a bell-shaped distribution
  called Gaussian distribution. t-tests and ANOVA
  tests assume that the population follows an
  approximately Gaussian distribution.
• For example, of we measure the height of everyone
  in 4th year and plot this, most people would fall
  in the middle of the curve, with a few at the
  bottom end, and a few at the top end of the
  curve.
• For Gaussian distribution, we use parametric tests

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Gaussian Distribution
Bell-shaped curve
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Outliers

• When analysing data, some values can be very
  different the rest.
• Tempting to delete it from analysis.
• Was the value typed in correctly?
• Was there an experimental problem with that
  value?
• Is it due to biological diversity?
• What if answers to these questions are no?

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Outliers

• If outlier is due to chance, keep it in the data
  set.
• If it is due to a mistake (e.g. bad pipetting,
  voltage spike, apparatus problem) then you must
  remove it from the analysis.
• If you want to be absolutely sure whether the
  outlier is due to chance or not, there are
  specific statistical tests you can do, but
  usually these basic checks are enough to decide.

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Mean

• Sample mean will probably not be exactly the
  population mean. Mean is more accurate if you
  have a bigger sample size with a low variability.
• You may calculate Confidence Intervals (CIs)
  telling you the area in which 95 of the
  population will fall.
• EXAMPLE Mean height of a student in BM4005 is
  1.71 metres. The 95 confidence limits for this
  value are 1.5 and 1.8 metres. These are the upper
  and lower heights between which 95 of the class
  will fall.

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Confidence Intervals

• Nothing magical about 95. You could do it for
  any value you liked 99, 90 etc.
• If you set a value of 99, then the intervals
  would be wider because 99 of the classs heights
  must fall within that range.
• 95 confidence limits mean you have a reasonable
  level of confidence that the true population mean
  lies within that range.

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Standard Deviation (SD)

• Quantifies variability
• If data follow Gaussian distribution, then 68 of
  values lie within one SD of mean (on either side)
  and 95 of values lie within 2 SDs of the mean.
• So, as a rule of thumb, if 2 points on a graph
  are more than 2 SDs away from each other, they
  are significantly different.
• Expressed in same units as data

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Standard Error of the Mean (SEM)

• Measure of how far sample mean is likely to be
  from the true population mean.
• SEM SD/?n
• Smaller than SD, so used more to give smaller
  error bars!
• SD quantifies scatter how much values vary from
  each other. Doesnt really change much even if
  you have a bigger sample size.
• SEM quantifies how accurately you know the true
  mean of the population. SEM gets smaller as
  sample gets larger

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P Values
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Students t-test

• Used to compare the means of two groups of data.
• Paired t-test control expt. and treatment done
  on same person, animal or cell etc.
• Unpaired t-test control done on 1 group of
  subjects, with the treatment being done on
  another separate group.
• Can be 1- or 2-tailed.

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Iron and zinc evoke electrogenic responses that
are pH-dependent
Krebs pH 6.0
Krebs pH 7.4
IRON (100mM)
ZINC (100mM)
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Iron- and zinc-evoked transport is
temperature-dependent
IRON
ZINC
? 4 oC ? 37 oC
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Paired or Unpaired?

• Choose paired if the 2 columns of data are
  matched, e.g.
• You measure weight before and after an
  intervention in the same subjects.
• You recruit subjects as pairs, matched for
  variables such as age, ethnic group, disease
  severity. One of the pair gets one treatment, the
  other gets an alternative treatment.
• You perform the control experiment in one cell or
  piece of tissue, and then apply a drug. You
  measure the effect of the drug in the same cell
  or tissue.
• Shouldnt be based on the variable you are
  comparing. For example, if measuring BP, you can
  match subjects based on their age or postcode,
  but not on their BPs.

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Students t-test

• You will probably always use a 2-tailed t-test.
• 2-tailed test just asks whether there is a
  difference between the 2 means.
• 1-tailed test predicts whether
• Mean 1 is bigger than Mean 2 or
• Mean 2 is bigger than Mean 1.
• For 1 tailed you must know which mean will be
  bigger before you start not usually possible
• Stick to a 2-tailed t-test to be safe!!!

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Analysis of Variance (ANOVA)

• Used to compare means of 3 or more groups.
• Again, can have matched (paired) or unmatched
  (unpaired) values.
• You will probably only use 1-way ANOVA
• EXAMPLE Your null hypothesis is that the average
  BP for 4 men is equal. ANOVA can compare each
  subjects BP and say if they are different or
  not.

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Features of ANOVA

• ANOVA produces an F value which tells you how
  much variation there is in your sample. Higher F
  value means more variation.
• Dunnetts post test allows you to compare against
  1 group e.g. A v B, A v C, A v D. Handy if A is
  the control group.
• Tukeys post test allows you to compare all
  columns against one another just to check for any
  differences between any groups. Good way of
  finding significant differences that you may not
  have expected.

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The effect of non-selective protein kinase
inhibition with staurosporine
IRON
ZINC
? 8-Br cGMP Staurosporine ? Staurosporine
(0.5 mM) ? 8-Br cGMP (100 mM) ? Control
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Non-Gaussian Distribution

• Use non-parametric tests for these unusual
  situations which rank data from low to high and
  analyse distribution of ranks.
• Less powerful than parametric but used when
  values are too low or high to measure by
  assigning arbitrary values. Also used if outcome
  is a rank or score with only a few categories.
• P values are usually higher.

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Skewness
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Correlation
ve correlation
-ve correlation
Correlation doesnt tell you about the cause of
the effect, it just tells you that there is a
link between value X and value Y. The nearer the
R value is to 1, the better the correlation.
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Regression
Regression calculates a line of best fit. Often
used to calculate a standard curve which you
could use to estimate value x if you know value
y. Unknowns must fall within your standard
curves range.
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Correlation and regression

• A word of caution about doing regression and
  finding correlations.
• Just because you can draw a line of best fit
  through some points and make quite a good
  straight line, it does not necessarily mean there
  is a relationship.
• Correlation does not necessarily imply causation!
• For example, the consumption of tropical fruit in
  the UK since WW2 has increased, and so has the
  birth rate in the UK. If I plot this on a graph,
  and did a regression, I would probably get a nice
  straight line as both increase together. I would
  probably also show there is a good correlation.
• This does not mean that I can say that eating
  tropical fruit improves your fertility!!!
• Use some common sense when interpreting your data!

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Summary

• This is just a basic introduction.
• For extra information, try the Help files on
  Graphpad Prism (on the University PCs)
• If you end up doing an Honours project with
  certain types of data (e.g. collecting
  psychological data, epidemiological studies
  etc.), your supervisor should inform you about
  any special tests/calculations they use for that
  type of data.
• Finally, if you are still unsure, make it clear
  to your supervisor that you do not understand why
  or what you are doing.