STATISTICS WORKSHOP - 2

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STATISTICS WORKSHOP - 2

• Contingency tables
• Correlation
• Analysis of variance

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Why relations between variables are important

• The ultimate goal of every research or scientific
  analysis is finding relations between variables.
• The philosophy of science teaches us that there
  is no other way of representing meaning except
  in terms of relations between some quantities or
  qualities either way involves relations between
  variables.
• The advancement of science must always involve
  finding new relations between variables.

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Qualitative Data (Contingency Table)
Example This test would be the one to use if we
have, say, different classes of patients (e.g.,
six types of cancers) and for a set of 1000
markers we can have either presence/absence of
each marker in each patient (this would yield
1000 contingency tables of dimensions 6x2 ---each
marker by each cancer type)
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Contingency Table
Question Is there evidence in the data for
association between the categorical variables?
For cross-classified data, the Pearson chi-square
test for independence and Fisher's exact test can
be used to test the null hypothesis that the row
and column classification variables of the data's
two-way contingency table are independent.
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Chi-Square test
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Odds Ratio (OR) (ad)/(bc) Relative risk
a(cd)/c(ab)
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Contingency Table
Chi-Square Test Example 3500 were observed
whether they snore or not Is there an
association between snoring and gender ?
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Contingency Table
Example - Is there an association between snoring
and gender?
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Contingency Table
Odds ratio 1.58 95 CI 1.39 to 1.81
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Contingency Table
Is there evidence of differences in smoking
pattern between the sexes?
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Contingency Table
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Measuring treatment differences with Y/N response

• For outcomes such as reduction in blood pressure
  there are obvious summaries of treatment effect
  such as the difference between the average of
  each group
• For yes/no outcomes like death or cure the choice
  of summary is not so obvious

Dead

Y N
aspirin
804
7783
9.4
placebo
1016
7584
11.8
TOTAL
1820
15367
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Relative Risk or Risk Ratio

• Relative risk or risk ratio risk of death in
  aspirin group divided by risk in placebo group
• Relative Risk 9.4 / 11.8 0.80
• mortality is reduced by 20
• Relative risk estimates are likely to generalise
  well from one population to another

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Absolute Risk Difference

• Absolute risk difference is the proportion of
  deaths in the aspirin group minus the proportion
  in the placebo group
• risk difference 9.7 - 11.8 -2.1
• "2.1 lives saved for each 100 patients treated"
• Risk difference has a more direct clinical
  interpretation, especially when considering cost-
  effectiveness

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Odds Ratio

• Odds ratio the odds of death in the aspirin
  group divided by the odds in the placebo group
• Odds Ratio (9.7/90.3) / (11.8/88.2) 0.77
• "reduction of 23 in the odds of death
• The odds ratio has some purely mathematical
  advantages. It is not much used in randomised
  studies

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Berksons Fallacy

• It is a treatment-seeking bias so called because
  Berkson indicated that individuals with more than
  one disorder are more likely to seek clinical
  services than are those with only one disorder. 
• This leads to an erroneously higher estimate of
  the prevalence of the association between these
  disorders than would be the case if each single
  disorder independently led the patient to seek
  care.

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Berksons Fallacy

• 2784 individuals were surveyed to determine
  whether each subject suffered from either a
  disease A or disease B or both. It is found that
  257 out of the 2784 patients were hospitalised
  for the condition.

Disease A Disease B Disease B Total
Disease A Yes No Total
Yes 7 29 36
No 13 208 221
Total 20 237 257
Disease A Disease B Disease B Total
Disease A Yes No Total
Yes 22 171 193
No 202 2389 2591
Total 224 2560 2784
P lt 0.025 There is some association between
having disease A and having disease B
P gt 0.1 There is no association between having
disease A and having disease B
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Gene Association Studies Typically Wrong

• Evolution of the strength of an association as
  more information is accumulated. The strength of
  the association is shown as an estimate of the
  odds ratio (OR) without confidence intervals.
• a, Eight topics in which the results of the first
  study or studies differed beyond chance (Plt0.05)
  when compared with the results of the subsequent
  studies.
• b, Eight topics in which the first study or
  studies did not claim formal statistical
  significance for the genetic association but
  formal significance was reached by the end of the
  meta-analysis.
• Each trajectory starts at the OR of the first
  study or studies. Updated cumulative OR estimates
  are obtained at the end of each subsequent year,
  summarizing all information to that time.

(Adapted from J.P.Ioannidis et al., Nature
Genetics 29306-9, 2001)
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Studies of disease association

• Given the number of potentially identifiable
  genetic markers and the multitude of clinical
  outcomes to which these may be linked, the
  testing and validation of statistical hypotheses
  in genetic epidemiology is a task of
  unprecedented scale

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Testing for equality of two proportions

• Example Two groups of genes 1. genes for
  transcription and translation 2. genes in the
  immune system Question Do they have similar
  purine-pyrimidine compositions? The question is
  asking whether the percentage of purines (or
  pyrimidines) in group 1 is the same as the
  percentage of purines (or pyrimidines) in group
  2. To form the null and alternative hypotheses
  we can say G1 the percentage of purines in
  group 1 G2 the percentage of purines in group
  2 H0 G1 G2 H1 G1 gt G2 or G2 gt G1

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Correlation

• Correlation can be used to summarise the amount
  of linear association between two continuous
  variables x and y.
• Let (x1, y1), (x2, y2), ..., (xn, yn) denote the
  data points.
• A scatter plot gives a "cloud" of points

Y
Y
X
X
Positive correlation
Negative correlation
No correlation
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Positive and Negative Association

• If the points are nearly in a straight line then
  knowing the value of one variable helps you to
  predict the value of the other.
• If there is little or no association, the "cloud"
  is more spread out and information about one
  variable doesn't tell you much about the other.

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A simple correlation formula

• Suppose there are n points altogether and that
  n(A) is the number in region A and similarly for
  n(B), n(C) and n(D
• Give a value of 1/n to every point in A or C and
  -1/n to every point in B or D
• Define
• Cor n(A)n(C)-n(B)-n(D)
• n
• What are the properties of cor?

D
A
C
B
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The Pearson product moment correlation coefficient

• The formula for cor works, but it is rather
  crude. For example both the diagrams below
  would give cor 1.

and
are positive or negative in the different regions
and so is the product

• Sum will not lie between -1 and 1. It depends
  on
• The scale of x and y
• The number of points

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Correlation formula
where
Partial correlation Correlation between 2
variables that controls for the effects of one or
more other variables. Rank Correlation
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Pearson Correlation Coefficient

• A measure of linear association between two
  variables, denoted as r.
• Values of the correlation coefficient range from
  -1 to 1.
• The sign of the coefficient indicates the
  direction of the relationship, and its absolute
  value indicates the strength, with larger
  absolute values indicating stronger relationships.

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Interpretation of correlation

• r measures the extent of linear association
  between two continuous variables.
• Association does not imply causation - both
  variables may be affected by a third variable.
• If r 0, there is no association between X and Y
  
• r does not indicate the extent of non linear
  associations
• The value of r can be affected by outliers
• Correlations Do Not Establish Causality
• Example When a gene is isolated that has some
  positive correlation to cancer, claim is often
  made that it enhances the susceptibility to the
  disease, and not cause it.

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Some misconceptions

• When the value of the correlation coefficient is
  large (small), the relation between the two
  variates is close to linear, thus, when r 0.9
  or 0.95 the relation is nearly linear
• When the value of the correlation coefficient is
  zero or near zero the two variates have no or
  almost no functional relation
• When the value of the correlation coefficient is
  positive (negative), the value of Y becomes
  larger (smaller) as a whole, as the value of X
  becomes large
• Example Let (X,Y) take (1,-1),(2,-2),(3,-3),(4,-
  4),(5,20) each with probability 1/5. Then we
  have Cor(X,Y) 0.62
• Concerning the first four points Y decreases as
  X increases. This example shows that even when
  the correlation coefficient between X and Y is
  positive, Y does not always increase as a whole
  as X increases.

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Examples

• Eg1. In Australia total alcohol consumption and
  the number of ministers of religion have both
  increased over time and would be positively
  correlated but the increase in one has not caused
  the increase in the other (both are related to
  the total population size)
• Eg2. In Japanese schoolchildren shoe size was
  reported to be correlated (positively) with
  scores on a test of mathematical ability.
• Eg3. Extracting informative genes with negative
  correlation for accurate cancer classification

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Effectiveness of the first Cold-War arms agreement

• "Most important, the negative correlation between
  the mutation rate and the parental year of birth
  among those born between 1950 and 1956 provides
  experimental evidence for change in human
  germline mutation rate with declining exposure to
  ionizing radiation and therefore shows that the
  Moscow treaty banning nuclear weapon tests in the
  atmosphere (August 1963) has been effective in
  reducing genetic risk to the affected
  population."

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Example - Heights and weights of 6 female students

• The table below shows the heights and weights of
  6 female students. How closely related are the
  heights and the weights?

The correlation coefficient 0.904
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Spearman Correlation Coefficient

• Commonly used nonparametric measure of
  correlation between two ordinal variables. For
  all of the cases, the values of each of the
  variables are ranked from smallest to largest,
  and the Pearson correlation coefficient is
  computed on the ranks.

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Rank Correlation

• 10 students, arranged in alphabetical order, were
  ranked according to their achievements in both
  the laboratory and lecture sections of a biology
  course. Find the coefficient of rank
  correlation.

Rank correlation 0.8545
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Thoughts
Patterns often emerge before the reasons for them
become apparent. - Vasant Dhar If you do not
expect, you cannot find the unexpected. -
Heracletes To consult the statistician after an
experiment is finished is often merely to ask him
to conduct a post mortem examination. He can
perhaps say what the experiment died of. -
R.A.Fisher