YY Teo

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Statistics for Medical Sciences Division

• YY Teo

Summer 07
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Resources

• Lectures
• Lecture Notes
• Ramsey and Schafer (2002) The Statistical Sleuth
  (2nd Edition) Thomson Learning
• Email teo_at_stats.ox.ac.uk

3
Lesson Structure

• 5 lectures practical sessions (1.5hr)
• Overview of statistics, exploratory data analysis
• Z-tests, t-tests, Analysis of Variance (ANOVA),
  post-hoc analysis
• Categorical analysis, odds ratio
• Linear regression
• Logistic regression

4
Overview of Statistics
5
Scientific Process
6
Analysing a set of data

• Look at the data (initial checks on the data)
• Downloading data, formatting, data collection,
  discrepant data, missing data
• Visualize the data (exploratory data analysis)
• Descriptive statistics, informative tables,
  well-constructed figures
• Analyse the data (definitive analysis)
• Formal statistical analysis
• Quantify any interesting results
• Report the findings

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Computers and Statistics

• Excel, SPSS, Minitab, Stata, Mathlab, R, etc
• Advantages
• Speed, accuracy, ease of data manipulation
• Easy to produce plots, cross-tabulation tables,
  summary statistics
• Disadvantages
• Inappropriate analysis / use of wrong tests
• Data dredging

8
Sample Selection

• Simple Random Sample
• Stratified Sample
• Cluster Sampling
• Multistage Sampling
• Multi-phase Sampling

9
Types of Variables

• Often, test to use depends on the type of
  variable at hand
• Two main classes of variables
• Categorical
• Numerical
• Categorical variables further divided into two
  sub-classes
• Nominal categorical (example gender, ethnic
  groups)
• Ordinal categorical (example size of a car,
  quality of teaching)

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Numerical Variables

• Distinguish between discrete or continuous
  numerical variables
• Discrete
• Integer values (number of male subjects, number
  of episodes of flu outbreaks)
• Continuous
• Takes a whole range of values (height, weight)
• Continuous variables treated as discrete (age)

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Exploratory Data Analysis
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EDA

• Tabular EDA
• Univariate tables, cross-tabulation of
  categorical variables
• Numerical EDA
• Location, spread, skewness, covariance and
  correlation
• Graphical EDA
• Frequency plots, histograms, boxplots,
  scatterplots
• The precise form of EDA depends on the data at
  hand.

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Tabular EDA

• Useful for summarising categorical data.

For example, the following table shows the
classification of 2,555 DNA samples in a
case-control study of Malaria onset into the
respective phenotypes
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Tabular EDA

• For two categorical variables i.e. the
  distribution of genotypes for a sub-sample of the
  individuals affected and unaffected by severe
  malaria

Question Appears to be more affected individuals
with the C allele (or conversely, more unaffected
individuals have the AA genotype). Does this mean
anything? (see later for test of independence)
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Numerical EDA

• Spread (range, standard deviation, interquartile
  range, mode)

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Numerical EDA

• Sample QuartilesQ1 25th quantile (or value of
  the 25 ranked data)Q2 50th quantile (also
  known as median of data)Q3 75th quantile (or
  value of the 75 ranked data)
• Interquartile range (IQR) IQR Q3 Q1
• Minimum, Maximum of data

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Numerical EDA

• Numbers can be informative to identify potential
  problems with the data
• Example Suppose the height for 1,496 individuals
  randomly sampled from the population produces the
  following summary

IQR Q3 Q1 188 172 16 Range Max Min
201 0 201
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Numerical EDA

• Skewness
• b1 is unit-free, where 0 indicates symmetry
  about sample mean. Positive values indicate
  right-skewness, and negative values indicate
  left-skewness.

19
Covariance and Correlation

• Two numerical variables height and weight
• Questions
• Are there any relationship between these
  variables?
• If there is, how do we quantify this
  relationship?
• Covariance and correlation Measures the degree
  of association between two numerical variables.

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Covariance and Correlation

• Covariance is scale-dependent, and correlation
  is unit-free.
• More intuitive to interpret correlation than
  covariance.
• Example Covariance for height and weight is 2.4
  when assessed using metres and kilograms, but
  240,000 when assessed using centimetres and
  grams. Correlation is a constant value at 0.83
  for both scenario.

• Correlation always bounded between 1 and 1
  inclusive.

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Example
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Graphical EDA

• Visual summaries of the data
• Flagging outliers, obvious relationships, check
  for distribution

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Boxplots

• Univariate boxplot for 1 numerical variable

Ends of box Q1 and Q3 Length of box IQR White
line Sample median Whiskers 1.5 times IQR Lines
outside whiskers Outliers Circles Extreme
outliers
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Boxplots

• Multivariate boxplots for 1 numerical variable
  across different levels of a categorical variable
• Graphical comparison

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Scatterplots

• Graphical representation for 2 numerical
  variables

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Scatterplots
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Scatterplots