• What is statistics? Why do we need it?
• Understanding data types
• Worked examples and exercises of various forms of "t-test"
  • using online tools

• Rarely feasible to study the whole population that we are interested in, so we take a sample instead
• Assume that data collected represents a larger population
• Use sample data to make conclusions about the overall population

• Which samples to include?
  • Randomly selected?
  • Generalisability
• Always think about the statistical analysis
  • Randomised comparisons, or biased?
  • Any dependency between measurements?
  • Data type?
  • Distribution of data? (Normally distributed? Skewed? Bimodal?…)

• How samples are selected affects interpretation
  • What is the population that the results apply to?
  • How widely applicable will the study be?
• Statistical methods assume random samples
• Do not extrapolate beyond range of the data
  • i.e. don’t assume results apply to anything not represented in the data
• Examples:
  • Males only, no idea about females
  • Adults only, no idea about children
  • 1 litter of mice, no idea about other litters

• Several different categorisations
• Simplest:
  • Categorical (nominal)
  • Categorical with ordering (ordinal)
  • Discrete
  • Continuous

• Most basic type of data
• Three requirements:
  • Same value assigned to all the members of level
  • Same number not assigned to different levels
  • Each observation only assigned to one level
• Boils down to yes/no answer
• e.g. Surgery type, smoker / non-smoker, eye colour, dead/alive, ethnicity.

• Mutually exclusive fixed categories
• Implicit order
• Can say one category higher than another
  • But not how much higher
• Example: stress level 1 = low … 7 = high
• Others: Grade, stage, treatment response, education level, pain level.

• Fixed categories, can only take certain values
• Like ordinal but with well-defined distances
  • Can be treated as continuous if range is large
• Anything counted (cardinal) is discrete
  • how many?
• Examples: number of tumours, shoe size, hospital admissions, number of side effects, medication dose, CD4 count, viral load, reads.

• Final type of data
• Anything that is measured, can take any value
• May have finite or infinite range
• Zero may be meaningful: ratios, differences
  • Care required with interpretation
• Given any two observations, one fits between
• Examples: Height, weight, blood pressure, temperature, operation time, blood loss, age.

• Several different categorisations
• Simplest:
  • Categorical (nominal) – yes/no
  • Categorical with ordering (ordinal) – implicit order
  • Discrete – only takes certain values; counts (cardinal)
  • Continuous – measurements; finite/infinite range

• Measurements of gene expression taken from each of 20 individuals
• Are any measurements more closely related than others?
  • Siblings/littermates?
  • Same individual measured twice?
  • Batch effects?
• If no reason, assume independent observations

• Measures of location and spread

• Mean and standard deviation

\(\bar{X} = \frac{X_1 + X_2 + \dots X_n}{n}\) \(s.d = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}\)

• Median: middle value
• Lower quartile: median bottom half of data
• Upper quartile: median top half of data

• e.g. No of Facebook friends for 7 colleagues
  • 311, 345, 270, 310, 243, 5300, 11

• Mean and standard deviation

  \(\bar{X} = \frac{X_1 + X_2 + \dots X_n}{n} = 970\)

  \(s.d = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}=1912.57\)

• Median and Interquartile range
  • 11, 243, 270, 310, 311, 345, 5300

• e.g. No of Facebook friends for 7 colleagues
  • 311, 345, 270, 310, 243, 530, 11

• Mean and standard deviation

  \(\bar{X} = \frac{X_1 + X_2 + \dots X_n}{n} = 289\)

  \(s.d = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}=153.79\)

• Median and Interquartile range
  • 11, 243, 270, 310, 311, 345, 530

• e.g. No of Facebook friends for 7 colleagues
  • 311, 345, 270, 310, 243, 530, 11

• Mean and standard deviation: low breakdown point

  \(\bar{X} = \frac{X_1 + X_2 + \dots X_n}{n} = 289\)

  \(s.d = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}=153.79\)

• Median and Interquartile range: robust to outliers
  • 11, 243, 270, 310, 311, 345, 5300

• Summarised by counts and percentages
• Examples
  • 19/82 (23%) subjects had Grade IV tumour
  • 48/82 (58%) subjects had Diarrhoea as an Adverse Event

• Formulate a null hypothesis, \(H_0\)
  • Example: the difference in gene expression before and after treatment = 0
• Calculate a test statistic from the data under the null hypothesis
• Compare the test statistic to the theoretical values
  • is it more extreme than expected?
  • the "p-value"
• Either reject or do not reject the null hypothesis
  • "Absence of evidence is not evidence of absence" (Bland and Altman, 1995)
• (Correction for multiple testing)

• The Lady Tasting Tea - Randomised Experiment by Fisher
• Randomly ordered 8 cups of tea
  • 4 were prepared by first adding milk
  • 4 were prepared by first adding tea
• Task: Lady had to select the 4 cups of one particular method

• \(H_0\): Lady had no such ability
• Test Statistic: number of successes in selecting the 4 cups
• Result: Lady got all 4 cups correct
• Conclusion: Reject the null hypothesis

• Many factors that may affect our results
  • significance level, sample size, difference of interest, variability of the observations
• Be aware of issues of multiple testing

Credit: Effect Size FAQs by Paul Ellis

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