Statistics

For help with statistics, Drs Andrea Giochini and Owen Tomlinson offer 1-to-1 stats help sessions.
You can find more information on the Medical Sciences Maths and Stats Support Engage (formerly Yammer) page

The peer mentors will offer at least one statistics session in the year, and monthly drop-ins where you can ask us any questions (statistics or otherwise). To find the date of the next drop-in, check Instagram or send us an email.

Here are some notes that may help you in statistics. 

These notes are here to complement the lectures you receive and should not be taken as a sole resource for revision.

If the figures don't show correctly, click here to download the information contained on this page as a pdf file.

Here is a link to a flowchart for choosing statistical tests (Exeter login required)

Fundamentals of Statistics Cheat Sheet

Definitions:
Population – the full set of units we are interested in
Sample – Subset of units that we experiment on or observe from which inferences about the population can be drawn.
P–Value – The probability that you might see something as extreme, or more extreme, if nothing was going on (under the null hypothesis)
Descriptive statistic – describes data in a sample (fact)
Inferential statistic – draw inferences about the population from the sample (estimate)

P – Values Strength of Evidence
0.1 Weak
0.05 Moderate
0.01 Strong
0.001 Very strong

Types of Data:

Numerical – Expressed in numbers

• Continuous – Theoretically can take any value

• Count – Only takes integer values

Categorical – No inherent numerical value 

• Nominal –No inherent order 

• Ordinal –Inherent order

Descriptive Statistics

Categorical :

• Frequency table

• Percentages

• Bar chart

Numerical:

• Many possible values therefore tabulation is often not practical. Solutions include:

  • Grouping data 

    • Data becomes categorical

    • Can’t see differences within groups

  • Histogram

    • Bin width = size of each group

      • Too many – data points in each bin too small so won’t be able to see shape.

      • Too few – lose the detail

    • Density = proportion in bin/bin width. Allows comparison of histograms with different bin widths (at the expense of interpretability)

      • Area = 1

    • Heights indicate relative frequency of observations in a bin

    • 50 observations for a decent histogram

    • No gaps between bars 

• • Summary statistics

    • Provide a quantitative description of the data.

    • Dispersion

      • Standard deviation – how far a typical value is from the mean

      • Interquartile range

        • Lower quartile is the point where 25% of the data is below

        • Upper quartile is the point where 25% of the data is above

      • Range

    • Location

      • Mean

      • Mode

      • Median

      • Equal if data is normally distributed

  • Box and whisker plots

    • Easier to compare between groups

    • Outliers, more than 1.5, +/- the appropriate quartile

Inferential test - T-Test

Used to test whether the means in two groups of continuous data are different from one another. As an inferential test the T-Test shows whether it is likely differences observed in sample data reflect a difference in the population

SEM – Strongly related to the t–test

• The standard deviation of data if we were to repeat our study over and over

• = SD/√n

  • SEM increases with increasing variability

  • SEM decreases with increasing sample size

• Null Hypothesis – two groups have the same mean in the population

• Assumptions:

  • Standard

    • Normal distribution

    • Equal variance across groups (SD²)

    • Independent data points

  • For data with different SDs a t-test assuming unequal variances can be used.

  • The paired t-test handles non-independent data

    • e.g. same people before/after or from the same group of people in a hospital

ANOVA (ANalysis Of VAriance)

Looks overall at the data to see if there are any differences between groups rather than comparing each group. THEN individual differences should be investigated.

• Means indicate which groups are higher or lower however not formally tested.

• Post-hoc pairwise comparisons.

  • Made after overall assessments

  • T – tests often used however fail to take into account multiple uses of the same information.

  • Alternatives include Tukey tests.

• Assumptions

  • Normal distribution

  • Equal variance across groups

  • Independent of the t-test

Wilcoxon Rank Sum tests

• Used instead of a t-test when data is not normally distributed.

• Null hypothesis – If we select a value from each group at random the value from the first group will be larger than the value from the second group 50% of the time.

• If the distribution in each group is equal it can be used to see if there is a difference in medians.

• Non - Parametric

  • Lacks assumptions (only that data points are independent)

  • Less powerful

• Alternatives

  • Paired data – Wilcoxon signed-rank test

  • 2+ groups – Kruskal Wallis test

Chi Squared

• Comparison of categorical data

• χ 2 = the sum of (observed-expected)² /expected

• Degrees of freedom = (number of columns – 1) x (number of rows – 1)

• Assumptions

  • Independent data

  • Data can be described by binomial/multinomial distribution

  • E >5 in each cell (Fishers exact test is an alternative) 

Correlation

• Two continuous variables

• Pearson’s correlation coefficient (rho or p)

  • 1 = strong positive correlation

  • 0 = no correlation

  • -1 = strong negative correlation

• Limitation

  • Certain non-linear relationships indicate no correlation

• Context

  • In physics a value <0.8 might be considered weak

  • Medical research:

  • Rho value >0.5 Strong

  • 0.3-0.5 Moderate

  • <0.3 Weak

• Assumptions

  • Independent data points

  • Relationship is linear

  • One set of data is normally distributed for any given value of the other with a constant variance

• Alternative

  • Spearman’s rank correlation coefficient

    • When the latter 2 assumptions above break down

    • Replaces values with their rank before calculating the correlation coefficient.

    • Still can’t cope with circular distribution

Linear Regression

• Captures how x depends upon y rather than the strength of their association (correlation)

• Normally given in the form y = bx +c

• y = outcome (dependent variable) 

• b = Gradient (Average change in unit of y per unit change of x) 

• x = Exposure (independent variable) 

• c = Intercept (x=0) 

• Other values 

  • P–value = a test against the null hypothesis that the slope = 0 

  • R²= How much of the variability of y can be explained by the variability of x (Rho²) 

  • R² close to 0 indicates huge variability of data points about the regression line

  • R² = 1 means data points are on the regression line

• What else can regression do?

  • Classify one category of variables with 0s and the other with 1s

    • T-test

    • ANOVA

  • Logistic regression can do Chi²

• Assumptions

  • Data points are independent

  • Outcome data is normally distributed for any given value of exposure (residuals are normally distributed)

    • A histogram of the residuals can be used to examine this.

  • Outcome data has constant variance (residuals are homoscedastic)

  • Linear relationship

How to choose a test (Summary)

Difference in means

• 2 groups – t-test/regression

• 3 + groups – ANOVA/regression

Difference in percentages

• Chi² /logistic regression

Association between 2 continuous variables

• Correlation coefficient/regression

Data Presentation

• Figure/table legends

  • Axis (what it shows)

  • Key

  • Significance

  • Statistics used

  • Type of trend line

  • n number

  • Standard error bars