• All about frequencies!
• Row x Column table (2 x 2 simplest)
• Categorical data
• Look for association (relationship) between row variable and column variable
• N.B. we have already seen an example of this in the Lady tasting tea experiment

• In a study, 372 people wearing a helmet received head injuries compared to 267 that were not
  • but this does not show the full picture.

• It turns out that far more people in the study were wearing helmets
• Analysis on the data shows that a much higher proportion of the cyclists not wearing a helmet have a higher proportion of head injuries

                   Head.Injury Other.Injury
Wearing Helmet             372         4715
Not Wearing Helmet         267         1391

• E.g. Research question: A trial to assess the effectiveness of a new treatment versus a placebo in reducing tumour size in patients with ovarian cancer.

          Tumour.Did.Not.Shrink Tumour.Did.Shrink
Treatment                    44                40
Placebo                      24                16

• Is there an association between treatment group and tumour shrinkage
• Null hypothesis, \(H_0\): No association
• Alternative hypothesis, \(H_1\): Some association

          Tumour.Did.Not.Shrink Tumour.Did.Shrink Total
Treatment                    44                40    84
Placebo                      24                16    40
Total                        68                56    68

\[E = \frac{row total \times col total}{overall total} \]

• e.g. for row 1, column 1 \[\frac{84}{124} \times \frac{68}{124} \times 124 = \frac{84\times68}{124} = 46.1\]

Observed frequencies:

          Tumour.Did.Not.Shrink Tumour.Did.Shrink
Treatment                    44                40
Placebo                      24                16

Expected frequencies:

          Tumour.Did.Not.Shrink Tumour.Did.Shrink
Treatment                  46.1              37.9
Placebo                    21.9              18.1

\[\chi^2_1 = \frac{(44-46.06)^2}{46.06} + \frac{(40-37.94)^2}{37.94} + \frac{(24-21.94)^2}{21.94} + \frac{(16-18.06)^2}{18.06}\]

Test statistic: \({\chi_1}^2\) = 0.43 df = 1 P-value = 0.43

Do not reject \(H_0\) (No evidence of an association between treatment group and tumour shrinkage)

• In general, a Chi-square test is appropriate when:
  • at least 80% of the cells have an expected frequency of 5 or greater
  • none of the cells have an expected frequency less than 1
• If these conditions aren’t met, Fisher’s exact test should be used.

• e.g. Research question: Is there an association between treatment group and tumour shrinkage?

          Tumour.Did.Not.Shrink Tumour.Did.Shrink Total
Treatment                     8                 3    11
Placebo                       9                 4    13
Total                        17                 7    17

• Null hypothesis: \(H_0\): No association
• Alternative hypothesis: \(H_1\): Some association

Expected frequencies:-

          Tumour.Did.Not.Shrink Tumour.Did.Shrink
Treatment                   7.8               3.2
Placebo                     9.2               3.8

• Test statistic: N/A
• P-value 1
• Interpretation: *Do not reject \(H_0\) (No evidence of an association between treatment group and tumour shrinkage)

• Chi-square test
  • Use when we have two categorical variables, each with two or more levels, and our expected frequencies are not too small.
• Fishers exact test
  • Use when we have two categorical variables, each with two levels, and our expected frequencies are small.
• (Chi-square test for trend)
  • Use when we have two categorical variables, where one or both are naturally ordered and the ordered variable has at least three levels, and our expected frequencies are not too small.
• (McNemar’s test)
  • Use when we have two categorical paired variables.

• Turn scientific question to null and alternative hypothesis

• Calculate expected frequencies

• Think about test assumptions

• Carry out chi-square or Fishers test if appropriate

• Complete contingency table practical

• Inside the folder mystery-data you will find 8 csv files containing data for analysis
  • details are given in the practical
• Each group of 3/4 people will be assigned a dataset to analyse
• On this interactive document, describe how you approached the analysis, what test you used and your conclusions
  • https://public.etherpad-mozilla.org/p/2018-10-24-intro-to-stats

Correlation does not equal causation

David Spiegelhalter, Chair, Winton Centre for Risk and Evidence Communication

"To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He [/she] can perhaps say what the experiment died of" - R.A. Fisher

Design consultations available at Sheffield Bioinformatics Core:- bioinformatics-core@sheffield.ac.uk