Statistics for biological data Significance tests for continuous variables Aya Elwazir Teaching assistant of medical genetics, FOMSCU PHD student, University of Sheffield

NOT normally distributed Choice of test Normally distributed NOT normally distributed Descriptives Mean ± SD Median (IQR) Significance tests Parametric tests Non-Parametric tests ≤2 groups T test Wilcoxon Kruskal Wallis Friedman > 2 groups ANOVA

T-test One sample t-test Independent samples t-test Dependent samples Group 1 Sample Group 2 Sample A Sample B Sample Population Same group

One sample t-test H0 x̄ = µ x̄ = 72 H1 x̄ ≠ µ x̄ ≠ 72 Compare the mean of the sample [x̄] with a pre-specified value (population mean [µ]) Continuous Average score of medical students in UK universities = 72 Student name Score John 63.5 Sue 71.2 Sarah 56.6 Nick 80.0 Ben 79.4 We think that the average score of medical students in the University of Sheffield will be different H0 x̄ = µ x̄ = 72 H1 x̄ ≠ µ x̄ ≠ 72

Independent sample t-test Compare the mean between 2 independent groups [x̄1 , x̄2] Categorical (grouping) Continuous Average score of medical students between University of Sheffield & University of Leeds Student name University Score John Sheffield 63.5 Marwa 71.3 Sarah 56.5 Nick 80.0 Ben 79.3 Ruby Leeds 83.3 Ahmed 73.5 Beth 55.0 Sue 67.0 Claire 46.5 H0 x̄1 = x̄2 H1 x̄1 ≠ x̄2

Independent sample t-test Assumptions Normality Independent groups Equal variance between groups Group 1 Group 2

Independent sample t-test Why does variance matter? Equal mean - Equal variance Different mean - Equal variance var.equal = T Group 1 Group 2 t- test Group 1 Group 2 Assumes Equal variance ‘R’ Default Equal mean - Different variance Different mean - Different variance Welch t- test Group 1 Group 2 Group 1 Group 2 Assumes Different variance

Dependent sample t-test Categorical (grouping) Also called paired t-test Continuous Compare the mean between 2 dependent groups [x̄ , x̄’] Student name Pre/ post Score John Pre 63.5 Sue 71.2 Sarah 56.6 Nick 80.0 Ben 79.4 Post 65.5 80.3 52.5 80.5 86.3 Average score of medical students at University of Sheffield before & after a ‘course revision’ module Continuous Student name Pre Post John 63.5 65.5 Sue 71.2 80.3 Sarah 56.6 52.5 Nick 80.0 80.5 Ben 79.4 86.3 H0 x̄ = x̄‘ H1 x̄ ≠ x̄‘ Wide format Long format

NOT normally distributed Choice of test Normally distributed NOT normally distributed Descriptives Mean ± SD Median (IQR) Significance tests Parametric tests Non-Parametric tests ≤2 groups T test Wilcoxon Kruskal Wallis Friedman > 2 groups ANOVA

One-Sample Wilcoxon Signed Rank Test Wilcoxon Signed-Rank Test Wilcoxon-test One-Sample Wilcoxon Signed Rank Test Wilcoxon– Mann–Whitney test Wilcoxon Signed-Rank Test Sample Group 1 Sample Group 2 Sample A Sample B Sample Population Same group

NOT normally distributed Choice of test Normally distributed NOT normally distributed Descriptives Mean ± SD Median (IQR) Significance tests Parametric tests Non-Parametric tests ≤2 groups T test Wilcoxon Kruskal Wallis Friedman > 2 groups ANOVA

Repeated measures ANOVA One-way ANOVA Two-way ANOVA Repeated measures ANOVA 1 categorical (grouping variable>2 levels) 1 numeric/continuous variable 2 categorical (grouping variables) 1 numeric/continuous variable Equivalent to dependant t-test But >2 repeated measures

One-way ANOVA H0 x̄1 = x̄2 = x̄3 H1 x̄1 ≠ x̄2 ≠ x̄3 Equivalent to independent t-test but for > 2 groups Categorical (grouping) Continuous Compare the mean between 3 or more independent groups [x̄1 , x̄2, , x̄3 ] Student name University Score John Sheffield 63.5 Marwa 71.3 Sarah 56.5 Nick Manchester 80.0 Ben 79.3 Ruby 83.3 Ahmed Leeds 73.5 Beth 55.0 Sue 67.0 Claire 46.5 Average score of medical students between University of Sheffield, University of Leeds and University of Manchester H0 x̄1 = x̄2 = x̄3 H1 x̄1 ≠ x̄2 ≠ x̄3

Two-way ANOVA 2 categorical (grouping variables) Continuous Student name University Gender Score John Sheffield Male 63.5 Marwa Female 71.3 Sarah 56.5 Nick Manchester 80.0 Ben 79.3 Ruby 83.3 Ahmed Leeds 73.5 Beth 55.0 Sue 67.0 Claire 46.5 Average score of medical students between University of Sheffield, University of Leeds and University of Manchester AND between males & females

Repeated measures ANOVA Categorical (grouping) Equivalent to paired t-test but for >2 repeated measures Compare the mean between > 2 dependent groups [x̄ , x̄’ , x̄’’] Continuous Student name Exam Score John Mid-term 63.5 Sue 71.2 Sarah 56.6 Term 65.5 80.3 52.5 Final 85.5 86.0 50.5 Average score of medical students at University of Sheffield for mid-term, term & final H0 x̄ = x̄‘ = x̄‘‘ Student name Mid-term Term Final John 63.5 65.5 85.5 Sue 71.2 80.3 86.0 Sarah 56.6 52.5 50.5 H1 x̄ ≠ x̄‘≠ x̄‘‘ Long format Wide format

Post Hoc test Only done if ANOVA result is significant (p<0.05) Indicates the significant result was due to differences in which groups Sheffield Manchester Leeds - 0.032 0.251 0.042 Sheffield ≠ Manchester Leeds ≠ Manchester

NOT normally distributed Choice of test Normally distributed NOT normally distributed Descriptives Mean ± SD Median (IQR) Significance tests Parametric tests Non-Parametric tests ≤2 groups T test Wilcoxon Kruskal Wallis Friedman > 2 groups ANOVA

Kruskal Wallis - Friedman Wallis test Friedman test Equivalent to one-way ANOVA for non-parametric data Equivalent to repeated measures ANOVA for non-parametric data

Statistics for biological data Course Objectives Introduction to statistics 1. Contingency tables & testing for categorial variables 2. Normality testing & Descriptive statistics 3. Testing for continuous variables Lots of practice!