Statistics for biological data Significance tests for continuous

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

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 > 2 groups ANOVA Kruskal Wallis Friedman

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

One sample t-test Compare the mean of the sample [x ] with a pre-specified value (population mean [µ]) Average score of medical students in UK universities = 72 We think that the average score of medical students in the University of Sheffield will be different H 0 x = µ H 1 x ≠ µ x = 72 x ≠ 72 Continuous Student name Score John 63. 5 Sue 71. 2 Sarah 56. 6 Nick 80. 0 Ben 79. 4

Independent sample t-test Compare the mean between 2 independent groups [x 1 , x 2] Average score of medical students between University of Sheffield & University of Leeds H 0 x 1 = x 2 H 1 x 1 ≠ x 2 Categorical Continuous (grouping) Student name University Score John Sheffield 63. 5 Marwa Sheffield 71. 3 Sarah Sheffield 56. 5 Nick Sheffield 80. 0 Ben Sheffield 79. 3 Ruby Leeds 83. 3 Ahmed Leeds 73. 5 Beth Leeds 55. 0 Sue Leeds 67. 0 Claire Leeds 46. 5

Independent sample t-test Assumptions 1. Normality 2. Independent groups 3. Equal variance between groups Group 1 Group 2

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

Dependent sample t-test Categorical (grouping) Continuous Also called paired t-test Compare the mean between 2 dependent groups [x , x ’] Average score of medical students at University of Sheffield before & after a ‘course revision’ module Continuous H 0 x = x ‘ H 1 x ≠ x ‘ Student Pre/ name post Score John Pre 63. 5 Sue Pre 71. 2 Sarah Pre 56. 6 Nick Pre 80. 0 Ben Pre 79. 4 Student Pre name Post John 63. 5 65. 5 John Post 65. 5 Sue 71. 2 80. 3 Sue Post 80. 3 Sarah 56. 6 52. 5 Sarah Post 52. 5 Nick 80. 0 80. 5 Nick Post 80. 5 Ben 79. 4 86. 3 Ben Post 86. 3 Wide format Long format

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 > 2 groups ANOVA Kruskal Wallis Friedman

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

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 > 2 groups ANOVA Kruskal Wallis Friedman

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

One-way ANOVA Equivalent to independent t-test but for > 2 groups Compare the mean between 3 or more independent groups [x 1 , x 2, , x 3 ] Average score of medical students between University of Sheffield, University of Leeds and University of Manchester H 0 x 1 = x 2 = x 3 H 1 x 1 ≠ x 2 ≠ x 3 Categorical Continuous (grouping) Student name University Score John Sheffield 63. 5 Marwa Sheffield 71. 3 Sarah Sheffield 56. 5 Nick Manchester 80. 0 Ben Manchester 79. 3 Ruby Manchester 83. 3 Ahmed Leeds 73. 5 Beth Leeds 55. 0 Sue Leeds 67. 0 Claire Leeds 46. 5

Two-way ANOVA 2 categorical (grouping variables) Average score of medical students between University of Sheffield, University of Leeds and University of Manchester AND between males & females Categorical 1 (grouping 1) Categorical 2 (grouping 2) Continuous Student name University Gender Score John Sheffield Male 63. 5 Marwa Sheffield Female 71. 3 Sarah Sheffield Female 56. 5 Nick Manchester Male 80. 0 Ben Manchester Male 79. 3 Ruby Manchester Female 83. 3 Ahmed Leeds Male 73. 5 Beth Leeds Female 55. 0 Sue Leeds Female 67. 0 Claire Leeds Female 46. 5

Repeated measures ANOVA Equivalent to paired t-test but for >2 repeated measures Compare the mean between > 2 dependent groups Categorical (grouping) Continuous [x , x ’’] Student name Exam Score Average score of medical students at University of Sheffield for mid-term, term & final John Mid-term 63. 5 Sue Mid-term 71. 2 Sarah Mid-term 56. 6 John Term 65. 5 Sue Term 80. 3 Sarah Term 52. 5 John Final 85. 5 Sue Final 86. 0 Sarah Final 50. 5 H 0 x = x ‘‘ H 1 x ≠ x ‘‘ Student Midname term Term John 63. 5 65. 5 85. 5 Sue 71. 2 80. 3 86. 0 Sarah 56. 6 52. 5 50. 5 Wide format Final Long 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 Sheffield 0. 032 0. 251 Manchester 0. 032 0. 042 Sheffield ≠ Manchester Leeds 0. 251 0. 042 -

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 > 2 groups ANOVA Kruskal Wallis Friedman

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

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