1. Stats Facts Mark Halloran

2. Diagnostic Stats Disease present Disease absent TOTALS Test positive aba+b Test negative cdc+d TOTALSa+cb+da+b+c+d

3. Formulae (1)  Sensitivity =a / (a+c)  Specificity =d / (b+d)  LR+ =sens / (1-spec)  LR- =(1-sens) / spec  PPV =a / (a+b)  NPV =d / (c+d) (LR+ = Likelihood ratio for a positive (+) result) (PPV = Positive Predictive Value, NPV = Neg predictive value)  Sensitivity =a / (a+c)  Specificity =d / (b+d)  LR+ =sens / (1-spec)  LR- =(1-sens) / spec  PPV =a / (a+b)  NPV =d / (c+d) (LR+ = Likelihood ratio for a positive (+) result) (PPV = Positive Predictive Value, NPV = Neg predictive value)

4. Formulae (2)  Prevalence = (a+c) / (a+b+c+d)  Pre-test odds = prev / (1-prev)  Post-test odds = pre-test odds x LR  Post-test probability = Post-test odds / Post-test odds + 1  Prevalence = (a+c) / (a+b+c+d)  Pre-test odds = prev / (1-prev)  Post-test odds = pre-test odds x LR  Post-test probability = Post-test odds / Post-test odds + 1

5. TB treatment RCT Death from TB Yes No total Control group (bed rest) 143852 Experimental Group (streptomycin+ bed rest 45155

6. Formulae (3)  Control event rate  = number of events/total for control group  14/52 =0.27 (CER) (the risk of dying in the control group is 27%)  Experimental event rate  =number of events/ total for experimental group  4/55 =0.07 (EER) (the risk of dying in the experimental group is 7%)  Control event rate  = number of events/total for control group  14/52 =0.27 (CER) (the risk of dying in the control group is 27%)  Experimental event rate  =number of events/ total for experimental group  4/55 =0.07 (EER) (the risk of dying in the experimental group is 7%)

7. Formulae (4)  Absolute risk reduction for the outcome - death:  ARR= risk of event in the control group – risk of event in the experimental group  ARR=CER-EER= 0.27 – 0.07 = 0.2 or 20%  Relative risk reduction for the outcome - death:  RRR= absolute risk reduction/ risk of event in control group  RRR =(CER-EER)/ CER = (0.27 – 0.07)/ 0.27 = 0.2/0.27 = 74%  Absolute risk reduction for the outcome - death:  ARR= risk of event in the control group – risk of event in the experimental group  ARR=CER-EER= 0.27 – 0.07 = 0.2 or 20%  Relative risk reduction for the outcome - death:  RRR= absolute risk reduction/ risk of event in control group  RRR =(CER-EER)/ CER = (0.27 – 0.07)/ 0.27 = 0.2/0.27 = 74%

8. Number Needed to Treat (NNT)  A more useful statistical expression for doctors and patients  NNT = 1 / ARR = 1 / 0.2 = 5 i.e. (in this study) five patients must be treated with streptomycin to prevent one death one death from TB  A more useful statistical expression for doctors and patients  NNT = 1 / ARR = 1 / 0.2 = 5 i.e. (in this study) five patients must be treated with streptomycin to prevent one death one death from TB

9. Number needed to harm (NNH)  What about non-maleficence?  NNH = NNT but for an undesirable event  To calculate the number needed to harm we need to construct another table, this time with the figures for the adverse outcome which was VIIIth nerve damage  What about non-maleficence?  NNH = NNT but for an undesirable event  To calculate the number needed to harm we need to construct another table, this time with the figures for the adverse outcome which was VIIIth nerve damage

10. Risk and Odds  9 horse race, all equal chance of winning.  The risk (probability) of your horse winning = 1 / total number of potential winners = 1/9.  The odds of your horse winning are 1 / number of horses not winning = 1/8  Using the example of a couple expecting a baby:  The risk (probability) of having a baby boy is calculated as the likelihood of that outcome/number of possible outcomes = ½  The Odds of having a boy is calculated as the likelihood of that outcome/likelihood of it not occurring = 1/1 =1  9 horse race, all equal chance of winning.  The risk (probability) of your horse winning = 1 / total number of potential winners = 1/9.  The odds of your horse winning are 1 / number of horses not winning = 1/8  Using the example of a couple expecting a baby:  The risk (probability) of having a baby boy is calculated as the likelihood of that outcome/number of possible outcomes = ½  The Odds of having a boy is calculated as the likelihood of that outcome/likelihood of it not occurring = 1/1 =1

11. Back to the streptomycin: risk and odds of death  Risk of death in control group= 14/52 = 0.27 (same as CER)  Risk of death in experimental group = 4/55 = 0.07 (same as EER)  Risk ratio (relative risk) for death in the experimental group compared to the control group= 0.07/0.27 = 0.26  Risk of death in control group= 14/52 = 0.27 (same as CER)  Risk of death in experimental group = 4/55 = 0.07 (same as EER)  Risk ratio (relative risk) for death in the experimental group compared to the control group= 0.07/0.27 = 0.26

12. Odds ratio  The odds of death = the number of people dying/ number of people not dying: Control group: odds of death= 14/38=0.37 Experimental group: odds of death 4/51= 0.078 Odds ratio = odds in experimental group/ odds in control group = 0.078/0.37 = 0.21  The odds of death = the number of people dying/ number of people not dying: Control group: odds of death= 14/38=0.37 Experimental group: odds of death 4/51= 0.078 Odds ratio = odds in experimental group/ odds in control group = 0.078/0.37 = 0.21

13. Formulae (23)  Standard Deviation: σ 2 = 1/n Σ(xi - μ) 2  Coefficient of Variation = (sd x 100) / mean)  Standard Error  Standard Deviation: σ 2 = 1/n Σ(xi - μ) 2  Coefficient of Variation = (sd x 100) / mean)  Standard Error

14. Standard Deviation

15. Confidence Interval  Single observation: 95% CI = mean ± 1.96sd  Mean of new sample: 95% CI = mean ± 1.96se  Single observation: 95% CI = mean ± 1.96sd  Mean of new sample: 95% CI = mean ± 1.96se

16. Study Designs

17. Types of Studies  Cross Sectional: Sample looked at at one point in time to attempt to find associations  Case-Control: Comparing subjects who have a condition to those who do not to identify factors that may contribute  Cohort: Group of people followed to see how variables affect outcome  Cross Sectional: Sample looked at at one point in time to attempt to find associations  Case-Control: Comparing subjects who have a condition to those who do not to identify factors that may contribute  Cohort: Group of people followed to see how variables affect outcome

18. Ia: Systematic review / meta-analysis of RCTs Ib: At least 1 RCT IIa: At least one well-designed controlled study (not randomised) IIb: At least one well-designed quasi-experimental study eg cohort III: Well-designed non-experimental descriptive studies eg case-control IV: Expert committee reports, opinions ± clinical experience of respected authorities Levels of Evidence