Introduction to Biomedical Statistics

1. Introduction to Biomedical Statistics

2. Signal Detection Theory • What do we actually “detect” when we say we’ve detected something?

3. Signal Detection Theory • What do we actually “detect” when we say we’ve detected something? • We say we’ve “detected” when a criterion value exceeds a threshold

4. Signal Detection Theory • examples: • the onset of a light or sound • the presence of an abnormality on x-ray

5. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent

6. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent

7. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent

8. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent

9. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent

10. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent This is the total # of Target Present Trials

11. Signal Detection Theory • There are 4 possible situations Target is: Present Absent Present You Respond: Absent This is the total # of Target Present Trials

12. Signal Detection Theory • Hit Rate (H) is the proportion of target present trials on which you respond “present”

13. Signal Detection Theory • Notice that H is a proportion, so 1 - H gives you the “miss” rate or…

14. Signal Detection Theory • False-Alarm Rate (FA) is the proportion of target absent trials on which you respond “present”

15. Signal Detection Theory • Notice that FA is a proportion. 1 minus FA gives you the correct rejections or …

16. Signal Detection Theory • Signal Detection can be modeled as signal + noise with some detection threshold

17. Signal Detection Theory Noise is normally distributed - Target Absent trials still contain some stimulus Frequency Target Absent Stimulus Intensity

18. Signal Detection Theory Target Present trials contain a little bit extra intensity contributed by the signal Noise is normally distributed - Target Absent trials still contain some stimulus Frequency Target Present Target Absent Stimulus Intensity

19. Signal Detection Theory This is the signal’s contribution Target Present trials contain a little bit extra intensity contributed by the signal Noise is normally distributed - Target Absent trials still contain some stimulus Frequency Target Present Target Absent Stimulus Intensity

20. Signal Detection Theory • We can imagine a static criterion above which we’ll respond “target is present” Criterion Frequency Target Present Target Absent Stimulus Intensity

21. Signal Detection Theory • Notice that H, FA, etc thus have graphical meanings Criterion Frequency Proportion Hits Stimulus Intensity

22. Signal Detection Theory • Notice that H, FA, etc thus have graphical meanings Criterion Frequency Proportion Misses Stimulus Intensity

23. Signal Detection Theory • Notice that H, FA, etc thus have graphical meanings Criterion Frequency Proportion False Alarms Stimulus Intensity

24. Signal Detection Theory • Notice that H, FA, etc thus have graphical meanings Criterion Frequency Proportion Correct Rejections Stimulus Intensity

25. Signal Detection Theory • Notice that as H increases, FA also increases Frequency Criterion H Stimulus Intensity

26. Signal Detection Theory • Notice that as H increases, FA also increases Frequency Criterion FA Stimulus Intensity

27. Signal Detection Theory • d’ (pronounced d prime) is a measure of sensitivity to detect a signal from noise and does not depend on criterion - it is the distance between the peaks of the signal present and signal absent curves • d’ is computed by converting from H and FA proportions into their corresponding Z scores and subtracting Zinv(FA) from Zinv(H) Frequency Stimulus Intensity

28. Some Common Biomedical Statistics • Sensitivity • Specificity • Positive Predictive Value • Negative Predictive Value • Likelihood Ratio • Relative Risk • Absolute Risk • Number needed to treat/harm

29. Sensitivity and Specificity • Consider a test for a condition • e.g. Pregnancy test • e.g. Prostate-specific Antigen (Prostate Cancer) • e.g. Ultrasound (Breast Cancer) • These are all signal detection problems

30. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent

31. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent

32. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent

33. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent

34. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent

35. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent This is Total # of Condition Present Cases

36. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: Absent This is Total # of Condition Absent Cases

37. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent This is Total # of “positive” tests Present Test Result: Absent

38. Sensitivity and Specificity • Four possible situations: Condition is: Present Absent Present Test Result: This is Total # of “negative” tests Absent

39. Sensitivity and Specificity • Sensitivity is the proportion of condition present cases on which the test returned “positive” • Analogous to the hit rate (H) in Signal Detection Theory

40. Sensitivity and Specificity • Specificity is the proportion of condition absent cases on which the test returned “negative” • Analogous to the Correct Rejection rate in Signal Detection Theory

41. Sensitivity and Specificity • Notice that 1 minus the Sensitivity is analogous to the FA of Signal Detection Theory

42. Sensitivity and Specificity • Notice that 1 minus the Sensitivity is analagous to the FA of Signal Detection Theory • Recall that in Signal Detection Theory, as criterion were relaxed both H and FA increased and as criterion were more stringent, H and FA decreased

43. Sensitivity and Specificity • Notice that 1 minus the Sensitivity is analagous to the FA of Signal Detection Theory • Recall that in Signal Detection Theory, as criterion were relaxed both H and FA increased and as criterion were more stringent, H and FA decreased • Sensitivity and Specificity have a similar relationship: as a cut-off value for a test becomes more stringent the sensitivity goes down and the specificity goes up…and vice versa

44. Sensitivity and Specificity • “For detecting any prostate cancer, PSA cutoff values of 1.1, 2.1, 3.1, and 4.1 ng/mL yielded sensitivities of 83.4%, 52.6%, 32.2%, and 20.5%, and specificities of 38.9%, 72.5%, 86.7%, and 93.8%, respectively.” JAMA. 2005 Jul 6;294(1):66-70.

45. Sensitivity and Specificity • Likelihood Ratio is the ratio of True Positive rate to False Positive rate • Loosely corresponds to d’ in that Likelihood ratio is insensitive to changes in criterion

46. Sensitivity and Specificity • If a test is positive, how likely is it that the condition is present? • Positive Predictive Value is the proportion of “positive” test results that are correct

47. Sensitivity and Specificity • Negative Predictive Value is the proportion of “negative” test results that are correct

48. Sensitivity and Specificity • Consider the influence of exposure to some substance or treatment on the presence or absence of a condition • e.g. smoking and cancer • e.g. aspirin and heart disease

49. Sensitivity and Specificity • A similar logic can be applied Disease: Yes No No Exposure: Yes

50. Sensitivity and Specificity • A similar logic can be applied Disease: Yes No This is total # exposure No Exposure: Yes