A lesson on statistics
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This document provides an overview of key concepts in statistics including different types of data, measures of central tendency, measures of dispersion, and methods for characterizing diagnostic tests. It discusses nominal, ordinal, interval and ratio data as well as discrete and continuous data. Measures of central tendency covered include mean, median and mode. Measures of dispersion include range, interquartile range, mean deviation, variance, standard deviation and coefficient of variation. The document also defines sensitivity, specificity, positive predictive value, negative predictive value and likelihood ratios for characterizing diagnostic tests. It introduces the receiver operating characteristic curve and how the area under the curve can be used to assess a test's accuracy.
A lesson on statistics
- 1. A lesson on Statistics: Data – Types, description and interpretation Dr Andrea Josephine R, 2nd year MD PG, Department of Pediatrics, ESIC Medical College & PGIMSR, Chennai.
- 2. Topics Types of data Measures of central tendency Measures of dispersion Measures of distribution Characterizing diagnostic tests – The test of a test
- 3. Types of data 1. Nominal: Qualitative data Characteristics of a variable – Categories Mutually exclusive, exhaustive No implied order E.g. Sex : Male/Female, Demographics (Urban/Suburban/Rural)
- 4. Types of data 2. Ordinal: Qualitative data – Categories Rank/Order into a progression, mutually exclusive, exhaustive Size of the interval not measurable or equal E.g. Satisfaction with treatment – Very satisfied / Somewhat satisfied / Somewhat dissatisfied / Very dissatisfied
- 5. Types of data 3. Interval: Quantitative data Meaningful intervals No absolute zero Ratio between 2 measurements not meaningful E.g. Temperature scale: In degrees Celsius, difference between 2 measurements quantifiable, but ratio not meaningful; 0⁰C does not imply a total absence of heat
- 6. Types of data 4. Ratio: Quantitative data Absolute zero Meaningful ratios E.g. Age (years), Weight(kg), Blood pressure(mmHg)
- 7. Types of data 1. Discrete: Only whole numbers possible / distinct categories E.g. Number of patients, number of syringes used, Gender, hair colour 2. Continuous: Any value in a continuum E.g. Weight, Height, Serum creatinine
- 8. Measures of central tendency 1. Mean: Used for interval & ratio data Summation of all values divided by number of values in the sample x = Ʃx n
- 9. Measures of central tendency 2. Median: Used for ordinal data Half of the values lie above it, half below it If n is odd, arrange in order: Middle value = median If n is even, arrange and take mean of middle 2 values
- 10. Measures of central tendency 3. Mode: Used for nominal data Most frequently appearing category If 2 categories appear equally, bimodal Can be multimodal
- 11. Measures of dispersion 1. Range: Difference between highest and lowest values E.g. A set of values 102, 105, 109, 111 and 120. Range is not 102-120. Range = 120-102 = 18.
- 12. Measures of dispersion 2. Interquartile range: Range of the middle 50% of the data Difference between the upper and lower quartile
- 13. Measures of dispersion 3. Mean deviation: Average of the absolute deviations from mean. Mean deviation = Ʃ ǀ x – x ǀ n
- 14. Example Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16 Step 1: Find the mean: (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16)/8 = 72/8 = 9 Step 2: Find the distance of each value from that mean:
- 15. Example (Contd.) Step 3. Find the mean of those distances: Mean Deviation = (6 + 3 + 3 + 2 + 1 + 2 + 6 + 7)/8 = 30/8 = 3.75 So, the mean = 9, and the mean deviation = 3.75 3.75 away from the middle Why take absolute value?
- 16. Measures of dispersion 4. Variance and Standard deviation: Variance(s2) = Mean of the squares of the deviation = Ʃ (x – x )2 n Standard deviation(SD) = Ʃ (x – x)2 √ n Smaller value of SD Closer the values cluster around the mean If a constant is added to all values, mean changes; Variance and SD remain the same.
- 17. Measures of dispersion 5. Coefficient of variation(CV): CV = SD/Mean The units of SD and mean are same, hence CV is an independent value. If both SD and mean are multiplied by a constant, CV remains the same (Useful in ratio measurements). Not useful in interval level data, as CV decreases with addition of a constant to each value.
- 18. Skewness Refers to the symmetry of the frequency-distribution curve. Value of 0 – Unskewed, Positive value – skewed to the right, Negative value – skewed to the left. Refers to the side of the longer tail, NOT that of the bulk of the data.
- 19. Kurtosis Refers to the peak of the frequency-distribution curve. Mesokurtosis – Normal distribution curve Leptokurtosis – Peaked; Platykurtosis - Flat
- 20. Sensitivity Ability of a test to correctly identify patients with disease Sensitivity = True positives True positives + False negatives Patients picked up, 80 Undiagnosed diseased population, 20 Sensitivity
- 21. Specificity Ability of the test to correctly identify patients who are disease-free/healthy Specificity = True negatives True negatives + False positives Healthy, 80 Healthy mis- labelled diseased, 20 Specificity
- 22. Positive predictive value Proportion of patients with positive test results who truly have disease. PPV = True positive True positive + False positive Answers the question: “I have tested positive. Am I really diseased?” Truly diseased 80% Healthy mislabelled diseased 20% PPV
- 23. Negative predictive value Proportion of patients with negative test results who are truly disease-free NPV = True negatives True negatives + False negatives Answers the question: “I have tested negative. Am I really disease-free?” Truly healthy 80% Diseased mis- labelled healthy 20% NPV
- 24. PPV and NPV Highly dependent on the prevalence of a disease in a given population. Less reliable in rare diseases. Less transferable from one population to another.
- 25. Likelihood ratio Combines sensitivity and specificity Positive likelihood ratio defines the extent to which a positive test result increases the likelihood of having disease. LR+ = Sensitivity 1 – Specificity If the LR + of a test is 1.36, a patient who tests positive is 1.36 times more likely to have the disease than a patient who tests negative.
- 26. LR – Interpretation: LR+ over 5 - 10: Significantly increases likelihood of the disease LR+ between 0.2 to 5 (esp if close to 1): Does not modify the likelihood of the disease LR+ below 0.1 - 0.2: Significantly decreases the likelihood of the disease
- 27. Likelihood ratio Negative likelihood ratio defines the extent to which a negative test result decreases the likelihood of having disease. LR- = 1 – sensitivity Specificity If LR- of a test is 1.5, it means a patient with a negative test result is 1.5 times more likely to be disease-free than a patient with a positive test result.
- 28. LR – Points: Independent of disease prevalence Specific to the test being used Can be applied to the individual patient to evaluate how worthwhile it is to perform a given test
- 29. Receiver Operator Characteristics Curve If the cut-off for a test is raised, both true and false positive rate would decrease. True positive rate = Sensitivity False positive rate = 1 – Specificity. A graph between the 2 is ROC curve.
- 30. ROC curve – Area under curve Area under curve – Used to assess overall accuracy of a test Value of 1 – High sensitivity and specificity Value of 0.5 – Zero diagnostic capability, Line of zero discrimination, no better than tossing a coin.
- 31. Using ROC curve and AUC to choose between tests ROC curves:
- 32. References Biostatistics: The bare essentials, 3e by Norman and Streiner Health services research methods, 2e by Leiyu Shi Bewick V, Cheek L, Ball J. Statistics review 13: Receiver operating characteristic curves. Crit Care. 2004;8(6):508-512. AG Lalkhen, A McCluskey. Clinical tests: sensitivity and specificity. Contin Educ Anaesth Crit Care Pain (2008) 8 (6): 221-223.
- 33. Thank You