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Statistics for Health Research Introduction to Distributions and Probability Peter T. Donnan Professor of Epidemiology and Biostatistics Overview • Distributions • History of probability • Definitions of probability • Random variable • Probability density function • Normal, Binomial and Poisson distributions Introduction to Probability Density Functions •Normal Distribution / •Gaussian / Bell curve •Poisson named after French Mathematician •Binomial related to binary factors (Bernoulli Trials) Early use of Normal Distribution • Gauss was a German mathematician who solved mystery of where Ceres would appear after it disappeared behind the Sun. • He assumed the errors formed a Normal distribution and managed to accurately predict the orbit of Ceres What is the relationship between the Normal or Gaussian distribution and probability? Probability “I cannot believe that God plays dice with the cosmos” Albert Einstein “The probable is what usually happens” Aristotle Origins of Probability • Early interest in permutations Vedic literature 400 BC • Distinguished origins in betting and gambling! • Pascal and Fermat studied division of stakes in gambling (1654) • Enlightenment – seen as helping public policy, social equity • Astronomy – Gauss (1801) • Social and genetic – Galton (1885) • Experimental design – Fisher (1936) Types of Probability Two basic definitions: 1) Frequentist 2) Subjectivist Classical Bayesian Proportion of Strength of belief times an event in event happening occurs in a long series of ‘trials’ Frequentists vs. Bayesians • Two entrenched camps • Scientists tend to use the frequentist approach • Bayesians gaining ground • Most scientists use frequentist methods but incorrectly interpret results in a Bayesian way! Frequentists • Consider tossing a fair coin • In any trial, event may be a ‘head’ or ‘tail’ i.e. binary • Repeated tossing gives series of ‘events’ • In long run prob of heads=0.5 THTTHHHHTHHHTHHHTTHTTTHHTTHTTHHHTTTHHTHHHTTTTTHHH 0.6 0.56 0.52 Frequentist Probability • Note the difference between ‘long run’ probability and an individual trial • In an individual trial a head either occurs (X=1) or does not occur (X=0) • Patient either survives or dies following an MI • Prob of dying after MI ≈ 30% based on a previous long series from a population of individuals who experienced MI Subjective Probability • Based on strength of belief • But more akin to thinking of clinician making a diagnosis • Faced with patient with chest pain, based on past experience, believes prob of heart disease is 20% • Person tossing coin believes prob of head is 1/2 Comparison of definitions of Probability • Problems of subjective probability • Probability for same patient can vary even with same clinician • Person can believe prob of head is 0.1 even if it is a fair coin • Subjectivists argue they are more realistic • This course sticks to ‘frequentist’ and ‘model-based’ methods of probability Random Variable • Consider rolling 2 dice and we want to summarise the probabilities of all possible outcomes • We call the outcome a random variable X which can have any value in this case from 2 to 12 • Enumerate all probabilities in sample space S • P (2) = 1/6x1/6 = 1/36, P (3)=2/36, P (4) = 3/36, etc….. Probability Density Function for rolling two dice 6/36 5/36 4/36 3/36 2/36 1/36 2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 2 3 4 5 6 7 8 9 10 11 12 Probability Density Function for rolling two dice What is probability of getting 12? Answer 1/36 What is probability of getting more than 8? Ans. 10/36 6/36 5/36 4/36 3/36 2/36 1/36 2 3 4 5 6 7 8 9 10 11 12 Probability Density Function for continuous variable 6/36 5/36 4/36 3/36 2/36 1/36 2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 2 3 4 5 6 7 8 9 10 11 12 Consider distribution of weight in kg; all values possible not just discrete Probability 20…….30……40…… 50 ……60…….70…….80…..90….100….110…… 120 Weight in kilograms 2 3 4 5 6 7 8 9 10 11 12 Probability Density Function in SPSS Use Analyze / Descriptive Statistics / Frequencies and select no table and charts box as below Probability Density Function in SPSS Data from ‘LDL Data.sav’ of baseline LDL cholesterol Normal Distribution Note that a Normal or Gaussian curve is defined by two parameters: Mean µ and Standard Deviation σ And often written as N ( µ, σ ) Hence any Normal distribution has mathematical form Impossible to be integrated so area under the curve obtained by numerical integration and tabulated! Normal Distribution As noted earlier the curve is symmetrical about the mean and so p ( x ) > mean = 0.5 or 50% And p ( x ) < mean = 0.5 or 50% And p (a < x < b) = p(b) – p(a) 50% 50% Normal Distribution and Probabilities So we now have a way of working out the probability of any value or range of values of a variables IF a Normal distribution is a reasonable fit to the data p (a < x < b) = p(b) – p(a) which is the area under the curve between a and b 50% 50% Normal Distribution Most of area lies between +1 and -1 SD (64%) The large majority lie between +2 and -2 SDs (95%) Normal Distribution Probability Density Function (PDF) = How well does my data fit a Normal Distribution? Statistics Baseline LDL N Valid 1383 Missing 0 Mean 3.454363 Median 3.506214 St d. Dev iation .9889157 Skewness .039 St d. Error of Skewness .066 Minimum .3345 Maximum 7.5650 Note median and mean virtually the same Skewness = 0.039, close to zero Skewness is measure of symmetry (0=perfect symetry) Eyeball test - fitted normal curve looks good! Try Q-Q plot in Analyze / Descriptive Statistics/ Q-Q plot Plot compares Expected Normal distribution with real data and if data lies on line y = x then the Normal Distribution is a good fit Note still an eyeball test! Is this a good fit? I used to be Normal until I discovered Kilmogorov-Smirnoff! One-Sample Kolmogorov-Smirnov Test Baseline LDL N 1383 Normal Parameters a,b Mean 3.454363 Std. Deviation .9889157 Most Extreme Absolute .043 Diff erences Positive .043 Negative -.043 Kolmogorov -Smirnov Z 1.617 Asy mp. Sig. (2-tailed) .011 a. Test distribution is Normal. b. Calculated f rom data. Eyeball Test indicates distribution is approximately Normal but K-S test is significant indicating discrepancy compared to Normal WARNING: DO NOT RELY ON THIS TEST Consider the distribution of survival times following surgery for colorectal cancer Statistics Time f rom Surgery N Valid 476 Missing 0 Mean 848.3908 Median 835.5000 Std. Dev iation 582.39657 Skewness 2.081 Std. Error of Skewness .112 Minimum 14.00 Maximum 5763.00 Note median=835 days and mean=848 Skewness = 2.081, very skewed (> 1.0) Strong tail to right! Approximately Normal? Try a log transformation for right positive skewed data? Statistics logtime N Valid 476 Missing 0 Mean 6.4346 Median 6.7286 Std. Deviation .95059 Skewness -1.504 Std. Error of Skewness .112 Minimum 2.67 Maximum 8.66 Better but now slightly skewed to left! Examples of skewed distributions in Health Research Discrete random variables – hospital admissions, cigarettes smoked, alcohol consumption, costs Continuous RV – BMI, cholesterol, BP 30% The Binomial Distribution • ‘Binomial’ means ‘two numbers’. • Outcomes of health research are often measured by whether they have occurred or not. • For example, recovered from disease, admitted to hospital, died, etc • May be modelled by assuming that the number of events n has a binomial distribution with a fixed probability of event p The Binomial Distribution • Based on work of Jakob Bernoulli, a Swiss mathematician • Refused a church appointment and instead studied mathematics • Early use was for games of chance but now used in every human endeavour • When n = 1 this is called a Bernoulli trial • Binomial distribution is distribution for a series of Bernoulli trials The Binomial Distribution • Binomial distribution written as B ( n , p) where n is the total number of events and p = prob of an event • This is a Binomial Distribution with 0.20 p=0.25 and n=20 Probability of R Successes 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Successes The Binomial Distribution The Poisson Distribution Poisson distribution (1838), named after its inventor Simeon Poisson who was a French mathematician. He found that if we have a rare event (i.e. p is small) and we know the expected or mean ( or µ) number of occurrences, the probabilities of 0, 1, 2 ... events are given by: R e P( R ) R! The Poisson Distribution Note similarity to Binomial In fact when p is small and n is large B(n, p) ~ P (µ = np) Also for large values of µ: P (µ) ~ N ( µ, µ ) Hence if n and p not known could use Poisson instead The Poisson Distribution In health research often used to model the number of events assumed to be random: Number of hip replacement failures, Number of cases of C. diff infection, Diagnoses of leukaemia around nuclear power stations, Number of H1N1 cases in Scotland, Etc. Summary •Many of variables measured in Health Research form distributions which approximate to common distributions with known mathematical properties 40 •Normal, Poisson, Binomial, etc… 30 •Note a relationship for all centred 20 10 Std. Dev = .96 around the exponential distribution Mean = -.04 0 N = 501.00 -2.95 -1.95 -.95 .05 1.05 2.05 RANNORM Where e = 2.718 • All belong to the Exponential Family of distributions • These probability distributions are critical to applying statistical methods SPSS Practical • Read in data file ‘LDL Data.sav’ • Consider adherence to statins, baseline LDL, min Chol achieved, BMI, duration of statin use • Assess distributions for normality • If non-normal consider a transformation • Try to carry out Q-Q plots