1. Introduction to Distributions and Probability
Statistics for Health Research
Introduction to Distributions and Probability
Peter T. Donnan
Professor of Epidemiology and Biostatistics

2. Overview Distributions History of probability
Definitions of probability
Random variable
Probability density function
Normal, Binomial and Poisson distributions

3. Introduction to Probability Density Functions
Normal Distribution /
Gaussian / Bell curve
Poisson named after French Mathematician
Binomial related to binary factors (Bernoulli Trials)

4. 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

5. What is the relationship between the Normal or Gaussian distribution and probability?

6. Probability “I cannot believe that God plays dice with the cosmos”
Albert Einstein
“The probable is what usually happens”
Aristotle

7. 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)

8. Types of Probability Two basic definitions: 1) Frequentist
Classical
Proportion of times an event occurs in a long series of ‘trials’
2) Subjectivist
Bayesian
Strength of belief in event happening

9. 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

10. 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

11. 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…..

12. Probability Density Function for rolling two dice
6/36
5/36
4/36
3/36
2/36
1/36
1/ /36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

13. 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

14. Probability Density Function for continuous variable
6/36
5/36
4/36
3/36
2/36
1/36
1/ /36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

15. Consider distribution of weight in kg; all values possible not just discrete
20…….30……40…… 50 ……60…….70…….80…..90….100….110…… 120
Probability
Weight in kilograms

16. Probability Density Function in SPSS
Use Analyze / Descriptive Statistics / Frequencies
and select no table and charts box as below

17. Probability Density Function in SPSS
Data from ‘LDL Data.sav’ of baseline LDL cholesterol

18. 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!

19. 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%

20. 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%

21. Normal Distribution Most of area lies between +1 and -1 SD (64%)
The large majority lie between +2 and -2 SDs (95%)

22. Normal Distribution
Probability Density Function (PDF) =

23. How well does my data fit a Normal Distribution?
Note median and mean virtually the same
Skewness = 0.039, close to zero
Skewness is measure of symmetry (0=perfect symmetry)
Eyeball test - fitted normal curve looks good!

24. 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?

25. I used to be Normal until I discovered Kilmogorov-Smirnoff!
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

26. Consider the distribution of survival times following surgery for colorectal cancer
Note median=835 days and mean=848
Skewness = 2.081, very skewed (> 1.0)
Strong tail to right! Approximately Normal?

27. Try a log transformation for right positive skewed data?
Better but now slightly skewed to left!

28. Examples of skewed distributions in Health Research
Discrete random variables – hospital admissions, cigarettes smoked, alcohol consumption, costs
Continuous RV – BMI, cholesterol, BP
30%

29. The Binomial Distribution
‘Binomial’ means ‘two numbers’.
Outcomes of health research are often measured by whether they have occurred or not i.e binary.
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

30. 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

31. 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
p=0.25 and n=20

32. The Binomial Distribution
Binomial distributions used for binary factors and so used to assess percentages or proportions
Utilised in Cross-tabulation and logistic regression
Note as N gets larger or
P ~0.5 then Binomial is
Equal to Normal Distr.
B(n,p) ~ N (np, np(1-p))

33. 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, events are given by:

34. 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

35. 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.

36. Summary
Many of variables measured in Health Research form distributions which approximate to common distributions with known mathematical properties
Normal, Poisson, Binomial, etc…
Note a relationship for all centred
around the exponential distribution
Where e = 2.718
All belong to the Exponential Family of distributions
These probability distributions are critical to applying statistical methods

37. 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