Statistics in Biology and Medicine

Richard Tseng

Pharmamatrix Workshop 2010

July 14, 2010

The goal of statistics is to analyze, interpret and present data collected to study systems of interest!!

Outline• Descriptive statistics• Inferential statistics

– Probability theory– Hypothesis test– Regression

• Some other tools– Tools for component analysis– Bayesian statistics

• Summary

Descriptive statistics

• Definitions– Set: A well-defined collection of objects and each

object is called an element– Operation of sets: union and intersection

For example, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

4} {3, BA

6} 5, 4, 3, 2, {1, BA

• Data type– Interval scale

• For example, body weight (g): 1, 1.5, 2, 3 …

– Ordinal scale• For example, scores for patient responses to treatment

– Nominal scale • Categorical data. For example, factors to influence

treatments

Response Much worse

Bit worse About same

Bit better Much better

Score -2 -1 0 1 2

• How large are the numbers?– Mean – Median

[1]

[1]

• How variable are the numbers?– Standard deviation (SD) – Coefficient of variance

(CV = SD/mean)

Inferential Statistics: Probability theory

• Law of large numbers– The mean of elements in a set converges to the

expected value when the number of elements close to infinite

• Law of small numbers– There are not enough small numbers to satisfy all

the demands placed on them

• Central limit theorem– states conditions under which the mean of a

sufficiently large number of independent random variable, each with finite mean and variance, will be approximately normally distributed

http://www.stat.sc.edu/~west/javahtml/CLT.html

• Probability– Meaning

• Frequency interpretation: A number are associated with the rate of occurrence of an event in a well defined random physical systems

• Bayesian interpretation: A number assigned to any statement whatsoever, even when no random process is involved, as a way to represent the degree to which the statement is supported by the available evidence

• Probability– Basic rules

• Subtraction

• Addition

• Multiplication

BAPBPAPBAP

'1 APAP

BAPAPBAP

• Probability– Bayesian rule

APABP

BPBAP

LikelihoodPriorPosterior

• Probability– Maximum entropy principle: The most honest

probability distribution assignment to a system is the one that maximizes the entropy of the system subject to any information available in hand.

Inferential Statistics: Regression

• Goal: To correlate the study outcomes of systems of interest and possible factors.

• Model: – Linear model

– Logistic model

bxaR

1exp

exp

bxa

bxaR

• OptimizationSuppose there are n outcomes di of a study

– Least-square method

– Maximum Likelihood estimate: Supoose a likelihood function is given by L(a,b|d)

i

iibbaa

dR 2

ˆ,ˆmin

dbaLbbaa

,maxˆ,ˆ

• Regression tests– Residual analysis

– Standard errors of regression coefficients

– Coefficient of determination

ii dR residual

n

ii

n

iii

dRnn

dRE

1

2

1

2

21S

2

2

)(

)(ˆ

dSD

RSDbR

• Example 1: Linear regression

• Example 2: MLE solution of Emax and EC50 in Michales-Menten equation

Likelihood function

MLE solution

Inferential Statistics: Hypothesis test

• Goal: Test of significance• Rationale

– Null hypothesis: H0, outcomes of a study purely result from chance

– Alternative hypothesis: H1, outcomes of a study are influenced from non-random sources

– Appropriate model: Normal distribution, t-distribution…

• Rationale– Appropriate analysis method

• P-value: The probability of observing a sample statistic as extreme as the test statistic, assuming the null hypothesis is true.

• Parametric method: t-test, F-test, Chi-square test• Non-parametric method: Kolmogorov-Smirnov test,

Mann-Whitney test

P-value for significant test: 1. What is the probability of a test value from a random

population? One or two tailed?

2. If p-value is less than the confidence level the null hypothesis is rejected

t-distribution

http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html

Test method Test statistic Null hypothesis

one sample t-test the means of two normally distributed populations are equal

two sample F-test the means of normally distributed populations, all having the same standard deviation, are equal

Pearson Chi-sqaure test whether theoretical population R and real population d are different

nSD

dRt

/

2

1

SD

SDF

n

i i

ii

d

dR

1

22

•Parametric test

Two sample t-test: (Online calculator http://www.usablestats.com/calcs/2samplet)

N Mean StDev SE Mean

Sample 1 15 0.633 0.2162 0.056

Sample 2 15 0.931 0.2021 0.052

Observed difference (Sample 1 - Sample 2): -0.298Standard Deviation of Difference : 0.0764

Unequal VariancesDF : 2795% Confidence Interval for the Difference ( -0.4548 , -0.1412 )T-Value -3.9005 Population 1 ≠ Population 2: P-Value = 0.0006 Population 1 > Population 2: P-Value = 0.9997 Population 1 < Population 2: P-Value = 0.0003 

Equal VariancesPooled Standard Deviation: 0.2093 Pooled DF: 2895% Confidence Interval for the Difference ( -0.4545 , -0.1415 ) T-Value -3.8992 Population 1 ≠ Population 2: P-Value = 0.0006 Population 1 > Population 2: P-Value = 0.9997 Population 1 < Population 2: P-Value = 0.0003

Some Statistics Worth to Know

• Tool for component analysis: – Principle Component Analysis (PCA): A way to

identify patterns in data, and express in a way to highlight their similarities and differences

– Independent Component Analysis (ICA): A way to separate independent components in data

– Variable and model selection: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC)

• Bayesian statistics

Summary

• What is “right” null hypothesis?• What is the appropriate distribution function?• What is the appropriate test statistics?

“Know” your data before analyze that!!

Information theory based statistics: Bayesian statstics

• Goal: Using Bayesian method to design and analyze data

• Bayesian inference– Appropriate distribution functions– Appropriate sampling techniques

• Maximum entropy method based inference– Appropriate form of entropy– Appropriate constriants

Information theory based statistics: Method of maximum

entropy

Reference

[1] P. Rowe, Essential Statistics for Pharmaceutical Sciences, Wiley 2007.