Advanced statistics

"there are three kinds of lies: lies, damned lies and statistics" Disraeli

Definition• procedure used in data exploration,

organisation, presentation, analysis and interpretation facilitating decision making

• separate real effect from randomvariation

• descriptive – tables– graphs– measures • inferential

– estimating size– sampling

• hypothesistesting

• makingmodels

Variability

repeated measurements18,2°C18,5°C19,1°C18,7°C intra-population variability

180cm175cm165cm157cm

ecological variabilityinter-population differences

ethnic differences

= BIODIVERSITY

temporal fluctuation

time

Data description

• data types: qualitative x quantitative• frequency: histogram• tendency: mean, range, median, mode,

quartiles, ….• variability: standard deviation, variance• distribution: symetric (sometimes

normal), asymetric

symetric

mean=median=modus

asymetric

median

mean

Distributions

Inferential statistics

• population

• sample

Hypothesis testing

• drawing conclusions about population by analyzing sample

research hypothesis

null hypothesis alternative hypothesis

Statistical significance???

by chanceDgenuine experimental effectC

observed difference

Statistical errors vs. measure of statistical significance

• Type 1: false positive• the study showed an effect which in reality does not exist

• Type 2: false negative• an effect was there but the study missed it

• P-value: probability of type I error (a)

Statistical tests

parametric (for normal,normal-likeor transformeddistribution)

non-parametric(distribution-free)

tests unpaired paired

• t-test independent(two-sample t-test)

• ANOVA / MANOVA• regression• correlation (Pearson)

• Mann-Whitney• median test• rank correlation (Spearman, Kendall)

• chi-squared • Kruskall-Walis

ANOVA

• t-test dependent(one-sample)• ANOVA

• Wilcoxon pairedtest

comparison of parameterbetween 2 or more groups of subjects

comparison of pair of parametersin subjects fromone group in timeintervals

ANOVA - Analysis of Variance

• determines the probability that two or more samples were drawn from the same parent population

• can be sub-classified by secondary or grouping variables (two way ANOVA)– eg. blood pressure before and after two different

treatments– if some of the measurements were made in the

same subjects then can be corrected for repeated measures

• comparison of two samples with one way ANOVA is very similar to performing a non-paired t- test

Regression and Correlation

• Regression indicates relationship(mathematical) between two or more variables

• parametric analysis rules apply so data must either be normally distributed or can be converted into normal distribution by transformation

• Correlation indicates an associationbetween two variables

• non-parametric rules apply as the association is calculated between the ranks of the variables, not the variables themselves

• correlation describes an association, not cause and effect!!!!

Regression model

• where a and b are parameters in the regression model

• the emphasis is on predicting one variable from the other

• the least-squares criteria for goodness-of-fit

Correlation

• a measure of the degree of linear relationshipbetween two variables (usually labeled X and Y)

• it is possible for two variables to be related (correlated), but not causing one another

• correlation coefficient (r)– the sign of the correlation coefficient (+ , -) defines the

direction of the relationship– the absolute value of the correlation coefficient measures

the strength of the relationship• r=0.0 indicates the absence of a linear relationship and

correlation• coefficients of r=+1.0 and r=-1.0 indicate a perfect linear

relationship

Scatterplots

Chi-Square test• a very useful, robust, simple test• handles classification data (eg. lived v died)• most frequently used for the 2 x 2 • cross-tabulation • table

• a null hypothesis that there is no significant difference between the survival rate in the two groups