BIOSTATISTICS Hypotheses testing and parameter estimation

INTRODUCTION 1.Hypotheses testing Errors related to testing Multiple testing 2.Parameter estimation Standard error Confidence interval Copyright ©2011, Joanna Szyda

STATISTICAL INFERENCE Freshwater mussel Hyridella menziesi n= mg/g n= mg/g 1. Observed the nature 2. Form a biological question 3. Form a statistical hypothesis 4. Sample 5. Estimate 6. Run an analysis 7. Statistical inference Copyright ©2011, Joanna Szyda

SAMPLE POPULATION STATISTICAL INFERENCE

Copyright ©2011, Joanna Szyda SAMPLE POPULATION PROBABILITY STATISTICAL INFERENCE

HYPOTHESES TESTINGPARAMETER ESTIMATION Copyright ©2011, Joanna Szyda STATISTICAL INFERENCE

HYPOTHESES TESTING H 0 - null hypothesis H 1 - alternative hypothesis Pr(H 0 ) + Pr(H 1 ) = 1 H 1 complementary to H 0 Hypotheses testing relates to rejecting / accepting H 0 Copyright ©2011, Joanna Szyda

e.g. 1 PARAMETER H 0 : lipid concentration in the air is equal to 25.0 mg/g H 1 : lipid concentration in the air is different from 25.0 H 1 H 0 H 1 H 0 : k = 25.0 H 1 : k  25.0 H0 H0 H1 H1 H 0 : k > 25.0 H 1 : k ≤ 25.0 H 0 : lipid concentration in the air exceeds 25.0 mg/g H 1 : lipid concentration in the air is less or equal 25.0 Copyright ©2011, Joanna Szyda HYPOTHESES TESTING

H 0 : k1 = k2 H 1 : k1  k2 k1 k2 H0H0 H1H1 H1H1 e.g. 2 PARAMETERS H 0 :lipid concentration in the air jest equal tolipid concentration in water H 1 :lipid concentration in the air and water are different H 0 : k1 > k2 H 1 : k1 ≤ k2 k1 k2 H0H0 H1H1 H 0 :lipid concentration in the air is higherthan in water H 1 :lipid concentration in the air is lower or equal than in water Copyright ©2011, Joanna Szyda HYPOTHESES TESTING

ERRORS IN HYPOTHESES TESTING ERROR TRUE HYPOTHESIS H0H0 H1H1 ACCEPTED HYPOTH. H0H0 H1H1  - type I error  - type II error   Copyright ©2011, Joanna Szyda

ERROR TRUE HYPOTHESIS H0H0 H1H1 ACCEPTED HYPOTH. H0H0 H1H1 probability of incorrect rejection of true H 0 significance level P value e.g. if  =0.05 then among 100 tests H 0 is incorrectly rejected can be controlled in testing TYPE I ERROR Copyright ©2011, Joanna Szyda ERRORS IN HYPOTHESES TESTING

ERROR TRUE HYPOTHESIS H0H0 H1H1 ACCEPTED HYPOTH. H0H0 H1H1 probability of incorrect rejection of true H 1 1-β – power estimation of power: computer simulations, numerical depends on sample informativeness: size, structure, difference between H 1 and H 0 TYPE II ERROR H 1 H 0 H 1 Copyright ©2011 Joanna Szyda ERRORS IN HYPOTHESES TESTING

MULTUIPLE HYPOTHESES TESTING … H 0 : k1≤k2 / H 1 : k1>k2 →  MAX =0.05 → t →  T → H 0 /H 1 → 5% OVERALL TYPE I ERROR  MAX  0.05*10 = 50% Copyright ©2011, Joanna Szyda

What to do  ? BONFERRONI CORRECTION → independent tests …  MAX* =  MAX / N →  MAX* = 0.05 / 10 →  MAX* = OVERALL TYPE I ERROR  MAX  0.005*10 = 5%  MAX* =  MAX / N →  MAX* = 0.05 / 10 →  MAX* = Copyright ©2011, Joanna Szyda MULTUIPLE HYPOTHESES TESTING

What to do  ? False Discovery Rate (FDR) ERROR TRUE HYPOTHESIS H0H0 H1H1 ACCEPTED HYPOTH. H0H0 H1H1 FDR = E + Copyright ©2011, Joanna Szyda MULTUIPLE HYPOTHESES TESTING

STATISTICAL INFERENCE HYPOTHESES TESTINGPARAMETER ESTIMATION Copyright ©2011, Joanna Szyda

HOW ACCURATE IS AN ESTIMATE ??? STANDARD ERROR CONFIDENCE INTERVAL Copyright ©2011, Joanna Szyda

STANDARD ERROR Copyright ©2012, Joanna Szyda Standard error of an estimate standard deviation of a distribution of the estimate

STANDARD ERROR OF A MEAN What is the distribution of ? How to calculate ? Copyright ©2012, Joanna Szyda

What is the distribution of the estimate of a mean? Large sample sizes (N): the distribution approaches the Normal distribution estimate of a mean approaches the true mean regardless of the disctibution of the data Copyright ©2011, Joanna Szyda STANDARD ERROR OF A MEAN

What is the distribution of ? How to calculate ? Copyright ©2012, Joanna Szyda

How to calculate the standard deviation for the distribution of a mean (without collecting multiple samples) ? Standard error in a sample Sample size STANDARD ERROR OF A MEAN Copyright ©2011, Joanna Szyda STANDARD ERROR OF A MEAN

STANDARD ERROR OF OTHER ESTIMATES Standard error of estimated probability Copyright ©2012. Joanna Szyda Standard error of regression coefficient What affects the standard error?

STATISTICAL INFERENCE Freshwater mussel Hyridella menziesi n= ± 0.9 mg/g n= ± 0.7 mg/g Copyright ©2011, Joanna Szyda

HOW ACCURATE IS AN ESTIMATE ??? STANDARD ERROR CONFIDENCE INTERVAL Copyright ©2011, Joanna Szyda

Confidence interval of a mean estimate: interval which with predefined propability contains the true mean confidence interval bounds CONFIDENCE INTERVAL FOR THE MEAN Copyright ©2011, Joanna Szyda

How to calculate bounds of a confidence interval ? 1.Known variance or large sample size 2.Unknown variance – variance calculated from a smaple Copyright ©20112 Joanna Szyda CONFIDENCE INTERVAL FOR THE MEAN

1.95% confidence interval Probability of containing the true mean and confidence interval length 2.99% confidence interval Copyright ©2011, Joanna Szyda CONFIDENCE INTERVAL FOR THE MEAN

STATISTICAL INFERENCE Freshwater mussel Hyridella menziesi n= ± 0.9 mg/g [23.2, 26.8] n= ± 0.7 mg/g [ 21.5, 24.3 ] Copyright ©2011, Joanna Szyda

Copyright ©2011 Joanna Szyda HYPOTHESES TESTING PARAMETER ESTIMATION