BIOSTATS lecture 8 and 9

normal quantile plots

straight line if perfectly normal

shape changes due to skew or different amount of probability in tails than a perfect normal

leptokurtic

too pointed

platykurtic

too flat

left skew

tail on left

right skew

tail on right

rule of thumb of normality

if absolute value of kurtosis/skew statistic > 2 X SE of the statistics

shapiro-wilks test

use when n < 50 (smaller sample sizes)

kolmogorov-smirnov test

use when n>50 (larger sample sizes)

low p value means

reject null hypothesis

what should we do when assumptions are strongly violated

• evaluate outliers

• transform the data to better approximate normality

• use non-parametric tests

evaluating outliers

• (is there legitimate reason to remove outlier? if no… do you get diff statistical test result with and without outlier. If outlier is only thing causing a result it should be removed)

common transformations

log10 and ln (similar effect)

square root

square

inverse

fixing positive skew

if slightly skewed, use square root transformation

if moderately skewed use log transformation

if extremely skewed use inverse

problem with square root transformation

negative numbers… add a constant to make all values greater than 0

numbers 0-1 increase while numbers >1 decrease… add a constant to make all values greater than

problem with log transformation

negative numbers and values between 0 and 1… add constant to make all values greater than 1

fixing negative skew

first try square transformation… if that fails… 2 step transformation

square transformation caveat

all values need to be same sign

2 step transformation

negative skew can first be reflected to be positively skewed and then use the previous transformations

Multiply by -1 then add constant to bring all values above 1

what if your groups are skewed in diff directions?

can’t use different transformation on each group

back transforming data

sqaure root —> square

log 10 —> anti log10

ln —> e^x

square —> square root

1/Y —> *Y

what parameters should you back transform

mean

standard error/95% confidence intervals

what does back transforming results look like

narrower confidence intervals and slightly different means

when transformations can’t fix it…

consider non-parametric tests

non-parametric

fewer assumptions about shape and spread of data

lower statistical power

2 sample t test non-parametric test

mann-whitney U test

mann-whitney U Test

converts data into ranks

tests for difference b/w medians

mann whitney u test assumption

similar shape and variance

is mann whitney u test good?

fairly powerful test at large sample sizes

paired and 1 sample t test non-parametric test

wilcoxon signed rank test or sign test

wilcoxon signed rank test

tests difference b/w sample median and hypothesizes median

turns difference data into ranks

wilcoxon signed rank test assumptions

data are symmetric around mean

sign test

tests difference b/w sample median and hypothesized median

turns differences into +1 and -1

sign test assumptions

non other than unbiased/random sample

is the sign test good?

low statistical power. throws out a lot of info

type 1 errors

incorrectly rejecting null hypothesis

p-value

is an estimate of your likelihood of committing a type 1 error

warnings of type 1 error

when assumptions of a statistical test are not met, the likelihood of a type 1 error may be greater or less than reported by the p-value

but still robust so even if modest deviation from assumptions, test statistic won’t change greatly

type 2 errors

fail to reject the null hypothesis even though the null hypothesis is false

difficult to quantify

why are type 2 errors hard to quantify

caused by small sample size or high variance

statistical power

your ability to reject a null hypothesis when there is a difference

statistical power of mann whitney u test and wilcoxon signed rank test when parametric assumptions are met

can be ~95% as powerful at large sample sizes

much less powerful at small sample sizes

statistical power when parametric assumptions are met for sign test

only 64% as powerful at large sample sizes

MUCH less powerful at small sample sizes

assumptions for 2 sample t test

independent

random

normal

equal variances

similar sample sizes