Range- difference between largest and smallest observation
standard deviation- measure of spread about the mean (SD= Square root of variance)
Shape (skewness/kurtosis/etc)
Various shape statistics exist:
Skewness (is it symmetrical or not)
Kurtosis (how concentrated is the data around the mean)
(and more)
Probablility Theory
Generally very poorly understood
describes outcomes that depend on chance
eg rolling a dice, tossing a coin, infected with disease, pups in a litter, etc.
can almost never predict an outcome w/ total accuracy, but can describe whay MIGHT happen, or the probability of different outcomes
Probability distributions
The probability of an outcome given that we know what happens in the 'system' (variability, predict the future)
What we believe about the 'system' given that we know the outcome (uncertanty, estimating the true population parameters)
Normal distribution
(Gaussian, Bell curve)
Described by mean, sd
data can be any continuous value
symmetrical distribution
mean=median=mode
ex: birth weights, heights, live weights gains, body temperatures, serum biochemistry parameters
Poisson distribution
used for count data (integer)
described by mean only
asymmetrical distribution
mean does not equal median does not equal mode
examples: pups in a litter, cars on the street, earthquaks in a year
Binomial distribution
used for binary outcomes (yes/no, pass/fail, m/f, dies/survvives)
described by the probability of a success at each trial, and the number of trials
ex: number of heads out of 10 tosses of a coin, number of female calves from sexed semen, number of you will pass exams
Hypothesis testing
used for research scientists:
does drug A kill mice faster than drug B
do a greater proportion of smokers than non-smokers get lung cancer?
(also relevant to vets)
5 Steps!!
5 steps to Hypothesis testing
Think of a question you want to ask
put the question into a testable format
collect the data
apply the correct statistical test
interpret the results of the test
Generating a hypothesis to test:
what do we want to find?
how many groups are we comparing?
typically a simple question with a yes or no answer
ex: are these 2 groups of calves growing at the same rate?, did pyoderma cases given synulox recover at the same speed as ampicillin?
The 'NULL' hypothesis (and alternative hypothesis)
The baseline belief- there is NO difference in groups/drugs (denoted H_{0})
the alternative hypothesis:
opposite of the baseline belief- there IS a difference in groups/drugs (denoted H_{1})
Hypothesis testing
Goal is to provide evidence that the 'Null' hypothesis is WRONG!- there is a difference between groups/drugs
BUT! we have to account for the effects of outcomes being uncertain
the difference between the groups/drugs is more than would be expected by chance
Rejecting the Null hypothesis
It is always possible that the difference between 2 sets of observations is entirely chance! (that the pops. are really the same even though the samples look diff.)
this becomes less and less likely as the magnitude of the differences increase and number of observations increases
Confidence intervals
use confidence intervals to look at the data in a more formal way
do the confidence intervals for the parameter of interest in each group overlap (95%- 2.5 high and 2.5%low)
The more data we have the small the confidence intervals become
Rejecting the null hypothesis with Confidence intervals
the amount of overlap in confidence intervals reflects the probability (p-value) with which we reject the null hypothesis
if there is LITTLE overlap, we reject Ho
how little is given by the p-value (0.05)
this makes no comment at all about the magnitude (or biological impact) of the difference
Failing to reject the null hypothesis
if there is not enough evidence to prove the groups are different we cannot reject the null hypothesis
(this does not necessarily mean that there really is no difference, only we couldn't find any difference in the samples obtained)
Statistical signifiance DOES NOT EQUAL biological relevance
Remember this!
Can compare means by?
Using a t-test
Camparison of means...?
can compare one mean or two or more than two
Compare ONE mean (with a fixed number)
Confidence interval approach
look at sample mean, size and sd. 95% confidence interval... does the fixed number overlap the conficence interval?
Null: population mean = XXX
alt: Population does not = XXX
Significance Testing
looks at how far the observed sample mean is from the population mean
if P value is lower thatn 0.05 than it is significant (reject null) if greater than 0.05 than it is NOT significatn and accept null
comparison of TWO means (with each other)
T-test
95% CI for difference between means
take mean of each group
null: means are the same
alt: means are not the same
P > 0.05 accept null
P< 0.05 reject null
Paired Values
Pre and post treatment (somatic cell count- sub clinical mastitis, createnine kinase- exertional rhabdomyolysis)
before on or after a certain date (hormone levels for oestrus detection)
compare the same thing at 2 different times in the same animal
Paired T-tests
(example)
weight before diet and weight at 3 months on diet
95% CI for mean difference
T-test of mean difference (=0 versus not = 0)
NOT independent!!! CI becomes tighter
Comparison of means (more than 2 means)
comparison of means may be extended into 3 groups
more complex- takes into account the variance between and w/in groups
ex: are the daily grouth rates of pigs in 3 rearing units different?
ANOVA (analysis of Variance)
null: all means are the same
alt: at least one mean is different
F (variance ratio)
P < 0.05 = evidence of diff. between population means (but which ones!!??)
Must compare each...
Compare Ranks
use a non-parametric equivalent of a t-test (or similar)
use if data is NOT normally distributed
Non-parametric tests
compare ranks
corresponding confidence intervals is for difference in population medians
tests work by ranking the scores and then computing the average rank and then testing:
Null: there is no diff. between sum of ranks of groups
Alt: a deff. exists between sum of ranks of groups
Parametric vs. non-parametric (equivalents)
Parametric: Non-Parametric
1-sample t-test----------1 sample Wilcoxon signed rank test
2-sample t-test----------Mann Whitney U test/Wilcoxon rank sum test
Paired t-test-------------Wilcoxon signed rank test
One Way ANOVA-----Kruskal Wallis Test
Comparison of proportions
use a Chi-square test (or equivalent)
these are used for categorical data
null: proportion of infection is the same for draped and undraped cases OR Null: That drape use and infection are independent
alt: proportion of infection is DIFFERENT between draped and undraped cases
eg:
252 surgical colic cases
102 used a drape, 150 did not
73 post-op infections
is there a difference in proportion of infection in those that used a drape compared to those who did not?
Chi-square test statistic is then based on the differences between observed and expected values
(remember this is a shown association/relationship- DOES NOT SAY: leaving undraped WILL cause infection/bias....)