1. Types of statistics
    • Descriptive stats (summary of the data, represent data in an easily understandable way/graphs, charts, tables, average, range)
    • inferential stats (sample vs. pop., experimental design, hypothesis testing)
  2. Sample vs population
    • samples should be:
    • representative and unbiases (males & females, all ages)
    • every type of subject should have the same chance of being included
    • (normal distribution/random sampling)
  3. Types of Data
    • Categorical
    • continuous
    • discrete
  4. Categorical Data
    • nominal (unordered) eg dog breeds
    • ordinal (ordered) eg body condition score of cattle
  5. Continuous data
    • Any positive value theoretically possible
    • eg: weight, height
  6. discrete data
    • can only be integer values (whole numbers)
    • eg: numbers of piglets in a litter
  7. Bar Charts
    Good for frequency (categorical or discrete)
  8. Scatterplot
    used for correlation and progression (regression)
  9. Descriptive Statistics
    • Data is collected so that we can obtain INFORMATION about a certain topic
    • when only a few observations are made, it might be easy to see a potential relationship
    • as more data is collected it's more difficult to obtain an overall picture
  10. Histogram
    Quantitive data/distribution
  11. Stem and leaf
    (not often used) number of observation above or below the median....??
  12. Boxplot
    • single and double to compair
    • line represents median value
  13. Numerical measures
    • numerical measures are used to summarise the position, spread and shape of a data distribution
    • used to describe the data so that we have a general idea of the data that we have and the population that it might have come from or represent
  14. Measures of central tendency
    • "averages" or the middle of data
    • Mean= sum of all observations/number of observations
    • median= middle observation (half smaller, half larger)
    • mode= value that occurs most frequently
  15. Measures of spread
    • Range- difference between largest and smallest observation
    • standard deviation- measure of spread about the mean (SD= Square root of variance)
  16. 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)
  17. 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
  18. 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)
  19. 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
  20. 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
  21. 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
  22. 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!!
  23. 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
  24. 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?
  25. The 'NULL' hypothesis (and alternative hypothesis)
    • The baseline belief- there is NO difference in groups/drugs (denoted H0)
    • the alternative hypothesis:
    • opposite of the baseline belief- there IS a difference in groups/drugs (denoted H1)
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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)
  31. Statistical signifiance DOES NOT EQUAL biological relevance
    Remember this!
  32. Can compare means by?
    Using a t-test
  33. Camparison of means...?
    can compare one mean or two or more than two
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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
  39. 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?
  40. 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...
  41. Compare Ranks
    • use a non-parametric equivalent of a t-test (or similar)
    • use if data is NOT normally distributed
  42. 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
  43. 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
  44. 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....)
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