1. Statistical basics Marian Scott Dept of Statistics, University of Glasgow August 2010

2. What shall we cover? Why might we need some statistical skills Statistical inference- what is it? how to handle variation exploring data probability models inferential tools- hypothesis tests and confidence intervals

3. Why bother with Statistics We need statistical skills to: Make sense of numerical information, Summarise data, Present results (graphically), Test hypotheses Construct models

4. statistical language variable- a single aspect of interest population- a large group of individuals sample- a subset of the population parameter- a single number summarising the variable in the population statistic- a single number summarising the variable in the sample

5. statistical language- Radiation protection- C-14 in fish variable- radiocarbon level (Bq/KgC) population- all fish caught for human consumption in W Scotland sample- 20 fish bought in local markets parameter- population mean C-14 level statistic- sample mean C-14 level

6. Questions Univariate: What is the distribution of results-this may be further resolved into questions concerning the mean or average value of the variable and the scatter or variability in the results? Bivariate: How are the two variables related? How can we model the dependence of one variable on the other? Multivariate: What relationships exist between the variables? Is it possible to reduce the number of variables, but still retain 'all' the information? Can we identify any grouping of the individuals on the basis of the variables?

7. Data types Numerical: a variable may be either continuous or discrete. – For a discrete variable, the values taken are whole numbers (e.g. number of invertebrates). – For a continuous variable, values taken are real numbers ( e.g. pH, alkalinity, DOC, temperature). Categorical: a limited number of categories or classes exist, each member of the sample belongs to one and only one of the classes. – Compliance is a nominal categorical variable since the categories are unordered. – Level of diluent (eg recorded as low, medium,high) would be an ordinal categorical variable since the different classes are ordered

8. Inference and Statistical Significance Sample Population inference Is the sample representative? Is the population homogeneous? Since only a sample has been taken from the population we cannot be 100% certain Significance testing

10. the statistical process A process that allows inferences about properties of a large collection of things (the population) to be made based on observations on a small number of individuals belonging to the population (the sample). The use of valid statistical sampling techniques increases the chance that a set of specimens (the sample, in the collective sense) is collected in a manner that is representative of the population.

11. Variation soil or sediment samples taken side-by- side, from different parts of the same plant, or from different animals in the same environment, exhibit different activity densities of a given radionuclide. The distribution of values observed will provide an estimate of the variability inherent in the population of samples that, theoretically, could be taken.

12. What is the population? The population is the set of all items that could be sampled, such as all fish in a lake, all people living in the UK, all trees in a spatially defined forest, or all 20-g soil samples from a field. Appropriate specification of the population includes a description of its spatial extent and perhaps its temporal stability

13. What are the sampling units? In some cases, sampling units are discrete entities (i.e., animals, trees), but in others, the sampling unit might be investigator-defined, and arbitrarily sized. Example- technetium in shellfish The objective here is to provide a measure (the average) of technetium in shellfish (eg lobsters for human consumption) for the west coast of Scotland. Population is all lobsters on the west coast Sampling unit is an individual animal.

14. Summarising data- means, medians and other such statistics

19. plotting data- histograms, boxplots, stem and leaf plots, scatterplots

25. median lower quartile upper quartile

26. Preliminary Analysis There is considerable variation –Across different sites –Within the same site across different years Distribution of data is highly skewed with evidence of outliers and in some cases bimodality

27. probability models- the Normal especially

33. checking distributional assumptions

35. Modelling Continuous Variables checking normality Normal probability plot Should show a straight line p-value of test is also reported (null: data are Normally distributed)

36. Statistical inference Confidence intervals Hypothesis testing and the p-value Statistical significance vs real-world importance

37. a formal statistical procedure- confidence intervals

38. Confidence intervals- an alternative to hypothesis testing A confidence interval is a range of credible values for the population parameter. The confidence coefficient is the percentage of times that the method will in the long run capture the true population parameter. A common form is sample estimator 2* estimated standard error

44. another formal inferential procedure- hypothesis testing

45. Hypothesis Testing Null hypothesis: usually no effect Alternative hypothesis: effect Make a decision based on the evidence (the data) There is a risk of getting it wrong! Two types of error:- –reject null when we shouldnt - Type I –dont reject null when we should - Type II

46. Significance Levels We cannot reduce probabilities of both Type I and Type II errors to zero. So we control the probability of a Type I error. This is referred to as the Significance Level or p-value. Generally p-value of <0.05 is considered a reasonable risk of a Type I error. (beyond reasonable doubt)

47. Statistical Significance vs. Practical Importance Statistical significance is concerned with the ability to discriminate between treatments given the background variation. Practical importance relates to the scientific domain and is concerned with scientific discovery and explanation.

48. Power Power is related to Type II error probability of power = 1 - making a Type II error Aim: to keep power as high as possible

52. relationships- linear or otherwise

53. Correlations and linear relationships pearson correlation Strength of linear relationship Simple indicator lying between –1 and +1 Check your plots for linearity

54. Interpreting correlations The correlation coefficient is used as a measure of the linear relationship between two variables, The correlation coefficient is a measure of the strength of the linear association between two variables. If the relationship is non-linear, the coefficient can still be evaluated and may appear sensible, so beware- plot the data first.

55. 0.167 0.134 0.380

56. what is a statistical model?

57. Statistical models Outcomes or Responses these are the results of the practical work and are sometimes referred to as dependent variables. Causes or Explanations these are the conditions or environment within which the outcomes or responses have been observed and are sometimes referred to asindependent variables, but more commonly known as covariates.

58. Specifying a statistical models Models specify the way in which outcomes and causes link together, eg. Metabolite ~ Temperature there should be an additional item on the right hand side giving a formula:- Metabolite ~ Temperature + Error

59. statistical model interpretation Metabolite ~ Temperature + Error The outcome Metabolite is explained by Temperature and other things that we have not recorded which we call Error. The task that we then have in terms of data analysis is simply to find out if the effect that Temperature has is large in comparison to that which Error has so that we can say whether or not the Metabolite that we observe is explained by Temperature.

60. summary hypothesis tests and confidence intervals are used to make inferences we build statistical models to explore relationships and explain variation a general linear modelling framework is very flexible assumptions should be checked.