Some statistical ideas

Marian ScottStatistics, University of Glasgow

June 2012

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

• Regression and relationships

Why quantify?

We need statistical skills to:•  Make sense of numerical

information, • Summarise data,•  Present results

(graphically),•  Test hypotheses•  Construct models

• Decision making- Which areas should be restricted?

• Prediction-What is the trend in temperature? Predict its level in 2050?

• Decision making-is it safe to eat fish?

• Regulatory- Have emission control agreements reduced air pollutants?

• Understanding -when did things happen in the past

Observed nitrogen signals in rivers, lakes

and groundwater in Europe (EEA).

• What is a trend and how should we evaluate it?

• How sure are we?

Trends in seasons over Europe (Global Change Biology, 2006)

• 21 countries, 125,000 studies, 542 plant and 19 animal species, 1971-2000

• Spring is on average 6 to 8 days earlier than it was 30 years ago

• Analysis of 254 national time series , pattern of observed change in spring matches measured national warming (correlation coefficient –0.69, P<0.001)

• What do the statistical terms mean?

Spatial patterns of change

• Spatial patterns of change may be important

• Changes in the start and end of the growing season between two years (1961, 2004)– heterogeneous

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.

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

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 shellfishThe 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.

Variability exists amongst the sampling units and hence within the population

Some other terminology

Parameter: this is a number that describes the population, usually what we want to know about

Example- population mean technetium level

Statistic: this is a number that we calculate from the sampleExample- sample mean technetium level

Variability exists amongst the sampling units and hence within the population, so we could imagine hypothetically drawing different samples, all would give a different sample mean.

• Summarising data- means, medians and other such statistics

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

• plotting data- histograms, boxplots, stem and leaf plots, scatterplots

median

lower quartile

upper quartile

Example -Bathing water quality

• All bathing water sites are classified as either ‘Excellent’, ‘Good’, ‘Sufficient’ or ‘Poor’ in terms of the quantities of 2 different microbiological indicator bacteria

• Faecal Streptococci (FS)• Faecal Coliforms (FC)

• ‘Sufficient’ is the minimum standard that bathing water sites are required to meet

• Classification for each site is based on the 90th & 95th percentiles of samples over the most recent 4 bathing seasons

joint work with Ruth Haggarty, Claire Ferguson

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

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Boxplots of FS: 114567

SEPA location code 114567Year

FS

• probability models- the Normal especially

• checking distributional assumptions

Histogram of FS

SEPA location code: 4556FS/100ml

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Normal Q-Q Plot

Theoretical Quantiles

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Histogram of log10(FS)

SEPA location code: 4556log10(FS)/100ml

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Theoretical Quantiles

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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)C1

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0.1211StDev 1.015N 100AD 0.361P-Value

Probability Plot of C1Normal

Statistical inference

• Confidence intervals

• Hypothesis testing and the p-value

• Statistical significance vs real-world importance

• Building statistical models

• a formal statistical procedure- confidence intervals

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

• another formal inferential procedure- hypothesis testing

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 shouldn’t - Type I– don’t reject null when we should - Type II

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)

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.

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 (also related to sample size calculations)

• relationships- linear or otherwise

Correlations and linear relationships

• pearson correlation

• Strength of linear relationship

• Simple indicator lying between –1 and +1

• Check your plots for linearity

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.

3210-1

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Scatterplot of log P vs log Fe

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Scatterplot of log N vs log Fe

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Scatterplot of log P vs log N

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• what is a statistical model?

Statistical models

• Outcomes or Responsesthese are the results of the practical work and are sometimes referred to as ‘dependent variables’.

• Causes or Explanationsthese are the conditions or environment within which the outcomes or responses have been observed and are sometimes referred to as ‘independent variables’, but more commonly known as covariates.

Specifying a statistical models• Models specify the way in which outcomes and

causes link together, eg. • Chl-a ~ Temperature• there should be an additional item on the right hand

side giving a formula:-

• Chl-a ~ Temperature + Error

• This says that Chl-a depends on temperature, but that there is also some random variability (error)

Example 1: are atmospheric SO2 concentrations declining?

• Measurements made at a monitoring station over a 20 year period

• Complex statistical model developed to describe the pattern, the model portions the variation to ‘trend’, seasonality, residual variation

so2 monitored in GB02

observations

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Plot of so2 against time, monitored in GB02Lines = Model 3

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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.

Statistics might be needed where?

• designing and evaluation monitoring and sampling networks; sampling strategies

• the analysis of observational records, (e.g. past climate indicators, water quality, pollutant trends); trends, spatio-temporal modelling, dealing with variation

• the study and modelling of extreme events (e.g. sea levels, flood prediction) for prediction and management of future occurrences; extremes, risk modelling, uncertainty

• evaluating the state of the environment;trends, uncertainty, prediction

Statistics might be needed where?

• the use of complex computer models to simulate the whole earth system (e.g. climate change and the carbon cycle); uncertainty, model evaluation

• the analysis of observational records, (e.g. past climate indicators, water quality, pollutant trends); trends, spatio-temporal modelling, dealing with variation

• the study and modelling of extreme events (e.g. sea levels, flood prediction) for prediction and management of future occurrences; extremes

• the evaluation and quantification of risk and uncertainty (e.g. volcanic or earthquake prediction);uncertainty, prediction

Statistics and the environment

• Appropriate statistical models can give – added value to your data, – better descriptions of complex change

behaviour and – begin to tease out climate change driven

effects in environmental quality – handle natural variation.

• Greater, innovative statistical analysis needed for environmental science

Statistics and the environment

As environmental scientists, we need to try and ensure that:

data are gathered under good statistical principles and that they are not left in the filing cabinet.

We need to ensure thatGood environmental science is served by good statistical science.

Environmental science should be “Data and information rich”

Statistics training

• we have chosen a few key statistical topics to cover- there are many others

• We will also have some practical examples for you to work through with guidance

• the main software tool will be R, which is freely available- this is very similar but not identical to S+

• there should be lots of opportunities to ask questions