MULTIVARIATE STATISTICS

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MULTIVARIATE STATISTICS

• A collection of techniques to help us understand
  patterns in and make predictions with large
  datasets with many variables
• Ordination find a (hopefully small) number of
  composite variables that capture most of the
  variability among data points
• Cluster Analysis discover natural groupings of
  similar data points

• Discriminant Analysis find a (hopefully small)
  number of composite variables that can be used to
  predict the levels of a categorical dependent
  variable
• Canonical Correlation Analysis find
  relationships between two groups of variables
• Dependent variable is multivariate

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WHAT CAN MULTIVARIATE STATISTICS DO?

• Reflect more accurately the true multidimensional
  nature of environmental systems
• Provide a way to handle large datasets with large
  numbers of variables by summarizing the
  redundancy
• Provide rules for combining variables in an
  optimal way

• Provide a means of detecting and quantifying
  truly multivariate patterns that arise out of
  correlational structure of the variable set
• Provide a means of exploring complex data sets
  for patterns and relationships from which
  hypotheses can be generated and subsequently
  tested experimentally

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Time series analysis

• A Time Series is a list of observations of a
  variable (or variables) through time.
• Most analyses require equal spacing in time
• Usually violate assumptions of OLS and ANOVA.
• Adjacent observations are not independent
• In contrast to regression models, Time Series
  Analysis exploits temporal patterns in data.
• Major uses (1) analyze trends cycles (2)
  describe relationships within time series and (3)
  forecasting

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ARIMA model

• Auto Regressive Integrated Moving Average
• Auto Regressive means current value depends on
  past values
• Integrated means take differences (to remove
  linear or seasonal trend)
• Moving Average means replace X(t) with average
  of points nearby in time
• Need to specify orders for each of these
• of AR coefficients
• of times differenced
• Window for moving average
• Possible seasonality in each
• Choosing ARIMA model used to be a black art
• Now just try lots of possibilities and select one
  with lowest AIC

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pH of Norwegian Lakes
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Questions about Norwegian lake data

• Is there spatial autocorrelation in pH values?
• How can we interpolate and smooth those values?

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Semivariogram

• Semivariance is half the average squared
  difference between pairs of lakes a certain
  distance apart
• Measures variance among sites as a function of
  distance
• Also called empirical variogram

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Theoretical variogram
Sill
Range
Nugget
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Kriging

• Minimum mean-squared-error method of spatial
  prediction
• Interpolates and smoothes
• Named after South African mining engineer D.G.
  Krige
• Uses a theoretical variogram
• Have to fit the theoretical variogram first
• Produces a smooth surface that fits the data

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pH
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Other complications

• Censored data
• Some sample units disappear before you get the
  data
• Rat dies early in study would it have gotten a
  tumor?
• Throwing that sample out of analysis may bias
  results
• Measurement error
• Creates bias in estimates of regression
  parameters
• There is a large set of tools for dealing with
  these
• Need to know something about the statistical
  properties of the measurement process or the
  causes of censoring

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Congratulations!!!
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WHAT IS STATISTICS GOOD FOR?

• Transforms data into information
• Describe patterns in data
• What is the trend in CO2 emissions over time?
• What is the covariation between temperature and
  rainfall?
• Enhance scientific understanding
• Is there a relationship between gasoline taxes
  and gasoline consumption?
• What is the nature of the relationship between
  fish stocks and fish recruits?

• Make predictions
• Given the observed relationship between
  investment in green technologies and share price,
  what will be the effect on a firms share price
  of a one-unit increase in its green technology
  investment?
• Need to distinguish between interpolation and
  extrapolation
• Make decisions
• Are we sufficiently confident that arsenic levels
  exceed a regulatory threshold that we should take
  action?

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Enhance understanding
Make predictions
Estimate parameters
Describe patterns and relationships in data
Select models
Test statistical hypotheses
Test theories
Make decisions
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A SYSTEMATIC APPROACH TO STATISTICAL ANALYSIS

• Clearly formulate the problem, question, or
  decision that you are facing
• What are the quantities that you need to
  estimate?
• Write down a statistical model that relates the
  quantities of interest to the data you will
  collect (or have collected)
• This model will include a random component that
  represents natural variability or sampling error

• Estimate the parameters of the statistical model
• In addition to the estimate of the most likely
  value, quantify your uncertainty in that estimate
• Use the results to address your problem,
  question, or decision
• Your report should include a quantification of
  the probability that your answer is incorrect

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HYPOTHESIS TESTING OVERVIEW
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Often start with OLS, but...