GG1011 MODULE D STATISTICS

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GG1011 MODULE DSTATISTICS

• Session 3 Confidence Intervals and difference
  tests

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Assumptions of the t-test

• Its a Parametric Statistic
• Parametric statistics are the most powerful BUT
• These statistics make the following key
  assumption
• 1. Background population from which the data is
  sampled is NORMALLY DISTRIBUTED
• What evidence would we have for this?

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Non-parametric tests for difference between two
samples

• If we cannot be confident that our sample data
  comes from a normally distributed population.
• Use a NON-PARAMETRIC TEST
• In this case, with TWO samples its called the
  Mann-Whitney Test.
• Non-parametric statistics make NO ASSUMPTIONS
  about population distribution.
• They compare the ranks and Median rather than the
  Mean standard deviation.

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Non-Parametric Test of Difference between TWO
samplesThe Mann-Whitney Test

• Has 95 power of the t-test.
• Can be applied to ordinal and higher data.
• Usual null/alternative hypotheses can be set
  upbut dont mention means
• The statistic, orders the combined samples and
  considers the relative ranks of the data in each
  sample. Sample 1 has n1 observations, Sample 2
  has n2 observations.
• The ranks of each sample are summed, R1 R2
• Statistic U, is calculated as Un1n2
  0.5n1(n11)-R1
• Significance of U is looked up on a table
  (Minitab will do this for you!) and a
  significance level (p) is given. plt0.05 is
  significant.

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What happens when we have MORE than two sample
means to test for difference?

• 1. We could use multiple t-testsbut this
  increases the chance of making a type 1 error.
• 2. Use a test that can compare SEVERAL samples at
  once.
• 3. This is called the F-test, or ANALYSIS OF
  VARIANCE.
• 4. The F-test asks are there any significant
  differences among the samples?
• 5. It compares the variability of values WITHIN
  groups with the variability of values BETWEEN
  GROUPS.

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The F-test assumes the variance of all our
samples is similar (Case i) NOT Case ii
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Analysis of Variance
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Analysis of Variance

• The F statistic is a ratio of BETWEEN GROUP
  VARIANCE/WITHIN GROUP VARIANCE.
• The F-sampling distribution is a series of curves
  like the t-distribution. It varies with sample
  SIZE and HOW MANY SAMPLES are being compared.
• As for the t-test, Minitab will give you a test
  statistic (F-Statistic) , and its associated
  significance level, p.
• These are interpreted in exactly the same way as
  for the t-test.

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Association between two variables How do
variables interact in natural and economic/social
processes?
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What is Correlation?

• Correlation is a measure of the degree of
  association between two variables.
• How does one variable control/affect another?
• This is the idea of process mechanisms in broad
  termsfinding links between phenomena.
• Three types of correlation POSITIVE, NEGATIVE
  AND ZERO.

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Correlation

• First steps Data must be PAIRED eg for each
  observation of variable 1, there must be an
  equivalent observation of variable 2.
• 1. Decide which of the two variables is the
  CONTROL or INDEPENDENT VARIABLE (X)
• 2. Decide which of the two variables is therefore
  the CONTROLLED or DEPENDENT VARIABLE (Y)
• 3. Plot the variables (paired x,y observations)
  on a SCATTERPLOT and look at the degree of
  scatter and any evidence of TREND in the data
  alignment.

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Correlation Scatterplot indications of STRENGTH.
Each of the dots represents a sample member,
for which there are TWO measurements, one for the
independent variable, one for the dependent
variable.
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Correlation factors to consider from the
scatter graph
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Correlation Linear non-linear. The middle
scatterplot has zero linear correlation but
clearly there IS a non-linear relationship
between the variables.
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Correlation Coefficient Parametric
StatisticsPearson Product Moment Coefficient
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Correlation Pearson Correlation coefficient
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Correlation

• Values of the Pearson Correlation Coefficient
• R 1.0. Perfect Positive correlation. The data
  is aligned perfectly on a straight line with a
  positive gradient. As x increases, y increases
• R -1.0. Perfect negative correlation. The data
  is aligned perfectly on a straight line with a
  negative gradient. As x increases, y decreases.
• R0.0. Zero correlation, no association between
  the x and y variables. Random scatter of points
  on scattergraph.

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Significance of the correlation coefficient

• Is the correlation found in our sample data
  reflecting a REAL association between the
  populations of the variables?
• Null Hypothesis The correlation coefficient
  between the two variables is NOT significant.
• Set the significance level 0.05
• T-test statistic t r. v(n-2)/(1-r2)
• Is Find significance level for t statistic.
• Is it lt0.05? Yes- Reject the Null hypothesis

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Pearson Product Moment Correlation Coefficient

• Assumptions
• Backround populations from which data sampled are
  NORMALLY distributed
• Data is on Interval or Ratio scale
• Look at the histograms of each sample variable.
• If the assumptions cannot be met? Skewed data?
• Transform the data to normalise the distributions
• Use a NON-PARAMETRIC STATISTIC OF CORRELATION
• Spearmans Rank Correlation Coefficient. (which
  can also be used for ordinal scale data)

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Trend lines

• Variation in the extent of sea-ice in the period
  sept 1979- sept 1988 as measured by satellites.
• What does the line allow us to do?

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Examples of trend lines
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What is a regression line?

• A regression line is the line of BEST FIT through
  a scatter of x,y data values.
• If correlation is 1 or -1, ALL the data will lie
  ON the regression line.
• Form of Regression Equation is Y-dependent
  variable, X-independent variable
• Y A BX
• (A is the y-intercept, where the line cuts the
  y-axis B is the gradient of the line, the rate
  of change of Y when X changes)

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Regression

• Direction of relationship
• Case 1 Positive gradient (correlation) Gradient
  (b) is ve
• Case 2 Negative gradient (correlation) Gradient
  (b) is ve.
• Case 3 No relationship (Correlation 0.0) b0.0

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Regression influence of UNITs of measurement

• Notice how changing the distance units from
  kilometers to Miles changes the GRADIENT of the
  regression line.

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Fitting a Regression Line
a
b

• a) Sum of perpendicular distances from the line
  are minimised
• b) area of triangles minimised
• c) Sum of distances in y variable minimised
• d) Least squares of distances minimised.

c
d
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Errors or residuals from a regression line

• How Residuals are found.

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Regression Errors or residuals

• Residuals measure the DIFFERENCE between the
  value of Y PREDICTED by the regression line and
  the ACTUAL value of Y.
• There is Explained variation (due to the line)
  and Unexplained variation due to other factors,
  not included in the regression equation.

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Regression

• How well does the regression line represent the
  data?
• Compare Observed variation of the data and the
  predicted variation given by the regression line.
  
• R2 the coefficient of variation is found to give
  this proportion.
• It is usually expressed as a percentage.

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Assumptions of Regression Analysis

• The residuals should be normally distrbuted
  around the regression line.
• Residuals above the line are Positive.
• Residuals below the line are Negative.

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Assumptions of Regression

• 1. Residuals should be randomly distributed
  around the line.
• 2. The size of residuals should be random along
  the regression line.

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What if my data is NON_LINEAR?

• Transformations
• These can be used to make the data linear so that
  correlation and linear regression can then be
  carried out.