ARCH 21266126

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ARCH 2126/6126

• Session 11 Relating variables to each other

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Bivariate statistics

• Considering 2 numerical variables observed on the
  same set of cases simultaneously
• For example, - length breadth of a set of
  scrapers - height weight of a set of
  people- spleen size malaria positivity-
  mothers education level use of traditional
  medicines

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Or sometimes

• Or some cases may have missing values for one
  variable, which you want to estimate from the
  other, e.g.
• You want to estimate stature from femur length

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Association of metric variables can be shown
visually

• Here is a simple hypothetical example of a
  positive association between variables
• They increase together
• Labels, units, caption, number, to be added

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And similarly

• Here is a simple hypothetical example of a
  negative association between variables
• As one increases, the other decreases

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And again

• Here is a simple hypothetical example of no
  (significant) association between variables
• There are other possibilities but lets consider
  these

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The approach is similar in many ways to univariate

• In the population out there, there is a
  measurable association (or lack of it) between
  these two variables the population parameter
• The association that we measure in the sample we
  have collected is an estimate of it the sample
  statistic
• The H0 is normally no association

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Last time correlation as a measure of association

• Correlation (positive or negative) is a specific
  term with a specific meaning
• Usual measure is Pearson or product-moment
  correlation coefficient
• Usual symbols are ? (rho) for population
  parameter r for sample statistic
• This is a quantity we can calculate

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Summary on calculation of r

• r SPxy/v(SSxSSy), where-
• SPxySxy ((Sx)(Sy)/n)
• SSxS(x2) ((Sx)2/n)
• SSyS(y2) ((Sy)2/n)
• n number of cases
• D.f. n - 2

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Some points about r

• Ranges from 1 through 0 to 1
• No units
• Extremes indicate perfect straight-line
  variation, upwards or downwards unlikely to be
  seen in real data
• Intermediate values indicate how tightly the
  points on a scatterplot cluster around a straight
  line

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Some further points

• r2 tells us the proportion of the variance in one
  variable that can be explained by straight-line
  dependence on the other
• Notice the stress on straight-line association r
  does not automatically measure other forms of
  association curvilinear, U- or J-shaped,
  bimodal

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Lets re-consider the same situation

• We have, for a sample of n cases, the values for
  each case of the 2 variables-x1, x2, x3, xn
  and y1, y2, y3, yn

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But now lets ask a differentquestion

• Not- how strong is the association between
  co-varying variables?
• But- how can we state the mathematical
  relationship between 2 variables?- how would you
  predict one, knowing the other?

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The appropriate approach for that question is
regression

• Also deals with 2 variables in 1 sample
• But the two are not treated in same way
• Mathematically related to correlation
• Statpacks will often give you both measures
  together for a bivariate data set
• Comes in a variety of versions, suited to
  different situations

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The relationship between variable x and variable y

• x, by convention plotted on the horizontal axis,
  is the independent, predictor, or controlled
  variable
• y, on the vertical axis, is the dependent or
  response variable
• This does not always literally mean a hypothesis
  of x causes y
• Linear regression finds the straight line that
  best fits this relationship

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Works in, but is not restricted to, experiments

• There is an interest in explaining the dependent
  (y) variable in terms of the independent (x) one
• Thus ideally x can be controlled or at least
  measured without error by researcher
• Distinction depends on purpose of research, not
  nature of variable

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Simple examples

• y x
• y x 2
• y x - 5
• y x 10
• y 2x
• y ½x
• y ½x 10
• y 1.5x 3.5

• All these equations symbolize straight lines that
  could be graphed, by hand or by computer
• In each case, they state a relationship whereby,
  if you know x, you can work out y

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y x - 5
y x
y x 2
y 2x
y x/2
y 1.5x 3.5
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y x0 6
y -0.8x
y 10 x/2
y 3x/4 - 4
y -2-x/2
y 29x/10
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The general form of the equation is y bx a,
where

• b is the coefficient of x and governs the slope
  of the linear relationship
• If b is positive, the line rises from left to
  right if negative, it falls
• a is the value of y when x 0, i.e. where the
  line crosses the y axis, is known as the y
  intercept
• If a is positive, the line crosses the y axis
  above the origin if negative, below it

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This is known as the regression line of y on x

• When the points on a scatterplot are a cloud
  rather than a row, how do we find the line of
  best fit through it?
• By statistics rather than by eye
• We take the values of x as given try to
  minimize the overall deviations (or residuals) of
  y from the line the sum of squares of all the
  residuals is least

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So this is sometimes called a least-squares
regression

• And again there is a not too complex formula for
  it, based on the observations and the means for
  each variable
• We need eight quantities-means of x and yn?x
  and ?x2?y and ?y2?xy

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Lets do it all components are already familiar

• Enter data as two columns
• Sum each column to get Sx Sy
• Also square each value, sum these, to get Sx2
  Sy2
• Also, for each case, multiply the two values,
  sum the products, to get Sxy

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Calculate sums of squaresas before

• SSxS(x2) ((Sx)2/n)
• SSyS(y2) ((Sy)2/n)
• and sum of products as before
• SPxySxy ((Sx)(Sy)/n)

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Now we are ready for the new and final steps

• b SPxy/SSx
• a y-bar (b x-bar)
• As a reminder the basic regression equation is
  that
• y bx a

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Now we are in a position to

• Review this in relation to r2 (calculated before)
  to see how much variation the equation can
  account for
• Test null hypothesis that the slope is not
  significantly different from 0 (by F or t test)
• Predict values of y from x (with confidence
  intervals)
• Analyse residuals to check fit of model (scatter
  should show no particular shape)

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Variations on regression

• There are other methods for defining the line of
  least fit, within the overall simple linear
  regression method
• There are also more complex regression methods
  which go beyond this one in a) fitting curves
  rather than straight lines and b) having a
  number of independent variables, not just one
  i.e. multiple regression