Correlation

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Correlation

• Forensic Statistics CIS205

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Introduction

• Chi-squared shows the strength of relationship
  between variables when the data is of count form
• However, many variables measured in a lab are on
  a continuous scale, such as concentrations of
  chemicals, time, and most machine responses
• The term for the strength of the relation between
  continuous variables is correlation
• Any continuous variables which have some sort of
  systematic relationship are said to covary, and
  any variable which covaries with another is said
  to be a covariate.
• A basic tool for the investigation of correlation
  is the scatterplot. Usually only two variables
  are plotted, but three can be accommodated.

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Correlation Coefficient

• A statistical measure of correlation is called
  the correlation coefficient, which can only take
  on values between -1 and 1.
• Both 1 and -1 mean that the variables are
  absolutely related
• 1 means that as one variable increases, so does
  the other
• -1 means that as one variable increases, the
  other decreases.
• 0 means that the variables are unrelated.
• The strength of relationship is independent of
  the form of relationship. Most commonly
  relationships are linear (plotting one variable
  against another yields a straight line), next
  most commonly loglinear (a graph of one variable
  against the logarithm of the other is linear).

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Ageing properties of the dye methyl violet (Grim
et al., 2002)

• This example will be used to demonstrate the
  process involved in the calculation of a linear
  correlation coefficient
• Laser desorption mass spectrometry was used to
  examine the ageing properties of the dye methyl
  violet, a dye used in inks from the 1950s.
• Documents written in methyl violet ink were
  artificially aged with ultra violet radiation.
• After various times the average molecular weight
  for the methyl violet compound was measured.
• The raw data is shown in table 6.1, and plotted
  in figure 6.2

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Table 6.1. Average molecular weight of the dye
methyl violet and UV irradiation time from an
accelerated ageing experiment.
Time (min) Weight (Da)
0.0 367.20
15.3 368.97
30.6 367.42
45.3 366.19
60.2 365.91
75.5 365.68
90.6 365.12
105.7 363.59
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Correlation coefficient r

• Visual inspection of Fig. 6.2 suggests that there
  is a negative linear correlation between time and
  mean molecular weight.
• A suitable measure of this linear correlation r
  is

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.
Time (min) x mean x (x mean x)² Weight (Da) y mean y (y mean y)² (x mean x)(y mean y)
0.0 -52.90 2798.41 367.20 0.94 0.883 -49.72
15.3 -37.61 1414.51 368.97 2.71 7.344 -101.92
30.6 -22.83 498.63 367.42 1.16 1.345 -25.90
45.3 -7.61 57.91 366.19 -0.07 0.005 0.53
60.2 7.33 53.73 365.91 -0.35 0.122 -2.57
75.5 22.61 511.21 365.68 -0.58 0.336 -13.11
90.6 37.67 1419.03 365.12 -1.14 1.300 -42.94
105.7 52.84 2792.06 363.59 -2.67 7.129 -141.08
mean x 52.89 S 9545.50 mean y 366.26 S 18.465 S -376.72
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Substituting these values into the equation for r
we have

• This means that as the irradiation time increases
  the average molecular weight of methyl violet
  ions decreases, and as -0.89 is close to -1, the
  negative linear relationship is quite strong

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Significance tests for correlation coefficients

• A linear correlation coefficient of -0.89 sounds
  quite high, but is it significantly high? Is it
  possible that such a coefficient would occur in
  data drawn randomly from a bivariate normal
  distribution?
• Also, what about the effect of sample size? It
  makes sense that a high coefficient based on lots
  of x,y pairs is somehow more significant than an
  equal correlation based on only a few
  observations.
• For the null hypothesis that the correlation
  coefficient is 0, a suitable test statistic is
• t r vdf / v (1 - r²).

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Substituting for the methyl violet example

• t r vdf / v (1 - r²).
• t is the ordinate (horizontal axis) on the
  t-distribution
• df is degrees of freedom equal to n 2 (here 6
  because we have 8 x,y pairs)
• The linear correlation coefficient was -0.89, so
• t -0.89 v6 / v (1 - -0.89²) -4.78
• If we look at the values of the t-distribution
  table for df 6 we see that 95 of the area is
  within 2.447.
• Our value of -4.78 is beyond -2.447, so we can
  say that the correlation coefficient is
  significant at 95 confidence.

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Correlation coefficients for non-linear data

• Andrasko and Ståhling measured three compounds
  associated with the discharge of firearms,
  napthalene, TEAC-2 and nitroglycerin over a
  period of time by solid phase microextraction
  (SPME) of the gaseous residue from the expended
  cartridge.
• They found that the concentrations of these
  compounds would decrease with time, and that this
  property would be of use in estimating the time
  since discharge for this type of cartridges.
• Table 6.3 is a table of the peak area for
  nitroglycerine and time elapsed since discharge
  for a Winchester SKEET 100 cartridge stored at
  7C, shown as scatterplots in Figure 6.3

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Time since discharge (days) Nitroglycerin (peak height)
1.21 218.34
2.42 216.16
3.62 100.00
4.69 75.55
7.49 56.52
9.42 50.62
11.60 31.00
14.69 41.44
21.50 15.53
25.70 14.63
29.86 10.41
37.20 5.16
42.42 7.26
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Log-linear relationships

• A common model for loss in chemistry (e.g.
  radioactive decay) is called inverse exponential
  decay, which entails a log-linear relationship
  between the two variables
• The right hand scatterplot of Figure 6.3 shows
  the log to the base e (or natural logarithm) of
  the nitroglycerine peak height against time. Here
  we can see that the data looks much more linear.
• The linear correlation coefficient is -0.95,
  which is quite high, and suggests that this may
  be a reasonable transformation of the variables
• The calculations for the log-linear correlation
  coefficient are exactly the same kind as in table
  6.2, only the log to the base e of the y variable
  has been used, rather than the untransformed y.

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The coefficient of determination

• The coefficient of determination is a direct
  measure of how much the variance in one of the
  covariates is attributed to the other.
• We can imagine that the total variance in the
  nitroglycerin peak is made up of two parts, that
  which is attributable to the relationship with x
  (time), and that which can be seen as random
  noise.
• The coefficient of determination describes what
  proportion of the variance is attributable to
  relationship with time.
• The coefficient of determination is simply the
  square of the correlation coefficient.
• If r - 0.95, r² 0.90.
• Often the coefficient of determination is
  described as a percentage, which in the example
  above would mean that 90 of the variance in
  nitroglycerin peak area is attributable to time.