Regression and Calibration

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Regression and Calibration

• Forensic Statistics CIS205

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Introduction

• More on the nature of the relationship between
  covariates which are continuous.
• This is useful because many variables are
  impossible to measure directly in a forensic
  context age at death for human identification,
  post-mortem interval, time since discharge for a
  firearm.
• However, changes occur which are covariates of
  these immeasurables, e.g. root dentine
  translucency, concentration of potassium in the
  vitreous humour, chemical residues in the barrels
  and chambers of firearms.
• From measurements of these variables and exact
  knowledge of the relationships with their
  immeasurable covariables, an estimate of the
  immeasurable covariate can be made.
• This process is called calibration.

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Values for post-mortem interval (PMI) and
vitreous potassium ion concentration K for a
sample of 8 cadavers (Munoz et al., 2001)
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Linear Models

• Figure 7.1 is a scatterplot of the data in Table
  7.1
• From Figure 7.1 it is easy to imagine a single
  straight line going through the cloud of x,y
  points.
• This line would represent a linear model of PMI
  and K. Such a line is marked as the straight
  line on Figure 7.2.

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Parameters

• Any linear model which describes the relationship
  between two covariates may be described by two
  parameters, the values of which are termed
  coefficients.
• The first is the gradient or slope, the second is
  the intercept with the y axis.
• The slope is dx /dy (the change in y divided by
  the change in x). The gradient of a simple linear
  model is conventionally denoted b.
• In Figure 7.2 the gradient is (9.35 7.07) / (20
  7) 2.28 / 13 0.17.
• Thus for an increase of 1h in PMI there is an
  increase of 0.17 mmol/l in vitreous K. This
  can be turned round to say that for every
  increase in 1mmol/l in K there is an increase
  of 1/0.17 5.88h in PMI.
• a is the value of K when PMI 0. From Figure
  7.2 this is 5.85.
• A general form for a linear model is y a bx.
• E.g. K 5.85 (0.17 PMI).

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Calculation of a linear regression model

• How are a and b calculated?
• Usually in linear regression we calculate a best
  fit model which minimises the sum of squared
  errors in either the x or y direction
• These errors are called the residuals, and are
  the difference between the model and the true
  data.
• Figure 7.3 is a detail of three of the points
  from Figure 7.2 showing these residuals. The
  objective of least squares regression is to
  select the model which minimises dy1² dy2²
  dy3²

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Estimating a and b

• Without going into mathematical detail, the
  estimate of the gradient b is b Sxy / Sxx,
• Where Sxx S(x mean x)²,
• Sxy S(x mean x)(y mean y)
• Sxx is related to variance, while Sxy is the
  covariance between x and y.
• An estimate of the intercept a is given by
• a mean y b(mean x)

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Table 7.2 Calculations for the regression y
(K) fitted to x (PMI)
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Calculation of a and b continued

• Sxx S(x mean x)² 371.73
• Sxy S(x mean x)(y mean y) 64.89
• b Sxy / Sxx 64.89 / 371.73 0.1745
• a mean y b(mean x) 7.47 (0.1745 9.28)
  5.85

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Testing goodness of fit

• One of the first things we may wish to know about
  our model is whether it is a good fit to the
  data. Measures of this type are known as
  goodness of fit statistics.
• A suitable test statistic is as follows, where df
  n 2, y is the estimate of y from the model

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Calculations for the goodness of fit statistic
for the regression y (K) fitted to x (PMI)
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Calculation of F continued

• F 11.33 / ( 1/6 4.92) 13.82
• See appendix F, the F-distribution, df n 2
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• For 6df at 5 significance, F 5.99. The
  calculated value of F is 13.82 which is greater
  than 5.99, so we act as though the model is an
  adequate fit at the 5 level of significance.
• Other assumptions include that the residuals must
  be normally distributed.

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Testing coefficients a and b
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Estimated Standard Errors

• Using the equations on the previous slide,
• s 0.9
• ESE(b) 0.047
• ESE(a) 0.54
• A confidence interval for both a and b is found
  using the t-distribution with n-2 df, so the 99
  confidence level occurs at 3.707 standard
  errors.
• This means the confidence level for b 0.1745
  (3.707 0.047) 0.0002 ? 0.3487. The lower
  limit is very close to 0, so it might be wise not
  to reject the null hypothesis that the gradient
  of the model is 0, and that there is no relation
  between K and PMI.
• The 99 confidence level for a 5.85(3.707
  0.54) 3.85 ? 7.85. This interval does not
  contain 0, so can be regarded as evidence for the
  hypothesis that the intercept is non-zero.

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Calibration

• We have now established that there is good
  evidence to suggest that there is a linear
  relationship between K and PMI, and we know
  the parameters for the relationship.
• In this case PMI is not directly observable, but
  a measurement of K is possible. From the
  regression model it should be possible to make an
  estimate of PMI from the measurement of K.
  This is calibration.
• From our equation K (0.17 PMI) 5.85, we
  derive PMI (K 5.85) / 0.17.
• Check this by drawing a residual plot of (PMI
  estimates of PMI) vs. PMI.
• We will see later how to calculate the standard
  errors of the estimates.

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Table 7.5. Calculation of estimated PMI values
and standard errors
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An approximation of standard error for the point
value (y0)
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Interpretation of Table 7.5

• The final column Table 7.5 gives the calculation
  on the previous slide for all 8 points.
• To arrive at a suitable confidence interval the
  standard error of the estimate has to be
  multiplied by the appropriate value from the
  t-distribution for n-2 degrees of freedom, i.e.
  x0 x0 t x Sx0
• From appendix C at 95 confidence and at n 2
  6 df, the value of t is 2.447.
• The standard error for the first value in Table
  7.5 is 5.91 and the point estimate for PMI is
  16.87, so a 95 confidence interval is 16.87
  (5.91 x 2.447) 16.87 14.46 2.41 ? 31.33.

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Points to remember

• Avoid using overly complex models use linear
  modelling unless background theory suggests a
  non-linear model, or unless a linear model does
  not meet goodness of fit criteria.
• If covariates really are related in a non-linear
  way, then usually a simple transformation such as
  taking logs of one or both of the variables will
  produce an adequate linear fit.
• Plot the independent variable (e.g. PMI) on the x
  axis, and the dependent variable (e.g. K) on
  the y axis. This is because of the notion of
  causation. It is PMI which causes K, but
  there is no way K could cause PMI. Minimise
  the residuals in the y direction (PTO).
• Finally, always plot covariates and residuals.
  Good eyeball statistics are more effective and
  give a better understanding of the relationships
  between variables than poorly understood and
  inappropriate tests.

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