Chap 8: Introduction to Logistic Regression

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Chap 8Introduction to Logistic Regression
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Logistic regression

• Models the relationship between a set of
  variables xi
• dichotomous (eat yes/no)
• categorical (social class, ... )
• continuous (age, ...)
• and
• dichotomous variable Y
• Dichotomous (binary) outcome most common
  situation in biology and epidemiology

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Logistic regression (1)
Table 2 Age and signs of coronary heart
disease (CD)
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How can we analyse these data?

• Comparison of the mean age of diseased and
  non-diseased women
• Non-diseased 38.6 years
• Diseased 58.7 years (plt0.0001)
• Linear regression?

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Dot-plot Data from Table 2
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Logistic regression (2)

• Table 3 Prevalence () of signs of CD
  according to age group

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Dot-plot Data from Table 3
Diseased
Age (years)
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The logistic function (1)
Probability of disease
x
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The logistic function (2)
logodds
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The logistic function (3)

• Advantages of the logit
• Simple transformation of P(yx)
• Linear relationship with x
• Can be continuous (Logit between - ? to ?)
• Known binomial distribution (P between 0 and 1)
• Directly related to the notion of odds of disease

• O

• LogO

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Interpretation of b (1)

• X1

• X0

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Interprepation

• Intercept is the point on the Y-axis (log odds)
  crossed by the regression line when X0.
• Slope is the rate at which the predicted log odds
  increases (or, in some cases, decreases) with
  each successive unit of X.
• Within the context of logistic regression, you
  will usually find the slope of the log odds
  regression line referred to as the "constant."
• The exponent of the slope  exp(slope) describes
  the proportionate rate at which the predicted
  odds changes with each successive unit of X.

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Example

• If X29 and the odds is 1.81, then we say
• that
• The predicted odds for x29 is
• 1.81 times as large as the one for X28
• the one for X30 is 1.81 times as large
• as the one for X29 and so on.

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Interpretation of b (2)

• b increase in log-odds for a one unit
  increase in x
• Test of the hypothesis that b0 (Wald test)
• Interval testing

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-0.123 is the rate at which the predicted CD odds
decreases with each successive unit of X. It
means also that the predicted CD odds for age30
is Exp(-0.123)0.9 times as large as the one for
age29 the one for X31 is 0.9 times as large
as the one for X30 and so on
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• Results of fitting Logistic Regression Model
• logO6.43(-.121 x 31)2.67The corresponding
  predicted odds would be exp(logO)exp(2.67)14.
  43
• And the corresponding predicted probability would
  be probabilityO/(1O)14.43/(114.43)0.93

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Link between Logistic regression and threshold
variable

• When y is continuous we may use
• yi?0 ?1 xi1.?k xik ei
• If we create a new variable y, called threshold
  variable such that
• yi 1 when yi ? 0
• 0 when yi lt 0
• if we use a logistic distribution on ei instead
  of a normal distribution, defined such that
• p(ei ?
  a)1/(1e-a).
• Then we have
• pi1/(1exp-(?0 ?1
  xi1.?k xik ) )

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Discrimination classification Rules

• Aim of this section is to classify a data into a
  given group.
• We need to find a rule which allows us to to
  determine whether or not an observation falls
  into a certain group/category/class.
• We define, a categorical variable and use this
  indicator as the response variable.
• We need to establish a classification rule for
  discriminating our populations.