Revision of last lecture

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Revision of last lecture
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Relative Risk

• For cohort studies the relative risk is used.
• The relative risk is the risk or rate of disease
  in exposed group relative to the risk or rate of
  disease in the unexposed group.
• The prevalence of disease can be found using a
  cohort study
• A RR of 1 means that the risk of the event is
  equal in the groups being compared.
• A RR gt1 suggests the event (disease) is more
  common in the exposed group than the non-exposed
• An odds ratio lt1 suggests the event is less
  common in the exposed group than the non-exposed

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Odds ratio

• For case-control studies the odds ratio is most
  often used to show relationships.
• The odds ratio is the ratio of the odds of
  exposure in the diseased group compared to the
  odds of exposure in the non-diseased group.
• An odds ratio of 1 means that the odds of the
  event is equal in the groups being compared.
• An odds ratio gt1 suggests the event (exposure) is
  more common in the diseased group than the
  non-diseased
• An odds ratio lt1 suggests the event is less
  common in the diseased group than the
  non-diseased

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General comments

• The odds, risks and rates are the probabilities
  or chances of an individual having a particular
  event such as death or an illness.
• The odds ratio and the relative risk are very
  similar when a disease or exposure is rare.

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Transformations and logistic regression

• Gordon Prescott

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Two separate topics

• Two entirely different methods for entirely
  different situations
• Transformations
• if continuous health outcome data are not
  Normally distributed
• if relationship between two continuous / discrete
  variables is not linear, e.g. curve
• Logistic regression
• if health outcome is binary

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Why transform?

• Normalise the distribution of the data
• Eliminate heterogeneity of variance
  (non-constant variance)
• To linearise a relationship between two
  continuous/ discrete variables

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Types of transformations

• Logarithmic (can use natural log also known as
  loge, or log to the base 10, log10).
• Square root (?x)
• Power (x2, x3)
• Inverse or reciprocal (1/x)
• Logit
• that is loge (p/1-p) which is the log of the odds
• There are some other transformations but these
  are the most common

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Advantage and disadvantage of transformations

• Advantage
• May allow the use of parametric methods
• More informative output
• Higher power
• Disadvantage
• Careful interpretation of the units of the
  transformed variable is required

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Transformations

• If the data is positively skewed then we can try
  either a logarithmic, reciprocal or square root
  transformation depending on the extent of the
  problem in order to remedy the situation
• Stretches small values squeezes large values
• If the data is negatively skewed the we can try a
  power transformation (x2 or x3) depending on the
  extent of the problem
• Squeezes small values stretches large values

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Logarithmic transformation

• Definition
• U log10 X X 10U 100 102
• U loge X X eU 100 e4.61

U ln X X invlog(U) 100 invlog(4.61)
expU exp4.61
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Logarithmic transformation - Normality
Histogram of waiting time
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Log (waiting time)
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Unequal variability
The standard deviation for group 1 is much larger
than the SD in group 2. T-test assuming equal
variances would be inappropriate
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Following log transformation

• Group Mean SD
• Waiting time (days)
• 1 108.2 63.45
• 2 57.4 37.65
• Log transformed waiting time (log days)
• 1 4.46 0.74
• 2 3.86 0.61

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Transformation and Interpretation

• It is important to note that if there are several
  problems with the data we would only need to use
  one transformation to attempt to remedy the
  situation
• The mathematical properties of the log
  transformation allow sensible interpretation of
  parameter and 95 CI
• The other transformations (x2, inverse, etc) do
  not have such interpretation

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Interpretation of logarithmic transformation

• Waiting time example
• After log transforming the waiting time data, we
  can compute the mean of the log waiting time
• We do not want to report the mean waiting time in
  log days
• Therefore we can back transform the data into the
  original units
• Following a logarithmic transformation, we
  antilog (or exponentiate) the mean of the logged
  data
• This mean is known as the geometric mean

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Geometric mean

• Waiting time (days)
• Mean 81.0 days median 53.9 days
• Log transformed waiting time (log days)
• Mean 4.17 log days
• Geometric mean exp (4.17) 64.7 days

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Comparison of two groups

• Two independent groups of patients are to be
  compared with respect to waiting time
• Waiting time is positively skewed
• A log transformation is applied to the data
• An independent groups t-test is then applied to
  the log transformed data

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Raw data and log transformed data
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Output from SPSS

• The mean log waiting time is 4.17 in group 1
  compared with 3.99 in group 2
• The geometric mean waiting time is exp(4.17)
  64.7 days in groups 1 compared to 54.1 days in
  group 2

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Independent t-test

• Ho mean (log waiting time) in group 1 mean
  (log waiting time) in group 2
• Is there a statistically significant difference
  in waiting time?

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Independent t-test

• Ho mean (log waiting time) in group 1 mean
  (log waiting time) in group 2
• Is there a statistically significant difference
  in waiting time?

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Independent t-test

• Ho mean (log waiting time) in group 1 mean
  (log waiting time) in group 2
• Is there a statistically significant difference
  in waiting time?
• Mean difference in log waiting time 0.1768 log
  days with 95 confidence interval for mean
  difference (0.08 to 0.27) log days

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Interpretation of t-test

• However, it would be more informative to report
  in natural units i.e. days.
• We therefore need to exponentiate (inverse log)
  results obtained.
• First need to consider some mathematics.
• Transformation has altered every value (x) into
  log x
• Estimate of difference from independent t-test
  relates to
• log x1 log x2 log (x1 / x2) mathematical
• x1 and x2 are the mean in groups 1 and 2
  respectively
• The inverse log (exponential) of the mean
  difference (in log scale) from the independent
  t-test is x1 / x2
• This denotes the ratio of waiting time in group 1
  relative to the waiting time in group 2

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• Mean difference 0.1768 log days
• Ratio of means exp(0.1789) 1.20
• 95 confidence interval for mean difference (0.08
  to 0.27) log days
• 95 confidence interval for ratio of means is
  (1.08 to 1.31)

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Transformations for non-linearity

• It is often found that the relationships between
  the dependent and explanatory (or independent)
  variables are non-linear
• There are two approaches to modelling the
  relationship
• Transform the variable(s) to produce a linear
  relationship
• Curve fitting techniques
• Polynomial regression

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Transformations for non-linearity

• When you have continuous variables any of the
  previous transformations described can be applied
  (e.g. logarithmic, square, inverse etc)
• When the variable is binary, a logit or probit
  transformation can be applied to the data

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Log transformation for non-linearity
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Non-linear relationships

• Curve fitting techniques
• Polynomial regression
• This approach incorporates the fact that the
  researcher is aware of the nature of the
  relationship between the two variables
• The researcher uses this knowledge to decide on
  the most appropriate relationship

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Curve fitting techniques

• If the researcher did not know the exact nature
  of the relationship between the variables but
  they knew that the relationship was non-linear
  then curve-fitting techniques could be adopted
• Many statistical packages have these facilities
  and there are packages whose sole function is to
  perform curve-fitting procedures

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Polynomial regression

• This type of regression is used if we have some
  idea about the nature of the relationship between
  the two variables
• It would not be sensible to try and fit a
  straight line to data that obviously was
  non-linear
• A polynomial regression model is of the form
• y ? ?1X ?2X2 ?3X3 ...

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Example

• The rate of photosynthesis of an Antarctic
  species of grass was determined for a series of
  environmental temperatures
• The aim of the researchers was to fit a line of
  best fit to this data so that they could use it
  to predict the temperature at which
  photosynthesis was maximum

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Net photosynthesis rate versus temperature
100
80
60
Photo. rate in
40
20
0
-5
0
5
10
15
20
25
30
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Temperature in Celsius
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Example quadratic regression
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Example quadratic regression
The coefficient for temperature squared
(-0.249) is statistically significant indicating
that a quadratic relationship appears to fit the
data
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A note of caution

• The regression equation we find for this data is
• y 46.37 6.77 x x - 0.249 x x2
• However when fitting this data in SPSS we will
  find that x and x2 are highly correlated
• This can lead to collinearity problems
• e.g. strange p-values
• What we need to do is to centre the x-variable
  first by subtracting the mean value from the
  individual xs first to reduce the high
  correlation between x and x2

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Collinearity and centering
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A typical non-linear relationship

• For example population growth can be represented
  by the equation
• N(t) N(0) ert
• Population at time t, N(t), is related to the
  size of the population at time 0, N(0),
  multiplied by the exponential rate of growth, r,
  and the time period involved, t.
• The relationship is obviously non-linear we can
  tackle this problem by taking a logarithmic
  transformation.

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Tackling non-linear relationships

• This can be linearised by taking the natural logs
  of both sides of the equation
• Taking the natural log of our population growth
  equation this gives
• loge N(t) loge N(0) r x t

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Logit transformation

• To examine the relationship between presence (or
  absence) of disease with a number of risk factors
• To compare two treatment groups with respect to
  re-treatment (yes/no) after correcting for
  characteristics of the patients (that may be
  unequally distributed across the two groups)
• To generate a prediction model for recurrence of
  hernia based on characteristics of patients,
  treatment, etc.

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Binary health outcomes

• Logit transformation
• Logistic regression
• regression for a binary outcome

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Probability of an event

• pi is the probability of a particular outcome
  (e.g. success)
• 0 ? pi ? 1
• Require a function that will transform pi onto
  the range (- ? to ?)

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Logit transformation

• - ? ? logit(p) ? ?
• That is the logit transformation will result in a
  variable that can take any value in a range (i.e.
  it is continuous)
• Logit(p) logep/(1-p)
• Recall, odds p/(1-p)
• Therefore this is the log of the odds
• It is known as the log odds

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Log odds form of the logistic model

• A very useful property of the logistic model is
  that by applying a transformation to both sides
  of the equation, it becomes linear
• pi eab1x1
• (1eab1x1)
• Logit(p) log odds a bixi
• To interpret the coefficients we need to
  transform them back to original units (i.e
  antilog / exponential)

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Logistic Regression

• This type of regression is used if the dependent
  variable is a binary (dichotomous) variable e.g.
  presence or absence of disease
• Under these circumstances classic multiple
  linear regression is unsuitable
• This method can be used to compare the
  characteristics of subjects with or without a
  particular disease
• Very commonly used in epidemiological studies

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Example of logistic regression

• Information was available on 111 consecutive
  patients admitted to ICU. The researcher wished
  to investigate the relationship between the
  patients vital status (lived/died) and the
  patients characteristics on entry to ICU
• Strategy for analysis
• Crosstabulations
• Calculation of odds of an event occurring
• Logistic regression

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Descriptive statistics
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Crosstabulation
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Interpretation of crosstabulation

• Borderline statistically significant association
  between type of admission and whether a patient
  leaves the ICU alive
• A greater proportion of emergency admission
  patients died (40) compared to elective
  admissions (12)
• You can also express the association between type
  of admission and vital status using an odds ratio

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Calculation of odds of dying

• The odds of an emergency admission patient dying
  whilst in ICU
• 38/56 0.679
• The odds of an elective admission patient dying
  whilst in ICU
• 2/15 0.133
• Therefore, the odds ratio for type of admission
• 0.679 / 0.133 5.11 
• The odds of a patient dying whilst in ICU is 5
  times greater if the patient was admitted as an
  emergency rather than an elective patient

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Logistic regression

• Type of surgery (TYPE) 0elective, 1emergency
• The coefficient for TYPE is 1.625. This
  indicates the difference in log odds between an
  elective and emergency admission is 1.625
• The increase in the ODDS for an emergency
  admission is by a factor of exp(1.625)
• That is the odds ratio (emergency/elective) is
  exp(1.625) 5.079.
• This is indicated in the final column of the
  table above.
• Recall, this is what was computed by hand for the
  2x2 table

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How good is the model at predicting outcome?

• The classification table indicates how good the
  logistic regression model is at predicting
  outcome
• The model is better at predicting those who died
  and overall, 65 of cases are correctly predicted
  using type of admission alone

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Multiple logistic regression

• Logistic regression can be extended to consider a
  number of explanatory variables simultaneously
• Explanatory variables can be continuous, ordinal
  or categorical
• SPSS will define dummy variables for you
• The relative contribution of each explanatory
  variable can be assessed
• Increasing the number of explanatory variables
  may improve the prediction capability of the
  model
• In our ICU example, age of patient is added to
  the model
• The coefficient for type of admission has now
  been adjusted for the effect of age
• This results in an adjusted odds ratio

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Multiple logistic model

• Including age in the model has resulted in an
  adjusted OR for type 6.5. Thus, after
  adjusting for age, the odds of dying for patients
  admitted as emergency cases is 6.5 times that of
  a patient admitted as an elective patient
• After adjusting for age, patients admitted as
  emergency admissions are approximately 6.5 times
  more likely to die in ICU compared with those
  admitted as an elective

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Why transform?

• Normalise the distribution of the data
• Eliminate heterogeneity of variance
  (non-constant variance)
• To linearise a relationship between two variables
• Log for positively skewed data the most useful
  and most easy to interpret

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What is logistic regression?

• Very similar to ordinary regression but
• Dependent variable is binary (instead of
  continuous)

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Uses of logistic regression

• Use regression equation to predict probability of
  an outcome given values of explanatory variables
• Determine which explanatory variables influence
  an outcome
• Adjust analyses for confounding variables (e.g.
  age, gender, etc.)