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Statistical Analysis SC504/HS927 Spring Term 2008. Introduction to Logistic Regression Dr. Daniel Nehring. Outline. Preliminaries: The SPSS syntax Linear regression and logistic regression OLS with a binary dependent variable Principles of logistic regression

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statistical analysis sc504 hs927 spring term 2008

Statistical AnalysisSC504/HS927Spring Term 2008

Introduction to Logistic Regression

Dr. Daniel Nehring

outline
Outline
  • Preliminaries: The SPSS syntax
  • Linear regression and logistic regression
  • OLS with a binary dependent variable
  • Principles of logistic regression
  • Interpreting logistic regression coefficients
  • Advanced principles of logistic regression (for self-study)
  • Source:

http://privatewww.essex.ac.uk/~dfnehr

the spss syntax
The SPSS syntax
  • Simple programming language allowing access to all SPSS operations
  • Access to operations not covered in the main interface
  • Accessible through syntax windows
  • Accessible through ‘Paste’ buttons in every window of the main interface
  • Documentation available in ‘Help’ menu
using spss syntax files
Using SPSS syntax files
  • Saved in a separate file format through the syntax window
  • Run commands by highlighting them and pressing the arrow button.
  • Comments can be entered into the syntax.
  • Copy-paste operations allow easy learning of the syntax.
  • The syntax is preferable at all times to the main interface to keep a log of work and identify and correct mistakes.
simple l inear regression
Simple linear regression
  • Relation between 2 continuous variables

Regression coefficient b1

    • Measures associationbetween y and x
    • Amount by which y changes on average when x changes by one unit
    • Least squares method

y

Slope

x

multiple l inear regression
Multiple linear regression
  • Relation between a continuous variable and a setof i continuous variables
  • Partial regression coefficients bi
    • Amount by which y changes on average when xi changes by one unit and all the other xis remain constant
    • Measures association between xi and y adjusted for all other xi
multiple linear regression
Multiple linear regression

Predicted Predictor variables

Response variable Explanatory variables

Dependent Independent variables

ols with a binary dependent variable
OLS with a binary dependent variable
  • Binary variables can take only 2 possible values:
    • yes/no (e.g. educated to degree level, smoker/non-smoker)
    • success/failure (e.g. of a medical treatment)
  • Coded 1 or 0 (by convention 1=yes/ success)
  • Using OLS for a binary dependent variable  predicted values can be interpreted as probabilities; expected to lie between 0 and 1
  • But nothing to constrain the regression model to predict values between 0 and 1; less than 0 & greater than 1 are possible and have no logical interpretation
  • Approaches which ensure that predicted values lie between 0 & 1 are required such as logistic regression
fitting equation to the data
Fitting equation to the data
  • Linear regression: Least squares
  • Logistic regression: Maximum likelihood
  • Likelihood function
    • Estimates parameters with property that likelihood (probability) of observed data is higher than for any other values
    • Practically easier to work with log-likelihood
maximum likelihood estimation mle
Maximum Likelihood Estimation (MLE)
  • OLS cannot be used for logistic regression since the relationship between the dependent and independent variable is non-linear
  • MLE is used instead to estimate coefficients on independent variables (parameters)
  • Of all possible values of these parameters, MLE chooses those under which the model would have been most likely to generate the observed sample
logistic regression
Logistic regression
  • Models relationship betweenset of variables xi
    • dichotomous (yes/no)
    • categorical (social class, ...)
    • continuous (age, ...)

and

    • dichotomous (binary) variable Y
logistic regression 1
Logistic regression (1)
  • ‘Logistic regression’ or ‘logit’
  • p is the probability of an event occurring
  • 1-p is the probability of the event not occurring
  • p can take any value from 0 to 1
  • the odds of the event occurring =
  • the dependent variable in a logistic regression is the natural log of the odds:
logistic regression 2
Logistic regression (2)
  • ln (.) can take any value, p will always range from 0 to 1
  • the equation to be estimated is:
l ogistic regression 3
{

logit of P(y|x)

Logistic regression (3)

Logistic transformation

predicting p
Predicting p

let

then to predict p for individual i,

l ogistic function 1
Logistic function (1)

Probability ofevent y

x

interpreting logistic regression coefficients
Interpreting logistic regression coefficients
  • intercept is value of ‘log of the odds’ when all independent variables are zero
  • each slope coefficient is the change in log odds from a 1-unit increase in the independent variable, controlling for the effects of other variables
  • two problems:
    • log odds not easy to interpret
    • change in log odds from 1-unit increase in one independent depends on values of other independent variables
  • but the exponent of b (eb) is not dependent on values of other independent variables and is the odds ratio
odds ratio
Odds ratio
  • odds ratio for coefficient on a dummy variable, e.g. female=1 for women, 0 for men
  • odds ratio = ratio of the odds of event occurring for women to the odds of its occurring for men
  • odds for women are eb times odds for men
general rules for interpreting logistic regression coefficients
General rules for interpreting logistic regression coefficients

if b1 > 0, X1 increases p

if b1 < 0, X1 decreases p

if odds ratio >1, X1 increases p

if odds ratio < 1, X1 decreases p

if CI for b1 includes 0, X1 does not have a statistically significant effect on p

if CI for odds ratio includes 1, X1 does not have a statistically significant effect on p

an example modelling the relationship between disability age and income in the 65 population
An example: modelling the relationship between disability, age and income in the 65+ population
  • dependent variable = presence of disability (1=yes,0=no)
  • independent variables:

X1 age in years (in excess of 65 i.e. 650, 70  5)

X2 whether has low income (in lowest 3rd of the income distribution)

  • data: Health Survey for England, 2000
odds odd ratios and probabilities
Odds, odd ratios and probabilities
  • pj= 0.2 i.e. a 20% probability
  • oddsj = 0.2/(1-0.2) = 0.2/0.8 = 0.25
  • pk = 0.4
  • oddsk= 0.4/0.6 = 0.67
  • relative probability/risk pj/pk = 0.2/0.4 = 0.5
  • odds ratio, oddsi/oddsj = 0.25/0.67 = 0.37
  • odds ratio is not equal to relative probability/risk
  • exceptapproximately if pj and pk are small………
points to note from logit example xls
Points to note from logit example.xls
  • if you see an odds ratio of e.g. 1.5 for a dummy variable indicating female, beware of saying ‘women have a probability 50% higher than men’. Only if both p’s are small can you say this.
  • better to calculate probabilities for example cases and compare these
predicting p1
Predicting p

let

then to predict p for individual i,

e g predicting a probability from our model
E.g.: Predicting a probability from our model
  • Predict disability for someone on low income aged 75:
  • Add up the linear equation

a(=-.912) + [age over 65 i.e.]10*0.078+1*-0.27

=-0.402

  • Take the exponent of it to get to the odds of being disabled

=.669

  • Put the odds over 1+the odds to give the probability

=c.0.4 – or a 40 per cent chance of being disabled

goodness of fit in logistic regressions
Goodness of fit in logistic regressions
  • based on improvements in the likelihood of observing the sample
  • use a chi-square test with the test statistic =
  • where R and U indicate restricted and unrestricted models
  • unrestricted – all independent variables in model
  • restricted – all or a subset of variables excluded from the model (their coefficients restricted to be 0)
statistical significance of coefficient estimates in logistic regressions
Statistical significance of coefficient estimates in logistic regressions
  • Calculated using standard errors as in OLS
  • for large n, t > 1.96 means that there is a 5% or lower probability that the true value of the coefficient is 0.

or p  0.05

95 confidence intervals for logistic regression coefficient estimates
95% confidence intervals for logistic regression coefficient estimates
  • For CIs of odds ratios calculate CIs for coefficients and take their exponents
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