Statistical Analysis SC504/HS927 Spring Term 2008

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# Statistical Analysis SC504/HS927 Spring Term 2008 - PowerPoint PPT Presentation

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 AnalysisSC504/HS927Spring 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
• Interpreting logistic regression coefficients
• Advanced principles of logistic regression (for self-study)
• Source:

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

The SPSS syntax
• Accessible through syntax windows
• Accessible through ‘Paste’ buttons in every window of the main interface
• Documentation available in ‘Help’ menu
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 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 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

Predicted Predictor variables

Response variable Explanatory variables

Dependent Independent variables

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
• 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)
• 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
• Models relationship betweenset of variables xi
• dichotomous (yes/no)
• categorical (social class, ...)
• continuous (age, ...)

and

• dichotomous (binary) variable Y
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)
• ln (.) can take any value, p will always range from 0 to 1
• the equation to be estimated is:
{

logit of P(y|x)

Logistic regression (3)

Logistic transformation

Predicting p

let

then to predict p for individual i,

Logistic function (1)

Probability ofevent y

x

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 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

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
• 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
• 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
• 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 p

let

then to predict p for individual i,

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
• 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
• 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
• For CIs of odds ratios calculate CIs for coefficients and take their exponents