1. Statistical Analysis SC504/HS927 Spring Term 2008
Introduction to Logistic Regression
Dr. Daniel Nehring

2. 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:

3. PRELIMINARIES

4. 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

5. 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.

6. PART I

7. Simple linear regression
Relation between 2 continuous variables
Regression coefficient b1
Measures association between y and x
Amount by which y changes on average when x changes by one unit
Least squares method y
Slope x

8. Multiple linear regression
Relation between a continuous variable and a set of 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

9. Multiple linear regression
Predicted Predictor variables
Response variable Explanatory variables
Dependent Independent variables

10. 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

11. 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

12. 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

13. Logistic regression Models relationship between set of variables xi
dichotomous (yes/no)
categorical (social class, ... )
continuous (age, ...)
and
dichotomous (binary) variable Y

14. PART II

15. 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:

16. Logistic regression (2)
ln (.) can take any value, p will always range from 0 to 1
the equation to be estimated is:

17. Logistic regression (3)
Logistic transformation
logit of P(y|x) {

18. Predicting p
let
then to predict p for individual i,

19. Logistic function (1)
Probability of event y x

20. PART III

21. 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

22. 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

23. 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

24. dependent variable = presence of disability (1=yes,0=no)
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

25. Example: logistic regression estimate for probability of being disabled, people aged 65+

26. PART IV

27. Odds, log odds, odds ratios and probabilities

29. 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
except approximately if pj and pk are small………

30. 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

31. Predicting p
let
then to predict p for individual i,

32. 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* *-0.27
=-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

33. 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)

34. 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

35. 95% confidence intervals for logistic regression coefficient estimates
For CIs of odds ratios calculate CIs for coefficients and take their exponents