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 Interpreting logistic regression coefficients Advanced principles of logistic regression (for self-study) Source: http://privatewww.essex.ac.uk/~dfnehr

PRELIMINARIES

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 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.

PART I

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

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

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

PART II

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:

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

Predicting p let then to predict p for individual i,

Logistic function (1) Probability of event y x

PART III

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

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

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

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

PART IV

Odds, log odds, odds 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 except approximately 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