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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 Analysis SC504/HS927 Spring Term 2008

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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

PredictedPredictor variables

Response variableExplanatory variables

DependentIndependent 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

### 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

• 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