Advanced Epi Methods

Linear, Logistic, Poisson, Survival Analysis

Regression Analysis

to determine if 1 or more independent variables is associated with a dependent variable

Independent variable

Explanatory variable

Predictor variable

X

Dependent variable

Response variable

Outcome variable

Y

What is a statistical model?

The equation that describes the putative relationship among variables.

Multivariable analysis

Inferences based on the parameter for any independent variable are conditional on the other independent variables in the model.

Avoid omitting potential confounding while not including variables of minimal sequence.

Linear Regression

Outcome is measured on a CONTINUOUS scale i.e. body weight

Predictors can be measures on a continuous or categorical (dichotomous) scale

Linear Regression Example

Is chest girth (cm) significantly associated with body weight (kg) among heifers?

How do we determine the line that best fits our data?

The method of least squares is used to estimate the parameters (in this case, β0 and β1) in such a way as to minimize the sum of the squared residuals

What is a residual?

Used to estimate error in the model

The difference between an observed value of Y & its predicted value for a given value of X

T-test LR

beta/SE

Used to evaluate whether the predictor is significantly associated with the outcome

Denotes that the predictor explains the variation in the outcome

R²

How well predictor explains the outcome

It always goes up if new predictors are added

Adjusted R²

Its value is adjusted for the number of predictor variables (k) in the model

Will go down new predictors are added and have minimal additional impact on the outcome

Model Assumptions (linear regression)

Independence: the values of the outcome variable are independent from one another, i.e. no clustered data

Linearity: the relationship between the outcome and any continuous predictor variables is linear

Normal distribution: the residuals are normally distributed

Homoscedasticity: the variance of the residuals is the same across the range of predicted values of y

What if underlying assumptions are not met in a linear regression?

Independence and linearity assumptions are the most important

do data transformation like logarithmic

Can proceed as planned if there are moderate departures from normality & homoscedasticity

Cook’s Distance (Di)

assesses the influence of each observation

Standardized measure of the change in regression parameters if the particular observation was omitted

Collinearity

presence of highly correlated predictor variables in the model

Leads to t-test statistics that are spuriously small and thus p-values that are misleading

Assessed using variance inflation factor (VIF)

Variance inflation factor (VIF)

Measures how much the variance of regression coefficients in the model is inflated by addition of a predictor variable that contains very similar information

Values of VIF > 10 indicate serious collinearity

The SE of a regression parameter will ↑ by a factor of about the square root of VIF when a collinear predictor variable is added to the model

VIF = 1/(1 – R2X)

where R2X is the coefficient of determination for describing the amount of variance in the incoming X that is explained by the predictors already in the model

Logistic Regression

Outcome of interest is measured on a categorical scale

Usually dichotomous: yes/no, negative/positive, 0/1

Predictors can be measured on a continuous or categorical

can we use regression model for logistic regression?

no. as we would be unable to interpret any predicted values of Y other than 0 or 1

Generalized linear models (GLM)

Random component: identifies the outcome variable Y & selects a probability distribution for it, e.g. normal, binomial, Poisson, negative binomial

Systematic component: specifies the linear combination of predictor variables, e.g. β0 + β1X1

Link function: specifies a function that relates the expected value of Y to the linear combination of predictor variables, i.e. it connects the random & systematic components

▪ Gives us a linear relationship between our outcome variable & predictor(s)

Interpreting OR for continuous predictors

The factor by which the odds are ↑ (or ↓) for each unit change in the predictor

Maximum likelihood estimation

used to estimate the regression parameters

Wald chi-squared test

used to evaluate the significance of individual parameters

Model assumptions logistic regression

Independence: the observations are independent from one another

Linearity: the relationship between the outcome (i.e. ln{p/(1 – p)}) and any continuous predictor variables is linear

Goodness-of-fit statistics address the differences between observed & predicted values or their ratio

Pearson χ2

Deviance χ2

Hosmer-Lemeshow test

Pearson & deviance χ²

Based on dividing the data into covariate patterns

Within each pattern, the predicted # of outcomes is computed & compared to the observed # of outcomes to yield the Pearson & deviance residuals

The Pearson & deviance chi-squared statistics represent the sums of the respective squared residuals

Hosmer-Lemeshow test

Based on dividing the data in more arbitrary fashion, e.g.percentiles of estimated probability

Predicted & observed outcome probabilities within each group are compared as before

More reliable if the # of covariate patterns is high relative to the # of observations

Poisson Regression

Outcome of interest is measured on a discrete scale. e.g. # of cases of disease, # of deaths

Predictors can be measured on a continuous or categorical (including dichotomous) scale

Model assumptions

Independence: the observations are independent from one another

Linearity: the relationship between the outcome, i.e. ln (μ/N), & any continuous predictor variables is linear

Mean = variance

Overdispersion

Greater variability than expected for a GLM

Count data often vary more than would be expected if the response distribution was truly Poisson, i.e. the variance of the counts >> the mean

Data as counts

nosocomial infections

cases of cvd

workplace injuries

Problem of overdispersion

Model-based estimates of variance are too small

Thus, estimates of SE for model parameters might not be appropriate

Inference (based on Wald statistics and corresponding p-values) is questionable

Methods of managing overdispersion

Account for clustering if present

Compute scaled SEs of parameter estimates

Assume a more flexible distribution, e.g. negative binomial distribution

Indicators of overdispersion

Can be quantified by dividing the Pearson or deviance χ2 by its df

Pearson x2 usually performs better

Values > 1 indicate overdispersion

Scaled SEs

One approach is to refit the model while allowing the variance to have a multiplicative scaling (dispersion) factor

o The χ2 statistic (Pearson is preferred) divided by its df is used

o Scaled deviance and scaled Pearson χ2 values (original values divided by the dispersion factor) are generated

The SEs of the parameter estimates are scaled through

multiplication by the square root of the Pearson dispersion factor

Negative binomial regression

allows for larger variance than Poisson, i.e. the variance is not constrained to = the mean

Survival Analysis

Class of statistical methods for studying the occurrence & timing of events

Length of time that elapses before an event happens

Factors that are associated with the timing of that event

Death is often the event of interest, but these methods are broadly applicable

Also called time-to-event analysis

Event

Any qualitative change (transition from 1 discrete state to another) that can be situated in time

Quantitative changes are fine too if the threshold has relevance to real life

Timing

It is best (but not always possible) to know the exact timing of the event

Time origin should mark the onset of continuous exposure to risk of the event

Often this origin is unavailable, so must use a proxy

Seconds to years

Risk factors

We generally want to know if the risk of an event depends on various exposures

Censored data

Censored observations have unknown event times, e.g. due to loss to follow-up or termination of the study before event occurrence

Right censoring

The event has not been observed to occur

Left censoring

The event has already occurred before a given time (e.g. the start of the study)

Interval censoring

A combination of right & left, typically occurring when observations are made at infrequent intervals

Type I

Censoring that occurs because observation was stopped at a fixed time

Type II

Censoring that occurs because observation was stopped after a fixed number of events

Random

Censoring that occurs for reasons not under the control of the investigator, e.g. death or loss to follow-up

Our assumption is that censoring times are non-informative

Perform sensitivity analysis to assess

Survivor Function

Probability that an event time is > t

S(t) = P{T > t}

If event of interest is death, it’s the probability of surviving beyond time t

Estimated by proportion of individuals still alive at t

Hazard function

Can be thought of as the instantaneous risk of event occurrence at t, conditional on the event not occurring up to t

Interpreted as expected # of events per interval of time (for repeatable events)

Non-parametric Methods (Kaplan-Meier)

No assumptions about the distribution of event times or the relationship between predictors & the time until event

Straightforward

Estimates survivor functions (also called product-limit estimates)

Parametric Methods

The distribution of event times is expected to be known

Characterized by unique hazard functions

Exponential

Gompertz

Weibull

Semi-parametric

Cox proportional hazards

Kaplan meier is ideal for

Preliminary examination of data

Computing derived quantities from regression models (e.g. median survival time, 3-year probability of survival, etc.)

Comparing survivor functions between groups

Exponential model

Hazard is assumed to be constant over time

h(t) = λ

ln h(t) = μ

Unrealistic, but potentially useful as a first approximation even when assumption is known to be false. e.g. hazard of winning the lottery

Gompertz Model

The log of the hazard is a linear function of time

ln h(t) = μ + αt (where α is a constant)

e.g. hazard of retirement ↑ linearly with age; hazard of making work-related errors ↓ linearly with time spent in your career

The hazard may ↑ or ↓ with time but may not change direction

Weibull Model

The log of the hazard is a linear function of the log of time

ln h(t) = μ + αln(t) (where α is a constant)

Again, the hazard may ↑ or ↓ with time but only in one direction

Cox proportional Hazard Model (def)

The distribution of event times does not need to be specified, i.e. no need to specify how the hazard depends on time

Doesn't assume specific shape for baseline hazard

But assumes proportional hazards between groups

Flexible like non-parametric (Kaplan-Meier)

Powerful like parametric (Weibull/Gompertz)

Cox proportional Hazard Model (def)

The model can only be used to compare hazards between subjects, not predict the value of the hazard for any one subject

Model fit (likelihood-ratio test (LRT)

Compares the likelihood of the model (with all your covariates included)

with that of the null model (only α(t)

log-rank

when survival differences matter

long term treatment effects

chronic disease studies

Wilcoxon

when early differences are important

Acute treatments

early deaths matter more

when many people drop out late

testing survival functions

log-rank

wilcoxon